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# SOLID STATE 2 PHZ 7427

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This 241 page Class Notes was uploaded by Maureen Roob IV on Friday September 18, 2015. The Class Notes belongs to PHZ 7427 at University of Florida taught by Staff in Fall. Since its upload, it has received 5 views. For similar materials see /class/206876/phz-7427-university-of-florida in Physics 2 at University of Florida.

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Date Created: 09/18/15

Contents 3 Quantum Magnetism 1 31 Introduction 1 311 Atomic magnetic Hamiltonian 1 312 Curie Law free spins 3 313 Magnetic interactions 5 32 lsing model 10 321 Phase transition critical point 11 322 1D solution by transfer matrix 14 323 Ferromagnetic domains 14 33 Ferromagnetic magnons 14 331 Holstein Primakoff transformation 14 332 Linear spin wave theory 15 333 Dynamical Susceptibility 19 34 Quantum antiferromagnet 22 341 Antiferromagnetic magnons 24 342 Quantum fluctuations in the ground state 27 343 Nonlinear spin wave theory 31 344 Frustrated models 31 35 1D amp 2D Heisenberg magnets 31 351 Mermin Wagner theorem 31 352 1D Bethe solution 31 353 2D Brief summary 31 35 ltinerant magnetism 31 37 Stoner model for magnetism in metals 31 371 Moment formation in itinerant systems 35 372 RKKY lnteraction 37 373 Kondo model 37 Reading 1 Chs 31 33 Ashcroft amp Mermin 2 Ch 4 Kittel 3 For a more detailed discussion of the topic see eg D Mattis Theory of Magnetism l amp ll Springer 1981 3 Quantum Magnetism The main purpose of this section is to introduce you to ordered magnetic states in solids and their spin waveelike77 elementary excitations Magnetism is an enormous eld and reviewing it entirely is beyond the scope of this course 31 Introduction 311 Atomic magnetic Hamiltonian The simplest magnetic systems to consider are insulators where electron electron interactions are weak lf this is the case the magnetic response of the solid to an applied eld is given by the sum of the susceptibilities of the individual atoms The mag netic susceptibility is de ned by the the 2nd derivative of the free energy1 2 x 31 lt1gt We would like to know if one can understand on the basis of an understanding of atomic structure why some systems eg some 1In this section I Choose units such that the system volume V 1 1 elements which are insulators are paramagnetic X gt O and some diamagnetic X lt O The onebody Hamiltonian for the motion of the electrons in the nuclear Coulomb potential in the presence of the eld is 1 e 2 Hawm 27 p 7Argt V01 gonBH S 2 m 2 C 2 Where E S is the total spin operator MB E ech is the Bohr magneton go 2 2 is the gyromagnetic ratio and A r X H is the vector potential corresponding to the applied uniform eld assumed to point in the 2 direction Expanding the kinetic energy Hawm may now be expressed in terms of the orbital magnetic moment operator L 21quot x p as 2 p H Z r H 3 atom 2m V02 6 7 lt gt 2 6H H 2 2363 143 4 6 877202 239 Given a set of exact eigenstates for the atomic Hamiltonian in MBltL 905 39 H zero eld ignore degeneracies for simplicity standard per turbation theory in 6H gives 39 lnlgtl2 62 2 2 2 6E H H r r 5 ltnlulngt 71 E17 En W W ax yzgtlngtlt gt Where 139 uBL 908 lt is easy to see that the rst term dominates and is of order the cyclotron frequency wc E eH me unless it vanishes for symmetry reasons The third term is of order wc 62 a0 smaller because typical electron orbits are con ned to atomic sizes a0 and is important in insulators only When the state ngt has zero magnetic moment L S 0 Since the coe icient of H 2 is manifestly positive the susceptibility2 in the u 0 ground state Ogt is X 826E08H2 which is clearly lt O ie diamagnetic3 ln most cases however the atomic shells are partially lled and the ground state is determined by minimizing the atomic en ergy together with the intraatomic Coulomb interaction needed to break certain degeneracies ie by Hund7s rules 4 Once this ground state in particlular the S L and J quantum numbers is known the atomic susceptibility can be calculated The simplest case is again the J 0 case where the rst term in 5 again vanishes Then the second term and third term compete and can result in either a diamagnetic or paramagnetic susceptibility The 2nd term is called the Van Vleck term in the energy lt is paramagnetic since its contribution to X is a2El02nd term 2 we 39 gt0 lt6gt W 8H2 n En EO 312 Curie Lawfree spins ln the more usual case of J y O the ground state is 2J 1 degenerate and we have to be more careful about de ning the susceptibility The free energy as T gt O can no longer be replaced by E0 as we did above We have to account for the entropy of the 2J 1 degenerate spin states as well Applying a magnetic eld breaks this degeneracy so we have a small statistical calculation 21f the ground state is nondegenerate7 we can replace F EiTS in the de nition of the susceptibility by the ground state energy E0 3This weak diamagnetism in insulators with lled shells is called Larrnor diamagnetism 4See AampM or any serious quantum mechanics book llm not going to lecture on this but ask you about it on homework to do The energies of the spin in a eld are given by H IJ 39 H7 and since 139 yJ Within the subspace of de nite J25 the 2J1 degeneracy in H O is completely broken The free energy is J F TlogZ Tlog Z emsz JZQVHUHm ee vHUHD TlOg e vHQ ei vHQ 8 so the magnetization of the free spins is M 1 VJBWVJH lt9 Where B is the Brillouin function i 2J 1 2J 1 1 1 3a 7 oth 6 i coth fur 10 2J C 2J 2J 2J Note l de ned 7 39 We were particularly interested in the H gt 0 case so as to compare With ions With lled shells or J O in this case one expands for T gtgt 7H ie cothsc N 13 333 333 2 J13J to nd the susceptibility 2 2 ie a Curie law for the high temperature susceptibility A 1T susceptibility at high T is generally taken as evidence for free paramagnetic spins and the size of the moment given by 2 72JJ1 5This is not obvious at rst sight The magnetic moment operator olt I gog is not proportional to the total angular momentum operator 3 I However its matrix elements within the subspace of de nite L7 5 J are proportional7 due to the WignerEckart theoremi One therefore may assume the proportionality Within the subspace7 Where the proportionality const 7 gJLSiBi g is called Land g factor7 and is independent of J according to WignerEckarti See eg Ashcroft amp Mermin pl 654 313 Magnetic interactions The most important interactions between magnetic moments in an insulator are electrostatic and inherently quantum mechanical in nature From a classical perspective one might expect two such moments to interact via the classical dipolar force giving a potential energy of con guration of two moments m and g separated by a distance r of UH139H2 3ltL 139fgtltH1fgt 7quot Putting in typical atomic moments of order MB ehmc and lt12 distances of order a Bohr radius 7 N do h2m62 we nd U 2 10 46V which is very small compared to typical atomic energies of order 6V Quantum mechanical exchange is almost aways a much larger effect and the dipolar interactions are there fore normally neglected in the discussions of magnetic interac tions in solids Exchange arises because eg two quantum me chanical spins 12 in isolation can be in either a spin triplet total S 1 or singlet total S O The spatial part of the two particle wavefunctions is antisymmetric for triplet and symmetric for singlet respectively Since the particles carrying the spins are also charged there will be a large energetic dif ference between the two spin con gurations due to the different size of the Coulomb matrix elements the amount of time the two particles spend close to each other in the two cases6 In terms of hypothetical atomic wave functions war and wbr for the two particles the singlet and triplet combinations are WW1 1 2 altr1gt bltr2gt i altr2gt bltr1gta SO the Singlettriplet 6Imagine moving 2 Hatoms together starting from in nite separation Initially the 3 S 1 states and l S 0 states must be degenerate As the particles start to interact Via the Coulomb force at very long range7 there will be a splitting between singlet and triplet 5 splitting is approximately J E E0 E1 ltOHOgt lt1H1gt 2 2d3r1d3T2 Zltr1WZUH1 11 2Waltr2gt bltlt 13l where V represents the Coulomb interactions between the parti cles and possible other particles in the problem7 Weld now like to write down a simple Hamiltonian for the spins which contains the physics of this exchange splitting This was done rst by Heisenberg check history here who suggested H27spm ng 39 S2 You can easily calculate that the energy of the triplet state in this Hamiltonian is J4 and that of the singlet state 3J4 So the splitting is indeed J Note that the sign of J in the H2 case is positive meaning the S 0 state is favored the interaction is then said to be antiferromagnetic meaning it favors antialigning the spins with each other8 The so called Heitler London model of exchange just reviewed presented works reasonably well for well separated molecules but for N atoms in a real solid magnetic interactions are much more complicated and it is in general not suf cient to restrict one7s consideration to the 4 state subspace singlet a9 3 components of 7For example in the H molecule Vr1r2 e 7 e 7 14 e2 7 J 7 7 lrl F2l lRl R2l lrl Rll lr2 R2l where R1 and R2 are the sites of the protons Note the argument above would suggest naively that the triplet state should be the ground state in the wellseparated ion limit because the Coulomb interaction is minimized in the spatially antisymmetric case However the true ground state of the H2 molecule is the HeitlerLondon singlet state 5r1r2 2 11 r1zbr2 ar2 br1 8Historically the sign convention for J was the opposite J gt 0 was usually taken to be ferromagnetic ie the Hamiltonian was de ned with another minus sign I use the more popular convention H J 25 Sj Be carefull triplet to calculate the effective exchange In many cases par ticularly when the magnetic ions are reasonably well separated it nevertheless turns out to be ok to simply extend the 2 spin form 15 to the entire lattice H J SM 16 2 6 where J is the exchange constant z39 runs over sites and 6 runs over nearest neighbors9 This is the so called Heisenberg model The sign of J can be either antiferromagnetic J gt 0 in this con vention or ferromagnetic J lt 0 This may lead at sufficiently low temperature to a quantum ordered state with ferromagnetic or antiferromagnetic type order Or it may not A good deal de pends on the type and dimensionality of the lattice as we discuss below gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt 4 gtlt gt gtlt gtlt lt gtlt gt gtlt gtlt 4 gtlt gt gtlt gtlt a b Figure 1 a Ferromagnetic ordering b antiferromagnetic ordering Although oversimplified the Heisenberg model is still very difficult to solve Fortunately a good deal has been learned about it and 9One has to be a bit careful about the counting J is defined conventionally such that there is one term in the Hamiltonian for each bond between two sites Therefore if 2 runs over all sites one should have 6 only run over eg for the simple cubic lattice 23 and 2 If it ran over all nearest neighbors bonds would be double counted and we would have to multiply by 12 7 once one has put in the work it turns out to describe magnetic ordering and magnetic correlations rather well for a wide class of solids provided one is Willing to treat J as a phenomenological parameter to be determined from a t to experiment The simplest thing one can do to take interactions between spins into account is to ask What is the average exchange felt by a given spin due to all the other spins77 This is the essence of the molecular eld theory or mean eld theory due to Weiss Split off from the Hamiltonian all terms connecting any spins to a speci c spin 3 For the nearest neighbor exchange model we are considering this means just look at the nearest neighbors This part of the Hamiltonian is A an S a H 17 J g St6 where we have included the coupling of the spin in question to the external eld as well We see that the 1 spin Hamiltonian looks like the Hamiltonian for a spin in an effective eld 511B l39Heff7 18 J A He H 7 S 19 ff WE 6 gt Note this effective eld77 as currently de ned is a complicated operator depending on the neighboring spin operators 8H5 The mean eld theory replaces this quantity With its thermal average J H6 H 7 s 20 ff 9MB y 6gt l zJ H M 21 mgr l l where the magnetization M a and z is the number of nearest neighbors again But since we have an effective one body Hamil 8 tonian thermal averages are supposed to be computed just as in the noninteracting system cf 9 but in the ensemble with effective magnetic eld Therefore the magnetization is M VSBWVSHeff 753 MSW Mi lt22 This is now a nonlinear equation for M which we can solve for any H and T lt should describe a ferromagnet with nite spon taneous H gt O magnetization below a critical temperature TC So to search for TC set H O and expand B for small x zJ S1zJ TC W44 24 So the critical temperature in this mean eld theory unlike the BCS mean eld theory is of order the fundamental interaction energy J We expect this value to be an upper bound to the true critical temperature which will be supressed by spin uctuations about the mean eld used in the calculation Below TC we can calculate how the magnetization varies near the transition by expanding the Brillouin fctn to one higher power in as The result is M N T TOY2 25 Note this exponent 12 is characteristic of the disappearance of the order parameter near the transition of any mean eld theory Landau 02 04 06 08 l 12 14 T Figure 2 Plot of Mathematica solution to eqn 22 for M vs T using 7 g 1 z 4 S 12 Tc1 for this choice Upper curve H 01 lower curve H 0 32 Ising model The lsing model10 consists of a set of spins 5 with z components only localized on lattice sites i interacting via nearest neighbor exchange J lt O H SZ39SJ39 278171 2 Note it is an inherently classical model since all spin commuta tors vanish 8 3 O lts historical importance consisted not so much in its applicability to real ferromagnetic systems as in the role its solutions particularly the analytical solution of the 2D model published by Onsager in 1944 played in elucidating the na ture of phase transitions Onsagerls solution demonstrated clearly that all approximation methods and series expansions heretofore used to attack the ferromagnetic transition failed in the critical regime and thereby highlighted the fundamental nature of the problem of critical phenomena not solved by Wilson and others until in the early 70s 1OThe lsing model77 was developed as a model for ferromagnetism by Wilhelm Lenz and his student Ernst lsing in the early 20s 321 Phase transitioncritical point We will be interested in properties of the model 26 at low and at high temperatures What we will nd is that there is a tempera ture TC below which the system magnetizes spontaneously just as in mean eld theory but that its true value is in general smaller than that given by mean eld theory due to the importance of thermal uctuations Qualitatively the phase diagram looks like this H A Tlt T C TTc TgtTC 1 MO gt M V Figure 3 Field magnetization curves for three cases M0 is spontaneous magnetization in ferromagnetic phase T lt Tc Below TC the system magnetizes spontaneously even for eld H gt O lnstead of investigating the Onsager solution in detail l will rely on the Monte Carlo simulation of the model developed by Jim Sethna and Bob Silsbee as part of the Solid State Sim ulation SSS project The idea is as follows We would like to minimize F T log Tr exp 5H for a given temperature and ap 11 plied eld Finding the con guration of lsing spins which does so is a complicated task but we can imagine starting the system at high temperatures where all con gurations are equally likely and cooling to the desired temperature T11 Along the way we allow the system to relax by sweeping through all spins in the nite size lattice and deciding in the next Monte Carlo time step whether the spin will be up or down Up and down are weighted by a Boltzman probability factor eiuHeffT pltS i12gt 27 where Hfff is the effective eld de ned in 19 The simulation picks a spin S in the next time step randomly but weighted with these up and down probabilities A single sweep time step consists of L X L such attempts to ip spins where L is the size of the square sample Periodic boundary conditions are assumed and the spin con guration is displayed with one color for up and one for down Here are some remarks on lsing critical phenomena some of which you can check yourself with the simulation 0 At high temperatures one recovers the expected Curie law X N 1 T o The susceptibility diverges at a critical temperature below the mean eld value12 Near but not too near the transition X has the CurieWeiss form X N T TC 1 0 With very careful application of the simulation one should obtain Onsager7s result that very near the transition critical 11This procedure is called simulated annealing 12This is given as a homework problem Note the value of J used in the simulation is 14 that de ned here7 since Sls are i1 12 reginie77 M ltTC Tr lt28gt With 6 18 The susceptibility actually varies as X N T Tel 29 with y 7 4 Other physical quantities also diverge near the transition eg the speci c heat varies as T TEFL With oz 0 log divergence There is no real singularity in any physical quantity so long as the system size rernains nite The critical exponents oz 5 7 get closer to their mean eld values 12 oz y as the number of nearest neighbors in the lattice increases or if the dimensionality of the system increases The mean square size of thenial magnetization uctuations gets very large close to the transition Critical opalescence so named for the increased scattering of light near the liquid solid critical point Magnetization relaxation gets very long near the transition Critical slowing down ln 1D there is no nite temperature phase transition although mean eld theory predicts one See next section 322 1D solution by transfer matrix 323 Ferromagnetic domains 33 Ferromagnetic magnons Let7s consider the simplest example of an insulating ferromagnet described by the ferromagnetic Heisenberg Hamiltonian H 39 S 5 2MEI 1T0 23227 26 i Where J lt O is the ferromagnetic exchange constant 139 runs over sites and 6 runs over nearest neighbors and H 0 is the magnetic eld pointing in the 2 direction lt is clear that the system can minimize its energy by having all the spins S align along the 2 direction at T O ie the quantum ground state is identical to the classical ground state Finding the elementary excitations of the quantum many body system is not so easy however due to the fact that the spin operators do not commute 331 HolsteinPrimakoff transformation One can attempt to transform the spin problem to a more stan dard many body interacting problem by replacing the spins with boson creation and annihilation operators This can be done 61 actly by the Holstein Primakoff transformation13 A A A 12 8 Sm sy2812 1 6122 a 31 A A A 12 S7 Sm z sw 2812al 1 6122 32 HT Holstein and H Primakoff7 Phys Rev 58 1098 1940 14 Verify for yourselves that these de nitions Sf give the correct commutation relations 5 if the bosonic commutation relations 1 all 1 are obeyed on a given lattice site Note also that the operators which commute with the Hamiltonian are 32 and 32 as usual so we can classify all states in terms of their eigenvalues SltS 1 and 82 To complete the algebra we need a representation for 32 which can be obtained by using the identity on a given site slts 1 g 3 33 Using 32 and some tedious applications of the bosonic commu tation relations we nd 32 S ala 34 Now since the system is periodic we are looking for excitations which can be characterized by a well de ned momentum crystal momentum k so we de ne the Fourier transformed variables 1 7239 x 1 ix where as usual the FT variables also satisfy the bosonic relations az bk bkl 6kk etc Looking forward a bit we will see that the operators b1 and bk create and destroy a magnon or spin wave excitation of the ferromagnet These turn out to be excitations where the spins locally deviate only a small amount from their ground state values 2 as the spin wave77 passes by This sug gests a posteriori that an expansion in the spin deviations ale see 34 may converge quickly Holstein and Primakoff therefore suggested expanding the nasty square roots in 32 giving 3 23W 0110404 gt i 43 2s 1 1 H e e N k7k7k k a alalaz 2 4S iikRibk eikk k 39Riblbkbkn lac 3 23W 28 12 i I I 1 i 7 quotI I k lebll m kgkn 6 klk k RZbLbLbkn A 1 I 7 7 Siz S alai S lgedkiklRiblbku Note that the expansion is formally an expansion in 18 so we might expect it to converge rapidly in the case of a large spin system14 The last equation is exact not approximate and it is useful to note that the total spin of the system along the magnetic eld is 1 82 m 82 NS i bibk 39gt 7 239 N k consistent with our picture of magnons tipping the spin away from its T O equilibrium direction along the applied eld 332 Linear spin wave theory The idea now is to keep initially only the bilinear terms in the magnon operators leaving us with a soluble Hamiltonian hoping that we can then treat the 4th order and higher terms pertur 14For spinlZ the case of greatest interest however it is far from obvious that this uncontrolled approximation makes any sense despite the words we have said about spin deviations being small Why should they be Yet empirically linear spin wave theory works very well in 3D and surprisingly well in 2D batively15 Simply substitute 36 38 into 30 and collect the terms rst proportional to SQ S 1 18 etc We nd 1 H Jst2 2MBHOS H mg w 91 40 Where z is the number of nearest neighbors eg 6 for simple cubic lattice and J3 Hsnagnan W 1 672k7kRiezk 6bkbll 62k7kRi672k 6bll bk eik7k Ribirbk eiik7k Ri6birbk 2M H0 Z eik7k Rib1rbk ikk J23 ykbkbl HbLbk 2bLbkl 2pBH0 g bLbk mango 7k 2iBHolbLbk 41 k Where 1 7k i Z 6d 42 Z 6 is the magnon dispersion function Which in this approximation depends only on the positions of the nearest neighbor spins Note in the last step of 41 l assumed yk 71 Which is true for lat tices With inversion symmetry For example for the simple cubic lattice in 3D with lattice constant a 7k cos kw cos kya cos kza 3 clearly an even function of k Under these assump tions the magnon part of the Hamiltonian is remarkably sim ple and can be written like a harmonic oscillator or phonon type Hamiltonian Hgmgm 2k nkwk Where nk bllbk is the num ber of magnons in state k and oak JSZ1 7k QuBHO 43 15physically these nonlinear spin wave77 terms represent the interactions of magnons7 and resemble closely terms representing interactions of phonons in anharmonic lattice theory 17 is the magnon dispersion The most important magnons will be those with momenta close to the center of the Brillouin zone k N 0 so we need to examine the small k dispersion function For a Bravais lattice like simple cubic this expansion gives 1 yk 2 2163 ie the magnon dispersion vanishes as k gt O For more complicated lattices there will be solutions with wk gt const There is always a gapless mode77 wk gt O as well however since the existence of such a mode is guaranteed by the Goldstone the orem17 The gure shows a simple 1D schematic of a spin wave llllll TTT777 Figure 4 Real space picture of spin deviations in magnon Top ordered ground state with wavelength 27rk corresponding to about 10 lattice sites The picture is supposed to convey the fact that the spin devia tions from the ordered state are small and vary slightly from site to site Quantum mechanically the wave function for the spin wave state contains at each site a small amplitude for the spin to be in a state with de nite 895 andor Sy This can be seen by inverting Eq 37 to leading order amp noting that the spin wave creation operator 1 lowers the spin with S S with phase e lk39Ri and amplitude N 18 at each site 239 16Check for simple cubicl 17For every spontaneously broken continuous symmetry of the Hamiltonian there is a wkmo 0 mode 18 333 Dynamical Susceptibility Experimental interlude The simple spin wave calculations described above and below are uncontrolled for spin l 2 systems and it would be nice to know to what extent one can trust them ln recent years numeri cal work exact diagonalization and quantum Monte Carlo tech niques have shown as noted that linear spin wave calculations compare suprisingly well with such exact results for the Heisen berg model But we still need to know if there are any physical systems whose effective spin Hamiltonian can really be described by Heisenberg models ln addition keep in mind that the utility of spin wave theory was recognized long before such numerical calculations were available mainly through comparison with ex periments on simple magnets The most useful probe of magnetic structure is slow neutron scattering a technique developed in the 40s by Brockhouse and Schull Nobel prize 1994 This section is a brief discussion of how one can use neutron scattering techniques to determine the dispersion and lifetimes of magnetic excitations in solids18 Neutrons scatter from solids primarily due to the nuclear strong force which leads to nonmagnetic neutron ion scattering and al lows structural determinations very similar to x ray diffraction analysis ln addition slow neutrons traversing a crystal can emit or absorb phonons so the inelastic neutron cross section is also a sensitive measure of the dispersion of the collective modes of the ionic system19 There is also a force on the neutron due to 13A complete discussion is found in Lovesey7 Theory of Neutron Scattering from Condensed Matter Oxford 1984 V 2 lgcf Ashcroft amp Mermin ch 24 the interaction of its small magnetic dipole moment with the spin magnetic moment of the electrons in the solid There are there fore additional contributions to the peaks in the elastic neutron scattering intensity at the Bragg angles corresponding to mag netic scattering if the solid has long range magnetic order they can be distinguished from the nonmagnetic scattering because the additional spectral weight is strongly temperature dependent and disappears above the critical temperature or through application of an external magnetic eld Furthermore in analogy to the phonon case inelastic neutron scattering experiments on ferro magnets yield peaks corresponding to processes where the neu tron absorbs or emits a spin wave excitation Thus the dispersion relation for the magnons can be mapped out20 l will not go through the derivation of the inelastic scattering cross section which can be found in Lovesey7s book lt is similar to the elementary derivation given by Ashcroft amp Mermin in Appendix N for phonons The result is d2 i 2 9 2 72Wq MOLE i amino e lt1bltwgtgt Xm gm Edi 1m Xa ltqgt WgtC44gt where do is the Bohr radius k and k are initial and nal wave vector q k k F q atomic form factor 6 2Wq the Debye Waller factor and bw the Bose distribution function N the number of unit cells and w is the energy change of the neuton 20Even in systems without long range magnetic order neutron experiments provide important infor mation on the correlation length and lifetime of spin fluctuations In strongly correlated systems eg magnets just above their critical temperature or itinerant magnets close to a magnetic transition as in Sec xxx these can be thought of as collective modes with nite lifetime and decay length This means the correlation function is not const as t lRi 7 le A 00 but may fall off very slowly as power laws77 t lRi 7 le v 20 k2 2m k 2 The physics we are interested in is contained in the imaginary part of the dynamic susceptibility Xq to For to lt 0 this measures the energy loss by neutrons as they slow down while emitting spin waves in the solid for w gt O the neu trons are picking up energy from thermally excited spin waves By spin rotational invariance all components X22 X My are equal lt is then most convenient to calculatwithin linear spin wave theoryithe transverse spin susceptibility ximwdmwz4HMQ Smww em and then its Fourier transform wrt momentum q and frequency w l wont do this calculation explicitly but leave it as an exercise Note it follows exactly the method used to calculate the charge susceptibility on p 22 of the previous section You express the S operators in terms of the bk7s whose time dependence is exactly known since the approximate Hamiltonian is quadratic At the end after Fourier transforming one recovers 28 1 i 46 hlt ltN wd ltgt Again as for the Fermi gas we see that the collective modes of the system are reflected as the poles in the appropriate response function The nal cross section is now proportional to hmm4a WN6Wwm ltmgt ie there is a peak when a magnon is absorbed neutron7s en ergy Isa2m is larger than initial energy kQQm gt w E kQQm Isa2m lt 0 There is another similar contribution 21 proportional to 5a wq emission coming from 99 Thus the dispersion tool can be mapped out by careful measurement ww I I I I I I I I 2quot 0011 op ul quotIIIMI Figure 5 Neutron scattering data on ferromagnet I searched a bit but couldn7t come up with any more modern data than this This is Fig 1 Ch 4 of Kittel Magnon dispersions in magnetite from inelastic neutron scattering by Brockhouse Nobel Prize 1994 and Watanabe The one magnon lines are of course broadened by magnon magnon interactions and by finite temperatures Debye Waller factor There are also multimagnon absorption and emission processes which contribute in higher order 34 Quantum antiferromagnet Antiferromagnetic systems are generally approached by analogy with ferromagnetic systems assuming that the system can be di Vided up into two or more sublattices ie infinite interpenetrating subsets of the lattice whose union is the entire lattice Classically it is frequently clear that if we choose all spins on a given judi ciously chosen sublattice to be aligned with one another we must achieve a minimum in the energy For example for the classical 22 AF Heisenberg model H J 25 S 8H5 with J gt O on a square lattice choosing the A B sublattices in the gure and making all spins up on one and down on another allows each bond to achieve its lowest energy of JSQ This state with alternating up and down spins is referred to as the classical Neel state Similarly it may be clear to you that on the triangular lattice the classical low est energy con guration is achieved when spins are placed at 1200 with respect to one another on the sublattices A B C However square lattice triangular lattice Figure 6 Possible choice of sublattices for antiferromagnet quantum magnetic systems are not quite so simple Consider the magnetization M A on a given sublattice say the A sites in the gure of the square lattice alternatively one can de ne the stag gered magnetization as MS E 1 ltS gt Note 1 means 1 on the A sites and 1 on the B sites Either construct can be used as the order parameter for an antiferromagnet on a bipartite lattice ln the classical Neel state these is simply M A N S 2 and M 5 N S respectively ie the sublattice or staggered mag netization are saturated ln the wave function for the ground 23 state of a quantum Heisenberg antiferromagnet however there is some amplitude for spins to be ipped on a a given sublattice due to the fact that for a given bond the system can lower its energy by taking advantage of the terms This effect can be seen already by examining the tvvo spin 12 case for the ferro magnet and antiferromagnet For the ferromagnet the classical energy is JS2 J4 but the quantum energy in the total spin 1 state is also J 4 For the antiferromagnet the classical energy is J82 J4 but the energy in the total spin 0 quan tum mechanical state is 3J4 So quantum uctuationsivvhich inevitably depress the magnetization on a given sublatticeilovver the energy in the antiferromagnetic case This can be illustrated in a very simple calculation of magnons in the antiferromagnet following our previous discussion in section 72 341 Antiferromagnetic magnons We Will follow the same procedure for the ferromagnet on each sublattice A and B de ning 2 AlA 1 Azr 48 AA7A quotA 12 SA SWJMSW 28 1 28 2 A A A AlA 1 554 83 i82lt28gt12A11 2Zsl 49 12 A A BlB 53 sg sg23121 9 Bi 50 12 A A A BlB i i T 2 2 SB 7 85 285 e lt2SgtWB 1 25 24 SA S AZlA 52 22 5 5 BlBi 53 ie we assume that in the absence of any quantum uctuations spins on sublattice A are up and those on B are down Otherwise the formalism on each sublattice is identical to What we did for the ferromagnet We introduce on each sublattice creation amp annihilation operators for spin waves With momentum k 1 mm T 1 T ez39kRZ ak Nl2 i214 ak N12 g1 Ale 54 i 1 mm i2 1 rez39kRZ bk NlleeBZe bki N12 2233 39 55gt ln principle k takes values only in the 1st magnetic Brillouin zone or half zone since the periodicity of the sublattices is twice that of the underlying lattice The spin operators on a given site are then expanded as A 2 851 2 Zezk39Rz aHHl 56 k A 2 12 85 2 ZeZk Rib 57 k A 12 Sfquot 2 ZeZk39Riall 58 k A 12 823 2 N ZeZkRibLJr 59 A 1 S Nl 62kikRialak A 1 Si S Zezlk klRialak 61 The expansion of the Heisenberg Hamiltonian in terms of these variables is now compare 40 H N2J82 WW 91 62 Hgiagnon AkBkgt lt63gt Unlike the ferromagnetic case merely expressing the Hamiltonian to bilinear order in the magnon variables does not diagonalize it immediately We can however perform a canonical transforma tion21 to bring the Hamiltonian into diagonal form checkl ozk ukAk UkBl Ozillt ukAiL UkBk 64 5k UkBk UkAllt 51 UkBil UkAka 65 where the coe icients uk vk must be chosen such that iii 1113 1 One such choice is uk cosh 6k and vk sinh 6k For each k choose the angle 6k such that the anomalous terms like A1131 vanish One then nds the solution tanh 26k 7k7 lt66 and HWW NEJ Wkltalak l k f 1 67 where c013 E30 71 lt68 and E J J28 Whereas in the ferromagnetic case we had wk N 1 7k N W it is noteworthy that in the antiferromagnetic case 21The de nition of a canonical transformation I remind you is one Which Will give canonical commu tation relations for the transformed elds This is important because it ensures that we can interpret the Hamiltonian represented in terms of the new elds as a now diagonal fermion Hamiltonian read off the energies etc 26 m an iiiieqiniea weiva cmMSTsec 30 p01 20x semi ha UK a mo 0 you 200 Tempemvme m Figuie 7 Incegiaced intensity of 100 Biagg peak V tempeiacuie f0 LaCuOM With TN 195K Afcei Shiiane et a1 1987 the result wk m l 7 7912 m 76 gives a linear magnon dispersion at long wavelengths a center of symmetry of the crystal must be assumed to draw this conclusion Note further that for each k there are two degenerate modes in H0 quot quot 342 Quantum uctuations in the ground state At T 07 there are no thermally excited spin Wave excitations Nevertheless the spin Wave theory yields an decrease of the ground state energy relative to the classical value iNJzSQ but an m crease over the quantum ferromagnet result of iNlJlSQS 1 due to the zeroepoint constant term in 6722 The groundestate energy is E iNE ijwk 69 quantum Heisenberg Hamlto nm n a Lazcu04 am am llllll wlw zoo ace m 0 oz n4 mlmev WIN 5 N m 0 on Figure 8 a Inelastic neutron scattering intensity v energy transfer at 296K near zone boundary Q its 5 for oriented Lacqu crystals with Neel temperature 26019 b spinrwave dispersion wq v mi where q IS inrplane wavevector after Hayden et a1 1991 The result is usually expressed in terms of a constant 3 de ned by EU E iNJzS 3 7 70 and 058 for 3D and xxx for 2D Quantum fluctuations have the further e ect of preventing the staggered magnetization from achieving its full saturated value of 57 as in the classical Neel state7 as shown rst by Anderson23 Let us consider the average zrcomponent of spin in equilibrium at temperature T7 averaging only over spins on sublattice A of a Dedimensional hypercubic lattice From 6O7 We have S 7 N 1 klt kAkgt Within linear spin Wave theory lnverting the transformation 657 We can express the As in terms of the 1 s and 3 s7 whose averages We can easily calculate Note the 0th order in 15 gives the classical result7 and the deviation is PW Andezson Phys Rev 85 69A 1952 the spin vvave reduction of the sublattice moment 5 i AA ii l N 7 lt zgt 8 N ltAkAkgt l N Zltltukoql 111quka Uk il k l N2u ltoz1lozkgtv12ltlt il kgty 71 k where N is the number of spins on sublattice B We have ne glected cross terms like because the oz and 6 are inde pendent quanta by construction However the diagonal averages 04on and are the expectation values for the number op erators of independent bosons With dispersion wk in equiblibrium and so can be replaced Within linear spin vvave theory by M E lm lt04llt04kgt Wk 72 where b is the Bose distribution function The tranformation coe icients uk cosh 6k and vk sinh 6k are determined by the condition 66 such that l 2 2 uk vk cosh 26k V1 713 v2 e 1 1 74 such that the sulattice magnetization 71 becomes 6MA l 1lt 1gt1 75 N 2 Nknk 2 13 Remarks 29 1 N 00 The correction 6MA is independent of S and negative as it must be see next point However relative to the leading classical term S it becomes smaller and smaller as S increases as expected The integral in 75 depends on the dimensionality of the system lt has a T dependent part coming from nk and a T independent part coming from the 12 At T 0 where there are no spin waves excited thermally nk O and we nd 0078 D 3 6M TA 2 O196 D 2 76 00 D 1 The divergence in D 1 indicates the failure of spin wave theory in one dimension as discussed further in section xx The low temperature behavior of 6MltTgt must be calculated carefully due to the singularities of the bose distribution func tion when wk gt O lf this divergence is cut off by introducing a scale k0 near k O and k 7ra 7ra one nds that 6MA diverges as 1 k0 in 1D and as log kg in 2D whereas it is nite as k0 gt O in 3D Thus on this basis one does not expect long range spin order at any nonzero temperature in two dimen sions see discussion of Mermin Wagner theorem below nor even at T O in one dimension 30 343 Nonlinear spin wave theory 344 Frustrated models 35 1D amp 2D Heisenberg magnets 351 MerminWagner theorem 352 1D Bethe solution 353 2D Brief summary 36 Itinerant magnetism ltinerant magnetism77 is a catch all phrase which refers to mag netic effects in metallic systems ie with conduction electrons Most of the above discussion assumes that the spins which interact with each other are localized and there are no mobile electrons However we may study systems in which the electrons which mag netize or nearly magnetize are mobile and situations in which localized impurity spins interact with conduction electron spins in a host metal The last problem turns out to be a very dif cult many body problem which stimulated Wilson to the develop of renormalization group ideas 37 Stoner model for magnetism in metals The rst question is can we have ferromagnetism in metallic sys tems with only one relevant band of electrons The answer is yes although the magnetizationelectron in the ferromagnetic state is typically reduced drastically with respect to insulating ferro magnets The simplest model which apparently describes this 31 kind of transition there is no exact solution in D gt 1 is the Hubbard model we have already encountered A great deal of at tention has been focussed on the Hubbard model and its variants particularly because it is the simplest model known to display a metal insulator Mott Hubbard transition qualitatively similar to what is observed in the high temperature superconductors To review the Hubbard model consists of a lattice of sites labelled by i on which electrons of spin T or i may sit The kinetic energy term in the Hamiltonian allows for electrons to hop between sites with matrix element 75 and the potential energy simply requires an energy cost U every time two opposing spins occupy the same site24 Longer range interactions are neglected 1 H t Z 0200 U2n10n10 ltz39jgt where lt ij gt means nearest neighbors only 4 ttut t In its current form the kinetic energy when Fourier transformed corresponds to a tight binding band in d dimensions of width 4dt d 6 2t 2 cos lead 78 041 24What happened to the long range part of the Coulomb interaction Now that we know it is screened we can hope to describe its effects by including only its matrix elements between electrons in wave functions localized on site 239 and site j with jl smaller than a screening length The largest element is normally the 239 j one so it is frequently retained by itself Note the Hamiltonian 77 includes only an on site interaction for opposite spins Of course like spins are forbidden to occupy the same site anyway by the Pauli principle 32 where a is the lattice constant The physics of the model is as follows lmagine rst that there is one electron per site ie the band is half lled lf U O the system is clearly metallic but if U gt 00 double occupation of sites will be frozen out Since there are no holes electrons cannot move so the model must correspond to an insulating state at some critical U a metal insulator transition must take place We are more interested in the case away from half lling where the Hubbard model is thought for large U and small doping deviation of density from 1 parti clesite to have a ferromagnetic ground state25 ln particular we would like to investigate the transition from a paragmagnetz39c to a ferromagnetic state as T is lowered This instability must show up in some quantity we can calcu late ln a ferromagnet the susceptibility X diverges at the tran sition ie the magnetization produced by the application of an in nitesimal external eld is suddenly nite ln many body lan guage the static uniform spin susceptibility is the retarded spin density 7 spin density correlation function for the discussion be low I take gnB 1 x Xltq o w 021310ltgt dgr if dtltiszltn a 82m 0gtigt lt79gt where in terms of electron number operators nU wlwg the magnetization operators are 82 12m m ie they just measure the surplus of up over down spins at some point in space 25At lQ lling one electron per site a great deal is known about the Hubbard model in particular that the system is metallic for small U at least for nonnested lattices otherwise a narrowgap spin density wave instability is present but that as we increase U past a critical value UC N D a transition to an antiferromagnetic insulating state occurs Brinkman Rice transition With one single hole present as U A 00 the system is however ferromagnetic Nagaoka state 33 16 0 k q o kq 0 kg Figure 9 1a Hubbard interaction 1b Spin susceptibility vs T for free fermions Diagramatically the Hubbard interaction Hint U Ein ml looks like gure 9a note only electrons of opposite spins interact The magnetic susceptibility is a correlation function similar to the charge susceptibility we have already calculated At time t0 we measure the magnetization 82 of the system allow the particle and hole thus created to propagate to a later time scatter in all possible ways and remeasure 82 The socalled Stoner model77 of ferromagnetism approximates the perturbation series by the RPA form we have already encountered in our discussion of screening26 which gives X X01 U X0 At su iciently high T X0 varies as 1 T in the nondegenerate regime Fig 9c we will have UX0T lt 1 but as T is lowered UX0T increases If U is large enough such that U No gt 1 there will be a transition at UX0TC 1 where X diverges Note for UX0T gt 1 T lt TC the susceptibility is negative so the model is unphysical in this region This problem arises in part because the ferromagnetic state has a spontaneously broken symmetry 82gt y 0 even for 26In the static7 homogeneous case it is equivalent to the selfconsistent eld SCF method of Weiss 34 hz gt 0 Nevertheless the approach to the transition from above and the location of the transition appear qualitatively reasonable lt is also interesting to note that for UXO UNO lt 1 there will be no transition but the magnetic susceptibility will be en hanced at low temperatures So called nearly ferromagnetic metals like Pd are qualitatively described by this theory Com paring the RPA form X0ltTgt l to the free gas susceptibility in Figure 9b we see that the system X 80 will look a bit like a free gas with enhanced density of states and reduced degeneracy temperature T2927 For Pd the Fermi temper ature calculated just by counting electrons works out to 1200K but the susceptibility is found experimentally to be N 10X larger than N0 calculated from band structure and the susceptibility is already Curie like around T N T 2300K 371 Moment formation in itinerant systems We will be interested in asking what happens when we put a lo calized spin in a metal but rst we should ask how does that local moment form in the rst place lf an arbitrary impurity is inserted into a metallic host it is far from clear that any kind of localized moment will result a donor electron could take its spin and wander off to join the conduction sea for example Fe impu rities are known to give rise to a Curie term in the susceptibility when embedded in Cu for example but not in Al suggesting 27Compare to the Fermi liquid form m X0 7 81 X m 1F6 35 that a moment simply does not form in the latter case Ander on28 showed under what circumstances an impurity level in an interacting host might give rise to a moment He considered a model with a band of electrons29 with energy 6k with an extra dispersionless impurity level E0 Suppose there are strong local Coulomb interactions on the impurity site so that we need to add a Hubbard type repulsion And nally suppose the conduc tion d electrons can hop on and off the impurity with some matrix element V The model then looks like 1 H ZEkCLUCkUEo Znog ZltclgcocljcmgtiU 2710071070 k0 7 ka 2 7 82 where n00 clJUCOU is the number of electrons of spin a on the impurity site 0 By the Fermi Golden rule the decay rate due to scattering from the impurity of a band state 6 away from the Fermi level EF tn the absence of the tntemetton U is of order Alt6gt 7Tl2 266 6k 2 7rV2N0 83 k ln the Kondo case shown in the gure where E0 is well below the Fermi level the scattering processes take place with electrons at the Fermi level 6 0 so the bare width of the impurity state is also A 2 7rV2N0 So far we still do not have a magnetic mo ment since in the absence of the interaction U there would be an occupation of 2 antiparallel electrons lf one could effectively prohibit double occupancy however ie if U gt A a single spin would remain in the localized with a net moment Anderson 2glE VV Anderson7 Phys Rev 124 41 1961 29The interesting situation for moment formation is when the bandwidth of the 77primary77 cond electron band overlapping the Fermi level is much larger than the bare hybridization width of the impurity state The two most commonly considered situations are a band of s electrons with d level impurity transition metal series and d electron band with localized f level rare earthsactinidesi heavy fermions 36 obtained the basic physics supression of double occupancy by doing a HartreeFock decoupling of the interaction U term Schri effer and Wolff in fact showed that in the limit U gt 00 the Virtual charge fluctuations on the impurity site occasional dou ble occupation are eliminated and the only degree of freedom left ln the so called Kondo regime corresponding to Fig 10a is a localized spin interacting With the conduction electrons via an effective Hamiltonian HKondo 39 07 Where J is an antiferromagnetic exchange expressed in terms of the original Anderson model parameters as v2 J 2 85 E07 lt gt S is the impurity spin 12 and l 0239 i E elm Wan Cm 86gt kka With 73 the Pauli matrices is just the conduction electron spin density at the impurity site 372 RKKY Interaction Kittel p 360 et seq 373 Kondo model The HartreeFock approach to the moment formation problem was able to account for the existence of local moments at defects 37 NE Figure 10 Three different regimes of large U Anderson model depending on position of bare level E0 ln Kondo regime E0 ltlt EF large moments form at high T but are screened at low T ln mixed valent regime occupancy of impurity level is fractional and moment formation is marginal For E0 gt EF level is empty and no moment forms in metallic hosts in particular for large Curie terms in the suscep tibility at high temperatures What it did not predict however was that the low temperature behavior of the model was very strange and that in fact the moments present at high tempera tures disappear at low temperatures 1e are screened completely by the conduction electrons one of which effectively binds with the impurity moment to create a local singlet state which acts at T 0 like a nonmagnetic scatterer This basic physical picture had been guessed at earlier by J Kondo30 who was trying to explain the existence of a resistance minimum in certain metallic alloys Normally the resistance of metals is monotonically decreas ing as the temperature is lowered and adding impurities gives rise to a constant offset Matthiesenls Rule which does not change the monotonicity For Fe in Cu however the impurity contribu SOJ Kondo7 Prog Theor Phys 32 37 64 38 tion 6pm increased as the temperature was lowered and even tually saturated Since Anderson had shown that Fe possessed a moment in this situation Kondo analyzed the problem perturba tively in terms of a magnetic impurity coupling by exchange to a conduction band ie adopted Eq 84 as a model lmagine that the system at t 00 consists of a Fermi sea Ogt and one ad ditional electron in state k a at the Fermi level The impurity spin has initial spin projection M so we write the initial state as cllw0 M gt Now turn on an interaction H1 adiabatically and look at the scattering amplitude31 between and GLUIO M 27rz39ltfH1 H1H1 90 6k H0 If H1 were just ordinary impurity potential scattering we would have H1 ZkkUUCkUkaCkU and there would be two distinct second order processes k gt k contributing to Eq 90 as shown schematically at the top of Figure 11 of type a 1 e cgvpkckcm Vkp flt Plvpk 91 6k 6p 1 l ltOCkCkVkpCpm 31Reminder when using Fermils Golden Rule see eg Messiah Quantum Mechanics p807 d0 27f i 7 T 2 E 87 5 i M gt lt gt we frequently are able to get away with replacing the full Tmatrix by its perhaps more familiar lst order expansion d0 27f 2 E lHll ME 88 Recall the T matrix is de ned by where the 457s are plane waves and is the scattering state with outgoing boundary condition In this case however we will not nd the interesting logT divergence until we go to 2nd orderl So we take matrix elements of T 7 H H 1 H 89 7 1 16k 7 H0 1 x x x This is equivalent and I hope clearer than the transition amplitude I calculated in class 39 and type b NowellWok cLVkpcpc O flt pgt V V pkek 6k 6p 61d kp 6 Vpk fl 1 Vkp 92 Ek 6p where l have assumed k is initially occupied and k empty with 6k k whereas p can be either the equalities then follow from the usual application of clc and cal to Careful checking of the order of the 07s in the two matrix elements will show you that the rst process only takes place if the intermediate state p is unoccupied and the second only if it is unoccupied Nnow when one sums the two processes the Fermi function cancels This means there is no signi cant T dependence to this order pmp due to impurities is a constant at low T and thus the exclusion principle does not play an important role in ordinary potential scattering Now consider the analagous processes for spin scattering The perturbing Hamiltonian is Eq 84 Lets examine the amplitude for spin flip transitions caused by H1 rst of type a OMSH1OMSgt 93 2 V 1 ZSMSVltOM5ICkUCLUTUUHCpoNm J2 1 flt pgt Z ltMgISVSMTUZUHTgHUIMSgt P T M T CpUHTUHUCkU 0kg 0M5 40 potential scatt VPquot Vk39p Vk39p Vpk gt x p gt5lt k39 k p k39 k a b spin scatt gtlt f I06quotElt k39G39 kc p6quot k39G39 k6 I a b Figure 11 2nd order scattering processes a direct and b exchange scattering Top potential scattering bottom spin scattering Two terms corresponding to Eq 90 and then of type b J2 i flt pgt ltMEISVSMTUTUMSgt 4 6k 6p Now note that app7 7 7 UU and use the identity TMTV 6W Macaw The 6W pieces clearly will only give contributions proportional to SQ so they aren7t the important ones which will distinguish between a and b processes compared to the potential scattering case The 6W terms give results of differing sign since a gives 6M and b gives 604 Note the basic difference between the 2nd order potential scattering and spin scattering is that the matrix elements in the spin scattering case ie the SM didn7t commute When we add a and b again the result is J2 1 733 1 MSaS0MS0gt 6k 6p 4 flt pgtltM 039S UlMsa 0gt 95 41 so the spin scattering amplitude does depend on T through fep Summing over the intermediate states p0 gives a factor flt pgt N flt5pgt 2 Nodgp gk p 6k 6p 5p 9 No 615p log Ia at on which is of order logT for states 5k at the Fermi surface Thus the spin part of the interaction to a rst approximation makes a contribution to the resistance of order J3 logT p involves the square of the scattering amplitude and the cross terms between the 1st and 2nd order terms in perturbation theory give this re sult Kondo pointed to this result and said Ahal here is a contribution which gets big as T gets small However this cant be the nal answer The divergence signals that perturbation the ory is breaking down so this is one of those very singular problems where we have to nd a way to sum all the processes We have discovered that the Kondo problem despite the fact that only a single impurity is involved is a complicated many body problem Why Because the spin induces correlations between the electrons in the Fermi sea Example suppose two electrons both with spin up try to spin ip scatter from a spin down impurity The rst electron can exchange its spin with the impurity and leave it spin up The second electron therefore cannot spin ip scatter by spin conservation Thus the electrons of the conduction band cant be treated as independent objects Summing an in nite subset of the processes depicted in Figure 11 or a variety of other techniques give a picture of a singlet bound state where the impurity spin binds one electron from the 42 conduction electron sea with binding energy TK De lJNO 97 where D is the conduction electron bandwidth The renormaliza tion group picture developed by Wilson in the 70s and the exact Bethe ansatZ solution of WiegmanTsvelick and Andrei Lowenstein in 198 give a picture of a free spin at high temperatures which hybridizes more and more strongly with the conduction electron sea as the temperature is lowered Below the singlet formation temperature of TX the moment is screened and the impurity acts like a potential scatterer with large phase shift which approaches 7r 2 at T O 43 Contents 5 Superconductivity 1 51 Phenomenology 1 52 Electron phonon interaction 3 53 Cooper problem 6 54 Pair condensate 85 BCS Wavefctn 11 55 BCS Model 12 551 BCS wave function gauge invariance and number con servation 14 552 ls the BCS order parameter general 16 56 Thermodynamics 13 561 Bogoliubov transformation 18 562 Density of states 19 563 Critical temperature 21 564 Speci c heat 23 57 Electrodynamics 25 571 Linear response to vector potential 25 572 Meissner Effect 27 573 Dynamical conductivity 32 6 Ginzburg Landau Theory 33 61 GL Free Energy 33 62 Type I and Type II superconductivity 42 63 Vortex Lattice 50 64 Properties of Single Vortex Lower critical eld H01 55 65 Josephson Effect 59 5 Superconductivity 51 Phenomenology Superconductivity was discovered in 1911 in the Leiden lab oratory of Kamerlingh Onnes when a so called blue boy local high school student recruited for the tedious job of mon itoring experiments noticed that the resistivity of Hg metal vanished abruptly at about 4K Although phenomenological models with predictive power were developed in the 30 s and 40 s the microscopic mechanism underlying superconductiv ity was not discovered until 1957 by Bardeen Cooper and Schrieffer Superconductors have been studied intensively for their fundamental interest and for the promise of technologi cal applications which would be possible if a material which superconducts at room temperature were discovered Until 1986 critical temperatures Tc s at which resistance disap pears were always less than about 23K In 1986 Bednorz and Mueller published a paper subsequently recognized with the 1987 Nobel prize for the discovery of a new class of materials which currently include members with Tc s of about 135K Normal metal Superconductor P P T Tc T A E B 1 E E T T gt Figure 1 Properties of superconductors Superconducting materials exhibit the following unusual be hayiors 1 Zero resistance Below a material s Tc the DC elec trical resistivity p is really zero not just very small This leads to the possibility of a related effect 2 Persistent currents If a current is set up in a super conductor with multiply connected topology eg a torus 2 it will flow forever without any driving voltage In prac tice experiments have been performed in which persistent currents flow for several years without signs of degrading 3 Perfect diamagnetism A superconductor expels a weak magnetic eld nearly completely from its interior screening currents flow to compensate the eld within a surface layer of a few 100 or 1000 A and the field at the sample surface drops to zero over this layer 4 Energy gap llost thermodynamic properties of a su perconductor are found to vary as eAkBT indicating the existence of a gap or energy interval with no allowed eigenenergies in the energy spectrum ldea when there is a gap only an exponentially small number of particles have enough thermal energy to be promoted to the avail able unoccupied states above the gap In addition this gap is visible in electromagnetic absorption send in a photon at low temperatures strictly speaking T 0 and no absorption is possible until the photon energy reaches 2A ie until the energy required to break a pair is available 52 Electronphonon interaction Superconductivity is due to an effective attraction between conduction electrons Since two electrons experience a re pulsive Coulomb force there must be an additional attrac tive force between two electrons when they are placed in a metallic environment In classic superconductors this force 3 is known to arise from the interaction with the ionic system In previous discussion of a normal metal the ions were re placed by a homogeneous positive background which enforces charge neutrality in the system In reality this medium is polarizablei the number of ions per unit volume can fluctuate in time In particular if we imagine a snapshot of a single electron entering a region of the metal it will create a net positive charge density near itself by attracting the oppositely charged ions Crucial here is that a typical electron close to the Fermi surface moves with velocity v1 hkp m which is much larger than the velocity of the ions 1 VFmM So by the time T N 27TwD N 10 13 sec the ions have polarized themselves 1st electron is long gone it s moved a distance UFT N 1080ms N 1000K and 2nd electron can happen by to lower its energy with the concentration of positive charge before the ionic fluctuation relaxes away This gives rise to an effective attraction between the two electrons as shown which may be large enough to overcome the repulsive Coulomb in teraction Historically this electron phonon pairing mech anism was suggested by Frolich in 1950 and con rmed by the discovery of the isotope effect wherein Tc was found to vary as M 1 2 for materials which were identical chemically but which were made with different isotopes The simplest model for the total interaction between two electrons in momentum states k and k with q E k k in teracting both by direct Coulomb and electron phonon forces is given by I 610 VF 0 t0 tt1 Figure 2 Effective attraction of two electrons due to phonon exchange77 47T 2 2 2 q 1 V w 7 q q2k q2k w2 w in the jellium model Here rst term is Coulomb interac tion in the presence of a medium with dielectric constant e 1k3q2 and tag are the phonon frequencies The screen ing length cs 1 is 11 or so in a good metal Second term is interaction due to exchange of phonons ie the mechanism pictured in the gure Note it is frequency dependent re ecting the retarded nature of interaction see gure and in particular that the 2nd term is attractive for w lt wq N wD Something is not quite right here however it looks indeed as though the two terms are of the same order as w gt 0 indeed they cancel each other there and V is seen to be al ways repulsive This indicates that the jellium approximation is too simple We should probably think about a more care ful calculation in a real system as producing two equivalent terms which vary in approximately the same way with lip and tag but with prefactors which are arbitrary In some ma terials then the second term might win at low frequencies depending on details The BCS interaction is sometimes re 5 ferred to as a residual attractive interaction ie what is left when the long range Coulomb 53 Cooper problem A great deal was known about the phenomenology of super conductivity in the 1950 s and it was already suspected that the electron phonon interaction was responsible but the mi croscopic form of the wave function was unknown A clue was provided by Leon Cooper who showed that the noninter acting Fermi sea is unstable towards the addition of a single pair of electrons with attractive interactions Cooper began by examining the wave function of this pair 01 73 which can always be written as a sum over plane waves 1W1 r2 uqu fm kqlf 2 where the ukq are expansion coefficients and C is the spin part of the wave function either the singlet l Ti iTgt or one of the triplet l TTgtl iigtl Ti iTgt In fact since we will demand that w is the ground state of the two electron system we will assume the wave function is re alized with zero center of mass momentum of the two elec trons ukq ukdqp Here is a quick argument related to the electron phonon origin of the attractive interaction1 The electron phonon interaction is strongest for those electrons with single particle energies 5k within COD of the Fermi level In the scattering process depicted in Fig 3 momentum is 1Thanks to Kevin McCarthy7 who forced me to think about this further 6 Figure 3 Electrons scattered by phonon exchange are con ned to shell of thickness rap about Fermi surface explicitly conserved ie the total momentum k p K 3 is the same in the incoming and outgoing parts of the diagram Now look at Figure 4 and note that if K is not N 0 the phase space for scattering attraction is dramatically reduced So the system can always lower its energy by creating K 0 pairs Henceforth we will make this assumption as Cooper did Then 01 73 becomes 2 ukelk lrrml Note that if uk is even in k the wave function has only terms oc cos 16 71 73 whereas if it is Odd only the sin 16 71 r2 will contribute This is an important distinction because only in the former case is there an amplitude for the two electrons to live quoton top of each other at the origin Note further that in order to 7 Figure 4 To get attractive scattering with nite cm momentum K7 need both electron energies to be within Lap of Fermi leveli very little phase space preserve the proper overall antisymmetry of the wave function uk even odd in 16 implies the wave function must be spin singlet triplet Let us assume further that there is a general two body interaction between the two electrons the rest of the Fermi sea is noninteracting in the model V7 17 2 so that the Hamiltonian for the system is V V3 H 7 7Vrr 4 2m 2m 1 2 lt Inserting the assumed form of 1D into the Schr dinger equation H p Ew and Fourier transforming both sides with respect to the relative coordinate 7 r1 73 we find E 26kuk Z VickUH 5 kgtkF where 6k 622771 and the VM fd3TV7 efkkf are the 8 matrix elements of the two body interaction Recall k k correspond to energies at the Fermi level 61 in the absence of V The question was posed by Cooper is it possible to nd an eigenvalue E lt 261 ie a bound state of the two electrons To simplify the problem Cooper assumed a model form for VW in which V Ekvgk lt wc VW 0 otherwise 6 where as usual 5 E 6k 61 The BCS interaction ka is sometimes referred to as a residual attractive interaction ie the attractive short distance part left when the long range Coulomb interaction has been subtracted out as in The bound state equation becomes Vgl uk 7 7 26k E lt where the prime on the summation in this context means sum only over 16 such that 6f lt 6k lt 6F we Now 11 may be eliminated from the equation by summing both sides 21 yielding W 1 1 7 i 8 V 2 k E 1 1 1 2 F2w E 271v FWCd 7N1 C9 2 091 E2e E 2 Dog 2eF E For a weak interaction NOV ltlt 1 we expect a solution if at all just below the Fermi level so we treat 26 E as a small 9 positive quantity eg negligible compared to 2we We then arrive at the pair binding energy 00 2NOV Acooper E 261 E 2 2wee 10 There are several remarks to be made about this result 1 Note for your own informationiCooper didn t know this at the time that the dependence of the bound state en ergy on both the interaction V and the cutoff frequency we strongly resembles the famous BCS transition temperature dependence with we identi ed as the phonon frequency wD as given in equation 11 the dependence on V is that of an essential singularity ie a nonanalytic function of the parameter Thus we may expect never to arrive at this result at any order in perturbation theory an unexpected problem which hin dered theoretical progress for a long time The solution found has isotropic or s symmetry since it doesn t depend on the f6 on the Fermi surface How would an angular dependence arise Look back over the calcula tion Note the integrand 2ee E 1 25k Aceeper1 peaks at the Fermi level with energy spread Aceeper of states involved in the pairing The weak coupling NOV ltlt 1 solution therefore provides a bit of a posteriom39 justi ca tion for its own existence since the fact that Aceepee ltlt we implies that the dependence of VW on energies out near 10 the cutoff and beyond is in fact not terribly important so the cutoff procedure used was ok 5 The spread in momentum is therefore roughly AcooperUF and the characteristic size of the pair using Heisenberg s uncertainty relation about vFTc This is about 100 1000A in metals so since there is of order 1 electron unit cell in a metal and if this toy calculation has anything to do with superconductivity there are certainly many electron pairs overlapping each other in real space in a superconductor 54 Pair condensate amp BCS Wavefctn Obviously one thing is missing from Cooper s picture if it is energetically favorable for two electrons in the presence of a noninteracting Fermi sea to pair ie to form a bound state why not have the other electrons pair too and lower the en ergy of the system still further This is an instability of the normal state just like magnetism or charge density wave for mation where a ground state of completely different charac ter and symmetry than the Fermi liquid is stabilized The Cooper calculation is a T0 problem but we expect that as one lowers the temperature it will become at some critical temperature To energetically favorable for all the electrons to pair Although this picture is appealing many things about it are unclear does the pairing of many other electrons alter the attractive interaction which led to the pairing in the rst 11 place Does the bound state energy per pair change Do all of the electrons in the Fermi sea participate And most im portantly how does the critical temperature actually depend on the parameters and can we calculate it 55 BCS Model A consistent theory of superconductivity may be constructed either using the full effective interaction or our approxima tion Vq w to it However almost all interesting questions can be answered by the even simpler model used by BCS The essential point is to have an attractive interaction for electrons in a shell near the Fermi surface retardation is sec ondary Therefore BCS proposed starting from a phenomeno logical Hamiltonian describing free electrons scattering via an effective instantaneous interaction a la Cooper H H0 V Z CLUCkq0CkqaCkla kkq 0039 where the prime on the sum indicates that the energies of the states 16 and k must lie in the shell of thickness cap Note the interaction term is just the Fourier transform of a completely local 4 Fermi interaction wlrwlrwrwr2 Recall that in our discussion of the instability of the nor mal state we suggested that an infinitesimal pair field could produce a finite amplitude for pairing That amplitude was the expectation value clacik We ignore for the moment 2Note this is not the most general form leading to superconductivity Pairing in higher angular momentum channels requires a bilocal model Hamiltonian7 as we shall see later 12 the problems with number conservation and ask if we can simplify the Hamiltonian still further with a mean eld ap proximation again to be justi ed a posteriorz39 We proceed along the lines of generalized Hartree Fock theory and rewrite the interaction as CLUCT kqa C k qa ck a ltCLUCT kqa gt X X ltCkqaCkagt 6CC where eg 600 CkqaCkla ltCkqaCklagt is the fluctu ation of this operator about its expectation value If a mean eld description is to be valid we should be able to neglect terms quadratic in the uctuations when we expand Eq 20 If we furthermore make the assumption that pairing will take place in a uniform state zero pair center of mass momentum then we put ltCkqaCklagt ltek0ckagt6q70 The effective Hamiltonian then becomes check H 2 Ho AClTCl u t hCl t AkltCLTCT kigtv 13 where A VltCkicmgt 14 What BCS actually Bogoliubov after BCS did was then to treat the order parameter A as a complex number and calculate expectation values in the approximate Hamiltonian 13 insisting that A be determined self consistently via Eq 14 at the same time 551 BCS wave function gauge invariance and number con servation What BCS actually did in their original paper is to treat the Hamiltonian 11 variationally Their ansatZ for the ground state of 11 is a trial state with the pairs 16 T k i occupied with amplitude 1 and unoccupied with amplitude ak such that lale My 1 This is a variational wave function so the energy is to be minimized over the space of at at Alternatively one can diagonalize the Hartree Fock BCS Hamiltonian directly to gether with the self consistency equation for the order param eter the two methods turn out to be equivalent 1 will follow the latter procedure but rst make a few remarks on the form of the wave function First note the explicit violation of par ticle number conservation W gt is a superposition of states describing 0 2 4 N particle systems3 In general a quan tum mechanical system with fixed particle number N like eg a real superconductor manifests a global U 1 gauge symmetry because H is invariant under elm gt ewclm The state W gt is characterized by a set of coefficients 2 at which becomes 2 2ka after the gauge transformation The two states W gt and wW where qt 28 are inequiva lent mutually orthogonal quantum states since they are not 3What happened to the odd numbers In mesoscopic superconductors there are actually differences in the properties of even and oddnumber particle systems7 but for bulk systems the distinction is irrelevant simply related by a multiplicative phase factor4 Since H is independent of qb however all states qb gt are contin uously degenerate ie the ground state has a U 1 gauge phase symmetry Any state o gt is said to be a bro ken symmetry state becaue it is not invariant under a U 1 transformation ie the system has chosen a particular qb out of the degenerate range 0 lt qb lt 27r Nevertheless the ab solute value of the overall phase of the ground state is not an observable but its variations 6gb7 t in space and time are It is the rigidity of the phase ie the energy cost of any of these fluctuations which is responsible for superconductivity Earlier I mentioned that it was possible to construct a num ber conserving theory It is now instructive to see how states of de nite number are formed Anderson 1958 by making co herent superpositions of states of de nite phase WV gt EldreiiNQlwo gt no The integration over qb gives zero unless there are in the ex pansion of the product contained in 1D gt precisely N 2 pair creation terms each with factor exp iqb Note while this state has maximal uncertainty in the value of the phase the rigidity of the system to phase fluctuations is retained5 It is now straightforward to see why BCS theory works The BCS wave function 1D gt may be expressed as a sum in gt EN aN N gt Convince yourself of this by calculat 4in the normal state it gt and we differ by a global multiplicative phase cm which has no physical consequences and the ground state is nondegenerate 5The phase and number are in fact canonically conjugate variables N2q i where N Zia8415 in the 45 representation 15 ing the my explicitlyll IF we can show that the distribution of coef cients aN is sharply peaked about its mean value lt N gt then we will get essentially the same answers as working with a state of de nite number N lt N gt Using the explicit form 23 it is easy to show ltNgt w gnaw 2 w ltltN ltNgtgt2gt 17 Now the uk and 1 will typically be numbers of order 1 so since the numbers of allowed k states appearing in the k sums scale with the volume of the system we have lt N gt V and lt N lt N gt2 gt V Therefore the width of the distribution of numbers in the BCS state is lt N lt N gt 2 gt12 lt N gt N 12 As N gt 1023 particles this relative error implied by the number nonconseryation in the BCS state becomes negligible 552 Is the BCS order parameter general Before leaving the subject of the phase in this section it is worthwhile asking again why we decided to pair states with opposite momenta and spin 6 T and k i The BCS argu ment had to do 1 with minimizing the energy of the entire system by giving the Cooper pairs zero center of mass mo mentum and 2 insisting on a spin singlet state because the phonon mechanism leads to electron attraction when the elec trons are at the same spatial position because it is retarded in timel and a spatially symmetric wayefunction with large 16 amplitude at the origin demands an antisymmetric spin part Can we relax these assumptions at all The rst require ment seems fairly general but it should be recalled that one can couple to the pair center of mass with an external mag netic eld so that one will create spatially inhomogeneous nite q states with current ow in the presence of a mag netic eld Even in zero external eld it has been proposed that systems with coexisting antiferromagnetic correlations could have pairing with nite antiferromagnetic nesting vec tor C Baltensberger and Strassler 196 The requirement for singlet pairing can clearly be relaxed if there is a pair ing mechanism which disfavors close approach of the paired particles This is the case in super uid 3H e where the hard core repulsion of two 3H e atoms supresses To for s wave sin glet pairing and enhances To for p wave triplet pairing where the amplitude for two particles to be together at the origin is always zero In general pairing is possible for some pair mechanism if the single particle energies corresponding to the states 160 and k a are degenerate since in this case the pairing interaction is most attractive In the BCS case a guarantee of this degen eracy for k T and k T in zero eld is provided by Kramer s theorem which says these states must be degenerate because they are connected by time reversal symmetry However there are other symmetries in a system with inversion sym metry parity will provide another type of degeneracy so 16 T k T k T and k T are all degenerate and may be paired with one another if allowed by the pair interaction 56 Thermodynamics 561 Bogoliubov transformation We now return to 13 and discuss the solution by canonical transformation given by Bogoliubov After our drastic ap proximation we are left with a quadratic Hamiltonian in the as but with clcl and cc terms in addition to Clc s We can di agonalize it easily however by introducing the quasipartz cle operators Yko and 71d by CkT U1ka 16de T Ckl T U1 Yk0 f Uk Yki 18 You may check that this transformation is canonical preserves fermion comm rels if lukl2 l1le 1 Substituting into 13 and using the commutation relations we get H303 ggkmule lvkl2lf7flt07ko 7117k1 2Mle 2ufltvfiik17ko 2Ukvk7flt17fltol f lltAkuka f Afiuiivk 7107k0 f de Yki 1 HAW Afiuff k ko Alina Akuli710711 AkltCLTCikigt which does not seem to be enormous progress to say the least But the game is to eliminate the terms which are not of the form quot y so to be left with a sum of independent number type 18 terms whose eigenvalues we can write down The coef cients of the ff and 77 type terms are seen to vanish if we choose 2 kukvk Akui 0 20 This condition and the normalization condition lukl2 l1le 1 are both satis ed by the solutions 2 1 Lle 7 1i 6 21 lvkl 2 Ek where I de ned the Bogolz39bov quasipartz cle energy Ek we w lt22 The BCS Hamiltonian has now been diagonalz zed H BCS Ek 7110714 7i17k1gt Ek Ek AkltclltTCT kigtgt lt23 Note the second term is just a constant which will be impor tant for calculating the ground state energy accurately The rst term however just describes a set of free fermion exci tations above the ground state with spectrum Ek 562 Density of states The BCS spectrum is easily seen to have a minimum Ak for a given direction k on the Fermi surface Ak therefore in addition to playing the role of order parameter for the su perconducting transition is also the energy gap in the 1 particle spectrum To see this explicitly we can simply do a change of variables in all energy integrals from the normal metal eigenenergies 5k to the quasiparticle energies Ek NltEgtdE weer lt24 If we are interested in the standard case where the gap A is much smaller than the energy over which the normal state dos N ME yaries near the Fermi level we can make the replace ment 2 E No so using the form of Ek from 22 we nd E No 0 E lt A This function is sketched in Figure 5 A N0 SC 1 Normal o 139 gt E Figure 5 a Normalized density of states b Quasiparticle spectrum 20 563 Critical temperature The critical temperature is de ned as the temperature at which the order parameter Ak vanishes We can now cal culate this with the aid of the diagonalized Hamiltonian The self consistency condition is A V EltCLITCTklgt V E ukUfQG wa Yko de Yk1gt q k lt1 2fltEkgtgt lt27 Since 1 2fE tanhE2T the BCS gap equation reads N Ek N k t h 1 VEZEk an 2T This equation may now be solved first for the critical temper ature itself ie the temperature at which A gt 0 and then for the normalized order parameter ATc for any temperature T It is the ability to eliminate all normal state parameters from the equation in favor of Tc itself which makes the BCS theory powerful For in practice the parameters cap N0 and particularly V are known quite poorly and the fact that two of them occur in an exponential makes an accurate rst prin ciples calculation of Tc nearly impossible You should always be suspicious of a theory which claims to be able to calculate Tc On the other hand To is easy to measure so if it is the only energy scale in the theory we have a tool with enormous predictive power 28 21 First note that at T0 the gap equation becomes 1 JD 1 5k d it h NOV 0 5 an 2T0 This integral can be approximated carefully but it is useful to get a sense of What is going on by doing a crude treatrnent Note that since Tc ltlt cap generally most of the integrand weight occurs for E gt T so we can estimate the tanh factor by 1 The integral is log divergent which is Why the cutoff cap is so important We find 1 to N V l 7 gt 70 2 1 0 30 NOVE 0g TWO WDe The more accurate analysis of the integral gives the BCS result 29 Tc 114wDe1N0V 31 We can do the same calculation near Tc expanding to lead ing order in the small quantity AT T to find ATTc 2 3061 TTc12 At T 0 we have 1 NOV 0WD dnglk D dEN E E 32 1 N W so that Am 2 2m exp lNOV or AOTc 2 176 The full temperature dependence of AT is sketched in Figure 6 In the halcyon days of superconductivity theory comparisons with the theory had to be compared With a careful table of ATc painstakingly calculated and compiled by Miihlschlegl DdE ln2wdA 33 22 176 TC AT Figure 6 B08 order parameter as fctn of T Nowadays the numerical solution requires a few seconds on a PC It is frequently even easier to use a phenomenological approximate closed form of the gap which is correct near T 0 and T 0 7T SOT AT scTc hi 770 17 4 H 6 tanrdscaCyT gt lt3 where 650 AOTc 176 a 23 and SCON 143 is the normalized speci c heat jump6 This is another of the universal ratios which the BCS theory predicted and which helped con rm the theory in classic superconductors 564 Speci c heat The gap in the density of states is re ected in all thermody namic quantities as an activated behavior eAT at low T due to the exponentially small number of Bogoliubov quasi 6Note to evaluate the last quantity7 we need only use the calculated temperature dependence of A near TC7 and insert into Eq 47 23 particles with thermal energy suf cient to be excited over the gap A at low temperatures T lt A The electronic speci c heat is particularly easy to calculate since the entropy of the BCS superconductor is once again the entropy of a free gas of noninteracting quasiparticles with modi ed spectrum Ek The expression 116 then gives the entropy directly and may be rewritten s 4 t dENltEgtfltEgt1nfltEgt1 fltEgti1ni1 fltEgti 35 where f is the Fermi function The constant volume spe ci c heat is just Cegy T dS dTlv which after a little alge bra may be written i f 2l LN 6EE 2TdT 36 A sketch of the result of a full numerical evaluation is shown in Figure 1 Note the discontinuity at Tc and the very rapid falloff at low temperatures 2 Cezy fdEZWEll It is instructive to actually calculate the entropy and speci c heat both at low temperatures and near Tc For T lt A f 2 eET and the density of states factor N in the integral cuts off the integration at the lower limit A giving C 2 N0A52T32eAT7 7To obtain this7 try the following o replace the derivative of Fermi function by expET 0 do integral by parts to remove singularity at Delta 0 expand around Delta E Delta delta E 0 change integration variables from E to delta E Somebody please check my answerl 24 Note the rst term in Eq 47is continuous through the transition A gt 0 and reduces to the normal state speci c heat 2W23N0T above Tc but the second one gives a dis continuity at T0 of ON CSCN 143 where 05 CTg and ON CTj To evaluate 36 we need the T depen dence of the order parameter from a general solution of 28 57 Electrodynamics 571 Linear response to vector potential The existence of an energy gap is not a suf cient condition for superconductivity actually it is not even a necessary onel Insulators for example do not possess the phase rigidity Which leads to perfect conductivity and perfect diarnagnetisrn which are the de ning characteristics of superconductivity We can understand both properties by calculating the cur rent response of the system to an applied magnetic or electric eld The kinetic energy in the presence of an applied vector potential A is just i 3 t 9 2 Ho 2md MTH 2V CAl 1Mr 37 and the second quantized current density operator is given by m gum2v Agtwltrgt W Agtwltrgtiwltrgti ea 2 wwwm ea 25 Where swang kwwovwcv ltvwwoxaoi em or in Fourier space ammgtggmtwa am We would like to do a calculation of the linear current re sponse jq w to the application of an external eld Aq w to the system a long time after the perturbation is turned on Expanding the Hamiltonian to rst order in A gives the interaction 7 3 7 e f H 7 d mpam A 7 k Aqckqacqa 42 The expectation value lt j gt may now be calculated to linear order Via the Kubo forrnula yielding ltjgt 1 w Km wAq w 43 With 7162 KltQla W ltjPara7jpa7 algtltq7 01 44 Note the rst term in the current ijMEj me as me is purely diagmagnem c ie these currents tend to screen the external eld note sign The second paramagnetic term is formally the Fourier transform of the current current corre lation function correlation function used in the sense of our 26 discussion of the Kubo formula8 Here are a few remarks on this expression Note the simple product structure of 43 in momentum space implies a nonlocal relationship in general between j and A ie j7 depends on the AM at many points 7quot around 7 9 Note also that the electric eld in a gauge where the elec trostatic potential is set to zero may be written Eq w ZwAq w so that the complex conductivity of the sys tem de ned by j QE may be written gm w gm w lt47 What happens in a normal metal The paramagnetic sec ond term cancels the diamagnetic response at w 0 leaving no real part of K lm part of a ie the conduc tivity is purely dissipative and not inductive at w q 0 in the normal metal 572 Meissner Effect There is a theorem of classical physics proved by Bohr10 which states that the ground state of a system of charged particles 8We will see that the rst term gives the diamagnetic response of the system7 and the second the temperature dependent paramagnetic response 91f we transformed back7 weld get the convolution jr d3T Krr Ar 46 10See The development of the quantummechanical electron theory of metals 19283377 Li Hoddeson and Gr Baym7 Rev Mod Phys7 597 287 1987 27 in an external magnetic eld carries zero current The essen tial element in the proof of this theorem is the fact that the magnetic forces on the particles are always perpendicular to their velocities In a quantum mechanical system the three components of the velocity do not commute in the presence of the field allowing for a nite current to be created in the ground state Thus the existence of the Meissner effect in superconductors wherein magnetic flux is expelled from the interior of a sample below its critical temperature is a clear proof that superconductivity is a manifestation of quantum mechanics The typical theorists geometry for calculating the penetra tion of an electromagnetic field into a superconductor is the half space shown in Figure 7 and compared to schematics of practical experimental setups involving resonant coils and mi crowave cavities in Figs 7 a c In the gedcmken experiment Figure 7 a Half space geometry for penetration depth calculation b Resonant coil setup 0 Microwave cavity 28 case a DC eld is applied parallel to the sample surface and currents and elds are therefore functions only of the coordi nate perpendicular to the surface A Az etc Since we are interested in an external electromagnetic wave of very long wavelength compared to the sample size and zero frequency we need the limit w 0 q gt 00 of the response We will assume that in this limit Km 0 gt const which we will call C47T2 for reasons which will become clear Equation 63 then has the form c J T 47F This is sometimes called London s equation which must be solved in conjunction with Maxwell s equation 4 V X B V2A lj 2A 49 C 2A 48 which immediately gives A N Z and B Bee 2M The currents evidently screen the fields for distances below the surface greater than about A This is precisely the Meissner effect which therefore follows only from our assumption that KO 0 const A BCS calculation will now tell us how the penetration depth depends on temperature Evaluating the expressions in 44 in full generality is te dious and is usually done with standard many body methods beyond the scope of this course However for q 0 the cal culation is simple enough to do without resorting to Green s functions First note that the perturbing Hamiltonian H may 29 be written in terms of the quasiparticle operators 18 as H izk Am mC k Ukukq ukvkq 71q0711 Ykq1 Yk0l 6 7 kA0 l l 0 m6 7k07k0 7k17k1 If you compare with the A 0 Hamiltonian 23 we see that the new excitations of the system are Ek0 H Ek it A0 720 Ek1 gt Ek 71 A0 52 me We may similarly insert the quasiparticle operators 18 into the expression for the expectation value of the paramagnetic current operator41 0gt ltquotYilc0 Yk0 7117k1 m fEk0 fltEk1 We are interested in the linear response A gt 0 so that when we expand wrt A the paramagnetic contribution becomes 2 0gt 262 2 19f m c k 9Ek Combining now with the de nition of the response function K and the diamagnetic current 45 and recalling 2k gt ltjpara q WSW 53 Z k E k ltjpmltq k Alt0l k lt54 30 50 ukukq Ukvkq 711q07k0 Vilm ki 51 Nofd kdQ47r with N0 3n26F and fdQ47rkk l 3 we get for the static homogeneous response is therefore Koo 75 1 daltggfgt1 lt55 2 mile 56 where in the last step I de ned the super uid density to be n5T E n nnT with normal fluid density 9f nnT nalEC 8Ekgt Note at T 0 9f9Ek gt 0 Not a delta function as in the normal state caseido you see why while at T To the integral nn gt 111 Correspondingly the super uid density as de ned varies between n at T 0 and 0 at Tc This is the BCS microscopic justi cation for the rather successful phenomeno logical two uid model of superconductivity the normal uid consists of the thermally excited Bogoliubov quasiparticle gas and the super uid is the condensate of Cooper pairs12 57 Now let s relate the BCS microscopic result for the static homogeneous response to the penetration depth appearing in the macroscopic electrodynamics calculation above We nd immediately 2 WC 12 T 58 lt gt cmmep lt gt 11The dimensioness function nn TTCn is sometimes called the Yoshida function YT and is plotted in FigB 12The BCS theory and subsequent extensions also allow one to understand the limitations of the two uid picture for example when one probes the system at suf ciently high frequencies w N A the normal uid and super uid fractions are no longer distinct 31 1 1 YIT exp T 9quotC 0 TTC 1 0 WT 1 TTC 1 a b c Figure 8 a Yoshida function b super uid density c penetration depth At T 0 the supercurrent screening excludes the eld from all of the sample except a sheath of thickness 0 At small but nite temperatures an exponentially small number of quasiparticles will be excited out of the condensate depeleting the supercurrent and allowing the eld to penetrate further Both nnT and T 0 may therefore be expected to vary as eAT for T lt Tc as may be con rmed by explicit expansion of Eq 57 See homework Close to Tc the pen etration depth diverges as it must since in the normal state the eld penetrates the sample completely 573 Dynamical conductivity The calculation of the full frequency dependent conductivity is beyond the scope of this course If you would like to read an old fashioned derivation I refer you to Tinkham s book The main point to absorb here is that as in a semiconductor with a gap at T 0 there is no process by which a photon can be absorbed in a superconductor until its energy exceeds 2A 32 the binding energy of the pair This threshold for optical absorption is one of the most direct measurements of the gaps of the old superconductors 6 Ginzburg Landau Theory 61 GL Free Energy While the BCS weak coupling theory we looked at the last two weeks is very powerful and provides at least a qualita tively correct description of most aspects of classic supercon ductors13 there is a complementary theory which a is simpler and more physically transparent although valid only near the transition and b provides exact results under certain circum stances This is the Ginzburg Landau theory VL Ginzburg and LD Landau Zh Eksp Teor FiZ 20 1064 1950 which received remarkably little attention in the west until Gor kov showed it was derivable from the BCS theory LP Gor kov Zh Eksp Teor FiZ 36 1918 1959 The the ory simply postulated the existence of a macrosopic quantum wave function 1M7 which was equivalent to an order param eter and proposed that on symmetry grounds alone the free energy density of a superconductor should be expressible in terms of an expansion in this quantity fsfni 2 4 T ealwl mm z e 7A 2 2m1V C W lt59 13In fact one could make a case that the BCS theory is the most quantitatively accurate theory in all of condensed matter physics 33 where the subscripts n and s refer to the normal and super conducting states respectively Let s see why GL might have been led to make such a guess The superconducting normal transition was empiri cally known to be second order in zero eld so it was natural to write down a theory analogous to the Landau theory of a ferromagnet which is an expansion in powers of the mag netization M The choice of order parameter for the super conductor corresponding to M for the ferromagnet was not obvious but a complex scalar eld 1b was a natural choice because of the analogy with liquid He where MP is known to represent the super uid density 71514 a quantum mechani cal density should be a complex wave function squared The biggest leap of GL was to specify correctly how electromag netic elds which had no analog in super uid He would couple to the system They exploited in this case the simi larity of the formalism to ordinary quantum mechanics and coupled the elds in the usual way to charges 6 associated with particles of mass m Recall for a real charge in a magnetic eld the kinetic energy is 1 Zea 11 mil 7 Nev 412x11 lt mm gt 12mm 6 60 3 16 2 7 V iAkl 1 2mdrK 6 l 6 after an integration by parts in the second step GL just re placed 6 m with 6 m to obtain the kinetic part of Eq 59 141 in the He case has the microscopic interpretation as the Bose condensate amplitude 34 they expected that 6 and m were the elementary electron charge and mass respectively but did not assume so 5f TgtTc TltT VV lwl Figure 9 Mexican hat potential for superconductor A system described by this free energy will undergo a second order phase transition in zero eld when a 0 clearly when a is positive the system can minimize 6 f by having it 0 no superconductivity whereas if a is negative 6f has a mini mum with p y 0 The free energy 59 is a functional of the order parameter it meaning the actual value of the order parameter realized in equilibrium satis es Sf61D 015 No tice f is independent of the phase qb of the order parameter in E Wlew and so the ground state for a lt 0 is equivalent to any state it related to it by multiplication by a pure phase This is the U 1 gauge invariance of which we spoke earlier 15Thus you should not be perturbed by the fact that f apparently depends on 1 even for a gt 0 The value of f in equilibrium will be fn 0 35 This symmetry is broken when the system chooses one of the ground states phases upon condensation Fig 1 For a uniform system in zero eld the total free energy F f dgr f is minimized when f is so one nd for the order parameter at the minimum Wleq PEP2 a lt 0 62 lwleq 0 a gt 0 63 When or changes sign a minimum with a nonzero value be comes possible For a second order transition as one lowers the temperature we assume that a and b are smooth functions of T near Tc Since we are only interested in the region near Tc we take only the leading terms in the Taylor series expansions in this region aT H a0T Tc and b constant Eqs 62 and 63 take the form Mama WP2 T lt To lt64 WTMeq 0 T gt TC 65 Substituting back into Eqs59 we nd MT MT aim m T lt T lt66 07 T gt Tc The idea now is to calculate various observables and de termine the GL coef cients for a given system Once they are determined the theory acquires predictive power due to its extreme simplicity It should be noted that GL theory is applied to many systems but it is in classic superconductors 36 that it is most accurate since the critical region where de viations frorn mean eld theory are large is of order 10 4 or less Near the transition it may be taken to be exact for all practical purposes This is not the case for the HTSC where the size of the critical region has been claimed to be as much as 10 20K in some samples Supercurrents Let s now focus our attention on the term in the GL free energy which leads to supercurrents the kinetic energy part 1 3 ka 7 T2Tln d rvwWwv emArwnee These expressions deserve several remarks First note that the free energy is gauge invariant if we make the transforma tion fl gt 15 VA where A is any scalar function of position while at the same time changing w gt 1D exp ZeAc Sec ond note that in the last step above I have split the kinetic part of f into a term dependent on the gradients of the order parameter rnagnitude MM and on the gradients of the phase qb Let us use a little intuition to guess what these terms mean The energy of the superconducting state below To is lower than that of the normal state by an amount called the condensation energy16 From Eq 59 in zero eld this is of order WP very close to the transition To make spatial vari ations of the magnitude of 1D rnust cost a signi cant fraction of the condensation energy in the region of space in which it 16We will see below from the Gorkov derivation of GL from BCS that it is of order N0A2i Kvfew2 we 37 occurs17 On the other hand the zero eld free energy is ac tually invariant with respect to changes in qb so uctuations of qb alone actually cost no energy With this in mind let s ask what will happen if we apply a weak magnetic eld described by A to the system Since it is a small perturbation we don t expect it to couple to Wl but rather to the phase qb The kinetic energy density should then reduce to the second term in Eq 69 and furthermore we expect that it should reduce to the intuitive two uid ex pression for the kinetic energy due to supercurrents mn5v Recall from the super uid He analogy we expect WP E n to be a kind of density of superconducting electrons but that we aren t certain of the charge or mass of the particles So let s put W5 equotCA2w2 E lt70gt Comparing with Eq xx we nd that the super uid veloc ity must be 1 2771 1 ie 2i 3 fle 2mltv C Us 1 v 61 71 e mg C gt lt gt Thus the gradient of the phase is related to the super uid velocity but the vector potential also appears to keep the entire formalism gauge invariant Meissner effect The lleissner effect now follows imme diately from the two uid identi cations we have made The 17We can make an analogy with a ferromagnet7 where if we have a domain wall the magnetization must go to zero at the domain boundary7 costing lots of surface energy 38 supercurrent density will certainly be just gtllt gtllt My i am 7 gtlt j5 718119 mgtllt Taking the curl of this equation the phase drops out and we nd the magnetic eld 2 gtlt vX FB m m c Now recall the Maxwell equation j iv X g 74 47T which when combined with 14 gives 2 c a c a e n a 4VXVXB 7WB 53 75 47139 47F mc lt or a a VWBR m where W m02 47re2n 77 Notice now that if we use what we know about Cooper pairs this expression reduces to the BCS London penetration depth We assume 6 is the charge of the pair namely 6 2e and similarly m 2m and WP n 7152 since 71 is the density of pairs Flux quantization If we look at the ux quantization described in Part 1 of these notes it is clear from our sub sequent discussion of the Meissner effect that the currents 39 which lead to ux quantization will only ow in a small part of the cross section a layer of thickness A This layer encloses the ux passing through the toroid Draw a contour C in the interior of the toroid as shown in Figure 10 Then v5 0 everywhere on C It follows that Figure 10 Quantization of ux in a toroid 1 e 0f0drv5 fcdrv 4 The last integral may be evaluated using f0 d qu 27r gtlt integer and deE Sd Vgtltl iSds 6 icb C 78 79 80 81 82 Here S is a surface spanning the hole and I the ux through 40 the hole Combining these results I 27rn 71 71 83 2e 26 where n is a integer Do is the flux quantum and I ve rein serted the correct factor of h in the first step to make the units right Flux quantization indeed follows from the fact that the current is the result of a phase gradient18 Derivation from Microscopic Theory One of the reasons the GL theory did not enjoy much success at rst was the fact that it is purely phenomenological in the sense that the parameters do I m are not given within any micro scopic framework The BCS theory is such a framework and gives values for these coefficients which set the scale for all quantities calculable from the GL free energy The GL theory is more general however since eg for strong coupling su perconductors the weak coupling values of the coefficients are simply replaced by different ones of the same order of mag nitude without changing the form of the GL free energy In consequence the dependence of observables on temperature field etc will obey the same universal forms The GL theory was derived from BCS by Gor kov The calculation is beyond the scope of this course but can be found in many texts 18It is important to note however that a phase gradient doesnlt guarantee that a current is owing For example in the interior of the system depicted in Fig 2 both V45 and A are nonzero in the most convenient gauge and cancel each otherl 41 62 Type I and Type II superconductivity Now let s look at the problem of the instability of the normal state to superconductivity in nite magnetic eld H A what magnetic eld to we expect superconductivity to be destroyed for a given T lt Tc19 Well overall energy is conserved so the total condensation energy of the system in zero eld f5 fnT of the system must be equal to the magnetic eld energy I dBTH287T the system would have contained at the critical eld H c in the absence of the Meissner effect For a completely homogeneous system I then have MT MT Jig8 84 and from Eq 8 this means that near Tc 27mg b Whether this thermodynamic critical eld He actually rep resents the applied eld at which ux penetrates the sample depends on geometry We assumed in the simpli ed treat ment above that the eld at the sample surface was the same as the applied eld Clearly for any realistic sample placed in a eld the lines of eld will have to follow the contour of the sample if it excludes the eld from its interior This means the value of H at different points on the surface will be different the homogeneity assumption we made will not quite hold If we imagine ramping up the applied eld from H0 Tc 19Clearly it will destroy superconductivity since it breaks the degenerace of between the two componenets of a Cooper pair 42 zero there will inevitably come a point Ham Happg where the eld at certain points on the sample surface exceeds the critical eld but at other points does not For applied elds Happhc lt Ham lt H0 part of the sample will then be normal with local eld penetration and other parts will still exclude eld and be superconducting This is the intermediate state of a type I superconductor The structure of this state for a real sample is typically a complicated striped pattern of su perconducting and normal phases Even for small elds edges and corners of samples typically go normal because the eld lines bunch up there these are called quotdemagnetiZing effects and must be accounted for in a quantitatively accurate mea surement of say the penetration depth It is important to note that these patterns are completely geometry dependent and have no intrinsic length scale associated with them In the 50 s there were several materials known however in which the ux in suf ciently large elds penetrated the sample in a manner which did not appear to be geometry de pendent For example samples of these so called quottype If superconductors with nearly zero demagnetiZing factors long thin plates placed parallel to the eld also showed ux pen etration in the superconducting state The type ll materials exhibit a second order transition at nite eld and the ux B through the sample varies continuously in the supercon ducting state Therefore the mixed state must have currents owing and yet the Meissner effect is not realized so that the London equation somehow does not hold 43 The answer was provided by Abrikosov in 1957 AAA Sov Phys JETP 5 1174 1957 in a paper which Landau apparently held up for several years because he did not be lieve it Let us neglect the effects of geometry again and go back to our theorist s sample with zero demagnetizing factor Can we relax any of the assumptions that led to the Lon don equation 72 Only one is potentially problematic that 710 W0 constant independent of position Let s examineias Abrikosov didithe energy cost of making spatial variations of the order parameter The free energy in zero eld is F dgrlalwlZ 221va bun lt86 OF I gap d3r lw12 521 SW 87 where I ve put 1 E 7 2ma0Tc T Clearly the length 5 represents some kind of stz ness of the quantitiy W12 the super uid density Check that it does in deed have dimensions of lengthl If E the so called coherence length is small the energy cost of 715 varying from place to place will be small If the order parameter is somehow changed from its homogeneous equilibrium value at one point in space by an external force 5 speci es the length scale over which it heals We can then investigate the possibility that as the kinetic energy of super uid ow increases with increasing 1 12 2ma 112 lt88 44 eld if is small enough it might eventually become favorable to bend WP instead In typical type 1 materials T 0 is of order several hundreds or even thousands of Angstrom but in heavy fermion superconductors for example coher ence lengths are of order 50 100A The smallest coherence lengths are attained in the HTSC where ab is of order 12 15A whereas 50 is only 2 3A The general problem of minimizing F when it depends on position is extremely dif cult However we are mainly in terested in the phase boundary where w is small so life is a bit simpler Let s recall our quantum mechanics analogy once more so as to write F in the form F d3rlalwl2 W H lt wl mlw gt 89 where 1 mm is the operator 1 29 a 2 2mV C A 90 Now note 1 suf ciently close to the transition we may always neglect the 4th order term which is much smaller 2 to minimize F it suf ces to minimize lt gt since the WP term will simply x the overall normalization The variational principle of quantum mechanics states that the minimum value of lt H gt over all possible in is achieved when w is the ground state for a given normalization of in So we need only solve the eigenvalue problem Hmwy39 E j 91 45 for the lowest eigenvalue Ej and corresponding eigenfunction 1 For the given form of Elmn this reduces to the classic quantum mechanics problem of a charged particle moving in an applied magnetic eld The applied eld H is essentially the same as the microscopic eld B since 1D is so small at the phase boundary onlyl l ll remind you of the solution due to to Landau in order to x notation We choose a convenient 8351118397 A H 31 92 in which Eq 44 becomes 2 2 1i3i2ii 2m 951 932 922 where M cequotH12 is the magnetic length Since the co ordinates I and 2 don t appear explicitly we make the ansatZ of a plane wave along those directions W7 ijjv 93 w dwelm kzza 94 yielding 1 y 92 27mka 82 9732 t 93779 EWle 95 But this you will recognize as just the equation for a one dirnensional harmonic oscillator centered at the point 3 l x w with an additional additive constant leg27m in the energy Recall the standard harmonic oscillator equation 1 a 1 77 7k 2 II EKI 96 46 with ground state energy 000 1 E0 3 Ism 12 lt97 where k is the spring constant and ground state wayefunc tion corresponding to the lowest order Hermite polynomial x110 exp ml 412L 2 98 Let s just take over these results identifying A eH H m 7 99 k Wm ZWCWW The ground state eigenfunction may then be chosen as fl 2 139 x 139 z m WTQ We M k2 expl y ear2a 3 100 where L3 is the size of the sample in the y direction LmLyLz V 1 The wave functions are normalized such that d37 l kxkzl2 3 101 since I set the volume of the system to 1 The prefactors are chosen such that 8 represents the average superfluid den stity One important aspect of the particle in a field problem seen from the above solution is the large degeneracy of the ground state the energy is independent of the variable km for example corresponding to many possible orbit centers AWe have now found the wayefunction which minimizes lt H mm gt Substituting back into 89 we find using 99 6Hld3rlwl2bd3rlwl4 102 F T Tc a0 2mc 47 When the coef cient of the quadratic term changes sign we have a transition The eld at which this occurs is called the upper critical eld H02 2 gtllt H02T m mom T 103 6 What is the criterion which distinguishes type l and type II materials Start in the normal state for T lt Tc as shown in Figure 3 and reduce the eld Eventually one crosses either He or H02 rst Whichever is crossed rst determines the nature of the instability in nite eld ie whether the sample expels all the eld or allows ux vortex penetration see section C I Normal state Meissner phase Figure 11 Phase boundaries for classic type ll superconductor In the gure I have displayed the situation where H02 is higher meaning it is encountered rst The criterion for the dividing line between type 1 and type II is simply dHc i dHcg dT dT 104 48 or using the results 38 and 56 gtlt 2 2 m C b 1 105 7re2 2 This criterion is a bit dif cult to extract information from in its current form Let s de ne the GL parameter I4 to be the ratio of the two fundamental lengths we have identi ed so far the penetration depth and the coherence length K 106 Recalling that m02 mc2b A2 1 47re2n 27re2a lt 07 and 1 52 2ma 108 The criterion 58 now becomes 2b2 2 22b 1 2 mc7ream20 7 109 12ma 7re 2 Therefore a material is type I ll if I4 is less than greater than ln type l superconductors the coherence length is large compared to the penetration depth and the system is stiff with respect to changes in the superfluid density This gives rise to the Meissner effect where 715 is nearly constant over the screened part of the sample Type ll systems can t screen out the eld close to H02 since their stiffness is too 49 small The screening is incomplete and the system must de cide what pattern of spatial variation and flux penetration is most favorable from an energetic point of view The result is the vortex lattice first described by Abrikosov 63 Vortex Lattice I commented above on the huge degeneracy of the wave func tions 53 In particular for fixed 62 0 there are as many ground states as allowed values of the parameter km At H 02 it didn t matter since we could use the eigenvalue alone to de termine the phase boundary Below H 02 the fourth order term becomes important and the minimization of f is no longer an eigenvalue problem Let s make the plausible assumption that if some spatially varying order parameter structure is going to form below H02 it will be periodic with period 27rq ie the system selects some wave vector q for energetic reasons The x dependence of the wave functions 7T 2 t x W73 We 1 expl ykx f422 f4l no 3 is given through plane wave phases eff If we choose km qnx with 7195 integer all such functions will be invariant under 1 gt 1 27rq Not all nx s are allowed however the center of the orbit kx w should be inside the sample Ly2 lt Iger q wnx lt LyZ 111 50 Thus 7 is restricted such that 7y A 2q 2 2q 2 and the total number of degenerate functions is Lyq w nmax2 lt 7 lt Tamar2 112 Clearly we need to build up a periodic spatially varying structure out of wave functions of this type with centers distributed somehow What is the criterion which determines this structure All the wave functions 110 minimize lt HM gt and are normalized to f dgrlle WOW They are all therefore degenerate at the level of the quadratic GL free energy F f dgrlwl2 lt gt The fourth order term must stabilize some linear combination of them We therefore write 1M7 7 Cm mlt v 113 with the normalization EM lleQ 1 which enforces f dgrlwr 2 D3 Note this must minimize lt 1113 gt Let s therefore choose the CM and q to minimize the remaining terms in F f d3ralwl2 WEE Substituting and using the normalization condition and orthogonality of the different MCsz we find f mp3 51 114 with aH T a0T TC ne H02T 115 bH 5b 116 and 2 mar MM 0 0 o o 1 i 7111 7112 71x3 14 y 11 77112 7711377114 51 dz d Q nx1 nx2nx3nz4 X 118 dy 7271Eyqnz1gl2yqnx2l2yqn13gl2yqnz4 w2llt1 19 The form of fh o is now the same as in zero eld so we immediately nd that in equilibrium d woleq 21312 120 and a2 4E 121 This expression depends on the variational parameters Cm q only through the quantity 3 appearing in 5 Thus if we mini mize s we will minimize i remember I gt 0 so f lt 0 The minimization of the complicated expression with con straint EM lleZ 1 is dif cult enough that A Abrikosov made a mistake the rst time he did it and I won t in ict the full solution on you To get an idea what it might look like however let s look at a very symmetric linear combination one where all the Cnx s are equal 0 72343 122 Then I 1W N qu expl y nq 222 imla 123 which is periodic in x with period 27rq W 27W 2 1W 2 124 and periodic in q with period q l up to a phase factor W56 2 MW lq fc7 y 125 52 Note if q m M WP forms a square lattice The area of a unit cell is 2 q gtlt q w 27T W and the flux through each one is therefore 0 H g eH 2e Where I inserted a factor of h in the last step We haven t performed the minimization explicitly but this is a charac teristic of the solution that each cell contains just one flux quantum The picture is crudely depicted in Fig 12a Note by symmetry that the currents must cancel on the boundaries 1061 2a MH 2a ltigt0 126 5118 5116 Figure 12 a Square vortex lattice b triangular vortex lattice of the cells Since jg en5175 integrating qu 14 0 around each square must give as in our previous discussion of flux quantization in a toroid 1061 71 n integer 127 Somehow the vortex lattice consists of many such rings The problem with this idea is that the only way i W d sat around the boundary can be nonzero and the usual argument about single valuedness of the wave function carried through is if there is a hole in the wave function If there is no hole or region from which the wave function is excluded the path can be shrunk to a point but the value of the integral must 53 remain the same since the integrand is the gradient of a scalar eld This is unphysical because it would imply a nite phase change along an in nitesimal path and a divergence of the kinetic energy The only way out of the paradox is to have the system introduce its own hole in itself ie have the am plitude of the order parameter density WP go to zero at the center of each cell lntuitively the magnetic eld will have an accompanying maximum here since the screening tendency will be minimized This reduction in order parameter ampli tude magnetic ux bundle and winding of the phase once by 27 constitute a magnetic vortex which I ll discuss in more detail next time Assuming On constant which leads to the square lattice does give a relatively good small value for the dimensionless quantity 3 which turns out to be 118 This was Abrikosov s claim for the absolute minimum of f But his paper contained a now famous numerical error and it turns out that the actual minimum 8 116 is attained for another set of the Cn s to wit On 71 12 71 even 128 On ingif n odd 129 This turns out to be a triangular lattice Fig 12b for which the optimal value of q is found to be 3147T12 r Again the area of the unit cell is 27r 2 and there is one ux q 130 54 quantum per unit cell 64 Properties of Single Vortex Lower critical eld H01 Given that the ux per unit cell is quantized it is very easy to see that the lattice spacing d is actually of order the coherence length near H02 Using 103 and 88 we have i C 1 7 DO 02 652 2W5 On the other hand as H is reduced d must increase To see this note that the area of the triangular lattice unit cell is A dQ 2 and that there is one quantum of ux per cell A 10 H Then the lattice constant may be expressed as 74777 H02 de a 131 12 132 Since gtgt 5 is the length scale on which supercurrents and magnetic elds vary we expect the size of a magnetic vor tex to be about A This means at H02 vortices are strongly overlapping but as the eld is lowered the vortices separate according to 126 and may eventually be shown to in uence each other only weakly To nd the structure of an isolated vortex is a straightforward but tedious exercise in minimizing the GL free energy and in fact can only be done numerically in full detail But let s exploit the fact that we are inter ested primarily in strongly type ll systems and therefore go back to the London equation we solved to nd the penetration 55 depth in the half space geometry for weak elds allow n5 to vary spatially and look for vortex like solutions For example equation 75 may be written 2V gtlt V X E B 133 Let s integrate this equation over a surface perpendicular to the eld B B x y spanning one cell of the vortex lattice 2VXVX d d 134 4 a a 274j5d 0 135 But we have already argued that 5 d should be zero on the boundary of a cell so the left side is zero and there is a contradiction What is wrong The equation came from assuming a two fluid hydrodynamics for the superconductor with a nonzero n5 everywhere We derived it in fact from BCS theory but only for the case where n5 was constant Clearly there must be another term in the equation when a vortex type solution is present one which can only contribute over the region where the superfluid density is strongly varying in space ie the coherence length sized region in the middle of the vortex where the order parameter goes to zero vortex core Let s simply add a term which enables us to get the right amount of flux to Eq 133 In general we should probably assume something like A2v gtlt V X e E 09033 136 where 97 is some function which is only nonzero in the core The flux will then come out right if we demand I dgrgf 56 1 But let s simplify things even further by using the fact that 5 lt A let s treat the core as having negligible size which means it is just a line singularity We therefore put 977 6 Then the modi ed London equation with line singularity acting as an inhomogeneous source term reads A2V2 052062 137 1 a an 1 A2pap 6p 32 10620 138 where p is the radial cylindrical coordinate Equation 91 has the form of a modi ed Bessel s equation with solution 10 P BZ K i 27m 0A The other components of g vanish If you go to Abramowitz 85 Stegun you can look up the asymptotic limits Do A 139 B2 l i 0116 A 14 10 7T B2 7 pA 2M2 2pc pgtgtA 141 Note the form 93 is actually the correct asymptotic solution to 91 all the way down to p 0 but the fact that the solution diverges logarithmically merely re ects the fact that we didn t minimize the free energy properly and allow the order parameter to relax as it liked within the core So the domain of validity of the solution is only down to roughly the core size p 2 E as stated In Figure 5 I show schematically the structure of the magnetic and order parameter pro les in 57 an isolated vortex The solution may now be inserted into the lwl BZ o x P core lt gt supercurrents Figure 13 Isolated vortex free energy and the spatial integrals performed with some interesting results 13 It is easy to get an intuitive feel for what this means since if we assume the eld is uniform and just ask what is the magnetic energy we get roughly 1 Fv 87 X vortex volume X B2 143 7r 1 g 87 X mm X cmmo 144 7r 12 Lz O 145 87T22 lt the same result up to a slowly varying factor Now the lower critical eld H01 is determined by requiring the Gibbs free energies of the Meissner phase with no vortex be equal to the Gibbs free energy of the phase with a vortex20 G differs from 20We haven t talked about the Gibbs vsi Helmholtz free energy but recall the Gibbs is the appropriate potential to use when the external eld H is held xed7 which is the situation we always have7 with a generator supplying work to maintain Hi 58 F through a term f BH47r In the Meissner state G F so we may put 1 F F ElmeLz 471101 Bd 146 7T 1 F ElmeLZ 7 0Lm 147 47F Where Elm is the free energy per unit length of the vortex itself Therefore 7 47TElme Do is the upper critical eld But the line energy is given precisely 2 by Eq 95 Elm loglt so H01 Helm logo 149 65 Josephson Effect 59 Contents 1 Introduction 2 11 Goals in this course 2 12 Statistical mechanics of free Fermi gas 2 121 T O Fermi sea 2 122 T gt 0 Free energy 4 123 Avg fermion number 4 124 Fermi gas at low T 5 125 Classical limit 8 13 Second quantization 8 131 Symmetry of many particle wavefunctions 9 132 Field operators 11 133 2nd quantized Hamiltonian 13 134 Schrodinger Heisenberg interaction representations 15 14 Phonons 17 141 Review of simple harmonic oscillator quantization 17 142 1D harmonic chain 18 143 Debye Model 20 144 Anharmonicity amp its consequences 22 1 Introduction 11 Goals in this course These are my hopes for the course Let me know if you feel the course is not ful lling them for you Be free with your criticism amp comments Thanks 0 Teach basics of collective phenomena in electron systems 0 Make frequent reference to real experiment and data 0 Use 2nd quantized notation without eld theoretical techniques 0 Get all students reading basic CM journals 0 Allow students to practice presenting a talk 0 Allow students to bootstrap own research if possible 12 Statistical mechanics of free Fermi gas 121 T O Fermi sea Start with simple model of electrons in metal neglecting 6 6 interac tions Hamiltonian is A h2v2 H Z 7 391N t39l 1 Mm 7 ltgt Eigenstates of each h2V22m are just plane waves ed labelled by k with k 27rmL in box with periodic BC Recall electrons are fermions which means we can only put one in each singleparticle state lncluding spin we can put two particles in each k state At zero temperature the ground state of N electron system is then formed by adding particles until we run out of electrons Energy is 6k h2k2 2m so start with two in lowest state k 0 then add two to next states with 95 or Icy 2 or k2 27TL etc as shown Energy of highest particle called Fermi energy 5F magnitude of corresponding wave vector called kp Typical Fermi energy for metal 5F 2 16V 2 104K At T 0 only states with k lt kp occupied Fermi sea or Fermi sphere so we can write density of electrons as 2 occupied states Volume 2 is for spin 2 kF d3k 1 z 2 16 EkOEZwamf O Mk2 2 SO 2 2 23 h 3 la 37r2n13 or 5F 2 W n 3 2m in other words nothing but the density of electrons controls the Fermi energy 8 A Figure 1 States of Fermi gas with parabolic spectrum 8 1622771 The total ground state energy of the Fermi gas must be of order 5F since there is no other energy in the problem If we simply add up the energies of all particles in states up to Fermi level get E 1 W2 m5 de 2 2 F 4 L3 7T2 0 2m 107T2m and the ground state energy per particle N nL3 is the total number is 5 55F 5 122 T gt 0 Free energy Reminder partition function for free fermions in grnd conical ensemble is Z Tr e mg m lt6 Z n1n2n00e H Nln1n2noogt 7 n1n2nkoo Z n1n2noole lxileini t mln1n2noogt 8 Where t labels singlefermion state eg t k a and 7 runs from O to 1 for fermions Since many fermion state in occ no representation is simple product n1n2noogt n1gtn2gtnoogt can factorize Z Z e lemimil Z eBlenooWoolgt 7 9 711 7100 so Z 1130 1 yaw 10 Since the free energy grand canonical potential is Q kBT log Z we get lo kBT 23 log 1 e 8i gtl n 123 Avg fermion number We may want to take statistical averages of quantum operators for which we need the statistical operator 5 Z 1e mH t Nl Then any operator 9 has an expectation value 9 Tr59 For example avg no of particles ltNgt mm A lt12gt T MH MNUV fle AA lt13 Trlte H MNgt 4 Now note this is related to derivative of Q wrt chem potential u 99 alogZ i kBT8 7 k T i 14 8 B an 2 a lt gt 1 A A 7TYWW ltNgt lt15 and using Eq 11 we see N 00 E 00 Q lt gt gilJreWErM 1712 16 where 71 is the avg number of fermions in a singleparticle state i in equilibrium at temperature T lf we recall 139 was a shorthand for k a but 6k doesn7t depend on a we get Fermi Dirac distribution function 1 1 665k l lt17 0 i nkai 124 Fermi gas at low T Since the Fermi energy of metals is so high N 104K its important to understand the limit kBT ltlt 61 where the Fermi gas is nearly degen erate and make sure the classical limit kBT gtgt 6F comes out right too Letls calculate for example the entropy and speci c heat which can be obtained from the thermodynamic potential Q via the general thermody namic relations 90 98 S 7 0 T 7 18 lt8TgtVM7 V lt8TgtVu l gt From 11 and 16 and including spin we have n 2kBTZlog i 6 50ka k 2kgTL3 d5Ng log 1 ew gt Cvi l 00 9 2 CV FiZkBiTA d5N5lt88gt5 p 2 kBiT Lids MS 2 19 5 where I introduced the density ofk states for one spin Ne L gzk e 6k The Fermi function is e 11exp gs M amp l de ned shifted energy variable 5 e u ln general the degenerate limit is characterized by k sums which decay rapidly for energies far from the Fermi surface so the game is to assume the density of states varies slowly on a scale of the thermal energy and replace N e by N e F E No This type of Sommer feld expansion1 assumes the density of states is a smoothly varying fctn ie the thermodynamic limit V gt 00 has been taken otherwise N e is too spikyl For a parabolic band ek thQQm in 3D the delta fctn can be evaluated to find2 Ne 912 95 22 This can be expanded around the Fermi level3 1 M5 NO N O infmg 24 ln a horrible misuse of notation NO NeF and N0 all mean the 11f you are integrating a smooth function of 5 multiplied by the Fermi function derivative 781685 the derivative restricts the range of integration to a region of width kBT around the Fermi surface If you are integrating something times itself7 itls convenient to do an integration by parts The result is see eg Ashcroft amp Mermin appendix C jo d5H5f5 M d5H5 Z ankBT2n dd11115M 20 where an 2 7 122 1 is Riemann Q fctn7 736 7r4907 etc 2Here7s one way to get this N6gtL 36Eeekgtell 16kegt aki sgggf 21gt 3When does the validity of the expansion break down When the approximation that the density of states is a smooth function does7 ie when the thermal energy kBT is comparable to the splitting between states at the Fermi leve h2kpdk 6k a 65 2 7 2 5 i 2 5 7 23 klap m F kF F L7 where a is the lattice spacing and L is the box size At T 1K requiring kBT N 65 and taking EFkB 2 104K says that systems boxes of size less than 1mm will show mesoscopic77 effects7 ie results from Sommerfeld type expansions are no longer valid Figure 2 Density of states for parabolic spectrum 8 k2 2m density of states at the Fermi level The leading order term in the low T speci c heat is therefore found directly by scaling out the factors of T in Eq 19 1 00 8f 00 8f CV 2 263 TNO LOO df39 g2 2cBTkOO dx 324 25 7T23 So 271392 3 This is the famous linear in temperature speci c heat of a free Fermi gas 4 4Note in 26 I extended the lower limit u of the integral in Eq 19 to oo since it can be shown that the chemical potential is very close to SF at low T Since we are interested in temperatures kBT ltlt 85 and the range in the integral is only several kBT at most this introduces neglible error Why At T0 the Fermi function nil gt step function 0u 8k so we know uT 0 must just be the Fermi energy 8F 71237T2n232m N 00 k 2 d5N5f5 2 u 2 d8N8 l BT2N8 M continued on next page 125 Classical limit l won7t calculate the classical limit All the standard results for a Boltzman statistics gas eg CVT gtgt 6F 32NkB follow immediately from noticing that the Fermi function reduces to the Boltzman distribution e gt e 8 T gt oo 27 You will need to convince yourself that the classical result nkBT gt 00 is recovered to make this argument 13 Second quantization The idea behind the term second quantization arises from the fact that in the early days of quantum mechanics forces between particles were treated classically Momentum position and other observables were represented by operators which do not in general commute with each other Particle number is assumed to be quantized as one of the tenets of the theory e g Einstein7s early work on blackbody radiation At some point it was also realized that forces between particles are also quantized because they are mediated by the exchange of other particles ln Schrodinger7s treatment of the H atom the force is just the classical static Coulomb force but a more complete treatment includes the interaction of the H atom and its constituents with the radiation eld which must itself be quantized photons This quantization of the elds mediat ing the interactions between matter particles was referred to as second quantization ln the meantime a second quantized description has been developed in which both matter elds and force elds are described 2 EF daN5 M 7 8FN8F 2kBT2N 5lEM 700 N EF N 5F 7T2 2 n 2 SF 7 F0631 Since N N is typically of order l8 corrections are small 8 by second quantized eld operators ln fact modern condensed matter physics usually does go backwards and describe particles interacting via classical Coulomb forces again5 but these particles are described by eld operators Once the calculational rules are absorbed calculating with the 2nd quantized formalism is easier for most people than doing 1st quantized calculations They must of course yield the same answer as they are rigorously equivalent l will not prove this equivalence in class as it is exceedingly tedious but merely motivate it for you below l urge you to read the proof once for your own edi cation however6 131 Symmetry of manyparticle wavefunctions Quantum mechanics allows for the possibility of indistinguishable parti cles and nature seems to have taken advantage of this as a way to con struct things No electron can be distinguished from another electron except by saying where it is what quantum state it is in etc lnter nal quantum mechanical consistency requires that when we write down a many identical particle state we make that state noncommittal as to which particle is in which singleparticle state For example we say that we have electron 1 and electron 2 and we put them in states a and b re spectively but exchange symmetry requires since electrons are fermions that a satisfactory wavefunction has the form W17 1 2 Alltwaltr2gtwbltr1gt altr1gt bltr2gtl 28gt lf we have N particles the wavefunctions must be either symmetric or antisymmetric under exchange7 IIBr1rZrjrN kllBr1rjrirN Bosons 29 llFr1rZrjrN llFr1rjrirN Fermions 30 5Q when does this approximation break down 6See7 eg Fetter amp VV39allecka7 Quantum Theory of ManyParticle Systems 7Recently7 a generalization of Bose amp Fermi statistics to particles called anyons has been intensely discussed Under exchange an anyon wavefunction behaves as 11 r1 Uri Hrj rN ei 11Ar1rjrirN for some 0 lt 9 lt 27f Given a set of single particle wave functions ngr where E is a quan tum number e g energy we can easily construct wave fctns which satisfy statistics e g B l I 112 2 i1 Semen mm 31 pEE1E2EN Remarks on Eq 31 0 sum all permutations of the Ei s in product E1r1ng2r2 ngNrN 8 o distinct Ei s occuring may be less than N some missing because of multiple occupation in boson case Example particles E3 11150100013917139271393 E2 0 E1ltr1gt altr2gt altr3gt E1 E31 1 E11 2 E11 3 E1r1 E3r2 E1r3 E0 N Fig 2 Possible state of 3 noninteracting Bose 0 Completely antisymmetric Fermionic wavefunction called Slater de terminant 1 12 Emmr1 Emaxr1 i1oor1 rN m s 32 CbEmmltFNgt EmaxI39N where there are N eigenvalues which occur between Emm and Em inclusive corresponding to N occupied states 8You might think the physical thing to do would be to sum all permutations of the particle labels This is correct but actually makes things harder since one can double count if particles are degenerate see example of 3 bosons below The normalization factor is chosen with the sum over all permutation of the Ei s in mind 10 132 Field operators 2nd quantization is alternative way of describing many body states We describe a particle by a eld operator wltrgt yamlt1 lt33gt where i runs over the quantum numbers associated with the set of eigen states gb a is a coef cient which is an operator l7m going to neglect the hats A which normally denote an operator for the as and abs and ngZ is a lst quantized wavefunction ie a normal Schrodinger wave function of the type we have used up to now such that for example H gb El Now we impose commutation relations erl lfr i 51 Iquot 34 erl rwi l lfrll lfr i 0 35 which implies an alli 5a lam ajli lly alli 039 36 The upper sign is for fermions and the lower for bosons in Eqs 35 and 36 Now construct many body states from vacuum no particles a called annihilation operator al creation operator see below Examples amp comments all properties follow from commutation rela tions m 0 one particle in state i aZlIO E 1 o annihilate vacuum a 0 O o Bosons9 alallO E 2gti gAnalogous state for fermions is zero7 by commutation relationsCheck 11 o Bosons two in i and one in j alalal0gt E 2gt 1gtj o alazr E m is number operator for state 239 Proof bosons dialMn e alazXahnlo all alaigtltalgt 1logt Int aIVaZcDHlm 2ngti aDSaZabr m nngti Similarly show bosons10 aim 71 112n 1gt o a ngt 711an 1gt o many particle state 1 12 n n W ll 1 ll2WIOgtE7quotL1n2noogt 37 9lt occupation numbers specify state completely exchange symmetry included due to commutation relations normalization factor left for problem set Fermions o Anticommutation relations complicate matters in particular note Pauli principle 38 10By now it should be clear that the algebra for the bosonic case is identical to the algebra of simple harmonic oscillator ladder operators ll0gt ID ID 0gt a a 39 al1gt0 a0gt0 l l o rnany particle state al 1a 2 0gt 2 m1 712 7100 40 9lt note normalization factor is 1 here 0 action of creation amp annilation operators suppose n5 1 asl ns lt1gtn1n2n371ltab711asaltaiogtnw0gt lt1gtn1n2n371ltabmlt1 aiogtnoo0gt 0 1mn2nsel H715 1gt also a50gt0 41 and similarly lt1gtn1n2n371 715 1 n5 O l a5n5gt 0 81 42 133 2ndquantized Hamiltonian 9lt Point Now it can be shown7711 that state vector Z fn1n2 nootn1n2 7100 43 n1 77121100 11Normally I hate skipping proofsi However7 as mentioned above7 this one is so tedious I just can t resist The braveheartecl can look7 eg in Chapter 1 of Fetter and Walleckai satis es the Schrodinger equation our old friend mghwc l wc ea if we take the 772nd quantized form of H ngmnntig hw nwmmk em rwwwmme ltm 1 A A A A d3quot dgr t WrWWr39llr HWYWI 47 where the 1st quantized Hamiltonian was H T V Translationally invariant system lt may be more satisfying if we can at least verify that this formalism works for a special case eg translationally invariant systems For such a system the momentum k is a good quantum number so choose singleparticle plane wave states r gbkgr L gQelk39rug 48 where my is a spinor like ui 1 etc 1st quantized T is V22m12 so 2nd quantized kinetic energy is A k2 TEZthmbum mm W k 2m 039 Since we showed allwakg is just number operator which counts of par ticles in state k0 this clearly just adds up the kinetic energy of all the occupied states as it should For general two particle interaction V let7s assume Vrr Vltr Iquot only as must be true if we have transl invariance ln terms of the Fourier transform 1 239 I39 Vq dgr 6 q Vr 50 121711 set E 1 from here on out7 unless required for an honest physical calculation 14 we have steps left as exercise A 1 V i Z alltaalltqUVltqgtakUakq0 51gt kkq 00 134 Schrodinger Heisenberg interaction representations Here we just give some de nitions amp reminders on different equivalent representations of quantum mechanics ln different reps time depen dence is associated with states Schrodinger operators Heisenberg or combination interaction 0 Schrodinger picture state 1 5t operators 55 St a A A A 25W H Slttgtgt has formal solution Wt emlttt0gtzislttogtgtl lt52gt Note H hermitian gt time evolution operatorm is uni tary 0 Interaction picture useful for pert thy 1 10 1 Where 10 usually soluble lttgtgt WW Def in terms of Schr picture A eiHot seiiHot willie H tll 1tgtgt 15 with H t elHOtIile mot Remarks states and ops t dependent in interaction picture but time de pendence of operators very simple eg H0 gekallak i iHQt A 2th zatak t 7 e ak7 Hole ekak1t gt ak1t ake lgkt Time evolution operator determines state at time t 1lttgtgt Wt togtlw1togtgt From Schrodinger picture we nd Ultt7 to H0t67 Htit0 H0t0 Note 10 y O o Heisenberg picture state t independent7 operators elHl ge lHl so operators evolve according to Heisenberg eqn of motion a A A ZEOHW OHM Hl 54gt 9lt Noteecompare three reps at t O we zgsltogtgt g 1ltogtgt lt55gt 95 9HO 910 56 16 14 Phonons 141 Review of simple harmonic oscillator quantization l Will simply write down some results for the standard SHO quantization from elementary QM using ladder operators We consider the Hamilto nian 2 K P and extract the relevant dimensions by putting K 2 i 7 w i M Ma 12 5 q in 8 712 1875 pth 58 so h 82 w i 7 7952 59 We recall soln goes like 6 522Hnf Where H n are Hermite polynomials and that eigenvalues are En hwltn 12 60 De ne ladder operators a al as a j ltggggt 61 Ladder ops obey commutation relations check aal1 aa0 alal0 63 Figure 3 Linear chain with spring couplings K Dynamical variables are 11 E xi 7 xi amp then H may be written check i 1 Hhw aa 64 a al connect eigenstates of different quantum nos 71 as al ngt EM2m 65 where Ogt is state which obeys a0gt 0 Operating on with al may be shown with use of commutation relations to give alngt n 1212n 1 aln n12n 1 66 so with these defs the ladder operators for SHO are seen to be identical to boson creation and annihilation operators de ned above in Sec 132 142 1D harmonic chain If we now consider N atoms on a linear chain each attached to its neighbor with a spring of spring constant K as shown in gure First letls consider the problem classically The Hamiltonian is p K H 7 7 2 67 2m 2W QZ1gt7 lt l 18 Where the qs are the displacements from atomic equilibrium positions Now Hamiltonls eqns or Newton7s 2nd law yield 4W1 MWQCIJ39 2617 1771 am 68 A standing sinusoidal wave qj Acosltkajgt satis es this equation if the eigenfrequencies have the form K to M20 cos ha 69 Where if a is the lattice constant k 27r Note that for small k wk is linear wk 2 KM12ka13 This is the classical calculation of the normal modes of oscillation on a 1D chain To quantize the theory let7s impose canonical commutation relations on the position and momentum of the Eth and jth atoms giapjl 1715M lt70 and construct collective variables Which describe the modes themselves recall k is wave vector E is position 1 Mai 1 iika Ge mge Qka QkW Ge 1 iika 1 Mai Which leads to canonical commutation relations in wave vector space 1 7239 a 239 am Qt PM N E e k 16 k qtpmi Am Hi I 7 Z e wlltk kgt M6 4 72 N g 7 Let7s novv express the Hamiltonian 67 in terms of the new variables We have With a little algebra and Eq 69 gr glam 73gt M 2 MW K g lz 15712 3 ZleinkQ 6m 67m gal3C2ka 13Note since k 27rnNa the wk are independent of system size 19 SO 74 l M 117 7 2 2M2ka k 2 ZklkakQ k Note that the energy is now expressed as the sum of kinetic potential energy of each mode k and there is no more explicit reference to the motion of the atomic constituents To second quantize the system we write down creation and annihilation operators for each mode k De ne a an which can be shown just as in the single SHO case to obey commutation relations 0 ILl 6kk laka akl O al7 all O 79 and the Hamiltonian expressed simply as 1 Z hwk alak 2 80 k which we could have probably guessed if we had realized that since the normal modes don7t interact we have simply the energy of all allowed harmonic oscillators Note this is a quantum Hamiltonian but the energy scale in H is the classical mode frequency wk 143 Debye Model Let us imagine using the Hamiltonian 80 as a starting point to calculate the speci c heat of a 3D solid due to phonons We take the Debye model for the dispersion to simplify the calculation ck k lt kD wk 0 k gt kip lt81 20 Where the Debye wave vector kD 6W2n13 is obtained by replacing the rst Brillouin zone of the solid by a sphere of radius kD which contains N wave vectors With N the number of ions in the crystal The average value of the Hamiltonian is 1 1 1 UH3 hwltala gt3 hw ltgt 82 since the average number of phonons in state k is simply the expectation value of a boson nurnber operator ak E Trpalak bhwk 83 Where 936 expwaj 1 1 is the free Bose distribution function The factors of 3 come from the 3 independent phonon polarizations Which we consider to be degenerate here Taking one derivative wrt temperature the spec heat per unit volume is i332 hck iggk kD k3 8Tn 8T k e hck 1 8T27r2 0 e hck 1 83kT4oo3 87r2kT4 2w kT3 B 330 B l kbltffcgtlt84gt CV7 2 97 2W2lth6 695 1 T hc3 T 5 415 So far we have done nothing Which couldnt have been done easily by or dinary 1st quantized niethods l have reviewed sonie Solid State l material here by way of introduction to problems of interacting particles to Which you have not been seriously exposed thus far in the condensed matter grad sequence The second quantization niethod beconies nianifestly useful for the analysis of interacting systenis l will now sketch the formulation not the solution of the problem in the case of the phonon phonon interaction in the anharmonic crystal 21 144 Anharmonicity 85 its consequences As you will recall from Solid State I most thermodynamic properties of insulators as well as neutron scattering experiments on most materials can be explained entirely in terms of the harmonic approximation for the ions in the crystal ie by assuming a Hamiltonian of the form 80 There are some problems with the harmonic theory First at higher tempera tures atoms tend to explore the anharmonic parts of the crystal potential more and more so deviations from predictions of the equilibrium theory increase The thermal expansion of solids in the harmonic approxima tion is rigorously zero14 Secondly some important qualitative aspects of transport cannot be understood for example the harmonic theory pre dicts in nite thermal transport of insulators See Aampll Ch 25 for a qualitative discussion of these failures The obvious way to go beyond the harmonic approximation is to take into account higher order corrections to the real space crystal potential systematically expanding15 1 1 U 5 2192 WCqu g 2 DWE m nngmqn 86 gm Zmn where 9 U Dln m 70 87gt 14This follows from the independence of the phonon energies in harmonic approx of the system volume see above Since pressure depends on temperature only through the volume derivative of the mode freqs see A amp M p 4907 W e 3T V 7 0 lt 8T gt Q 85 1 3V T 151 have dropped polarization indices everywhere in this discussion so one must be careful to go back and put them in for a 2 or 3D crystal 22 are the so called dynamical matrices16 Using Eqs 7176 we nd 1 h 12 39k Z T 39k Z Z a Z a qg m Zngke kltak aikgte lt gt Note that the product of 3 displacements can be written 1 i 2k1a k2amk3an m n i e so the cubic term may be written H3 2 vlt3gtltk1k2k3gtoklckgok3 lt90 klkgkg with V3ltk1k2k3gt Z 9607 m7 n expik1 kgm kgml 91 mm Note now that the indices E m 71 run over all unit cells of the crystal lattice Since crystal potential itself is periodic the momenta k1 kg and kg in the sum are not really independent ln fact if we shift all the sums in 91 by a lattice vector j we would have V3ltk1k2k3gt Z 177 m 177 n jgt k1aik2amk3an k1k2k3aj mm 2 m7 ngt k1aik2amk3an k1k2k3aj Zmn where in the last step l used the fact that the crystal potential U in every lattice cell is equivalent Now sum both sides over j and divide by N to nd V3k1k2k3 Dlt3gtr m n expik1 kgm kgnAkJ1 k2 k3 mm 93 16Recall that the theory with only harmonic and cubic terms is actually formally unstable7 since arbitrarily large displacements can lower the energy by an arbitrary amount If the cubic terms are treated perturbatively7 sensible answers normally result It is usually better to include quartic terms as shown in gure below7 however 23 where 1 1 A k i ltk1k2k3gtaj i 6 k G 94 lt gt N e N g lt gt lt gt and G is any reciprocal lattice vector Return now to 90 We have ascertained that Vlglw 77171 is zero unless k1 k2 k3 G ie crystal momentum is conserved lf we expand 90 we will get products of 3 creation or annihilation operators with coe icients Vlg The values of these coef cients depend on the elastic properties of the solid and are unimportant for us here The momenta of the three operators must be such that momentum is conserved up to a reciprocal lattice vector eg if we consider the term aklalkgalkg we have a contribution only from 1 k2 k3 G Note this term should be thought of as corresponding to a physical process wherein a phonon with momentum k1 is destroyed and two phonons with momenta kg and k3 are created lt can be drawn 77diagrammatically77 a la Feynman as the 1st of 2 3rd order processes in the gure below k p q k q p kgtltq p 339 ql D D Figure 4 Diagrams representing phonon phonon collision processes allowed by energy and momen tum conservation in 3rd and 4th order 17As usual7 processes with G 0 are called normal processes7 and those with nite G are called Umklapp processes 24 Questions 0 How does energy conservation enter What is importance of processes involving destruction or creation of 3 phonons o lf one does perturbation theory around harmonic solution does cubic term contribute to thermal averages 0 Can we calculate thermal expansion with cubic Hamiltonian 25 Contents 5 Superconductivity 1 51 Phenomenology 1 52 Electron phonon interaction 3 53 Cooper problem 6 54 Pair condensate 85 BCS Wavefctn 11 55 BCS llodel 12 551 BCS wave function7 gauge invariance7 and number con servation 14 552 ls the BCS order parameter general 16 56 Thermodynamics 13 561 Bogoliubov transformation 18 562 Density of states 19 563 Critical temperature 21 564 Speci c heat 23 57 Electrodynamics 25 571 Linear response to vector potential 25 572 Meissner Effect 27 573 Dynamical conductivity 32 6 Ginzburg Landau Theory 33 61 GL Free Energy 33 62 Type I and Type II superconductivity 42 63 Vortex Lattice 50 64 Properties of Single Vortex Lower critical eld H01 55 65 Josephson Effect 59 5 Superconductivity 51 Phenomenology Superconductivity was discovered in 1911 in the Leiden lab oratory of Kamerlingh Onnes when a so called blue boy local high school student recruited for the tedious job of mon itoring experiments noticed that the resistivity of Hg metal vanished abruptly at about 4K Although phenomenological models with predictive power were developed in the 30 s and 40 s the microscopic mechanism underlying superconductiv ity was not discovered until 1957 by Bardeen Cooper and Schrieffer Superconductors have been studied intensively for their fundamental interest and for the promise of technologi cal applications which would be possible if a material which superconducts at room temperature were discovered Until 1986 critical temperatures Tc s at which resistance disap pears were always less than about 23K In 1986 Bednorz and Mueller published a paper subsequently recognized with the 1987 Nobel prize for the discovery of a new class of materials which currently include members with Tc s of about 135K ll r r T TC T Figure 1 Properties of superconductors Superconducting materials exhibit the following unusual be haViors 1 Zero resistance Below a material s TC the DC elec trical resistivity p is really zero not just very small This leads to the possibility of a related effect 2 Persistent currents If a current is set up in a super conductor with multiply connected topology e g a torus 2 it will flow forever without any driving voltage In prac tice experiments have been performed in which persistent currents flow for several years without signs of degrading 3 Perfect diamagnetism A superconductor expels a weak magnetic eld nearly completely from its interior screening currents flow to compensate the eld within a surface layer of a few 100 or 1000 A and the field at the sample surface drops to zero over this layer 4 Energy gap llost thermodynamic properties of a su perconductor are found to vary as eAkBT indicating the existence of a gap or energy interval with no allowed eigenenergies in the energy spectrum ldea when there is a gap only an exponentially small number of particles have enough thermal energy to be promoted to the avail able unoccupied states above the gap In addition this gap is visible in electromagnetic absorption send in a photon at low temperatures strictly speaking T 0 and no absorption is possible until the photon energy reaches 2A ie until the energy required to break a pair is available 52 Electronphonon interaction Superconductivity is due to an effective attraction between conduction electrons Since two electrons experience a re pulsive Coulomb force there must be an additional attrac tive force between two electrons when they are placed in a metallic environment In classic superconductors this force 3 is known to arise from the interaction with the ionic system In previous discussion of a normal metal the ions were re placed by a homogeneous positive background which enforces charge neutrality in the system In reality this medium is polarizablei the number of ions per unit volume can fluctuate in time In particular if we imagine a snapshot of a single electron entering a region of the metal it will create a net positive charge density near itself by attracting the oppositely charged ions Crucial here is that a typical electron close to the Fermi surface moves with velocity v1 hkp m which is much larger than the velocity of the ions 1 VFmM So by the time T N 27TwD N 10 13 sec the ions have polarized themselves 1st electron is long gone it s moved a distance UFT N 1080ms N 1000K and 2nd electron can happen by to lower its energy with the concentration of positive charge before the ionic fluctuation relaxes away This gives rise to an effective attraction between the two electrons as shown which may be large enough to overcome the repulsive Coulomb in teraction Historically this electron phonon pairing mech anism was suggested by Frolich in 1950 and con rmed by the discovery of the isotope effect wherein T0 was found to vary as M 1 2 for materials which were identical chemically but which were made with different isotopes The simplest model for the total interaction between two electrons in momentum states k and k with q E k k in teracting both by direct Coulomb and electron phonon forces is given by t0 tt1 Figure 2 Effective attraction of two electrons due to phonon exchange 47m 2 2 q 1 7 q2k q2k wQ wg 2 Vq w in the jellium model Here first term is Coulomb interac tion in the presence of a medium with dielectric constant 6 l 12 and tag are the phonon frequencies The screening length CS 1 is 1A or so in a good metal Second term is interaction due to exchange of phonons ie the mech anism pictured in the figure Note it is frequency dependent re ecting the retarded nature of interaction see gure and in particular that the 2nd term is attractive for w lt wq N cap Something is not quite right here however it looks indeed as though the two terms are of the same order as w gt O indeed they cancel each other there giving Vw gt O 0 Further more V is always attractive at low frequencies suggesting that all metals should be superconductors which is not the case These points indicate that the jellium approximation is too simple We should probably think about a more care ful calculation in a real system as producing two equivalent terms which vary in approximately the same way with km and wq but with prefactors which are arbitrary In some ma 5 terials then the second term might win at low frequencies depending on details 53 Cooper problem A great deal was known about the phenomenology of super conductivity in the 1950 s and it was already suspected that the electron phonon interaction was responsible but the mi croscopic form of the wave function was unknown A clue was provided by Leon Cooper who showed that the noninter acting Fermi sea is unstable towards the addition of a single pair of electrons with attractive interactions Cooper began by examining the wave function of this pair 01 m which can always be written as a sum over plane waves 1W1 r2 Ukqelk e kqlf2 2 where the ukq are expansion coefficients and C is the spin part of the wave function either the singlet l Ti iTgt or one of the triplet l TTgtl iigtl Ti iTgt In fact since we will demand that w is the ground state of the two electron system we will assume the wave function is re alized with zero center of mass momentum of the two elec trons ukq ukdqp Here is a quick argument related to the electron phonon origin of the attractive interaction1 The electron phonon interaction is strongest for those electrons with single particle energies 5k within COD of the Fermi level In the scattering process depicted in Fig 3 momentum is 1Thanks to Kevin McCarthy7 who forced me to think about this further 6 kq Figure 3 Electrons scattered by phonon exchange are con ned to shell of thickness rap about Fermi surface explicitly conserved ie the total momentum k p K 3 is the same in the incoming and outgoing parts of the diagram Now look at Figure 4 and note that if K is not N 0 the phase space for scattering attraction is dramatically reduced So the system can always lower its energy by creating K 0 pairs Henceforth we will make this assumption as Cooper did Then 01 73 becomes 2 ukelk lrrml Note that if uk is even in k the wave function has only terms oc cos 16 71 73 whereas if it is Odd only the sin 16 71 73 will contribute This is an important distinction because only in the former case is there an amplitude for the two electrons to live quoton top of each other at the origin Note further that in order to 7 Figure 4 To get attractive scattering with nite cm momentum K7 need both electron energies to be within Lap of Fermi leveli very little phase space preserve the proper overall antisymmetry of the wave function uk even odd in 16 implies the wave function must be spin singlet triplet Let us assume further that there is a general two body interaction between the two electrons the rest of the Fermi sea is noninteracting in the model V7 17 2 so that the Hamiltonian for the system is V V3 H 7 7Vrr 4 2m 2m 1 2 lt Inserting the assumed form of 1D into the Schr dinger equation H p Ew and Fourier transforming both sides with respect to the relative coordinate 7 r1 m we find E 26kuk Z kaUka 5 kgtkF where 6k 622771 and the WW fd3rV7 elkkf are the 8 matrix elements of the two body interaction Recall 616 correspond to energies at the Fermi level 61 in the absence of V The question was posed by Cooper is it possible to nd an eigenvalue E lt 261 ie a bound state of the two electrons To simplify the problem Cooper assumed a model form for VW in which V Ek kl lt we V W 0 otherw1se 6 where as usual 5 E 6k 61 The BCS interaction is some times referred to as a residual attractive interaction ie what is left when the long range Coulomb interaction has been absorbed into renormalized electron phonon coupling constants etc as in The bound state equation becomes VkEuk 7 7 26k E lt where the prime on the summation in this context means sum only over 16 such that 6f lt 6k lt 61 we Now 11 may be eliminated from the equation by summing both sides 21 yielding 1 1 1 7 i 8 V 2 k E 1 1 1 2 F2w E 271v FWCd 7N1 C9 2 091 E2e E 2 Dog 2eF E For a weak interaction NOV ltlt 1 we expect a solution if at all just below the Fermi level so we treat 26 E as a small 9 positive quantity eg negligible compared to 2we We then arrive at the pair binding energy 00 2N0V Aceeper E 26F E 2 2wee 10 There are several remarks to be made about this result 1 Note for your own informationiCooper didn t know this at the time that the dependence of the bound state en ergy on both the interaction V and the cutoff frequency we strongly resembles the famous BCS transition temperature dependence with we identi ed as the phonon frequency wD as given in equation 11 the dependence on V is that of an essential singularity ie a nonanalytic function of the parameter Thus we may expect never to arrive at this result at any order in perturbation theory an unexpected problem which hin dered theoretical progress for a long time The solution found has isotropic or s symmetry since it doesn t depend on the f6 on the Fermi surface How would an angular dependence arise Look back over the calcula tion Note the integrand 2ee E 1 25k Aceeper1 peaks at the Fermi level with energy spread Aceeper of states involved in the pairing The weak coupling NOV ltlt 1 solution therefore provides a bit of a posterior justi ca tion for its own existence since the fact that Aceepee ltlt we implies that the dependence of We on energies out near 10 the cutoff and beyond is in fact not terribly important so the cutoff procedure used was ok 5 The spread in momentum is therefore roughly AcooperUF and the characteristic size of the pair using Heisenberg s uncertainty relation about vFTc This is about 100 1000A in metals so since there is of order 1 electron unit cell in a metal and if this toy calculation has anything to do with superconductivity there are certainly many electron pairs overlapping each other in real space in a superconductor 54 Pair condensate amp BCS Wavefctn Obviously one thing is missing from Cooper s picture if it is energetically favorable for two electrons in the presence of a noninteracting Fermi sea to pair ie to form a bound state why not have the other electrons pair too and lower the en ergy of the system still further This is an instability of the normal state just like magnetism or charge density wave for mation where a ground state of completely different charac ter and symmetry than the Fermi liquid is stabilized The Cooper calculation is a T0 problem but we expect that as one lowers the temperature it will become at some critical temperature To energetically favorable for all the electrons to pair Although this picture is appealing many things about it are unclear does the pairing of many other electrons alter the attractive interaction which led to the pairing in the rst 11 place Does the bound state energy per pair change Do all of the electrons in the Fermi sea participate And most im portantly how does the critical temperature actually depend on the parameters and can we calculate it 55 BCS Model A consistent theory of superconductivity may be constructed either using the full effective interaction or our approxima tion Vqw to it However almost all interesting questions can be answered by the even simpler model used by BCS The essential point is to have an attractive interaction for electrons in a shell near the Fermi surface retardation is sec ondary Therefore BCS proposed starting from a phenomeno logical Hamiltonian describing free electrons scattering via an effective instantaneous interaction a la Cooper H H0 CLOCkq0CkqaCkm q 7039 where the prime on the sum indicates that the energies of the states 16 and k must lie in the shell of thickness cap Note the interaction term is just the Fourier transform of a completely local 4 Fermi interaction wlrwlrwrwr2 Recall that in our discussion of the instability of the nor mal state we suggested that an infinitesimal pair field could produce a finite amplitude for pairing That amplitude was the expectation value czacik We ignore for the moment 2Note this is not the most general form leading to superconductivity Pairing in higher angular momentum channels requires a bilocal model Hamiltonian7 as we shall see later 12 the problems with number conservation and ask if we can simplify the Hamiltonian still further with a mean eld ap proximation again to be justi ed a posteriorz39 We proceed along the lines of generalized Hartree Fock theory and rewrite the interaction as CJLaclL kqa C k qa ck a ltCLUCkqa gt 6ltCTCUl X X ltCkqaCkagt 6CC where eg 600 CkqaCkla ltekqalek0gt is the fluctu ation of this operator about its expectation value If a mean eld description is to be valid we should be able to neglect terms quadratic in the uctuations when we expand Eq 20 If we furthermore make the assumption that pairing will take place in a uniform state zero pair center of mass momentum then we put ltekqalek0gt lte HackQ6613 The effective Hamiltonian then becomes check H 2 H0 ACLTCTM he AkltCLTCl kigta 13 where A VltCkicmgt 14 What BCS actually Bogoliubov after BCS did was then to treat the order parameter A as a complex number and calculate expectation values in the approximate Hamiltonian 13 insisting that A be determined self consistently via Eq 14 at the same time 551 BCS wave function gauge invariance and number con servation What BCS actually did in their original paper is to treat the Hamiltonian 11 variationally Their ansatZ for the ground state of 11 is a trial state with the pairs 16 T k i occupied with amplitude 1 and unoccupied with amplitude at such that law My 1 a gt Hm vkcLTCTMMO gt 15 This is a variational wave function so the energy is to be minimized over the space of at vk Alternatively one can diagonalize the Hartree Fock BCS Hamiltonian directly to gether with the self consistency equation for the order param eter the two methods turn out to be equivalent 1 will follow the latter procedure but rst make a few remarks on the form of the wave function First note the explicit violation of par ticle number conservation W gt is a superposition of states describing 0 2 4 N particle systems3 In general a quan tum mechanical system with fixed particle number N like eg a real superconductor manifests a global U 1 gauge symmetry because H is invariant under elm gt ewclm The state W gt is characterized by a set of coefficients am vk which becomes am 2ka after the gauge transformation The two states W gt and wW where qb 28 are inequiva lent mutually orthogonal quantum states since they are not 3What happened to the odd numbers In mesoscopic superconductors there are actually differences in the properties of even and oddnumber particle systems7 but for bulk systems the distinction is irrelevant simply related by a multiplicative phase factor4 Since H is independent of qb however all states qb gt are contin uously degenerate ie the ground state has a U 1 gauge phase symmetry Any state qb gt is said to be a bro ken symmetry state becaue it is not invariant under a U 1 transformation ie the system has chosen a particular qb out of the degenerate range 0 lt qb lt 27r Nevertheless the ab solute value of the overall phase of the ground state is not an observable but its variations 6gb7 t in space and time are It is the rigidity of the phase ie the energy cost of any of these fluctuations which is responsible for superconductivity Earlier I mentioned that it was possible to construct a num ber conserving theory It is now instructive to see how states of de nite number are formed Anderson 1958 by making co herent superpositions of states of de nite phase MN gt flaiqbeWQlwo gt no The integration over qb gives zero unless there are in the ex pansion of the product contained in 1D gt precisely N 2 pair creation terms each with factor exp iqbl Note while this state has maximal uncertainty in the value of the phase the rigidity of the system to phase fluctuations is retained5 It is now straightforward to see why BCS theory works The BCS wave function 1D gt may be expressed as a sum in gt EN aN N gt Convince yourself of this by calculat 4in the normal state it gt and we differ by a global multiplicative phase cm which has no physical consequences and the ground state is nondegenerate 5The phase and number are in fact canonically conjugate variables N2q5 i where N Zia8415 in the 45 representation 15 ing the my explicitlyll IF we can show that the distribution of coef cients aN is sharply peaked about its mean value lt N gt then we will get essentially the same answers as working with a state of de nite number N lt N gt Using the explicit form 23 it is easy to show ltNgt w gnaw 2 W ltltN ltNgtgt2gt lt17 Now the uk and 1 will typically be numbers of order 1 so since the numbers of allowed k states appearing in the k sums scale with the volume of the system we have lt N gt V and lt N lt N gt2 gt V Therefore the width of the distribution of numbers in the BCS state is lt N lt N gt 2 gt12 lt N gt N 12 As N gt 1023 particles this relative error implied by the number nonconseryation in the BCS state becomes negligible 552 Is the BCS order parameter general Before leaving the subject of the phase in this section it is worthwhile asking again why we decided to pair states with opposite momenta and spin 6 T and k i The BCS argu ment had to do 1 with minimizing the energy of the entire system by giving the Cooper pairs zero center of mass mo mentum and 2 insisting on a spin singlet state because the phonon mechanism leads to electron attraction when the elec trons are at the same spatial position because it is retarded in timel and a spatially symmetric wayefunction with large 16 amplitude at the origin demands an antisymmetric spin part Can we relax these assumptions at all The rst require ment seems fairly general but it should be recalled that one can couple to the pair center of mass with an external mag netic eld so that one will create spatially inhomogeneous nite q states with current ow in the presence of a mag netic eld Even in zero external eld it has been proposed that systems with coexisting antiferromagnetic correlations could have pairing with nite antiferromagnetic nesting vec tor C Baltensberger and Strassler 196 The requirement for singlet pairing can clearly be relaxed if there is a pair ing mechanism which disfavors close approach of the paired particles This is the case in super uid 3H e where the hard core repulsion of two 3H e atoms supresses To for s wave sin glet pairing and enhances To for p wave triplet pairing where the amplitude for two particles to be together at the origin is always zero In general pairing is possible for some pair mechanism if the single particle energies corresponding to the states 160 and k a are degenerate since in this case the pairing interaction is most attractive In the BCS case a guarantee of this degen eracy for k T and k T in zero eld is provided by Kramer s theorem which says these states must be degenerate because they are connected by time reversal symmetry However there are other symmetries in a system with inversion sym metry parity will provide another type of degeneracy so 16 T k T k T and k T are all degenerate and may be paired with one another if allowed by the pair interaction 56 Thermodynamics 561 Bogoliubov transformation We now return to 13 and discuss the solution by canonical transformation given by Bogoliubov After our drastic ap proximation we are left with a quadratic Hamiltonian in the as but with clcl and cc terms in addition to Clc s We can di agonalize it easily however by introducing the quasipartz cle operators Yko and 71d by CkT Mk0 Uk Ylil T Cki T U17k0 Ukai 18 You may check that this transformation is canonical preserves fermion comm rels if lirle l1le 1 Substituting into 13 and using the commutation relations we get H303 gamma lvkl2lf7flt07ko 7117k1 f 2Mle 2UfltUf 7k17ko 2ukvk71l fcol f ngkukUi Aiuivk 7107k0 f 711mm 1 Akvfi2 Afiu Wki YkO A1342 Akuli710711 AkltCLTCikigt which does not seem to be enormous progress to say the least But the game is to eliminate the terms which are not of the form quot y so to be left with a sum of independent number type 18 terms whose eigenvalues we can write down The coef cients of the ylyl and 77 type terms are seen to vanish if we choose ZEkUkUk A v Akui 0 20 This condition and the normalization condition lukl2 l1le 1 are both satis ed by the solutions lule 1 kgt l 7 21 vk Q 2 Ek 7 lt where I de ned the Bogolz39bov quasipartz cle energy Ek v5 lAle 22 The BCS Hamiltonian has now been diagonalz zed HBCS Ek 7110714 7i17k1gt Ek Ek AkltclltTCT kigtgt lt23 Note the second term is just a constant which will be impor tant for calculating the ground state energy accurately The rst term however just describes a set of free fermion exci tations above the ground state with spectrum Ek 562 Density of states The BCS spectrum is easily seen to have a minimum Ak for a given direction k on the Fermi surface Ak therefore in addition to playing the role of order parameter for the su perconducting transition is also the energy gap in the 1 particle spectrum To see this explicitly we can simply do a change of variables in all energy integrals from the normal metal eigenenergies 5k to the quasiparticle energies Ek NEdE NNgdg 24 If we are interested in the standard case where the gap A is much smaller than the energy over which the normal state dos N ME yaries near the Fermi level we can make the replace ment 2 E No so using the form of Ek from 22 we nd E No 0 E lt A This function is sketched in Figure 5 NE A N0 SC 1 Normal o 1 F E Figure 5 a Normalized density of states b Quasiparticle spectrum 20 563 Critical temperature The critical temperature is de ned as the temperature at which the order parameter Ak vanishes We can now cal culate this with the aid of the diagonalized Hamiltonian The self consistency condition is A VEltCLITCklgt V E amtlt1 rlorko vlwk A1 V 1 2 E 27 2Eklt f k lt gt Since 1 2fE tanhE2T the BCS gap equation reads gtlt A Ek Ak T V E 2Ek tanh 2T This equation may now be solved rst for the critical temper ature itself ie the temperature at which A gt 0 and then for the normalized order parameter ATc for any temperature T It is the ability to eliminate all normal state parameters from the equation in favor of Tc itself which makes the BCS theory powerful For in practice the parameters cap N0 and particularly V are known quite poorly and the fact that two of them occur in an exponential makes an accurate rst prin ciples calculation of Tc nearly impossible You should always be suspicious of a theory which claims to be able to calculate Tc On the other hand To is easy to measure so if it is the only energy scale in the theory we have a tool with enormous predictive power 28 21 First note that at Tc the gap equation becomes 1 wD 1 5k W 7 0 This integral can be approximated carefully but it is useful to get a sense of What is going on by doing a crude treatrnent Note that since Tc ltlt cap generally most of the integrand weight occurs for E gt T so we can estimate the tanh factor by 1 The integral is log divergent which is Why the cutoff cap is so important We find 1 w 1 i T02 1N0V 30 NOVO 0ch W l l The more accurate analysis of the integral gives the BCS result 29 Tc 114wDe1N0V 31 We can do the same calculation near Tc expanding to lead ing order in the small quantity ATT to find ATTc 2 3061 TTc12 At T 0 we have 1 NOV U 1 CU 0 D alng AD dENfEVE 32 1 m so that Am 2 2m exp 1N0V or AOTc 2 176 The full temperature dependence of AT is sketched in Figure 6 In the halcyon days of superconductivity theory comparisons with the theory had to be compared With a careful table of ATc painstakingly calculated and compiled by Miihlschlegl DdE 2ln2wdA 33 22 176 TC AT Figure 6 B08 order parameter as fctn of T Nowadays the numerical solution requires a few seconds on a PC It is frequently even easier to use a phenomenological approximate closed form of the gap which is correct near T 0 and T 0 7T 60 Tc NT scTctanM 107 1 34 where 650 AOTc 176 a 23 and SCON 143 is the normalized speci c heat jump6 This is another of the universal ratios which the BCS theory predicted and which helped con rm the theory in classic superconductors 564 Speci c heat The gap in the density of states is re ected in all thermody namic quantities as an activated behavior eAT at low T due to the exponentially small number of Bogoliubov quasi 6Note to evaluate the last quantity7 we need only use the calculated temperature dependence of A near TC7 and insert into Eq 47 23 particles with thermal energy suf cient to be excited over the gap A at low temperatures T lt A The electronic speci c heat is particularly easy to calculate since the entropy of the BCS superconductor is once again the entropy of a free gas of noninteracting quasiparticles with modi ed spectrum Ek The expression 116 then gives the entropy directly and may be rewritten S kB t dENEfltEgt1nfltEgt1 fltEgti1nu fltEgtii 35 where f is the Fermi function The constant volume spe ci c heat is just Ody T dS dTlv which after a little alge bra may be written af 2 1dA2 A sketch of the result of a full numerical evaluation is shown in Figure 1 Note the discontinuity at T0 and the very rapid falloff at low temperatures 2 Cezy fdEZWEll It is instructive to actually calculate the entropy and speci c heat both at low temperatures and near Tc For T lt A f 2 eET and the density of states factor N in the integral cuts off the integration at the lower limit A giving C 2 N0A52T32eAT7 7To obtain this7 try the following o replace the derivative of Fermi function by expE T 0 do integral by parts to remove singularity at Delta 0 expand around Delta E Delta delta E 0 change integration variables from E to delta E Somebody please check my answerl 24 Note the rst term in Eq 47is continuous through the transition A gt 0 and reduces to the normal state speci c heat 2W23N0T above Tc but the second one gives a dis continuity at T0 of ON CSCN 143 where 05 CTg and ON CTj To evaluate 36 we need the T depen dence of the order parameter from a general solution of 28 57 Electrodynamics 571 Linear response to vector potential The existence of an energy gap is not a suf cient condition for superconductivity actually it is not even a necessary onel Insulators for example do not possess the phase rigidity Which leads to perfect conductivity and perfect diarnagnetisrn which are the de ning characteristics of superconductivity We can understand both properties by calculating the cur rent response of the system to an applied magnetic or electric eld The kinetic energy in the presence of an applied vector potential A is just 7 i 3 t g 2 Ho 7 2m d war N CA we lt37 and the second quantized current density operator is given by m gum2v Agtwltrgt W Agtwltrgtiwltrgti ea 2 wwwm ea 25 Where jpam XWWWWT VWWDWNL 40 or in Fourier space e JParaltq 1 kCL qacka We would like to do a calculation of the linear current re sponse jq w to the application of an external eld Aq w to the system a long time after the perturbation is turned on Expanding the Hamiltonian to rst order in A gives the interaction 7 3 7 e f H 7 d mpam A 7 k Aqckqacqa 42 The expectation value lt j gt may now be calculated to linear order Via the Kubo forrnula yielding ltjgtltqwgt KltqwgtAltqwgt lt43 With 2 moi w 1 ltLipamajpamlgtltqa w lt44 Note the rst term in the current 2 ne Jdialtq7 00 E Altq7 00 45 is purely diagmagnem c ie these currents tend to screen the external eld note sign The second paramagnetic term is formally the Fourier transform of the current current corre lation function correlation function used in the sense of our 26 discussion of the Kubo formula8 Here are a few remarks on this expression Note the simple product structure of 43 in momentum space implies a nonlocal relationship in general between j and A ie j7 depends on the AM at many points 7quot around 7 9 Note also that the electric eld in a gauge where the elec trostatic potential is set to zero may be written Eq w ZwAq w so that the complex conductivity of the sys tem de ned by j QE may be written gm w gm w lt47 What happens in a normal metal The paramagnetic sec ond term cancels the diamagnetic response at w 0 leaving no real part of K lm part of a ie the conduc tivity is purely dissipative and not inductive at w q 0 in the normal metal 572 Meissner Effect There is a theorem of classical physics proved by Bohr10 which states that the ground state of a system of charged particles 8We will see that the rst term gives the diamagnetic response of the system7 and the second the temperature dependent paramagnetic response 91f we transformed back7 weld get the convolution jr d3TKrrAr 46 10See The development of the quantummechanical electron theory of metals 19283377 Li Hoddeson and Gr Baym7 Rev Mod Phys7 597 287 1987 27 in an external magnetic eld carries zero current The essen tial element in the proof of this theorem is the fact that the magnetic forces on the particles are always perpendicular to their velocities In a quantum mechanical system the three components of the velocity do not commute in the presence of the eld allowing for a nite current to be created in the ground state Thus the existence of the Meissner effect in superconductors wherein magnetic flux is expelled from the interior of a sample below its critical temperature is a clear proof that superconductivity is a manifestation of quantum mechanics The typical theorists geometry for calculating the penetra tion of an electromagnetic field into a superconductor is the half space shown in Figure 7 and compared to schematics of practical experimental setups involving resonant coils and mi crowave cavities in Figs 7 a c In the gedanken experiment Figure 7 a Half space geometry for penetration depth calculation b Resonant coil setup 0 Microwave cavity 28 case a DC eld is applied parallel to the sample surface and currents and elds are therefore functions only of the coordi nate perpendicular to the surface A Az etc Since we are interested in an external electromagnetic wave of very long wavelength compared to the sample size and zero frequency we need the limit w 0 q gt 00 of the response We will assume that in this limit Km 0 gt const which we will call C47T2 for reasons which will become clear Equation 63 then has the form c J T 47F This is sometimes called London s equation which must be solved in conjunction with Maxwell s equation 4 V X B V2A lj 2A 49 C 2A 48 which immediately gives A N Z and B Boe A The currents evidently screen the fields for distances below the surface greater than about A This is precisely the Meissner effect which therefore follows only from our assumption that KO 0 const A BCS calculation will now tell us how the penetration depth depends on temperature Evaluating the expressions in 44 in full generality is te dious and is usually done with standard many body methods beyond the scope of this course However for q 0 the cal culation is simple enough to do without resorting to Green s functions First note that the perturbing Hamiltonian H may 29 be written in terms of the quasiparticle operators 18 as H izk Am mC k Ukukq ukvkq 71q0711 Ykq1 Yk0l e gt 7 Z k A071llt07k0 7117k1 me k q gt0 If you compare with the A 0 Hamiltonian 23 we see that the new excitations of the system are Ek0 H Ek it A0 720 Ek1 gt Ek 71 A0 52 me We may similarly insert the quasiparticle operators 18 into the expression for the expectation value of the paramagnetic current operator41 0gt ltquotYilc0 Yk0 71271 m fEk0 fltEk1 We are interested in the linear response A gt 0 so that when we expand wrt A the paramagnetic contribution becomes 2 0gt 262 2 19f m c k 9Ek Combining now with the de nition of the response function K and the diamagnetic current 45 and recalling 2k gt ltjpara q WSW 53 Z k E k ltjpmltq k Alt0l k lt54 30 50 ukukwi JV Ukvkq fl1q0flk0 71q17k1 51 Nofd kdQ47r with N0 3n26F and fdQ47rkk l 3 we get for the static homogeneous response is therefore mo 1 1 daltg 3jgt1 lt55 2 wileh 56 where in the last step I de ned the super uid density to be n5T E n nnT with normal fluid density 9f nnT nalEC 8Ekgt Note at T 0 9f9Ek gt 0 Not a delta function as in the normal state caseido you see why while at T To the integral nn gt 111 Correspondingly the super uid density as de ned varies between n at T 0 and 0 at T0 This is the BCS microscopic justi cation for the rather successful phenomeno logical two uid model of superconductivity the normal uid consists of the thermally excited Bogoliubov quasiparticle gas and the super uid is the condensate of Cooper pairs12 57 Now let s relate the BCS microscopic result for the static homogeneous response to the penetration depth appearing in the macroscopic electrodynamics calculation above We nd immediately MT AMY2 lt58 11The dimensioness function nn TTCn is sometimes called the Yoshida function YT and is plotted in FigB 12The BCS theory and subsequent extensions also allow one to understand the limitations of the two uid picture for example when one probes the system at suf ciently high frequencies w N A the normal uid and super uid fractions are no longer distinct 31 1 1 WT expAT 7 0 0 TT 1 0 TT 1 TTC 1 a b c Figure 8 a Yoshida function b super uid density c penetration depth At T 0 the supercurrent screening excludes the eld from all of the sample except a sheath of thickness 0 At small but nite temperatures an exponentially small number of quasiparticles will be excited out of the condensate depeleting the supercurrent and allowing the eld to penetrate further Both nnT and T 0 may therefore be expected to vary as eAT for T lt Tc as may be con rmed by explicit expansion of Eq 57 See homework Close to Tc the pen etration depth diverges as it must since in the normal state the eld penetrates the sample completely 573 Dynamical conductivity The calculation of the full frequency dependent conductivity is beyond the scope of this course If you would like to read an old fashioned derivation I refer you to Tinkham s book The main point to absorb here is that as in a semiconductor with a gap at T 0 there is no process by which a photon can be absorbed in a superconductor until its energy exceeds 2A 32 the binding energy of the pair This threshold for optical absorption is one of the most direct measurements of the gaps of the old superconductors 6 Ginzburg Landau Theory 61 GL Free Energy While the BCS weak coupling theory we looked at the last two weeks is very powerful and provides at least a qualita tively correct description of most aspects of classic supercon ductors13 there is a complementary theory which a is simpler and more physically transparent although valid only near the transition and b provides exact results under certain circum stances This is the Ginzburg Landau theory VL Ginzburg and LD Landau Zh Eksp Teor Fiz 20 1064 1950 which received remarkably little attention in the west until Gor kov showed it was derivable from the BCS theory LP Gor kov Zh Eksp Teor FiZ 36 1918 1959 The the ory simply postulated the existence of a macrosopic quantum wave function 1M7 which was equivalent to an order param eter and proposed that on symmetry grounds alone the free energy density of a superconductor should be expressible in terms of an expansion in this quantity fsfni 2 4 T ealwl mm z e 7A 2 2m1V C W lt59 13In fact one could make a case that the BCS theory is the most quantitatively accurate theory in all of condensed matter physics 33 where the subscripts n and s refer to the normal and super conducting states respectively Let s see why GL might have been led to make such a guess The superconducting normal transition was empiri cally known to be second order in zero eld so it was natural to write down a theory analogous to the Landau theory of a ferromagnet which is an expansion in powers of the mag netization M The choice of order parameter for the super conductor corresponding to M for the ferromagnet was not obvious but a complex scalar eld 1b was a natural choice because of the analogy with liquid He where MP is known to represent the super uid density 71514 a quantum mechani cal density should be a complex wave function squared The biggest leap of GL was to specify correctly how electromag netic elds which had no analog in super uid He would couple to the system They exploited in this case the simi larity of the formalism to ordinary quantum mechanics and coupled the elds in the usual way to charges 6 associated with particles of mass m Recall for a real charge in a magnetic eld the kinetic energy is 1 Zea 11 mq 7 3 q 71421 lt lHkl gt 12mdr Vac 60 i 3 E 2 7 2mdrlv CANl 61 after an integration by parts in the second step GL just re placed 6 m with 6 m to obtain the kinetic part of Eq 59 141 in the He case has the microscopic interpretation as the Bose condensate amplitude 34 they expected that 6 and m were the elementary electron charge and mass respectively but did not assume so 5f TgtTc TltT VV lwl Figure 9 Mexican hat potential for superconductor A system described by this free energy will undergo a second order phase transition in zero eld when a 0 clearly when a is positive the system can minimize 6 f by having it 0 no superconductivity whereas if a is negative 6f has a mini mum with p y 0 The free energy 59 is a functional of the order parameter it meaning the actual value of the order parameter realized in equilibrium satis es Sf61D 015 No tice f is independent of the phase qb of the order parameter in E Wlew and so the ground state for a lt 0 is equivalent to any state it related to it by multiplication by a pure phase This is the U 1 gauge invariance of which we spoke earlier 15Thus you should not be perturbed by the fact that f apparently depends on 1 even for a gt 0 The value of f in equilibrium will be fn 0 35 This symmetry is broken when the system chooses one of the ground states phases upon condensation Fig 1 For a uniform system in zero eld the total free energy F f 37 f is minimized when f is so one nd for the order parameter at the minimum Wleq 373 a lt 0 62 Wleq 0 a gt 0 63 When or changes sign a minimum with a nonzero value be comes possible For a second order transition as one lowers the temperature we assume that a and b are smooth functions of T near Tc Since we are only interested in the region near Tc we take only the leading terms in the Taylor series expansions in this region aT H a0T Tc and b constant Eqs 62 and 63 take the form WTMeq WP2 T lt To 64 WTMeq 0 T gt Tc 65 Substituting back into Eqs59 we nd MT MT Z ltTc Tr T lt To lt66 07 T gt Tc The idea now is to calculate various observables and de termine the GL coef cients for a given system Once they are determined the theory acquires predictive power due to its extreme simplicity It should be noted that GL theory is applied to many systems but it is in classic superconductors 36 that it is most accurate since the critical region where de viations frorn mean eld theory are large is of order 10 4 or less Near the transition it may be taken to be exact for all practical purposes This is not the case for the HTSC where the size of the critical region has been claimed to be as much as 10 20K in some samples Supercurrents Let s now focus our attention on the term in the GL free energy which leads to supercurrents the kinetic energy part Fm dBer w few lt68 d3r2iltviwigt2 w ecAgt2Wi lt69 These expressions deserve several remarks First note that the free energy is gauge invariant if we make the transforma tion fl gt 15 VA where A is any scalar function of position while at the same time changing w gt 1D exp ZeAc Sec ond note that in the last step above I have split the kinetic part of f into a term dependent on the gradients of the order parameter rnagnitude MM and on the gradients of the phase qb Let us use a little intuition to guess what these terms mean The energy of the superconducting state below To is lower than that of the normal state by an amount called the condensation energy16 From Eq 59 in zero eld this is of order WP very close to the transition To make spatial vari ations of the magnitude of 1D rnust cost a signi cant fraction of the condensation energy in the region of space in which it 16We will see below from the Gorkov derivation of GL from BCS that it is of order N0A2i 37 occurs17 On the other hand the zero eld free energy is ac tually invariant with respect to changes in qb so uctuations of qb alone actually cost no energy With this in mind let s ask what will happen if we apply a weak magnetic eld described by A to the system Since it is a small perturbation we don t expect it to couple to Wl but rather to the phase qb The kinetic energy density should then reduce to the second term in Eq 69 and furthermore we expect that it should reduce to the intuitive two uid ex pression for the kinetic energy due to supercurrents mn5v Recall from the super uid He analogy we expect WP E n to be a kind of density of superconducting electrons but that we aren t certain of the charge or mass of the particles So let s put 1 2771 N a 2 i 3 gtk 2 2 1 gtk gtk 2 fkin C 39 i T e 2m 511539 70 Comparing with Eq xx we nd that the super uid veloc ity must be Us 1 6 a Vb FA 71 Thus the gradient of the phase is related to the super uid velocity but the vector potential also appears to keep the entire formalism gauge invariant Meissner effect The lleissner effect now follows imme diately from the two uid identi cations we have made The 17We can make an analogy with a ferromagnet7 where if we have a domain wall the magnetization must go to zero at the domain boundary7 costing lots of surface energy 38 supercurrent density will certainly be just gtllt gtllt 6wa m mgtllt Taking the curl of this equation the phase drops out and we nd the magnetic eld 7 gtlt jS TLSUS 2 gtlt VX B m m c Now recall the Maxwell equation j iv X g 74 47T which when combined with 14 gives 2 C C a e n a 4V v B 7WB 5B 75 47139 X X 47 mc 7 lt or a a VWB3 m where W m02 47re2n 77 Notice now that if we use what we know about Cooper pairs this expression reduces to the BCS London penetration depth We assume 6 is the charge of the pair namely 6 2e and similarly m 2m and WP n 7152 since 71 is the density of pairs Flux quantization If we look at the ux quantization described in Part 1 of these notes it is clear from our sub sequent discussion of the Meissner effect that the currents 39 which lead to ux quantization will only ow in a small part of the cross section a layer of thickness A This layer encloses the ux passing through the toroid Draw a contour C in the interior of the toroid as shown in Figure 10 Then v5 0 everywhere on C It follows that Figure 10 Quantization of ux in a toroid 1 e Ofcd v5 fcd V EA The last integral may be evaluated using f0 d qu 27r gtlt integer and MEE Sd Vgtltl 33d B 6 go C 78 79 80 81 82 Here S is a surface spanning the hole and I the ux through 40 the hole Combining these results he he I 7 27r2 n 7 7126 7 71 83 where n is a integer Do is the flux quantum and I ve rein serted the correct factor of h in the first step to make the units right Flux quantization indeed follows from the fact that the current is the result of a phase gradient18 Derivation from Microscopic Theory One of the reasons the GL theory did not enjoy much success at rst was the fact that it is purely phenomenological in the sense that the parameters do I m are not given within any micro scopic framework The BCS theory is such a framework and gives values for these coefficients which set the scale for all quantities calculable from the GL free energy The GL theory is more general however since eg for strong coupling su perconductors the weak coupling values of the coefficients are simply replaced by different ones of the same order of mag nitude without changing the form of the GL free energy In consequence the dependence of observables on temperature field etc will obey the same universal forms The GL theory was derived from BCS by Gor kov The calculation is beyond the scope of this course but can be found in many texts 18It is important to note however that a phase gradient doesnlt guarantee that a current is owing For example in the interior of the system depicted in Fig 2 both V45 and A are nonzero in the most convenient gauge and cancel each otherl 41 62 Type I and Type II superconductivity Now let s look at the problem of the instability of the normal state to superconductivity in nite magnetic eld H A what magnetic eld to we expect superconductivity to be destroyed for a given T lt Tc19 Well overall energy is conserved so the total condensation energy of the system in zero eldf5 fnT of the system must be equal to the magnetic eld energy I d3rH287r the system would have contained at the critical eld H c in the absence of the Meissner effect For a completely homogeneous system I then have Hc287T7 and from Eq 8 this means that near Tc7 27mg b Whether this thermodynamic critical eld He actually rep resents the applied eld at which ux penetrates the sample depends on geometry We assumed in the simpli ed treat ment above that the eld at the sample surface was the same as the applied eld Clearly for any realistic sample placed in a eld the lines of eld will have to follow the contour of the sample if it excludes the eld from its interior This means the value of H at different points on the surface will be different the homogeneity assumption we made will not quite hold If we imagine ramping up the applied eld from He Tc T 85 19Clearly it will destroy superconductivity since it breaks the degenerace of between the two componenets of a Cooper pair 42 zero there will inevitably come a point Ham Happhc where the eld at certain points on the sample surface exceeds the critical eld but at other points does not For applied elds Happg lt Ham lt H0 part of the sample will then be normal with local eld penetration and other parts will still exclude eld and be superconducting This is the intermediate state of a type I superconductor The structure of this state for a real sample is typically a complicated striped pattern of su perconducting and normal phases Even for small elds edges and corners of samples typically go normal because the eld lines bunch up there these are called quotdemagnetiZing effects and must be accounted for in a quantitatively accurate mea surement of say the penetration depth It is important to note that these patterns are completely geometry dependent and have no intrinsic length scale associated with them In the 50 s there were several materials known however in which the ux in suf ciently large elds penetrated the sample in a manner which did not appear to be geometry de pendent For example samples of these so called quottype If superconductors with nearly zero demagnetiZing factors long thin plates placed parallel to the eld also showed ux pen etration in the superconducting state The type ll materials exhibit a second order transition at nite eld and the ux B through the sample varies continuously in the supercon ducting state Therefore the mixed state must have currents owing and yet the Meissner effect is not realized so that the London equation somehow does not hold 43 The answer was provided by Abrikosov in 1957 AAA Sov Phys JETP 5 1174 1957 in a paper which Landau apparently held up for several years because he did not be lieve it Let us neglect the effects of geometry again and go back to our theorist s sample with zero demagnetizing factor Can we relax any of the assumptions that led to the Lon don equation 72 Only one is potentially problematic that 710 W0 constant independent of position Let s examineias Abrikosov didithe energy cost of making spatial variations of the order parameter The free energy in zero eld is 1 2m Fd3ra1w12 1W12blw14t lt86 or 1 b 3F d3ri iw12 21vw jaw lt87 where I ve put 1 12 1 12 88 E 2ma 2ma0Tc T lt Clearly the length 5 represents some kind of stz ness of the quantitiy W12 the super uid density Check that it does in deed have dimensions of lengthl If E the so called coherence length is small the energy cost of 715 varying from place to place will be small If the order parameter is somehow changed from its homogeneous equilibrium value at one point in space by an external force 5 speci es the length scale over which it heals We can then investigate the possibility that as the kinetic energy of super uid ow increases with increasing 44 eld if is small enough it might eventually become favorable to bend WP instead In typical type 1 materials T 0 is of order several hundreds or even thousands of Angstrom but in heavy fermion superconductors for example coher ence lengths are of order 50 100A The smallest coherence lengths are attained in the HTSC where gab is of order 12 15A whereas 50 is only 2 3A The general problem of minimizing F when it depends on position is extremely dif cult However we are mainly in terested in the phase boundary where w is small so life is a bit simpler Let s recall our quantum mechanics analogy once more so as to write F in the form F d3rlalwl2 W H lt MHzaw gt 89 where Hm is the operator 1 29 a 2 2mV C A 90 Now note 1 suf ciently close to the transition we may always neglect the 4th order term which is much smaller 2 to minimize F it suf ces to minimize lt Hm gt since the WP term will simply x the overall normalization The variational principle of quantum mechanics states that the minimum value of lt H gt over all possible in is achieved when w is the ground state for a given normalization of in So we need only solve the eigenvalue problem Efka 37 91 45 for the lowest eigenvalue E and corresponding eigenfunction it For the given form of Elmn this reduces to the classic quantum mechanics problem of a charged particle moving in an applied magnetic eld The applied eld H is essentially the same as the microscopic eld B since 1D is so small at the phase boundary onlyl l ll remind you of the solution due to to Landau in order to x notation We choose a convenient 8351118397 A H 31 92 in which Eq 44 becomes 2 2 1lt z 6 gt2 a a 2771 91 M 932 922 where M cequotH12 is the magnetic length Since the co ordinates I and 2 don t appear explicitly we make the ansatZ of a plane wave along those directions W7 ijjv 93 w dwelm kzza 94 yielding 1 y 92 27mka 82 9732 93779 EWle 95 But this you will recognize as just the equation for a one dirnensional harmonic oscillator centered at the point 3 l x w with an additional additiye constant kEZmquot in the energy Recall the standard harmonic oscillator equation 1 a 1 77 7k 2 II EKI 96 46 with ground state energy we 1 E0 3 km127 97 where k is the spring constant and ground state wayefunc tion corresponding to the lowest order Hermite polynomial x110 exp ml 412L 2 98 Let s just take over these results identifying A eH H m 7 99 k Wale 2W6 km The ground state eigenfunction may then be chosen as 7T 2 139 x 139 2 Wm M739 146 I k2 GXDl Z k i la y 100 where L3 is the size of the sample in the y direction LmLyLz V 1 The wave functions are normalized such that d37 l kzkzl2 3 101 since I set the volume of the system to 1 The prefactors are chosen such that 8 represents the average superfluid den stity One important aspect of the particle in a field problem seen from the above solution is the large degeneracy of the ground state the energy is independent of the variable km for example corresponding to many possible orbit centers AWe have now found the wayefunction which minimizes lt Hkm gt Substituting back into 89 we find using 99 6Hid3riwi2bd3riwi4 102 F T Tc a0 2mc 47 When the coef cient of the quadratic term changes sign we have a transition The eld at which this occurs is called the upper critical eld H02 H02T 2mca0Tc T 103 6 What is the criterion which distinguishes type l and type II materials Start in the normal state for T lt Tc as shown in Figure 3 and reduce the eld Eventually one crosses either He or H02 rst Whichever is crossed rst determines the nature of the instability in nite eld ie whether the sample expels all the eld or allows ux vortex penetration see section C Normal state Meissner phase Figure 11 Phase boundaries for classic type ll superconductor In the gure I have displayed the situation where H02 is higher meaning it is encountered rst The criterion for the dividing line between type 1 and type II is simply dHc i dHcg dT dT 104 48 or using the results 38 and 56 gtlt 2 2 M l 105 7re2 2 This criterion is a bit dif cult to extract information from in its current form Let s de ne the GL parameter I4 to be the ratio of the two fundamental lengths we have identi ed so far the penetration depth and the coherence length K 106 Recalling that m02 mc2b A2 1 47re2n 27re2a lt 07 and 1 52 2ma 108 The criterion 58 now becomes 2 m02b27re2a m202b 109 12ma 7T 2 2 Therefore a material is type I ll if I4 is less than greater than ln type l superconductors the coherence length is large compared to the penetration depth and the system is stiff with respect to changes in the superfluid density This gives rise to the Meissner effect where 715 is nearly constant over the screened part of the sample Type ll systems can t screen out the eld close to H02 since their stiffness is too 49 small The screening is incomplete and the system must de cide what pattern of spatial variation and flux penetration is most favorable from an energetic point of view The result is the vortex lattice first described by Abrikosov 63 Vortex Lattice I commented above on the huge degeneracy of the wave func tions 53 In particular for fixed 62 0 there are as many ground states as allowed values of the parameter km At H 02 it didn t matter since we could use the eigenvalue alone to de termine the phase boundary Below H 02 the fourth order term becomes important and the minimization of f is no longer an eigenvalue problem Let s make the plausible assumption that if some spatially varying order parameter structure is going to form below H02 it will be periodic with period 27rq ie the system selects some wave vector q for energetic reasons The x dependence of the wave functions 7T 2 t x W73 We 1 expl ykx f422 f4l no 3 is given through plane wave phases eff If we choose km qnx with 7195 integer all such functions will be invariant under 1 gt 1 27rq Not all nx s are allowed however the center of the orbit kx w should be inside the sample Ly2 lt mil q wnx lt LyZ 111 50 Thus 7195 is restricted such that Ly L 4 max 2 lt x lt max 2 y and the total number of degenerate functions is Lyq w Clearly we need to build up a periodic spatially varying structure out of wave functions of this type with centers distributed somehow What is the criterion which determines this structure All the wave functions 110 minimize lt lmn gt and are normalized to f dBTWlQ WOW They are all therefore degenerate at the level of the quadratic GL free energy F f dgrlwl2 lt lmn gt The fourth order term must stabilize some linear combination of them We therefore write W 712 anwn 113 with the normalization EM lCWlZ 1A7 which enforces f dBTWO 2 1amp3 Note this must minimize lt H km gt Let s therefore choose the CW and q to minimize the remaining terms in F f d3ralwl2 blwlA Substituting and using the normalization condition and orthogonality of the different 1mgsz we find f mp3 51 114 with i 6H i 6 aH 7 a0T TC m i H62ltTgtgt bH 5b 116 and 2 7TEM 1 m 7 7 7 0 0 C C lt Z gt 7111771127711377114 n11 n12 n13 n14 51 dz d Q nxl nz2n13nx4 X 118 Tj yqnzl12yqn12l2yqn13 2yqnz4g 2llt119gt The form of fh o is now the same as in zero eld so we immediately nd that in equilibrium amp wOleq 25 120 and a2 4E 121 This expression depends on the variational parameters CM q only through the quantity 3 appearing in 5 Thus if we mini mize s we will minimize i remember I gt 0 so f lt 0 The minimization of the complicated expression with con straint Eng lCWlZ 1 is dif cult enough that A Abrikosov made a mistake the rst time he did it and I won t in ict the full solution on you To get an idea what it might look like however let s look at a very symmetric linear combination one where all the Cm s are equal 0 71243 122 Then 1W N qu expl y mgr2m 123 which is periodic in x with period 27rq W 27W 1 W56 2 124 and periodic in q with period q l up to a phase factor W56 2 M14 e mwwa y 125 52 Note if q m M WP forms a square lattice The area of a unit cell is 2 q gtlt q w 27T W and the flux through each one is therefore 2 0 he 0611 7 27T MH 7 27T HH 7 2e 7 Where I inserted a factor of h in the last step We haven t performed the minimization explicitly but this is a charac teristic of the solution that each cell contains just one flux quantum The picture is crudely depicted in Fig 12a Note by symmetry that the currents must cancel on the boundaries cigt0 126 5118 5116 Figure 12 a Square vortex lattice b triangular vortex lattice 7 7 26 7 of the cells Since jg 7 en8v5 integrating qu A 7 0 around each square must give as in our prev10us discussion of flux quantization in a toroid 1061 71 n integer 127 Somehow the vortex lattice consists of many such rings The problem with this idea is that the only way Na d 7 sat around the boundary can be nonzero and the usual argument about single valuedness of the wave function carried through is if there is a hole in the wave function If there is no hole or region from which the wave function is excluded the path can be shrunk to a point but the value of the integral must 53 remain the same since the integrand is the gradient of a scalar eld This is unphysical because it would imply a nite phase change along an in nitesimal path and a divergence of the kinetic energy The only way out of the paradox is to have the system introduce its own hole in itself ie have the am plitude of the order parameter density WP go to zero at the center of each cell lntuitively the magnetic eld will have an accompanying maximum here since the screening tendency will be minimized This reduction in order parameter ampli tude magnetic ux bundle and winding of the phase once by 27 constitute a magnetic vortex which I ll discuss in more detail next time Assuming On constant which leads to the square lattice does give a relatively good small value for the dimensionless quantity 3 which turns out to be 118 This was Abrikosov s claim for the absolute minimum of f But his paper contained a now famous numerical error and it turns out that the actual minimum 8 116 is attained for another set of the Cn s to wit On 71 12 71 even 128 On ingif n odd 129 This turns out to be a triangular lattice Fig 12b for which the optimal value of q is found to be 3147T12 r Again the area of the unit cell is 27r 2 and there is one ux 7 130 54 quantum per unit cell 64 Properties of Single Vortex Lower critical eld H01 Given that the ux per unit cell is quantized it is very easy to see that the lattice spacing d is actually of order the coherence length near H02 Using 103 and 88 we have C 1 7 DO 6 2 7 2715 On the other hand as H is reduced d must increase To see this note that the area of the triangular lattice unit cell is A ap 2 and that there is one quantum of ux per cell A 10 H Then the lattice constant may be expressed as 7477 H02 de a 02 12 132 Since gtgt 5 is the length scale on which supercurrents and magnetic elds vary we expect the size of a magnetic vor tex to be about A This means at H02 vortices are strongly overlapping but as the eld is lowered the vortices separate according to 126 and may eventually be shown to in uence each other only weakly To nd the structure of an isolated vortex is a straightforward but tedious exercise in minimizing the GL free energy and in fact can only be done numerically in full detail But let s exploit the fact that we are inter ested primarily in strongly type ll systems and therefore go back to the London equation we solved to nd the penetration 55 depth in the half space geometry for weak elds allow n5 to vary spatially and look for vortex like solutions For example equation 75 may be written 2V gtlt V X E E 133 Let s integrate this equation over a surface perpendicular to the eld B B x y2 spanning one cell of the vortex lattice 2VXVX d d 134 4 a a 274j5d 0 135 But we have already argued that 5 d should be zero on the boundary of a cell so the left side is zero and there is a contradiction What is wrong The equation came from assuming a two fluid hydrodynamics for the superconductor with a nonzero n5 everywhere We derived it in fact from BCS theory but only for the case where n5 was constant Clearly there must be another term in the equation when a vortex type solution is present one which can only contribute over the region where the superfluid density is strongly varying in space ie the coherence length sized region in the middle of the vortex where the order parameter goes to zero vortex core Let s simply add a term which enables us to get the right amount of flux to Eq 133 In general we should probably assume something like AZV gtlt V X E E lt1gtOg773 136 where 97 is some function which is only nonzero in the core The flux will then come out right if we demand I d3rgf 56 1 But let s simplify things even further by using the fact that 5 lt A let s treat the core as having negligible size which means it is just a line singularity We therefore put 977 6 Then the modi ed London equation with line singularity acting as an inhomogeneous source term reads 2V2 E 062F2 137 1 a an 1 2papp 9p t B2 10520 138 where p is the radial cylindrical coordinate Equation 91 has the form of a modi ed Bessel s equation with solution 10 P B2 K i 27m 0A The other components of g vanish If you go to Abramowitz 85 Stegun you can look up the asymptotic limits 139 10 B2 1 7 11 14 2210gltpgt0 61 5ltpltltA lt 0 BZ 232 ZapA pgtgt A 141 Note the form 93 is actually the correct asymptotic solution to 91 all the way down to p 0 but the fact that the solution diverges logarithmically merely re ects the fact that we didn t minimize the free energy properly and allow the order parameter to relax as it liked within the core So the domain of validity of the solution is only down to roughly the core size p 2 E as stated In Figure 5 I show schematically the structure of the magnetic and order parameter pro les in 57 an isolated vortex The solution may now be inserted into the lwl BZ l l l o l P core supercurrents Figure 13 Isolated vortex free energy and the spatial integrals performed with some interesting results 2 D0 It is easy to get an intuitive feel for what this means since if we assume the eld is uniform and just ask what is the magnetic energy we get roughly 1 Fv 87 X vortex volume X B2 143 7r 1 g 87 X 7T2LZ gtlt op7m2 144 7r 12 Lz O 145 87T22 lt the same result up to a slowly varying factor Now the lower critical eld H01 is determined by requiring the Gibbs free energies of the Meissner phase with no vortex be equal to the Gibbs free energy of the phase with a vortex20 G differs from 20We haven t talked about the Gibbs Vsl Helmholtz free energy but recall the Gibbs is the appropriate potential to use when the external eld H is held xed7 which is the situation we always have7 with a generator supplying work to maintain l 58 F through a term f BH47r In the Meissner state G F so we may put 1 F F ElmeLz 471101 Bd 146 7T 1 F EmeLz 71gt L2 147 1 4w 0 lt l Where Elma is the free energy per unit length of the vortex itself Therefore 7 47TElme Do is the upper critical eld But the line energy is given precisely 2 by Eq 95 Elm loglt so H01 Helm 432 logo 149 65 Josephson Effect 59 PHZ 7427 SOLID STATE II Electronelectron interaction and Fermiliquid theory D L Maslov Dated April 37 2006 I ELECTROSTATIC SCREENING A ThomasFermi model For Thomas Fermi model7 see hand written notes B Effective strength of the electronelectron interaction Parameter rs The ratio of the Coulomb energy at a typical inter electron distance to the Fermi energy Uc e2ltrgt ETD EF ltrgt is found from 4 3 13 g7rltrgt3n 1 a ltrgt nil3 1 eznlgm 2 223 62m 62m E710 U0 lt47rgt13 i 4 3 3712232gt 7123 3 7r 7113quot nlB39 Lower densities correspond to stronger e ective interactions and Vice versa Parameter T5 is introduced as the average distance between electrons measured in units of the Bohr radius ltrgt rsaB rsmez Expressing T5 in terms of n and relating density to kp7 we nd 3 13 1 97139 13 mez T5 7 E n13a3 7 I M 39 In terms of T 2 U0 6 T5043 and l 9 2 3 l 1 EF 7 lt1 2 7 2 4 maB 7 U0 4 23 7 2 7 s m 54 5 EF 9 T T C Full solution Lindhard function4 In the Thomas Fermi model7 one makes two assumptions a the e ective potential acting on electrons is weak and b the e ective potential and corresponding density varies slowly on the scale of the electrons wavelength Assumption a allows one to use the perturbation theory whereas assumption b casts this theory into a quasi classical form In a full theory7 one discards assumption b but still keeps assumption a So now we want to do a complete quantum mechanical no quasi classical assumptions form Let the total electrostatic potential acting on an electron be Z5 ext ind7 where at is the potential of external charges and And is that of induced charges Corre spondingly7 the potential energy v 76 7 ext 7 ind Because we are doing the linear response theory7 the form of the external perturbation does not matter Let7s choose it as a plane wave a 1 i F7wt 1 mt ivqe 1 00 1 Before the perturbation was applied7 the wavefunction was 1 iEFia t 110 W6lt k The wavefunction in the presence of the perturbation is given by standard expression from the rst order perturbation theory 2 q Fiwt Uq ei sziwt 7 v IJII 1 1 0 25k75E w Zekiegiiiw where the last term is a response to a cc term in Eq1 The Fourier component of the wavefunction 1 1 pk10k 1 25kiegliw 25197515717744 The induced charge density is related to the wavefunction is ind 2619 Woklz7 2 k where fk is the Fermi function7 factor of 2 is from the spin summation and the homogeneous unperturbed charge density was subtracted o Keeping only the rst order terms in 1 Eq2 gives 1 1 1 in 72 7 3 p d engfkvq18ki8Eq w18198157 7700 dgk e Qevq 3M7 4 2 5k 817W where we shifted the variables as I 7 fa I and I a I j in the last term The charge susceptibility X7 is de ned as 12 2 2 in 7 7 7 7 5 pa e47rxvq EMMA where g is the Fourier component of the net electrostatic potential Comparing Eqs 4 and 57 we see that 7 47139 Fig fk fg X E Amm The meaning of X becomes more clear7 if we write down the Poisson equation in a Fourier transformed form q2 q 47139 pext 1 pind External charges and potentials satisfy a Poisson equation on their own q2 ext 47rpext 7 2 q q2 q q2 ext 1 4717711111 q2 ext 7 47T ZEX Q ext 1 Xez39 511 Using a de nition of the dielectric function we see that 4713962 72 dgk fk fgirg e 7W1 621 q X qz 271 381678EJF 1 JVW This is the Lindhard s expression for the dielectric function 4 1 Check Let7s make sure that the general form of e qua does reduce to the Thomas Fermi one in the limit to 0 and q lt kp 47W 13 fk f eltqvogt172 lt72 3 q lt27rgt swam For small q7 f q f 513 f gaff 5k 5k Na gawk and 2 3 H42 val2170 At T 07 7 6 5k 7 EF The density of states at the Fermi energy VF 2 W 368k 7 47w n q701 qz VF17 Where n2 E 47r 2VF7 Which is just the Thomas Fermi form D Lindhard function As you have shown in the homework 2 the static form of the Lindhard7s dielectric function is given by 4713962 1 1 7 2 1 m 01 7 1 q7 qz 2 456 n liw Where m E qZkF Notice that the derivative of e q70 is singular for q 21mm7 ie7 m 1 This singularity gives rise to a very interesting phenomenoniFrz39edel oscillations in the induced charge density and corresponding potentials Mathematically7 it arises because of the property of the Fourier transform To nd the net electrostatic potential in the real space we need to Fourier transform back to real space Eq6 Lets say that the external perturbation is a single point charge Q Then in the q7 space 7 om 47rQq2 and dgq 72 47m r We q2 q70 lt8 1 A discourse properties of Fourier transforms Fourier transforms have the following property Suppose we want to nd the large 75 limit of For 61 gt lt9 76 w 27139 If function Fw is analytic the integral in Eq6 can be done by closing the contour in the complex plane Ft for t a 00 will be then given by an exponentially decaying function exp7uI inf7 where Lug n is the imaginary part of that pole of Fw which is closest to the real axis For exarnple7 if F to 002 371 7 F 75 0c exp fat Thus7 the large t asyrnptotes of analytic functions decay exponentially in time On the other hand7 if Fw is non analytic Ft decays rnuch slowerionly as a power law For exarnple7 for Fw exp 7a 7 we have 1 00 d F t 67Wtexp 7a 0 67 61 00 d 1 1 1 1 2Re ieiwteiw 2Re77 7L 0c 7 for t a 00 0 27139 27ra2t 7ra2 752 t2 In addition7 if Fw has a divergent derivative of order n at nite W7 eg7 for w 1410 that F t oscillates in t This can be seen by doing the partial integration in Eq6 n 1 times 27139F t dwe th w O 1006th w 1406th w 700 we 7 1 iiwt we 1 iiwt co 7 1 7239th iiwe Fwlim7we Fwlwo ind dwe dwFw7 ie7 until the boundary terrns gives the divergent expression d F 1410 th which oscillates as eiiwot 2 End of discourse Coming back to Eq97 we can now understand why the induced density around the point charge oscillates as cos 2kpr and falls off only as a power law of the distance cos 2kpr pind QC 3 7 Both of these effects are the consequences of the singularity of e q70 at q 2kp E Friedel oscillations The physics of F riedel oscillations is very simple they arise due to standing waves formed as a result of interference between incoming and backscattered electron waves For the sake of simplicity let7s analyze a 1D case Suppose that at m 0 we have an in nitely high barrier wall For each k the wavefunction is a superposition of the incoming plane wave L llzemz and a re ected wave L llze i 239 ll LilZed 7 Lil2e72 7122 sin km The probability density Mlle 4L sin2 1m oscillates in space If the probability that the state with momentum k is occupied is a smooth function of k as it is the case for the Maxwell Boltzmann or Bose Einstein statistics then summation over k would smear out the oscillations However for the Fermi statistics fk has a sharp at T 0 boundary between the occupied and empty states As a result oscillations survive even after the summation of k The pro le of the density is described by dk 2 kHzc dek 2E7 was7 2164717 2k k 27rfkl l 0 27rs1n m 0 27T cos n sin2kF n077 7 7T where no 2kF7r is the density of the homogeneous electron gas Away from the barrier oscillations die o as 4 At the barrier 72 0 0 A 3D case is di erent in that the 4 decay changes to a 7 3 one In general D dimensional case the F riedel oscillations fall o as T D F riedel oscillations were observed in STM experiment see attached gures 1 3 L t of the L 39 u 391 y r L LT due to Friedel oscillations As it was discussed in the previous Section Friedel oscillations arise already in the single particle picture However they in uence scattering of interacting electrons at impurities and other imperfections Once a Friedel oscillation is formed the e ective potential barrier seen by other electrons is the sum of the bare potential plus the potential produced by the Friedel oscillation Consider a simple example when 1D electrons interact via a contact potential U u6 The potential produced by the Friedel oscillation is VF x dx n 5 7 no v x 7 5 sin 2kp n 7 no 7 7117 Backscattering at an oscillatory potential is enhanced due to resonance In the Born ap proximation7 the backscattering amplitude for an electron with momentum k is 72 m u X LT lms A d 6 gt VF 77 7s1n2kFe 7r 0 v 7E ltgto dim lt62ikkpz 22 k7kpzgt I 7r 0 z The rst term gives a convergent integral we remind that our k gt 07 so forget about it Its only role is to guarantee the convergence at m 7 07 but we will take this into account by cutting the integral at m 2 k 1The second term becomes log divergent at large distances if k kp To estimate the integral7 notice that it diverges for k kp and converges for k a kp Thus A 3 VHF d3 3 UL 7r k m 7r lk 7 kpl Precisely at the Fermi surface k kp7 the backscattering amplitude and thus the prob ability blows up which means that impurity becomes impenetrable This e ect in 1D is usually described as arising due to the non Fermi liquid nature of the ground state How ever7 as we just saw that this e ect can be simply understood in terms of backscattering from the Friedel oscillationsl Higher orders in the e e interaction can be summed up see Refl It turns out that each next order in u brings in an additional log of lk 7 kpl The transmission amplitude of the barrier becomes H t 1717 72127773137 0 gnlk7kpl2g n lk7kpl 69 n lk7kpl kp lk7kpl 9 t 717 t 7 oexPgn kkF kF where g 117er is the dimensionless coupling constant and to is the transmission amplitude in the absence of 76 interaction At k kp7 the transmission amplitude vanishes Suppose one measures a tunneling conductance of the barrier inserted into a 1D system Then typical lk 7 kpl 2 max TUF76VUF7 where V is the applied voltage With the help of the Landauer formula G 262h ltolz we conclude that the tunneling conductance exhibits a a 390 WWW 390 1quot eno oo amp a FIG 1 Carbon nanotube dlIdV I T a crossing 0 1 II 1 0 130 ewigr FlGi 2 Currentvoltage characteristics for tunneling between two nanotubesi powerlaw scaling in the voltage or temperature C olt max TeV29i Such a powerlaw scaling was indeed found in carbon nanotubes Which are essentially quan tum Wires With two channels II SPECTRUM OF AN INTERACTING SYSTEM SELFENERGY ln Notes on Scattering mechanisms and electronic transport in conductors we found that an electron With energy a above the Fermi level scatters at other electrons With a 9 rate which is quadratic in 5 7 EF At nite T l5 7 Epl 2 T so the thermally averaged scattering rate goes as T2 From the quantum mechanical point of view a nite scattering rate of a state means that this state is not an eigenstate of the Hamiltonian This is very natural as the plane wave or Bloch states if lattice e ects are important which are the eigenstates of a free Hamiltonian are not the eigenstates of a Hamiltonian which includes e e interactions Due to the interaction the plane waves decay A decaying state is described by an eigenenergy which has both real and imaginary part think of a well known QM problem about the 047 decay of a nucleus so Ep 5p 2T where P A 5p 7 EF2 at T 0 However we cannot be sure anymore that the real part of the spectrum 5p is still p22m as it is for free particles In fact we should expect the interaction changes the electron spectrum so that 5p p22m 7 5 Our next task is to nd the renormalized ie changed by the interaction electron spectrum This is achieved by introducing the concept of self energy A Selfenergy A ground state energy of a free electron system is p2 E0 2 pr p where fp is the Fermi function At T 0 3 3 E0 gL nEF where L3 is the system volume see A M Suppose that the interaction part of the ground state energy can be written as Eint 2 p7 p where 21 is some function of the electrons momentum called the self energy which depends on the interaction and vanishes in the limit when the interaction goes to zero The total energy is then 2 p Etot E0 Eint 2 Epgt fp E 25fp p p 10 where i p2 5p 21 So now it looks like we have electrons with a new spectrum which is modi ed by the in teractions Our goal is to nd 21 in a couple of simple models The rst one is the Hartree Fock model in which electrons are assumed to interact via the bare unscreened Coulomb potential This model will not be of much practical use as we will see later it will produce a meaningless result a vanishing electron7s mass However this model sets up a stage for more sophisticated approaches B Hamiltonian of a jellium model The model system we study is a bunch of electrons in the presence of a positively charged ions lonic charge is assumed to be spread uniformly over the system volume jellium model The net electron charge is equal in magnitude and opposite in sign to the that of ions so that overall the system is electroneutral This part of the model is common for all tractable models of e e interactions in solids including but not limited to the Hartree Fock approximation The classical energy of the system electrons ions is 1 a a a a a a a Em i d3r1d3r2n r1 V58 T1 7 r2n m d3r1d3r2n r1 V5 r1 7 r2n 10 1 d3T1d3T27Z l F1 F2 7 where n F is the non uniform electron density m is the uniform density of ions VEWW are the potentials of electron electron electron ion and ion ion interactions In the Hartree Fock approximation Vii 7717772 11 Using 11 Eq10 can be re arranged as Emt 7 g d3 d37 2 71071 7 71 Va A 7 F2 7103 7 77 In this form the energy has a very simple meaning At point F1 the electron density deviates from the ionic density so that the net charge density is 72 F1 7 m Similarly at point F2 the net charge density is 72 F2 7m These local uctuations in density interact via the Coulomb 11 potential In the absence of any external perturbations boundaries and impurities the average electron density is uniform and equal to the density of ions 72 771 m However the ground state energy involves the product of densities at di erent points 72 F n a correlation function which is not uniform In classical systems local uctuations in the charge density are due to thermal motion of electrons In a quantum mechanical system at T 0 they are due to the zero point motion of electrons in a Fermi system the kinetic energy is nite at T 0 To treat the QM system we need to pass from the classical energy to a Hamiltonian A 1 7 7 g d3r1 cm W m 7 7221145017 72 m a 7 71217 lt12 where now fl is a number density operator Densities of ions do not uctuate so we can leave them as classical variables c numbers Also the potential is the same as in the classical system It is convenient to re write Eq10 in a Fourier transformed form A 1 1 A A Hint 7 Z n 7 niljlee n 7 niliq v q where a 1 A 7217 71007722 EZWVFMEE q a 1 72 87 FZVBBQDE qv with V58 q 47reZq2 and where we took into account that V58 q depends only on the magnitude of q The Fourier transform of a uniform density of ions is niLg qjo so A 1 1 Hint Z 7341 nil36110 V58 q 7347 7 nil36110 7 1 1 A A g E 7117 N qol Veg q 72177 N510 17 where N is the total number of ions equal to that of electrons On the other hand 7370 d3r F N where N is an operator ofthe total number of particles Because the total number of particles is xed this operator is simply equal to its expectation value7N Thus the f 0 term gives no contribution to the sum Physically it means that the uctuation of in nite size in real 12 space corresponding to f 0 do not interact as they are compensated by uniform charge of ions With that7 we re write Hint as 1 A A ijzn 5qniq 13 At T 07 the ground state energy is simply an expectation value of Hint Eint Hint l l A A 0gt g ZVEE 1 lt0 lnq niq l 0gt 0 As we simply do the perturbation theory with respect to V857 the expectation value is cal culated using the wave functions of the free system vacuum average Now we need the second quantized from of g To this end7 we recall that in real space W Zwtltmltrgt Operator 11 Ila creates annihilates a particle with spin projection on at point F Using the plane waves as our basis set7 1 dd m a 732 p 1 rr mm P gt A H R 02 x m d A gt Q J l R 02 x m d A it Q M 6 Q d A 3 6 Q Q J 1 dd dd dd idBTFmTZ Cgaew Tc ae 2 7 L 0 1510quot 7 l n ZZCptq aCa 0 15 Substituting this last result into Eq137 we obtain Hint Z Z ZVEE q oldidacp acld c 2 L3 0 5 W p q 19 5 It is convenient to re write the second quantized Hamiltonians in the normal ordered form7 when all creation operators are positioned to the right of the annihilation ones lnterchanging the positions of two fermionic operators twice no sign change and discarding the terms which result from the 6 function part of the anti commutation relations those will give only a shift of the chemical potential7 we arrive at A l l 7 T T Hint 7 Z Z Z We Cfdjacgi c c a 0 5 W3 13 Thus7 Hint 0gt i i Eint 7 lt0 Cfill yDlCEq BCEBc a The expectation value of the product of of operators can be calculated directly However7 it is much more convenient to use the result known as the Wick s theoremzfl The Wick theorem states that an expectation value of a product of any number of of operators splits into products of expectation values of pairwise averages called contractions 0gt T 0p m Cm m M1 T M2 0H 1Cp2ainj1cp jaj 0gt Engag ffmej lt0 P for M1 M2 and is equal to zero otherwise The sum goes over permutations The factor E is equal to one if it takes an even number of permutations to bring Ogi mcp jm together no permutation is an even permutation of order zero and it is equal to minus one if takes an odd number of permutations The The pairwise averages are nothing more then occupation numbers 0 c T p m CW 1 0gt daiajaj i15jfpi39 where we suppress the spin index of fpi assuming that the ground state is paramagnetic For our case7 we can pair operators in two ways lt0 cgiacgwcmcm 0gt Olcgtjacp almOlcg mcmm iltOlcltmcmlOgtltOlcgmc a10gt 6170fpfk 7 6a56 5715fpfp The rst term drops out from the sum7 whereas the second one gives 1 l Eint Zznelt page 0 57131 mi 7 Z Mire31m 57kee where 2373919 i dgk V54 217 ZVEJ 6 Calculate the integral over I at T 0 1 kF 1 1 2 if 2 dkkz d 6 14 p 271396 0 71 COS p2 k2 7 Zpk cos 6 14 1 2 k k 7 773dekin p 0 2M lpikl 62k 17y2 1y 7 1 ln 7 27139 2y 17y where y E pkp Near the bottom of the band when 1 ltlt kp7 ie7 y lt 17 we have 62k 2 62k ezpz g 7 2 7 7 2 7 p 27139 lt 3y 7r 37rkp Adding this up with the free spectrum7 we obtain 8 762k Ii ezpz 762k p2 p 7r 2m 37rkp 7r 277v A constant term means that the chemical potential is shifted by the interaction The effect we are after is the change of the coef cient in front of p2 1 1 1 62m m m 37rkp m OT 52m 1 3ka Near the bottom of the band7 the effective mass of electrons is smaller than the band mass However7 this is not the part of the spectrum we are interested in thermodynamic and transport phenomena In this phenomena7 the vicinity of kp plays the major role Away from the bottom of the band the self energy is not a quadratic function of p7 so the resulting spectrum is not parabolic see Fig 3 We need to understand what is the meaning of the effective mass in this situation C Effective mass near the Fermi level In a Fermi gas7 the Fermi momentum is obtained by requiring that the number of states within the Fermi sphere is equal to the number of electrons 27rkf p n 2703 This condition does not change in the presence of the interaction Therefore7 kp of the interacting system is the same as in a free one This statement can be proven rigorously 15 FIG 3 Electronic spectrum in the Hartree Fock approximation as a function of pkp Solid with interaction 62m27rkp 05 and m 1 Points bare spectum Notice a kink in the full spectrum atpkp and is known as the Luttmger theoremg The energy of a topmost state is the chemical potential In a Fermi gas7 2 F g 739 M pp 2m When the spectrum is renormalized by the interaction7 At the Fermi surface this equality reduces to 2 PF 0 7 E 7 a 2m pF M PF 7 E M 2m pF The chemical potential is changed due to the interaction Near the Fermi surface ie for lP PFl ltlt va 2 6 77 2 PF p7 2m2PPFPFTTLEPF 2 Near the Fermi surface the spectrum can always be linearized 2 p ipipipippppp 7 2m 2m 7 2m NUF p pF39 Expand the self energy in Taylor series 2pipppp 720910 W Swim Where 2 E azaplpw The renormalized spectrum can be also linearized near the Fermi surface Eimv p pivb Where 1 is the renormalized Fermi velocity Now we have v pipp vFppFE pipF 1 vp Y The effective mass near the Fermi surface is de ned as L E 771 7 PF so that i i Ly 771 PF or 772 m 15 1 XYUF39 772 is determined by the derivative of the self energy at the Fermi surface 1 E ective mass in the Hartree Fock approximation Calculating the derivative of the self energy in the Hartree Fock approximation Eq14l7 we arrive at an unpleasant surprise 2 00 According to Eq15 this means that m 0 This unphysical result is the penalty we pay for using an oversimpli ed model in Which electrons interact via the unscreened Coulom potential 2 Beyond the Hartree Fock approximation The de ciency of the Hartree Fock approximation is cured by using the screened Coulomb potential Because electrons exchange energies the polarization clouds of induced charges around them are dynamic ie the two body potential depends not only on the distance but also on time In the Fourier space it means the interaction is a function not only of q but also of frequency w 4713962 V w 7 ea 17 126 q7w7 Where 6 qw is the full Lindhard function A calculation of the self energy in this case is a rather arduous task so I Will give here only the result for the effective mass near the Fermi leve 13 m4lt 2 6 2PF 7 7 lm 7 m 7er 6H 7 Where 6 2718 Where a is the screening wavevector n2 4713962 VF m4lt 1 1 4 13 1 7 4 13 1 E 7 7 H H 27182 7 022 17 08rsln T5 In terms of T5 This expression is valid for a weak interaction ie Ts ltlt 1 For very small rs m decreases with T5 At if m 008 m has a minimum and it becomes equal to m at rs m 022 111 STONER MODEL OF FERROMAGNETISM IN ITINERANT SYSTEMS Consider a model of fermions interacting via a delta function potential V071 7 F2 96 F1 7 F2 a good model for He3 atoms The interaction part of Hamiltonian reads in 18 real space 1 a a a a a a d3r1 d3r2VT1 a w n w n w n m n For a delta function interaction we need to take into account that 110 F 115 F a 0 only if or a Hint then reduces to g dwl F W F m r m m An expectation value of the interaction energy per unit volume is given by L73lt01Himiogt L gg d3rlto M r w c m F m ml 216 using Wicks theorem r39 d3rlt01111 F xix F10gtlt01111 FIIlF10gt 17 L Bgdgrnml 9mm 18 Where mm is the expectation value of number density of spin up down fermions Which is position independent In a paramagnetic state 72 ni 722 Where n is the total number density Let7s analyze the possibility of a transition into a state With a nite spin polarization ferromagnetic state in Which n a ml The kinetic energy of a partially spin polarized system is E 1 d3k 75 75 0 0lt1cltlcjv 271393 k 0lt1clt1cfv 271393 k7 Where kiln are the Fermi momenta of the spin up down fermions related to 72M via gvr W gt3 lt 72 2mg M 1 3 a we Fig 8 7 4w fl 2K7 kgif 0ltkltk 271393 k 7 2703 0 2m 7 207r2m 0 6W253 53 53 207r2m 1711 n1 1 Introducing the full density n 72 72 and the di erence in densities 6n n 772 so that 72M 12 72 i 672 the equation for E0 takes the form 6W253 ll 6 n 670531 0 19 Obviously E0 has a minimum at 672 0 ie in a paramagnetic state The interaction term changes the balance gm 94 W 7 W For a repulsive interaction 9 gt 0 the interaction part of the energy is lowered by spin polarization The total energy is the sum of two contributions WW3 ll 6 n Wgl 94 722 e 671 EE0Eint lntroducing dimensionless quantity Q 67272 and using the relation between the Fermi energy and the density the expression for energy is simpli ed to E 130nm 1 053 lt1 7 053 72294 17 lt2 Suppose that Q is small then the rst term can be expanded to the second order the rst order contribution vanishes using 1 53 17 53 2 g lt4 0 5 Elt gnEF gnEFCZ 71294 lt17 2 711919 E0alt2 nEFC4 where E0 gnEF 72294 and i l 39 i l 1 a i gnEFlt17 i gnEFlt179VF 7 with V19 3n4EF being the density of states per one spin orientation If a gt 0 the ferromagnetic state is energetically unfavorable If a is negative the ferromagnetic state is energetically unfavorable The transition occurs when a crosses zero The critical value of the coupling constant is go 1 Viol For g gt go polarization is stabilized by the quartic term An equilibrium value of polariza tion is determined from the condition 2E ac 0 20 2 g 4 3 7E 177 7E 0 371 Flt 96gtC817l FC m 12 9790 2447 lt gc Unfortunately7 the critical value for g is outside the weak coupling regime and7 therefore7 the Stoner model cannot be considered as a quantitative theory of ferromagnetism However7 it does provide with a hint as to how ferromagnetism occurs in real system The Stoner model is also probably the rst example of zero temperature or quantum phase transitions ie7 phase changes in the ground state of the system driven by varying some control parameter in this case7 the e ective interaction strength lncidentally7 it is also a second order or continuous phase transition the energy of the system is continuous through the critical point The equilibrium value of the order parameter Q is zero for g lt 96 and becomes nite but still small for g gt go A square root dependence of Q on the devitation of the control parameter from its critical value is a characteristic feature of the mean eld theories The Stoner model also predicts that the spin susceptibility of the paramagnetic state is enhanced by the repulsive interaction To this end we need to introduce magnetization the magnetic moment per unit volume MmmMmw where 9L is the Lande gifactor not to be confused with the coupling constant and MB is the Bohr magneton7 and switch from the ground state energy to the free energy at T 0 F E 7 MH Retaining only quadratic in M terms7 we have FE gjg iMH 91371 An equlibrium magnetization is obtained from the condition 8i 0 a 8M 2 M 91371 H 2a Recalling that the spin susceptibility is de ned by the following relation7 we read o the susceptibility as X LIL13702 LIL13702 2a lt23gtnEF17 M 21 2 QLMB VF X 7 3 701 3X0 1 yVF 1 yVF where VF is the density of states per two spin orientations7 X0 is the spin susceptibility of a free electron gas at T 0 Pauli susceptibility7 and 1 1 7 9V S E is called the Stoner enhancement factor For g gt 07 X gt X0 Also7 X diverges at the critical point A divergent susceptibility is another characteristic feature of both nite and zero temperature phase transitions What is the magnetic response of the system at the critical point7 where the linear susceptibility X 00 To answer this7 we need to restore the quartic term in the free energy 1 E FE02 27MHi9 FZM4 LIL 371 81 QLMB 1 At the critical point7 g go and a 0 The equation of state then reads 8F 4 E 7 7H iQ FM393 8M 81 gLuB2n Therefore7 at the critical point M scales as Hlg Note that a magnetic eld smears the phase transition because now there is a nite magnetization even above the critical point The order parameter now changes continuously through the transition which is de ned as a critical value of y where the linear susceptibility diverges IV WIGNER CRYSTAL As we now understand the properties of a weakly interacting electron gas7 let7s turn to the opposite limit7 when the interaction energy is much larger than the kinetic one At T 07 the kinetic energy is the Fermi energy so the condition for the strong interaction is rs gtgt 1 What happens in this limit The answer was given by Wigner back in 1934 Because the Coulomb energy is very high7 electrons would try to be as further away from each other as possible ldeally7 they would all move to the sample boundaries However7 this would create an enormous uncompensated positive charge of ions The next best thing for electrons to do is to themselves into a lattice Wigner crystal of spacing comparable to the average inter electron distance in the liquid phase 22 Let7s estimate When the formation of the Wigner crystal is possible In a liquid7 electrons can come to each other at arbitrarily small distances7 Where the Coulomb energy is large In a crystal7 electrons are separated by the lattice spacing7 a The gain in the potential energy PL 7 PC 2 i a for the sake of simplicity I assume that the dielectric constants of ions is unity On the other hand7 kinetic energy in a liquid is KL 2 1ma2 Whereas that in a crystal KC 2 1mrg7 Where r0 lt a is the rms displacement of an electron about its equilibrium position due to the zero point motion For the crystal to be stable7 one must require that To ltlt 1 Thus Kc gtgt KL and KC 7 KL m 1mr3 Crystallization is energetically favorable When the gain in the potential energy exceeds the loss in the kinetic one7 ie7 PL 7 PC gt Kc 7 KL or 62 3 gtgt 1mr3 19 Now I want to show that the condition above is equivalent to T5 gtgt 1 To estimate To consider an oscillatory motion of an electron in a 1D lattice interacting With its nearest neighbors via Coulomb forces The potential energy of the central electron When it is moved by distance m from its equilibrium position m 0 is 62 62 Um 17 a Expanding the expression above for m lt a we obtain 62 62 U m 27 27x2 a 13 The harmonic part of the potential is reduced to the canonical form by equating 62 1 2732 E 7mw x2 7 a 2 2 e w 473 ma 3 Notice that because 1 2 717 4410 is of order of the plasma frequency 47m62 m Which is quite a natural result Thus the Debye frequency77 of a Wigner crystal is the plasma frequency 23 The quantum amplitude is related to the frequency via 1 N W mrg T 0 or 2 1 2 62 7 2 2 20 ltmr3gt W0 ma3 Squaring Eq22 and using Eq20 we nd or T5 gtgt 1 Thus crystallization is energetically favorable for T5 gtgt 1 Quantum Monte Carlo simulations show that the critical value of T5 at which the liquid crystallizes is 150 in 3D and 37 in 2D Why so huge numbers This one can understand by recalling that crystals melt when the amplitude of the oscillations is still smaller than the lattice spacing The critical ratio of the amplitude to spacing is called the Lindemann parameter A For all lattices A is appreciably smaller than unity this helps to understand why the melting temperatures are signi cantly smaller than cohesive energies Typically A 2 01 7 03 Strictly speaking one has to distinguish between classical and quantum Lindemann parametersisince entropy plays a role for the former but not for the latteribut we will ignore this subtlety With the help of Eq20 we nd that TQl 1L4 Ts Wigner crystal melts when TQl A a 7 i gt 1 A4 Wigner crystals were observed in layers of electrons adsorbed on a surface of liquid helium that makes the smoothest substrate one can think of The search for Wigner crystalliza tion in semiconductor heterostructures is a very active eld which so far has not provided a direct evidence for a crystal structureisuch as Bragg peaksialthough a circumstantial evi dence does exist One of the problem here is that e ects of disorder in solid state structures 24 becomes very pronounced at lower densities so there is no hope to observe Wigner crys tallization in its pure form What one can hope for is to get a distorted crystal Coulomb glass V FERMILIQUID THEORY A General concepts A detailed theory of the Fermi liquid FL and its microscopic justi cation in particular goes far beyond the scope of this course Standard references345 provide an exhaustive if not elementary treatment of the FL theory In what follows I will do a simpli ed version of the theory FL lite and illustrate main concepts on various examples 1 Motivation All Fermi systems metals degenerate semiconductors normal He3 neutron stars etc belong to the categories of either moderately or strongly interacting systems For example in metals T5 in the range from 2 to 5 There are only few exceptions of this rule for example bismuth in which the large value of the background dielectric constant brings the value of T5 to 03 and GaAs heterostructures in which the small value of the effective mass 7007 of the bare massileads to the higher value of the Fermi energy and thus to T5 lt 1 in a certain density range On the other hand as we learned from the section on Wigner crystallization the critical T5 for Wigner crystallization is very high 7150 in 3D and 37 in 2D Thus almost all Fermi systems occuring the Nature are too strongly interacting to be described by the weak coupling theory Hartree Fock and its improved versions but too weakly interacting to solidify In short since they are neither gases nor solids the only choice left is that they are liquids A liquid is a system of interacting particles which preserves all symmetries of gas A difference between usual classical gases and liquids is of quantitative but not qualitative nature Following this analogy Landau put forward a hypothesis that an interacting Fermi system is qualitatively similar to the Fermi gas6 Although original Landau7s formulation refers to a translationally invariant system of particles interacting via short range forces eg normal He3 later on his arguments were extended to metals which have only discrete symmetries and to charged particles 25 Experiment gives a strong justi cation to this hypothesis The speci c heat of almost all fermionic systems in solids one need to subtract o the lattice contribution to get the one from electrons scales linearly with temperature C T 7T Some systems demonstrate the deviation from this law and these systems are subject of an active studies for the last 10 years see more in Non fermi lz39quid behavior In a free Fermi gas 7 7 7123 VF 13mkp In a band model when non interacting electrons move in the presence of a periodic potential one should use the appropriate value of the density of states at the Fermi level for a given lattice structure In reality the coe icient 7 can di er signi cantly from the band value but the linearity of CT in T is well preserved In those cases when one can change continously the interaction for example by applying pressure to normal He3 7 is found to vary One is then tempted to assume that the interacting Fermi liquid is composed of some e ective particles quasi particles that behave as free fermions albeit their masses are di erent from the non interacting values B Quasiparticles The concept of quasi particles central to the Landau7s theory of Fermi liquids The ground state of a Fermi gas is a completely lled Fermi sphere The spectrum of excited states can be classi ed in terms of how many fermions were promoted from states below the Fermi surface to the ones above For example the rst excited state is the one with an electron above the Fermi sphere and the hole below The energy of this state measured from the ground state is e p22m 7 EF The net momentum of the system is 1 The next state correspond to two fermions above the Fermi sphere etc If the net momentum of the system is 13 then 171 52 where 131 and g are the momenta of individual electrons We see that in a free system any excited macrosopic state is a superposition of single particle states This is not so in an interacting system Even if we promote only one particle to a state above the Fermi surface the energy of this state would not be equal to p22m 7 EF because the interaction will change the energy of all other fermions However Landau assumed that excited states with energies near the Fermi level that is weakly excited states can be described as a superposition of elementary excitations which behave as free particles although the original system may as well be a strongly interacting one An example of such 26 a behavior are familiar phonons in a solid Suppose that we have a gas of sodium atoms which are fermions which essentially don7t interact because of low density The elementary excitations in an ideal gas simply coincide with real atoms Now we condense the gas into a metal lndividual atoms are not free to move on their own any longer Instead they can only participate in a collective oscillatory motion which is a sound wave For small frequencies the sound wave can be thought of consisting of elementary quanta of free particlesiphonons The spectrum of each phonon branch is to q siq where 32 is the speed of sound and the oscillatory energy is E ngwmi 2 27F where n is the number of excited phonons at given temperature If the number of phonons is varied so is the total energy dgq 6E 7w16n1 2 2703 Z Z Quite similarly Landau assumed that the variation of the total energy of a single component Fermi liquid or single band metal can be written as 6E 3351 672 p 21 For the sake of simplicity I also assume that the system is isotropic ie the energy and n depend only on the magnitude but not the direction of the momentum but the argument extends easily to anisotropic systems as well In this formula 721 is the distribution function of quasi particles which are elementary excitations of the interacting system In analogy with bosons these quasi particles are free One more iand crucial assumptioniis that the quasi particles of an interacting Fermi system are fermions which is not at all obvious For example regardless of the statistics of individual atoms which can be either fermions or bosons phonons are always bosons Landau7s argument was that if quasi particles were bosons they could accumulate without a restriction in every quantum state That means that an excited state of a quantum system has a classical analog lndeed an excited state of many coupled oscillators is a classical sound wave Fermions don7t have macrosopic states so quasi particles of a Fermi systems must be fermions to be precize they should not be bosons proposal for particles of a statistics intermediate between bosons and fermionsianyonsihave been made recently Thus 1 27 On quite general grounds7 one can show that quasi particles must have spin 12 regardless of half integer spin of original particles374 Thus7 quasi particles of a system composed of fermions With spin S 32 still have spin 12 Generally speaking7 the quasiparticle energy is an operator in the spin space 5 a Q and7 correspondingly7 1 exp 17 nan 22 Where and fl are 2gtlt2 matrices If the system is not in the presence of the magnetic eld and not ferromagnetic7 gut5 86003772043 7746043 In a general case7 instead of 21 we have dgp A A 6E Wrrapwmpx Which7 for a spin isotropic liquid7 reduces to 6E 2 535 p 672 p The occupation number is normalized by the condition 6N Where N is the total number of real particles d3 2703 T367109 2 113571 p 07 2 At T07 the chemical potential coincides With energy of the topmost state u 5 pp E EF Another important property known as Luttmger theorem is that the volume of the Fermi surface is not a ected by the interaction For an isotropic system7 this means the Fermi momenta of free and interacting systems are the same A simple argument is that the counting of states is not a ected by the interaction7 ie7 the relation 4 3 3 N2L F 27F holds in both cases A general proof of this statement is given in Ref 28 1 Interaction of quasiparticles Phonons in a solid do not interact only in the rst harmonic approximation Anhar monism results in the phonon phonon interaction However the interaction is weak at low energies not really because the coupling constant is weak but rather because the scattering rate of phonons on each other is proportional to a high power of their frequency ip 0c 4415 As a result at small to phonons are almost free quasi particles Something similar happens with the fermions The nominal interaction may as well be strong However because of the Pauli principle the scattering rate is proportional to 5 7 EF2 and weakly excited states interact only weakly In the Landau theory the interaction between quasi particles is introduced via a phenomenological interaction function de ned by the proportionality coef cient more precisely a kernel between the variation of the occupation number and the corresponding variation in the quasi particle spectrum 58 Lp3fm5657175717617 27F Function fm g 13 describes the interaction between the quasi particles of momenta and 17 notice that these are both initial states of the of the scattering process Spin indices 04 and 5 correspond to the state of momentum whereas indices 7 and 6 correspond to momentum In a matrix form ampn where Tr denotes trace over spin indices 7 and 6 For a spin isotropic FL when 60565 f 5 13 7 23 d3p 2703 and 6n 605672 external spin indices 04 and B can also be traced out and Eq23 reduces to where f M 2 gm M For small deviations from the equilibrium 6n is peaked near the Fermi surface Function f can be then estimated directly on the Fermi surface ie for pp Then f depends only on the angle between and The spin dependence of f can be established on quite general grounds In a spin isotropic FL f can depend only on the scalar product of spin operators but no on the products of the individual spin operators with some other 29 vectors since there are no more vectors in the spin space Thus the most general form of f for a spin isotropic system is WWW F9fG95 5 7 where f is the unity matrix lt3 is the vector of three Pauli matrices 6 is the angle between 13 and 17 and the density of states was introduced just to make functions F and G dimensionless Star in M means that we have used a renormalized value of the e ective mass so that M mkp7r2 but this is again just a matter of convenience Explicitly foams 17717 F 9 5048576 G 9 Ta8 39 576 24 In general the interaction function is not known However if the interaction is weak one can relate f to the pair interaction potential In a microscopic theory it can be shown that if particles interact via a weak pair wise potential U q such that U 0 is nite which excludes eg a bare Coulomb potential then to the rst order in this potential 4 4 1 4 4 1 4 4 fam pvp 5576 10 Ulp p l 3M5 39 MUG 1 D On the Fermi surface nd that 13 2pp sin 62 Comparing this expression with Eq24 we 1w V1 U 0 4 U2stme2l 25 cw i wzstmez Notice that repulsive interaction U gt 0 corresponds to the attraction in the spin exchange channel G lt 0 C General strategy of the Fermiliquid theory One may wonder what one can achieve introducing unknown phenomenological quantities such as the interaction function or its charge and spin components FL theory allows one to express general thermodynamic characteristic of a liquid e ective mass compressibility spin susceptibility etc via angular averages of Landau functionsiF 6 and CG Some quantities depend on the same averages and thus one express such quantities via each other The relationship between such quantities can be checked by comparison with the experiment 30 D Effective mass As a rst application of the Landau theory7 let7s consider the e ective mass In a Galilean 85 A invariant system7 the momentum per unit volume coincides with the ux of mass d3 8A Trizmim 27139 81 d3 7 TrTz fpn i 26 where m is the bare mass Taking the variation of both sides of this equation7 we obtain 85 d3 d3 86 T 6A T 7 7A 76A T27r3p TZwgm 813 813 n d3 8A ear 12 fP7P6 p 39 Now7 865 i r f 8p 2w3 8p For a spin isotropic liquid7 Eq26 reduces to d3 d3 88 dng dgp af 1717 a6n7mi 6n m 672 4 n 2mg 3 8p 2M3 2703 8p 1 In the second term re label the variable 17 177539 a p 7 use the fact that f 13 7 f and integrate by parts d3 d3 85 d3 dgp 8n a6 mi n 7 a 839 6n 2703 2mg 8p 2mg 2703f 17 p a Because this equation should be satis ed for an arbitrary variation 672 7 Eii dgp fa m 8 2m3 M 8 Near the Fermi surface7 the quasi particle energy can be always written as 81 v pipp so that 35 7 l A 7 PF A f va 197 where 13 is the unit vector in the direction of 13 and thus I7 p I3 dgp a an 1339 i 3fltp7 m m 27139 31 Now aw 85w an 817 7 817 85 Near the Fermi surface7 an 776 7E 85 8 F and 31pr 1 cm H m m7 W 2VF 47TfP7P W or E 7 p171 l 4lt dQ a 4 pF i 17F3 19 a 4 A mi W 2 T fmp W i W 2W2 Tm pmmpp Setting pp and multiplying both sides of the equation by 137 we obtain 1 7 1 lpp do Ringi Ef6cos6 or 1 1 pp 7 7 7 7 7 6 6 27 m m 271392 47rflt COS This is the Landau7s formula for the e ective mass Noticing that f 6 12TrTr f 6 2ng F 6 27139277 pp7 the last equation can be reduced to d9 m7 17F6cos631F1 m 47139 Although we do not know the explicit form of f 67 some interesting conclusions can be derived already from the most general form Eq27 If f 6 cos 6 is negative7 1m gt 1m a m lt m conversely7 if f 6 cos 6 is positive7 m gt m For the weak short range interaction7 we know that m gt m7 whereas for a weak long range Coulomb interaction m lt m Now we see that both of these cases are described by the Landau7s formula Recalling the weak coupling form of F 6 Elt11257 we nd d9 d9 1 6 F1 EF6cos6 U 07 U lt2pps1n gt cos6 dQl 6 7 EgU lt2pps1n gt cos6 If U is repulsive and peaked at small 67 where cos6 is positive7 F1 lt 0 and m lt m This case corresponds to a screened Coulomb potential If U is repulsive and depends on 6 only weakly short range interaction7 the angular integral is dominated by values of 6 close to 71397 where cos6 lt 0 Then m gt m In general7 forward scattering reduces the effective mass7 whereas large angle scattering enhances it E Spin susceptibility 1 Free electrons We start with free electrons An electron with spin 3 has a magnetic moment u 2mm7 where MB 671ch is the Bohr magneton Zeeman splitting of energy levels is AE ET 7 El MBH 7 fuBH ZuBH The number of spin up and down electrons is found as an integral over the density of states 1 EpinBH 71M iO day The Fermi energy now also depends on the magnetic eld but the dependence is only quadratic why7 whereas the Zeeman terms are linear in H For weak elds7 one can ne glect the eld dependence of EF Total magnetic moment per unit volumeimagnetizationiis found by expanding in H MB EFMBH EF MBH M MBnT7ni7 A d5u57 d5u5 MB EF EF 7 d8Vlt8gtMBHVF d8Vlt8gtMBHVF 0 0 MZBHVF Spin susceptibility 8M 8H is nite and positive7 which means that spins are oriented along the external magnetic eld X NZBVF 28 Notice that if7 for some reason the 9 factor of electrons were not equal to 27 then Eq28 would change to X QNZBVF 29 2 Fermi liquid In a Fermi liquid7 the energy of a quasi particle in a magnetic eld is changed not only due to the Zeeman splitting as in the Fermi gas but also because the occupation number is changed This e ect brings in an additional term in the energy as the energy is related to the occupation number This is expressed by following equation p A a a a 3fp7p5np d3 87 H T 7 5 MB 0 r 2 33 The rst term is just the Zeeman splitting7 the second one comes from the interaction Now7 6A 762 n 88 5 and we have an equation for d d p A 87 6 7 H n rriasr 8 3 0 2W3fppag 81 Replacing by the delta function and setting pp A A A 4lt dQ A A A A A 66 pr fugH 0 7 T1 VF E pmm 58 W 30 Lets try a solution in the following form 6gJLBgHa7 31 2 where g has the meaning of an effective 97 factor For free electrons7 g 2 Recalling that V13f G amp 6 and substituting 31 into 317 we obtain NB 7 A dQ A A A MB A 79H077MBH07TrEF IG aallt79H0gt The term containing F in the integral vanishes because Pauli matrices are traceless Tra 0 The term containing G is re arranged using the identity Tramp 6 6 26 upon which we get g 1 7 gGm 2 2 where if G 7G 6 0 47139 Therefore7 i 2 g T 1 Co Substituting this result into Eq29 and replacing VF by its renormalized value V we obtain the spin susceptibility of a Fermi liquid 4lt MTBV 1 F1 X 1G0X1G0 where X is the susceptibility of free fermions Notice that x di fers from X both because the effective mass is renormalized and because the g factor di fers from 2 When G0 717 the g factor and susceptibility diverge signaling a ferromagnetic instability However7 even 34 if G0 0 x is still renormalized in proportion to the effective mass If m diverges which is a signature of a metal insulator transition of Mott type x diverges as well Recall that Eq25 in the weak coupling limit G 7U 2pp sin62 Thus for repulsive interactions 9 gt 2 which signals a ferromagnetic tendency This is another manifestation of general principle that repulsively interacting fermions tend to have spin aligned to minimize the energy of repulsion For normal He3 G0 m 723 F Zero sound All gases and liquids support sound waves Even ideal gases have nite compressibilities 8 83 8 For example in an ideal Boltzmann gas P nT pTm and 5quot T711 m 73 where UT MSTm is the rms thermal velocity In an ideal Fermi gas P 23E where and therefor nite sound velocities E 25 nEF is the ground state energy and 1 9 El n It seems that interactions are not essential for sound propagation This conclusion is not true In a sound wave all thermodynamic characteristicsidensity n r t pressure P rt temperature Trt if the experiment is performed under adiabatic conditions etcrvary in space and time in sync with the sound wave Thus we are dealing with a non equilibrium situation To describe a non equilibrium situation by a set of local and time dependent quantities one needs to maintain local and temporal equilibrium When the sound wave arrives to an initially unperturbed region this region is driven away from its equilibrium state Collisions between molecules or fermions in a Fermi gas has to be frequent enough to establish a new equilibrium state before the sound wave leaves the region For sound propagation the characteristic spatial scale is the wavelength A and characteristic time is the period 27rw If the mean free path and time are l and 739 respectively the conditions of establishing local and temporal equilibrium are A gt l 35 27rw gtgt 739 Therefore7 collisions interactions are essential to ensure that the sound propagation occurs under quasi equilibrium conditions For classical gases and liquids these conditions are satis ed for all not extremely high frequencies For example7 the mean free path of molecules in air at P 1 atm and T 300 K is about l 2 10 5 cm and 3 2 300 ms3gtlt104 cms Condition A gt Z takes the form W ltlt g 3 X104cms l 10 5cm In a liquid7 l is of order of the inter molecule separation few 10 7 cm As long as A gt 10 7 3 gtlt109 s71 cm7 equilibrium is maintained In a Fermi liquid7 the situation is di erent because l and 739 increase as temperature goes down Local equilibrium is maintained as long as 7Tw ltlt 1 Now7 739 T 1AT2 so at xed temperature sound of frequency Ad gtgt 7 1 T AT2 cannot proceed in a quasi equilibrium manner What happens when we start at small frequency and then increase it As long as 7Tw ltlt 17 a Fermi liquid supports normal sound in the FL theory it is called rst sound The velocity of this sound is linked to the sound velocity in the ideal Fermi gas but di ers from because of renormalization ixg where F0 f F The rst sound corresponds to the oscillations of fermion number 91 1 F0121 Fly27 density in sync with the wave Because the number density xes the radius of the Fermi sphere7 pp oscillates in time and space As the product 441739 increases7 sound absorption does so too7 at when 441739 becomes impossible However7 as 441739 increases further and becomes gtgt 17 another type of sound wave emerges These waves are called zero sound These waves can propagate even at T 07 when 739 00 as they do not require local equilibrium Another di erence between zero and rst sound is that in the rst sound wave the shape of the Fermi surface remains spherical whereas its radius changes lnstead7 in the zero sound wave the shape of the Fermi surface is distorted In a simplest zero sound wave 7 the occupation number is 672 6 5 7 EF V eiohtm A cos V USOUp 7 cos 36 where 30 gt Up is the zero sound velocity The Fermi surface is an egg shaped spheroid with the narrow end pointing in the direction of the wave propagation The condition 30 gt vp always satis ed for zero sound means absence of Landau damping As well as 31 the zero sound velocity 30 depends on Landau parameters F0 and F1 although the functional dependence is more complicated At the same time F1 can be extracted from the e ective mass measured from the speci c heat and F0 is extracted from the compressibility Having two experimentally determined parameters F0 and F1 one can substitute them into theoretical expressions for 31 and 30 and compare them with measured sound velocities The agreement see the plot is quite satisfactory This comparison provides a quantitative check for the Fermi liquid theory VI NON FERMILIQUID BEHAVIORS Majority of simple metals con rm to the Fermi liquid behavior He3 is a classical Fermi liquid system However there is quite a few of situation when the expected Fermi liquid behavior is not expected Sometimes its because the system is disordered and the Fermi liquid theory is constructed for a translationally invariant system Even more interestingly there is quite a few of systems most notably HTC superconductors in their normal phase and heavy fermion materials where e ects of disorder can be ruled out yet the Fermi liquid behavior is not observed In what follows I will give a brief overview of deviations from the Fermi liquid behavior A Dirty Fermi liquids 1 Scattering rate According the Fermi liquid theory the scattering rate of quasi particles behaves as T2 Since these are inelastic collisions they break quantum mechanical phase and one might have expected that the phase breaking rate observed in the experiments on weak localization goes also as T2 at low enough temperatures when phonons are already frozen out In fact a T2 is never observed at low temperature Instead the phase breaking rate in thin lms goes as T and in thin wires as T23 What went wrong Well we forgot to take into account that in a disordered metal electrons move along di usive trajectoris rather than along straight 37 lines This is going to change T27 law One can understand very easily why di fusive motion enhances the scattering rate of electron electron interaction Diffusing electrons move slowly and thus spend more time lingering around each other hence the interaction is effectively enhanced In what follows I will give a simple argument due to Altshuler and Aronov to why and how the scattering rate is changed in the presence of disorder In a clean system electrons exchange energies of order A5 2 T in the course of interaction By the uncertainty principle the duration of the collision event not to be confused with the mean free time is given by 1 1 At 7 AtltltTigtTTigtgt1 where Ti is the impurity mean free time electrons don7t have time to scatter at impurities during a single act of interaction and interaction between them proceeds as if there is no 1 disorder In this case we are back to the FL result 7quot oc T2 In the opposite limit when AtgtgtTigtTTiltlt1 electrons experience many collisions with impurities while interacting with each other To see how this is going to affect the scattering rate let7s come back to the calculation of 7 1 in a clean system at T 0 Notes 1 There we showed that 1 a 0 1 1 20 dwlwdeldqq2wq1l dcos 6w iqvpcos i1L dcos 6w iqvpcosGf Angular integral of each of the delta function gives a factor 1qvp Re writing this factor as we see that it has units of time The meaning of this time is the duration of the interaction act in which electrons change their momenta by q Thus At g In a clean system the q integral is convergent at the lower limit and simply gives a constant prefactor in front of the 627 dependence which comes from the double energy integral In a dirty system momentum trasfer will take much longer Time and space are related via the diffusion law If1 m a 1 AtW 38 Now instead of the factor q 2 from the delta functions7 we have A752 1D2q4 The matrix element is nite at q a 0 However7 the integral now diverges at the lower limit 1 dq 1 d 27777777 32 qq lt14 lt12 lt1 To regularize the divergence we have to recalling that the uncertainty in the transferred energy has to be smaller than the transferred energy itself A7571 lt w a qu gt w a q gt MuD Thus the lower limit in Eq32 is MooD and dq 1 7 oc 7 xwD q W The energy integral now becomes 1 6 0 1 7 A 140 deli AegZ 739 0 7o W instead of the FL form 62 Repeating the same steps in 2D7 we nd for the q7 integral dqq 1 7 OC xwD q4 u and thus 1 7 Ae 7 Upon thermal averaging7 this gives us a linear in T dependence of the dephasing rateM Does a change from the T2 to the T7 dependences means a breakdown of the FL Not really Restoring all dimensional prefactors and for the dimensionless coupling constant nkp 2 17 the scattering rate in a dirty FL can be written as 1 1T2 EFF recall that T73 ltlt 1 an thus 7 1 gtgt TZEF or 1 T 7 i 33 739 EFTi A quasi particle is a well de ned excitation as long as the width of energy level P 127 is much smaller than the energy itself For thermal quasi particles7 typical energy counted 39 from the Fermi level 5 2 T Using Eq33 we see that the condition F lt 5 is satis ed as long as EFTi gtgt 1 which coincides with the general validity of the approach to disordered systems Therefore quasi particles are well de ned even in a disordered metal provided that it is still a good metal in a sense that EFTi gtgt 1 2 T dependence of the resistivity One can often see or hear the statement that the resistivity of a Fermi liquid goes as T2 As such this statement is incorrect A FL is by construction a translationally invariant system Electron electron collisions in such a system conserve the total momentum and thus the current which is proportional to the momentum cannot be relaxed by these collisions Hence the resistivity of FL per se is in nite When the FL is coupled to an external system lattice impurities the resulting resistivity is going to be determined by what the FL is coupled to We know for example that the resistivity of Fermi gas interacting with lattice vibrations goes as T and T5 in the regime of high and low temperatures respectively A FL would show the same dependence Electron electron collisions can lead to a nite resistivity only if Umklapp processes are involved lfthe Fermi surface lies within the Brillouin zone and is separated from the BZ boundaries by a nite momentim qmin then at lower temperatures the probability of an Umklapp scattering is exponentially small oc exp iquminT beacause it7s proportional to the number of electrons with momentum qmin above the Fermi surface Consequently p oc exp iquminT which is quite different from the expected T2 Such a behavior is indeed observed in ultra pure alkaline metals whose almost spherical Fermi surfaces lie within the Brillouin zone On the other hand if the Fermi surface intersects the boundaries of the Brillouin zone Umklapp processes are allowed at all temperatures and go at the rate comparable to that of normal processes In this case indeed p oc T2 Although most of the conventional metals belong to the second category cases of unambigous determinations of the T27 dependence as coming from the e e interactions are quite rare The problem is that the prefactor in the phonon T5 is numerically very large explain why and this dependens extends to quite low temperatures where saturation of the T dependence occurs Also other processes can mimic the T2 law For example when the FS is strongly anisotropic eg an ellipsoid with vastly different semi major axises kllm gtgt 161 the linear in T dependence is observed for T gtgt 3k whereas the T5 law sets in for T lt sk 40 In the intermediate range ski lt T ltlt 316 the resistivity is quadratic in T Also interference between electron impurity and electron phonon scattering results in a T2 term in the resistivity2 So far the most well established observations of the T2 law and its identi cation with the e e mechanism are limited to a very special class of compoundsi heavy fermion materials which are alloys of rare earth metals UPtg etc see the two comprehensive reviews by Greg Stewart on this subject Conduction electrons in these material come from very narrow f 7 bands and the e ective masses are enormously high 7100 200 of the bare electron masses whereas the densities are at the normal metal scale Consequently the e e scattering rate 1 N T2 N mT2 739 7 EF 7 7123 7 which is proportional to the mass is also very high and dominate the resistivity The Fermi surfaces of these materials tend to be very complicated which means that there is no problem with Umklapp scattering What happens at even low temperatures when the dominant scattering mechanism is the impurity scattering A scattering rate of a Fermi gas electron at an impurity is T7 independent and thus the resistivity saturates at the residual value ls this also true in a FL The answer is NO First of all as we learned when studying weak localization the resistivity of lms and wires does not really saturate at T a 0 but continues to increase Suppose that we applied a relatively weak magnetic eldijust enough to kill the weak localization e ect without causing any more changesiso this e ect is suppressed ls that7s all No because now we have to consider scattering of interacting quasi particles at impurities Back in 1979 Altshuler and Aronov7 predicted that the conductivity of such a system acquires an additional temperature dependence of the type similar to the one coming from weak localization 2 e 1 60 7E0 ln T773 which is valid for T73 ltlt 1 when log is positive In contrast to weak localization prefactor 34 C is generally speaking not universal but depends on the strength of the electron electron interaction Eq34 is known as interaction correction77 or Altshuler Aronov correction In the model of weakly interacting electron gas 0 is a universal positive number and thus sign of the correction coincides with the weak localization one However if spin exchange processes are strong so that the metal is close to the ferromagnetic instability C lt 0 and 41 thus the interaction increases the conductivity Up until recently only positive localizing signs of C were observed Altshuler Aronov theory was developed for the case of low tem peratures in a sense that T73 ltlt 1 ie when electrons move di usively during a single act of interaction Although this theory was veri ed in a number of independent derivations and countless number of experiments it was only recentlt realized that the condition T73 ltlt 1 is a suf cient but not necessary for the occurrence of the e ect In the opposite limit T73 gtgt 1 electrons interact with one impurity at a time thus the problem can be formulated as a scattering theory for interacting electrons We have already seen an application of such a scattering theory to the 1D case see Section Friedel oscillations There we found that a resonant scattering at a Friedel oscillation surrounding a potential barrier leads to a non analytic log dependence of the transmission coef cient on 5 7 EF In 2D and 3D it is more appropriate to speak of a scattering amplitude f 6 not to be confused with the Landau interaction function A modi cation of the reasoning for the 2D cases leads to the result that f acquires a non analytic dependence m 2 xT in a narrow angular interval around 7r 6 7 7rl 2 m 2 Thus the correction to the cross section and hence to conductivity goes as T 60 720T739i The sign and magnitude of C depends on the interaction strength for strong interactions the correction is of the metallic ie anti localizing sign Thus a metal with interactions is a better metal than without This e ect is a prime suspect for the recently observed metallic T dependences in semiconductor heterostructures which are associated with the phenomena of metal insulator transition in 2D779 lO occurring at odds with the single parameter scaling theory of localization VII KONDO EFFECT H J Schulz Fermi liquids and non Fermi liquids in Proceedings of Les Houches Summer School LXl edited by E Akkermans et al Elsevier Amsterdam 1995 p 553 Available at XXXlanlgovabscomd mat9503150 1 D Yue L l Glazman and K A Matveev Phys Rev B 49 1966 1994 42 u 00 H o H H H N Mahan Martybody physics A A Abrikosov L P Gorkov and l E Dzyaloshinski Methods of quantum eld theory in statistical physics Dover Publications New York 1963 EM Lifshitz and LP Pitaevskii Statistical Physics ll Course of Theoretical Physics V IX Pergamon Press Oxford 1980 D Pines and P Nozieres The Theory of Quantum Liquids Benjamin New York 1969 L D Landau Zh Eksp Teor Fiz 30 1058 19567Sov Phys JETP 3 920 1957 Reprinted in D Pines The MartyBody Problem Benjamin Reading MA B L Altshuler and A G Aronov in El t l t 39 t quot edited by A L Efros and M Pollak Elsevier 1985 p 1 G Zala B N Narozhny and l L Aleiner Phys Rev B 64 214204 2001 E Abrahams S V Kravchenko and M P Sarachik Rev Mod Phys 73 251 2001 B L Altshuler D L Maslov and V M Pudalov Physica E 9 209 2001 This argument is not entirely correct because what we really evaluated is the quasi particle scattering rate which does not coincide with the dephasing rate The point is that the scattering rate is controlled by energy transfers of order T whereas the dephasing rate is controlled by smaller transfers The typical value of these transfers is determined self consistently from the condition w 731 In 2D w is smaller than T by a factor ln kpl and the 7 1 diifers fro 7 1 by the same factor In 1D thin wires the difference is more signi cant An argument similar to that given for 2D and 3D in the main text would lead to 7 1 olt T12 in 1D whereas 7 1 olt T23 This argument breaks down if the current is not proportional to the momentum Such a situation occurs in semi metals which contains two or more bands lled by electrons and holes The net current ePem8 7 ePhmh 7 P8 Ph where Pam and meh are the total momenta and masses of electrons and holes respectively In this case electron hole collisions lead to nite resistivity which indeed scales as T2 a well known example of such a behavior is found in bismuth 43 Contents 4 Electronphonon interaction 1 41 Hamiltonian 1 411 Derivation of e ph coupling 3 412 Jelliurn model 5 413 Screening 6 42 Polarons 9 43 Bloch resistivity 9 44 Effective 8 interaction 9 Reading 1 Ch 26 Ashcroft amp Mermin 2 Ch 7 Kittel 4 Electronphonon interaction 41 Hamiltonian The subtle interplay of electrons and phonons was explained in the 50s by some of the earliest practitioners of quantum many body theory leading eventually to an understanding of the mechanism underlying superconductivity Recall that the ions in a metal have two basic effects on the electronic states 1 the static ionic lat tice provides a periodic potential in which conduction electrons must move leading to the evolution of plane wave states in the Fermi gas into Bloch waves in the crystal and 2 the scatter ing of electrons by lattice vibrations and vice versa The rst effect will be ignored here as we are essentially interested in long wavelength phenomena where the differences between proper cal culations using Bloch waves and simpler ones using plane waves are negligible lt suf ces then to consider the phonons in a lattice interacting with a Fermi gas in which the most important effects of the long range Coulomb interaction have been accounted for Without the Coulomb interaction the phonon frequencies are just those we would get from a classical model of balls of mass M ionic mass connected by springs For a 3D solid with 1 atom per unit cell there are 3N normal modes comprising 3 acoustic phonon branches of When one includes the long range Coulomb interaction but neglects the electron phonon coupling one nds 1 that the longitudinal acoustic mode has been lifted to the ionic plasma frequency tag 2 47rZ262nM12 The terms of the Goldstone theorem which insists on the existence of an acoustic mode for each spontaneously broken continuous symmetry are vi olated by the long range nature of the Coulomb force and the sloshing back and forth of the ion fluid at wig occurs for the same reason and at the same frequency up to the mass difference that it does in the electron case At this point we are seriously worried that we dont understand how acoustic phonons ever ex ist in charged systems lf one now includes the electron phonon coupling however the electronic medium screens the long range Coulomb interaction leading to a nite interaction length and the recovery of the Goldstone acoustic mode Letls give a brief overview of where were going l rst want to get to the point where we can write down the full Hamiltonian for the problem We want to show that it makes sense to write the Hamiltonian describing the electron phonon system as H H501H19hHcaulHmt7 where HE kE kClUCka 2 H0 l l 3 ph ZWkAlakAakA l lt l M 2 1 Howl i Z VltqgtcLqUclgckqgckU 4 2 ckq lt5 where ab creates a phonon with wave vector q E k k and polarization and gkk oc M 71 2 is the bare electron phonon 2 coupling The unperturbed phonon Hamiltonian Hp is of course just the sum of 3N independent harmonic oscillators in 2nd quan tized form and the bare Coulomb matrix element in HOW is Vltqgt 47w2 q2 The derivation of the electron phonon Hamilto nian Hm and its quantization is relatively straightforward and l will sketch it here 411 Derivation of eph coupling Assume the ion is located at position m at a displacement u from its equilibrium position R3 lf the potential of the ion is assumed to be rigid the interaction energy of the electronic charge density with the ions is simply1 Hm dgrwlrgtwgltrgtVr R2 6 For small amplitude vibrations we can expand in powers of u Hm dgrwlltrgtwaltrgtvltr R dwarmnm vRvltr R4le 7 Now expand the eld operators 1 in terms of Bloch waves wmmm o where MHRWmm lta so the quantity which appears in Eq 7 may be recast by per forming a shift by a Bravais lattice vector and using the periodicity 1This is for a Bravais lattice If there is a basis one has to be a bit careful about labelling the lattice sites with additional intracell indices7 ie Rm 1 l m where m is number of atomscell of ngwr Hg d3r Ultrgt kgltrgtVRgVltr R d3rgbiUr Rgmge ngngr R3 e ltkkgtR9 dgrq iUrq kgrVRglr R3 10 WW 11gt Now let us 2nd quantize the displacement u as we did when we were discussing the isolated phonon system2 l r 0 it kt Ak Rz39 13 no my gteltgte ltgt with we mm alt qgtgt lt14gt xQCMCI so interaction Hamiltonian can be rewritten 1 Hm T U W zk7kR9 X 39 Q X ZQAltqgteAltqgt Zq R qA Z 20L Cka Wkk eAqgtgt QAltqgtl kkUA U M E kgAgkkAcldgcka aAQgtallt Qgtgt 15gt 2Before we dealt primarily with the 1D chain7 so I supressed the polarization indicesi It is important to recall that in a 3 dimensional system there are 3N normal modes SmN if there are m atoms per unit cell For each value of k there are 3 acoustic optical only if there are addll atoms per unit cell requiring index 1 modes denoted by different values of the branch index A The vectors e k are the polarization vectors of the modes and satisfy the orthogonality condition 292400 624 k 5w 12 where now q due to momentum conservation 6 function from summing over j is to be interpreted as q k k G 16 with G is a vector of reciprocal lattice arose because q was de ned to lie in 1st B zone The electron phonon coupling constant is i 2Mw i q The nal result then is that an electron in state k a can un gkkA Wkk 39 eAltQgtgt 17gt dergo a scattering process with amplitude gkk ending up in nal state k a by absorption emission of a phonon of momentum q This form is useful but calculating gkk from rst principles is di icult because V is poorly known 412 Jellium model We can get some dimensionally reasonable results in the so called 77jellium77 model where the ions are represented as a featureless positively charged elastic continuum3 we will simply replace the eigenfrequencies a of the neutral system by the constant 00 according to the arguments given above Again we expand the crystal potential V0quot Rj around the equilibrium sites RE The dispacements uR in the jellium gives a charge density uctua tion4 nZeV 11 This interacts with the electron gas through the Coulomb interaction leading to the interaction Hjillwm 262 g d3rd3r r grv ur 18 r 1 l 3 justi ed by the large masses and correspondingly long timescales for ionic motion Born Oppenheimer 4Recall from Eamp M the polarization charge density is pp 7V P7 where P is the polarization and the polarization due to a density n of dipole moments p Zeu is therefore nZeu 5 and then quantiZing the ionic displacements u R R as in Eq13 one nds 47rz39q eq Z6271 M12 q2 Hiiflwm k am a1ltqgtgt lt19gt Comparing with Eq 15 we see that the effective eph coupling constant in an isotropic system is i 47riZ62n12 9a in 20gt 413 Screening The rst point l would like to review is the renormalization of the electron phonon coupling which leads to screening and the recov ery of the acoustic spectrum The main point is to realize that the singular behavior is due to the long range Coulomb interac tion which will be screened Any time a phonon of momentum q is excited it creates charge density fluctuations because the ions are positively charged These then interact with the electron gas The bare potential created by the charge fluctuation is pro portional to the electron phonon coupling constant so screening the ionic charge amounts to replacing the bare coupling 9 with a screened coupling gltQ7 W71 gltQgt ltQ7 W71 where q k k and eqwn 1 VqXqwn is the RPA dielectric constant The frequency dependence of the dielectric constant will become important for us on scales of the Debye frequency cap or smaller and this is normally a very small scale by 6 electronic standards So for most purposes we keep a frequency independent screened coupling gltqgt 2 gq6q 0 We would like to see that this screening recovers acoustic modes as observed and expected The bare interaction Hamiltonian may be written in the Jelliuni model as see Eq 15 Hm 9qquq 22 with nq 201 C 010k Consider now the entire Hamiltonian for the phonon coordinates including the coupling to the electron gas recall derivation of normal modes for linear chain in Section 1 l l r 2 th Hint MPOIP01 Maui QqQq QQqunqgt 23 The Heisenberg equation of motion for the operator Qq beconies check r 2 Qq 00 6201 gqnq O 24 We noted above that the ionic charge density uctuation induced ion by an ionic displacenient u was en nZeV u in Fourier space with Eq 13 this reads z on n nq zzqc2q lt25gt Also recall the de nition of the dielectric constant external char e77 e g 26 total charge 2071 Now the total charge uctuation is just electronic ionic nqnq SO nq mama w zqult1 1Egt lt27gt

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