LANDSCAPE ECOLOGY ESM 215
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ECE215BMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 Maggetostatics3 n1 n to quot39 Energv Densitv In Notes2 we predicted a very large energy density up to 01 Jcm3 or 100 kJm3 for the energy density in a garden variety ferromagnetic material 7 AlNiCo In reality the energy density is less than this as shown in Table I The reason for this is that the hysteresis curve sags as the magnitude of H is reducedtoward zero because of the magnetic domains in all ferromagnetics more on this later So the guresof merit for ferromagnets become the coercive eld Hc and the remanent magnetic induction BR as shown in the more realistic hysteresis curve of Fig 712 Magnetic Ene O r MGOe Com arison FMAg E IC MATEg p IALS NOD0103 00000 0 6 3 39O 2 n gt 9 a C Lu 2 H magnetic field I I Hc coercive field 13 Fig 712 a Energy Density for Common Ferromagnetic Materials at Room temperature The vertical aXis is in units of KJm3 b more realistic hysteresis curve for ferrogmanetics SQinSgin Interaction To explain ferromagnetism we needed to make the term bM roughly 104 larger than a classical magnetic dipole haVing the same angular momentum as S Heisenberg explained this in terms of the exchange phenomenon Ummi Blocal gt2IESi Sj 763 where I E is the exchange integral Z 12 meVin iron for example Exchange is strictly a quantum mechanical effect The Pauli exclusion principle prevents particles in the same spin state from being in the same location in space Therefore the spins must approach counteralignment as d decreases as shown in Fig 713 Ifthe particles are charged the decrease in dis equivalent to an increase in energy because electrostatic potential increases as d decreases recall V qzd for Coulomb potential like charges Again we have an example of how important the Pauli exclusion principle is in the microscopic behaVior of solids In a nearest neighbor approximation U239IE39SP39SP1SPH p gtatomicindex 764 For spins m p g uBSp 765 TiTTTvTT l alh39lh lt l d l Fig 713 The physical behavior of neighboring spins as a function of their spatial separation d called the exchange interaction At close separation the spin vectors must be opposite in direction to comply with the Pauli exclusion principle At greater separation they can be parallel So we can rewrite the energy expression as U 21Esp sp1 SW U mp gZ Pfyspil SW1 766 where BE is the exchange magnetic induction We can now derive the dynamic equation of motion From classical mechanics dL dt r torque 767 In magnetic materials 139 m 39 Be We make the correspondence L gt h S to get 61S 21 P E 7 dt gyBSp gtltSp71spl gluB dS 21 p E 7 7 sp gtltSp71 SW 768 To proceed further we decompose this equation into Cartesian coordinates making the z aXis the nominal alignment aXis of each spin We get iwwwwwww dt W 769 i mawmmawm 770 615 SSH sSX 321 ZN 771 Now in classical sense we can assume that S and S are much less in magnitude than S p and that Sp z SIM m SpH E Sp to rst order Thus we can ignore terms where S andSpy occur together We nd x 15 21E y z y y 2sp sp SPSP1SP1 772 dt 77 dSy 21 P E j 7513s1s1 S2Sp 773 dSZ dS F F 0 dt dt 774 l Sgin Wave Equation Ogtitmal or 20082 As equation 774 is a crosscoupled set of linear difference equations we seek latticewave like traveling waves for the solution S SP S Vepr pka exijan S Wepr pka expijl 775 Substituting the trial solutions of 775 into 772 above we get a Vexp j pka a 0 2SpWejpkarat SFW eUp rlymrat eJUFeleraz 776 or 39 V 21 23 W S W 17 7 1w h p p e e 777 or 41 ja 7ESPW 17 cos ka 778 Similar substitution of 775 into 773 yields after simple algebra 4 E aW 7SPVCOS 61 1 779 The preceding two equations can be written in matrix form 411981 ja l coska V 0 4IES W P 1 k 780 T cos a ja From linear algebra we know that a nontrivial solutions exist only if the matrix is singular ie does not have an inverse But a lack of inverse also means that the determinant must vanish 41 S 2 3 02 licoska 0 4 S 781 a l coska h Also note that by substituting 779 into 778 4 2 aw Sp 1 coska ijV V 782 which implies VjW or W jV 783 So that S and S are 90 out ofphase gt circular motion in X y plane gt precession The spinwave dispersion relation 41 S a 17 cos ka 784 h is very similar to that derived for lattice waves m2 0 cos ka 785 m But note the different exponent on the to term in each The oz term for spin waves makes the dependence on k near k 0 stronger and nonlinear for spin waves We can see this analytically by Taylor expansion for small k 41 S 41 S 2 21 S k2a2 a E P 1 cos ka m E p l 1 ka Lk ltlt a h h 2 h 39 786 Similarly as ka approaches 11 Nyquist wave vector doadk gt 0 for the spin waves just as for the lattice waves But there is faster dependence on k Right at the Nyquist wave 160 Spin Wave 12mev sp1 120 X 80 m x10 12 39 K39 40 Lattice Wave I C 30 Wm M 28mp o 0 05 1 15 2 25 3 35 ka 1 Fig 714 Representative latticewave and spinwave dispersion curves for the following parameters Lattice wave C 30 Nm M 28 rnp where rnp is the proton rest mass Spin wave 1E 12 meV iron Sp 1 vector k na we get a 81 ES P h a convenient fact for doing statistical mechanics with the dispersion curve as we will see below These points are illustrated in Fig 714 that shows representative latticewave and spinwave dispersion curves for the following parameters lattice wave C 30 Nm M 28 mp where mp is the proton rest mass spin wave IE 12 meVir0n Sp 1 Spin Wave Energv and f 39 39 M ommv Optional for 2 As in the analysis of lattice waves it becomes very useful to analyze the total energy of spin waves kinetic plus potential with a thought towards their quantization and statistical mechanics To do this we continue our classical thinking by applying an expression from classical rotational dynamics L2 Kinetic Energy 787 21m Where L is the orbital angular momentum and 1m is the moment of inertia Also we have KEllmm2 2 21m IL lsKElLm a 2 If we make correspondencelLl lt gt mpl gyBSp then 1 2 IE KE guBSpa 2gluBSp l cos ka 2 h Potential energy E PE 21ESP 39 Spl SP U total This looks very much like total energy for lattice waves in Chapter 44 mcozugv C l cos ka uAv KE E 2 KEPE 2guB 71 coskasp 21Esp Sp1 S p1 788 789 790 791 792 793 Following a derivation similar to that for phonons we can now show that the spin total energy has the form of a harmonic oscillator in Fourier transform k space This is yet another example in solidstate of a collective excitation 7 a wave in real space that takes on the form of a collection of independent harmonic oscillators in k space after Fourier transformation As before we can associate a fictitious massless particle with each independent mode In the case of spin waves we call this particle a magnon The energy of each kspace mode or magnon is then given by 794 where nk is the number of magnons excited or equivalently the excitation amplitude of the associated spin wave llll of Mmmnm Optional for 2 Since IE can be so large in ferromagnets the energy stored in magnon modes is signi cant As in phonons magnons are indistinguishable and have zero mass Therefore they abide to photon and phonon statistics ie the Planck function The mean number of magnons in each mode at a bath temperature T is therefore given by 1 lt gt 795 quotK exphakkTil And the total number of magnons in the solid is wmax Nmml ZltnkgtI0 Da lt nK gt da 796 K where Dn is the magnon density of states and mm 81 ES P h As usual Dn is needed ax to convert the summation over k to an integral over 03 So it is best defined by a chain rule expansion Dadco dw 797 where NkL3i7239k3 3 szz 798 211 3 dk 211 In the longwavelength approximation 60K 21Espk2a2 h 799 I h k mdwg 7100 E P l h 2mg 7101 da 2 ZIESFa 01 V ha D a k 3250 de 7102 477221ESpa2 k In the low temperature limit th gt kBT the Planck function has signi cant weight only at low k where the dispersion is quadratic Hence we can extend mm gt 00 32 2 N j V R alde 7103 mm 0 4112 ZIESpa2 exphakkBT71 I let xthkBT 1 k T w x12 NaN V B32 dx 7104 M2 HEW 0 6H lt gt Total energy IO DaKlt nK gt 1 2mmde U J V h 32 halfZak W 0 4112 ZIESpa2 exphaKkBT 1 Again letxhaKkBT dxhaKkBT so that 32 2 2 2 1 kT kT x U B B 32 dx 411 ZIESa 0 F ex l 2 S a2 To simplify this we de ne a quantityA through wK Ak2 E 5 M where 3 2 00 dx 39 x I0 ex 1 H130 from integral tables So in the end we get a rather simple expression for the heat capacity dU39 k k T 5 CV 5 2180 dT MA 871 32 0114per unit volume It is informative to do a numerical example for iron at room temperature 12 7105 7106 7107 k2 This leads to 7108 7109 7110 ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 Transport Theory 7 Quick Review of Classical Scattering Theory Classical mechanics teaches us of two types of particle scattering elastic and inelastic In elastic scattering momentum and energy are conserved and the scattering process between a particle and a scatterer can be understood in terms of the total geometric scatterercross section 6c the density of scatterers n and the particle velocity which together yield a collision rate 1 fc nov E 7 TC where 1c is the collision time In general the total collision cross section can be related to a differential cross section 69 that is almost always independent of the azimutal angle 1 so that 0C Ha6dQ27rio6sin6d6 8 In transport we are always concerned about the ux of particles which also constitutes a ux of momentum even for massless particles like phonons So another cross section the momentum scattering cross section 6m is used to weight 69 in the integrand of l by the degree to which the incident particle has its 1 f incident momentum de ected As seen in the a sketch to the right with incident velocity 71 and a nal after collision velocity 7 the de ection V I factor should go to zero at 9 0 zero and to a quotquotquotquotquotquotquot quot maximum at 9 11 Clearly a de ection factor ofl cosG makes sense in all ways leading to the expression 0quot Hey60 cos0dQ Z jo Q cos0sin0d0 9 Although originally classical 9 applies to quantum mechanical scattering as well and when combined with 7 yields the useful expression 11quotquot namv nvJJU0l cos0dQ 27rnvJU0l cos0sin0d0 10 0 Fig1 Quantum Mechanical Scattering Theory The Basics p pun waves having wave vector 12 This occurs of course when the particles are free and have the associated energy eigenfunction WV expljUE 1 n where the normalizationis satis ed over a reference volume v In crystals we have the corresponding Bloch energy eigenfunction 42 s WOO V urexp1k7 12 where v is now the volume ofthe primitive unit cell Intuitively we expect such a defect or perturbation ifsmall enough will scatter the incident 39 39 A schematic r l Fnrtunatel L 39 39 r r I 1 than the incident I i but the cellperiodic function 141 is nearly unchanged Such k r direction not vector amplitude ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 The problem of calculating the transition of a freeparticle quantummechanical state 132k 2 is one of the oldest and most important in modern physics It reduces to from k kl to nding the probability rate R13 with two key assumptions 1 that there are so many possible k2 states and so much coupling between them that after the k1 t transition from k2 back to k1 1 transition from 0 k2 there is negligible likelihood of the same particle making the 2 the particle energies before and after the transition are exactly equal consistent with elastic scattering 2 In th1s case the trans1t10n rate can be wr1tten R11 277r39H 12212 392 5W IvaU 1601 E 13 L r09 Historically 13 was rst derived by Fermi and is so useful in solid state and other branches of physics that is called the Golden Rule The last step in 13 de nes rigorously the scattering time added earlier to the semiclassical equations of motion as part of the singleparticle relaxation time approximation The perturbation Hamiltonian is de ned by the usual quantum mechanical expectation value Hm lt WW IH39UEJblvM gt 14 Semiclassical Scattering Theorem As in the classical analysis 139 is inherently a function of k sometimes a strong function So the best approach is to incorporate 13 into the semiclassical Boltzmann formalism through two collision terms 1 This is distinctly different than the analogous problem in isolatedatom ie atomic physics whereby an external perturbation acts on an atomic wave function Vl causing a transition to V2 In this case after some time called the Rabi opping time the atomic wave function can re cycle back to Vl This assumes of course that no signi cant scattering or dephasing of the wave function occurs during the process 2 See any good book on Quantum Mechanics eg l H Kroemer Quantum Mechanics Prentice Hall New York 1994 or 2 C CohenTanneudji et al Quantum Mechanics Wiley Interscience New York 1977 ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 0N V 3 a a a a at Mimi27 I d k2R12k1k2fk11fk2 15 and aflt 3gt 2 V 11 collision 2gt1 Jd3k2R2ltI J 2gtfltI 2gt1 fab 16 am 2 60 60 at 17 collision 2gt1 at collision collision 1gt 2 The factor V is the volume of the sample and V2113 is the volume per state in k space Both 15 and 16 display the occupancytodeoccupancy principle addressed earlier Note that 15 and 16 do not violate the irreversibletransition assumption behind 13 since a particle that contributions to 15 and one that contributes to 16 will always be two dz rent particles In general 15 and 16 are very difficult to solve But just as in the case of Boltzmann classical and semiclassical transport life gets relatively simple in the special case of low particle concentrations Then we can approximate 1fk1 1 and l fk2ml am V 3 a a a a a a at 7yld k2R21kpk2fk2R12k1 k2fk1 18 collision total for the nonequilibrium distribution function f The second simplifying step is to invoke the principle of detailed balance one of the most profound principles in all of transport theory and quantum physics for that matter It states that in the equilibrium state any microscopic processes involving the same two states must balance each other in detail In other words the total number of particles tranferring into a quantum state must equal the number of particles transferring out Mathematically this can be stated in the present context as R21ltI 1J 2mltl 2gt 121201995031 19 where f0 is the FermiDirac function Substitution of 19 back into 18 leads to ECE215BMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 am 2 V at collision 3 total 3 v I d k2R12ltk1k2gtfltk2gt f0 E2 fk1 20 A 0 where we assume k z 1 elastic approximation f0 2 am N V at collixion total I d3sz1zltII zgtfltI zgt flt1 1 21 where the last step follows from the assumption of elastic or nearelastic scattering To go further we need to apply our solution derived previously for the semiclassical Boltzmann equation Eqn 20 in previous section 1f01 f0vgqE qrfovgE f 2 f0 m f0 22 k T kBT B We can apply 22 to simplify 21 through the use of trigonometry applied to the scattering diagram in Fig lb3 with gvl de ning the polar axis in spherical coordinates 17g having polar angle 92 and azimuthal 1 E having polar angle 91 and azumuthal 11 k2 having polar angle 93 and azumuthal 13 and a relative azumuthal angle between E and 17g of I 11 71 In this same coordinate system the volume differential is d3kz kz2sin63d63d 3 23 Given these definitions we can rewrite 22 as a a qrfo 71 E a qrfov ylEcos l fk1zfok1kB fok1z f 24 3 Note that this is diagram is written in terms of group velocities rather than wave vectors and that the group velocity being proportional 6U 1 is not necessarily parallel to k except in the special case of a spherical band ECE215BMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 i i 611mgsz 6 6 396396 f 2f0 2 T cos 1cos 2s1n 1s1n 2cosgb 25 k3 Now utilizing the elastic condition once again f0kz m f0k1 k2 m k1 and vgl m vgz Hence substitution of 23 24 and 25 into 22 yields T E z Id3k m COS 61 cos 62 l sin 61 sin 62 cos sin 63d63d 3 26 col 3 total where the last step uses an elegant theorem from spherical trigonometry for the dot product between two vectors spherical law of cosines each oriented away from the polar axis4 Depending on the geometry 4 and 13 will be different by just a constant so that the last term in the integrand of 26 that depends on coscl will be zero in the d 3 integral from 0 to 211 Thus carrying out the d 3 integral over the first term and substitution of 24 back into 26 yields 0219 NVfl1fol1 2 at NT j 12de IRlyzcos62 1s1n63d63 27 collision total which can be rewritten in the elegant form afltkgt mic Dag 1 28 at Collision Tmk1 Dial 1 m V 61ij 1 cos6sin6d6 Imago 2 2 2 2 12 2 3 3 29 The development culminating in 28 is sometimes called the semiclassical scattering theorem but also constitutes a proof of the relaxationtime approximation we adopted previously for the semiclassical Boltzmann transport equation But now we have a rigorous expression for calculating the relaxation time And because we assumed that the scattering was elastic or quasielastic 39cm is the time required to change the crystal momentum Hence it is usually called the momentum relaxation time and hence the subscript m This is the most useful form of relaxation time in solidstate transport theory The energy relaxation time 7 the other one commonly used 7 is generally much greater than the momentum relaxation time but becomes 4 See for example CRC Standard Mathematical Tables 253911 Ed CRC Press W Palm Beach FL 1978 p 176 ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 more and more important as external fields get larger and drive the distribution function f further away from equilibrium Calculation of Momentum Relaxation Time with Quantum Mechanics The most common application of 29 occurs when the carriers lie in a band that is spherical or at least spherical enough that g is approximately parallel to k at all points of the constant energy surface In that case 93 m 92 and l V a H 2 ENE 27 As we shall see shortly 30 is very handy in direct band gap semiconductors such as GaAs as j k22de j R110 cos 62sin62d62 30 InP in both the conduction band exactly spherical or spheroidal and valence band warped sphere The factor in the integrand sine lcose is reminiscent of the same factor in the classical scattering theory 9 and 10 To make the correspondence we switch the order of the integration in 30 to get N L rm 1 N 272 This is the momentum relaxation time for one particle and one scatterer But the expression 10 j j Rlyzkzzdkz1 cos 62sin62d62 31 was for one particle and a large number N nV of scatterers For one scatterer we can rewrite 10 as l v If T 27I7IU6l COS9Slnt9dt9 32 m 0 where again V is the volume of the entire sample and v is the incident particle velocity 7 now necessarily a group velocity Direct comparison of 31 and 32 yields V2 06 m R kzdk 33 2703 12 2 2 an elegant expression that is a bit confusing until we remember that k2 is the radial variable in the sphericalcoordinate basis of k space with k1 defining the polar axis and 9 is the polar angle in ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 Transport Theory 9 The Dri Di quotnsi0n F armalism One of the key assumptions that the transport theory has rested upon to this point is the conservation of particles This is what allowed us to relate the nonequilibrium distribution function to the condition k dkdf N 1 But in solidstate devices there is necessarily a deviation from the equilibrium particle concentration because devices by de nition must exchange energy with the outside environment to be useful This is true whether the device is electrical electron and or hole mechanical acoustical thermal ie phononic radiative ie photonic or some combination of these ie mixed domain So it is incumbent on any practical transport theory to address deviations from 1 at every point in the device First we will seek and derive fundamental constraints and conservation laws that can be used instead of l to help in solving for the nonequilibrium state of the solid Then we will inspect what common types of external forces or in uences can create significant deviation from equilibrium Finally we will generalize the Boltzmann transport formalism to include the fundamental principles and forces The result is driftdiffusion equations 7 arguably the most important formalism in the world today for simulating and designing semiconductor devices today Fundamental Constraints Conservation of Charge Along with energy and momentum a third important conserved quantity in many natural processes is electric charge This is generally true in solids and can be analyzed simply using the most scalable of all physical laws 7 Maxwell s equations We start with the sample cube of volume Vc inside the solid as shown in Fig I assumed electrically homogeneous so that the unipolar ie free charge density pf t is continuous throughout Then the total free charge inside is Q J pde 2 Vc And the free charge can be related to Maxwell s displacement vector by v1 pf 3 As shown in our coverage of electrostatics a second form of charge must generally be considered in solids particularly in solids having strong paraelectric response such as semiconductors This is the dipolar charge density which we showed was generally neutral inside the solid but could be described in finite samples by the unneutralized or unpaired surface charge The magnitude is always describable by a surface charge 63 which is related to an equivalent volumetric bound charge density pb V P where P is the polarization density vector Hence the total surface charge on the cube in Fig l is just 1 Assuming of course that there are no nuclear reactions occurring ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 dx I H dz Igt gt Jxdx Fig l Q ltjgtasds ltjgtPd jv13dVjpde 4 s s V V where the step from the third term to the fourth term follows from Gauss divergence theorem arguably the most useful integral theorem of vector calculus The conservation of charge means that any change of the total free charge of 2 with time must be accountable to transport of an equivalent amount of charge into or outof the cube Mathematically we can state this conditions using 2 in the following way an a apf a a a I 2 dV dV J d 2 Va dV 5 at f atipf J at is f 5 f where If is the total free electrical current leaving the cube From the third and sixth terms we get a vJ 6 at f an extremely important relation in device physics called the freecharge current continuity equation Similarly any time dependent change of the bound charge can be written as r 3jpdV jdI 3jv dV jv dl 7 6t 0t Vat at V 0t where Ib is the total bound electrical current leaving the cube From the fourth and sixth terms we get ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 6p 4 613 4 4 vEVJ 8 at at P where JP is the polarization current density Eqn 8 is called the polarization current continuity relation and it has a central role in quantum electronics for connecting atomic and molecular quantum transitions back to Maxwell s equations Students who have had a thorough course in quantum electronics or electromagnetism may think that this is the only purpose of 8 But it has a central role in device physics too de ning a key component of the reactive or capacitive current current2 once we add it to the vacuum term The vacuum current is given by the time derivative of the vacuum energy storage term apo a T T T 6E a a V E 5V VJ 9 at 6T 50 so 6T T where JT is the total current A good way to think about 8 is the current associated with the change of charge outside the sample that is needed to create E0 Adding up all three terms we have a a a a a EVJ V5Pgv0E 10 at at at at f at 0 at now by using the electrostatic relation 13 80 BE this becomes a 444 0 444 0157 444013 i VJf V 50 1 VJf V ergo VJf V at at at at The right side is just the divergence of one side of Maxwell s dynamic equations 4 4 4 613 v x H J f at By taking the divergence of both sides and utilizing a basic theorem of vector calculus we get vex zozvjf5D6Jr at at 2 Another current could be added to the mix corresponding to the magnetic dipole or magnetization current This is very important in magnetic devices but is left out of the present analysis for brevity Since there are no magnetic monopoles there is no magnetic current analogous to 5 and 6 ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 This expression emphasizes two very important facts 1 the conserved quantity in spacecharge neutrality is really the total charge density m and 2 the corresponding total current Jf aD at must be a constant independent of location in the solid These are powerful concepts that come up repeatedly in solidstate device analysis They are particularly useful when the electric eld in the solid is a pure sinusoid or a sum of sinusoids as often occurs in electrical engineering Then since Ohm s lanf 2 oquot E is so often valid in solids we can write off 0 which means that the quantityoE J39ng C a constant independent of position in the solid Local Thermodynamic Equilibrium Devices force us to abandon the notion of equilibrium over the entire volume of the device However they are often operated under conditions that are close enough to equilibrium that the non equilibrium distribution function throughout the device is a linear deviation from the equilibrium function or at least traceable back to the equilibrium function is some way This justifies a second pervasive constraint in device physics and engineering in which the device is globally in non equilibrium but locally in thermodynamic equilibrium In other words at any point in the solid we can define local macroscopic thermodynamic variables that can be considered constant over some dimension much greater than the atomic scale The most important such variables are the intensive ones3 the temperature the chemical potential the electric field and the elastic strain Being able to define these uniquely at each and every point in the solid has profound consequences We have already seen an example of this in 39 J where 39 39 Ag of the and chemical potential synonymous with chemical potential in electronic devices allowed us to write expressions for the electron and hole densities in the conduction and valence bands respectively UF UCVkT U U kT no NCT6 and pvaVe V These result in a very useful constraint Ucery 41 k T k T 2 rchv NCNVe 5 NCNVe 5 nl were ni is the intrinsic carrier concentration This is called the law of mass action External Forces and In uences for the Special Case of a Semiconductor There are a variety of external forces that can significantly alter the charge carrier densities from their equilibrium values 7 a process called charge injection With metals or semimetals these are all frustrated by the fact the are many ways to do this but the following two are the most common 3 See Chapter 1 for review of thermodynamics in solids ECE215BMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 Direct gap eg GaAs Indirect gap eg Si I UG h Fig 2 Qualitative behavior of absorption coefficient in semiconductors or insulators as a function of radiation frequency v Once above threshold the a of directbandgap semiconductors eg GaAs is generally much larger than it is for indirect bandgap semiconductors Much larger for directgap materials eg GaAs than indirectgap semiconductors eg Si Radiation Perhaps the easiest to understand is crossgap generation in semiconductors Semiconductors have the have the wonderful property of internal photoemission which is very similar to surface photoemission or photoelectric effect that led to the original concept of representing electromagnetic radiation as photons4 The absorption coefficient of radiation is represented by a threshold absorption coef cient a av6hv UG where 9 is the Heaviside step function and 0cv is generally sublinear as shown in Fig 2 Once in the solid the intensity associated with the radiation tends to vary according to Beer s universal exponential decay law I 1087 Every photon making up I is absorbed in a corpuscular fashion meaning that the power absorbed per unit volume at each point in the solid is just P z 05106 0512 and the corresponding photon absorption rate is P 11v Each photon generates an exciton 7 an electronhole pair 7 that is initially highly correlated or even bound but then rapidly decorrelates and annihilates to an independent electron and hole Assuming that this happens to every exciton generated the electron and hole generation rates GH and GP are given by 4 A representation that won Einstein his only Nobel prize in 1921 ECE215BMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 angpzm hv By the principle of quantum reversibility photon absorption has an inverse process called spontaneous emission whereby a free electron and free hole annihilate or recombine emitting a photon in the process This process has associated recombination rates for electrons and holes RH and RP respectively each of which is proportional to the natural radiative lifetime l cel1 In very pure semiconductors these two rates are approximately equally Rum Rp In impure semiconductors having many energy levels in the band gap these two rates can be substantially different Electrical Contacts Much more common in devices but also more complicated from a solidstate physics standpoint is electrical injection When the solid is an insulator or semiconductor the injection is usually done by contacts They are usually made with metal and fall into one of the following categories 1 ohmic contacts for injecting the same type charge carriers as are predominant in the solid in the equilibrium state5 and 2 rectz ing contacts for injecting the opposite type carrier The are several types of rectifiying contacts with two being very common n semiconductor devices 1 pn junctions and 2 Schottky barriers The pn junctions are easy to fabricate internally to the solid creating the ability to do the minority carrier injection that is crucial to the function of bipolar transistors and semiconductor injection lasers The Schottky barriers are easy to fabricate at a surface of the solid creating the ability to do majority carrier injection in a manner controlled by the difference in the metal and semiconductor or insulator Fermi energies6 and the difference in their electron affinities Like radiation electrical injection can also be characterized by generation rates for majority and minority carriers Gn for electrons and GF for holes But these rates are not necessarily equal at each point in the solid as they are with photonic injection However they generally have well defined boundary conditions that get used to solve real devices There are corresponding recombination rates RH and Rp which again are not necessarily equal Generalization of Current Continuity Equations Given a semiconductor solid and the existence of generation and recombination mechanisms that perturb both free carrier types away from their equilibrium densities we need to generalize the continuity equations The first issue we face is how to generalize relation 6 apf at jf for two types of free carriers The total electrical current can be thought of as independent electron and hole currents taken as statistical averages over the conduction and valence bands respectively ie 5 In a semiconductor or semim etal the predominant charge carrier type in equilibrium is called colloquially the majority carrier The other carrier type is called the minority carrier 6 A quantity called the work function ECE215BMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 if jquot jp Hence a necessary but not sufficient condition to satisfy 6 is to decouple the electron and hole densities too leading to an a a AJ G R eat e n n 11 617 a eE AJpeGp Rp 12 where we have paid careful attention to the signs of the chargepolaritydependent terms To apply 1 l and 12 we need expressions for j jp G R GP and RF in terms ofn and p For in and jp we go back to the semiclassical Boltzmann equation in a uniform electric eld of forceFeqE Ed 9 d a d 6 d t my izl 21 dvlekU dt h h di h df 45f Feaf af so V T T 13 all 6r h 6k 61 mm This can be reduced to special forms by various methods all dependent on ascribing a temperature to the carriers consistent with local thermodynamic equilibrium The following discussion only reviews these methods For details the student is referred to an excellent book by C M Snowden Semiconductor Device Modelling Chapter 3 Carrier Transport Equations Peter Peregrinus Ltd on behalf of IEE 1998 Method of Moments The Moment Method multiplies both sides the Boltzmann equation 13 by the group velocity electron or hole and then integrates over each band in k space The resulting equations are a a a E vnkr av Electrons e Ve Vve q e 14 at me men scan a i a E v kT 6 Holes hvhVvhq wh 15 mlquot ml lp malt ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER BrownSpring 2008 Te gt local temperature electrons T h gt local temperature holes n gt nonequilibrium concentration electrons p gt nonequilibrium concentration holes me gt effectlve mass electrons conduct1v1ty mass m gt effective mass holes conductivity mass Drz39 ft Dz usi on Aggroximation The application of 14 and 15 are fairly straightforward on computers but not simple by analytic means So approximations are usually made pertaining to the spatial and time dependence of the temperatures and velocities a gt 1 Uniform temperature 3 T0 Th T0 V m 0 VTh 0 2 Nearlyuniform velocities V2 VVE and Vh Vvh bOth small 6 v 3 Quas1 steady state 6 v e m h m 0 6t ave N 7e 6 7 7 at scat N 7 I scat T 4 Relaxation time approximation 9 h These approximations yield 4 er a k T 1 a a k T a V2 E Bf e Vquot E eE B 0 V 16 me men ne LquotVquotquot LquotVquotquot Drift term Diffusion term Je And with the understanding that all the terms in 16 are averaged over the distribution function by the Moment Method we can write the electrical current directly as enve engeE kBToy n Similarly we get for the holes 17 ECE215BMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 115 6 mh th and Z1 2 epvh 2 e17th kBTOuh p 18 The key results 17 and 18 can be written in eDn n eynnf 19 JF erVp eyppE 20 k T where Dquot MJDP Z UPB e e and where we have changed the labeling from e and h to n and p for electrons and hole 39 39 the 39 device literature For RH and R p we make the assumption that recombination always occurs in pairs and is exponential similar in spirit to the relaxationtime approximation so that Rn m Rp Substitution of j j Rn 85Rp into the current continuity relations 1 l and 12 yields Fl 6n 2 a a n E DHV nunVnEGn T n 21 0p e e e p Eszvzp ypVpEGP T 22 P Eqns 21 and 22 are the famous driftdiffusion equations The are so pervasive in semiconductor device analysis and modeling that are also sometimes called the master equations The can be further simpli ed with a theorem from vector calculus for the divergence of a product of a scalar and a vector T l 21 The leads to the forms an 2 e e e a 7 1 a DHV nynEVnnynVEGn 23 t r n ECE215BMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 6p 72 7 7 7 e p EDPV p ypEVp pupVEGp 71 24 P Even with all the assumptions 23 and 24 are dif cult to solve analytically but relatively easy to solve numerically by modern nite element techniques However analytic results become simple for small deviations from equilibrium 7 the analog of the smallsignal approximation in circuit theory 5 p p p0 5p lt majority carrier concentration 5n n no 5n lt majority carrier concentration We further assume spacecharge neutrality and complete ionization of any donors or acceptors concentrations NA and ND respectively so that so that nNApND or n06nNApo6pND Now we know that in thermodynamic equilibrium n0 NA 170 ND which yields the crucial relation 5 5 25 Substitution into the driftdiffusion equations 23 and 24 result in WDn zmo 5nunE n0 5nunn0 5n EGn 7M 26 Tquot a 5 a a 7 a a 5 DPV2p0 6p EVp0 6p p0 6pVEG W 27 P Since no and p0 are spatial and time independent by de nition we thus get Dn 25nyn 5nynno5nWEGn 7M 28 In 65 a a 7 a a 5 a fDV26p EV6p po6pgtVEG 29 P Inspection of 28 and 29 indicate that we can utilize the constraints 5p 5n and p 7 n po 7 no to eliminate 5p in favor of Sn or vice versa We choose the rst option here getting ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 66 a a e a a a tnDHV26nynEV6nunnVEGn l 3o Tn 667 a a a a 7Dpvz n pEyEn yppVEGp 31 F We now carry out a step aimed at eliminating the subtle V E term 7 a term that is not necessarily zero in the present analysis even though we started with a uniform eld in the Boltzmann equation We will come back to this term later To eliminate it for now we multiply 30 by Map and 31 by y n and then add them to get pp M ppDn nnDsz5n n povo quot05 g 5n 32 at n I7 G G By dividing both sides by unp pen we nally approach a beautiful result n gt 6511 PpDnnnDP 25 nlupp0n0EV n 33 at A1 W1 A5 WI p Gn WIquot WMGP 171 mp WI mp WI We can eliminate simplify the rst term on the right side with Einstein s relation D kBTy e D D n a n a a SO 0611 Z n pp 26 170 Mm Evan 34 5t Dpp Dnn pp m ppGn quotTn HIKING prp pp Wt DMDp p n The term D E 35 Dpp Dnn is called the ambipolar diffusion coef cient and labeled D The term 17 n mu Hgtllt E 36 pp Mn 11 ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 Magnetostatics 2 Paramagnetism o Seldom is there the spherically symmetric distribution of charge in solids that creates zero builtin magnetic moment and just a small diamagnetic response 0 Usually there is a builtin magnetic moment Two types are prevalent l orbital dipole m1 and 2 spin dipole r71 0 Atoms with net orbital or spin angular momentum can have large paramagnetic effect e g atoms with partially lled inner shells transition elements and rare earths Best example is atoms with an odd number of electrons so one spin is unpaired Fundamental unit for magnetic dipoles is the Bohr magneton magnetic moment for ground state of hydrogen 7239a A ilzlea0132 But for Bohr quantization rules murLnh n123 So for ground state mUV mUUi0 71 and h m8 iAeva02e ELLB 093X10 23A m2 2m So for builtin magnetic dipoles we write mfg15 j Note minus sign because m de ned in terms of current L in terms of electron motion g gt gyromagnetic ratio Z 2 for electron spin Z l for orbital angular momentum ECE215BMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 Potential energy 47112060 gyBjl lomj jgyBBlomj That is for Jpointed anywhere in the upper hemisphere polar axis de ned by direction of Blocal the potential energy is positive And for J pointed anywhere in the lower hemisphere the potential energy is negative l a For a s1ngle spin m i g 2 eg s1ngle electron I m I 8 j 2 U U2 Hence UPE ill 39Bloml the sign meaning that the spin is pointed along the E axis For spin 12 we have two possible energy values U1 U1 uBB U2 LLBB These are conveniently represented by the energy diagram to the right Fig 2 Since spins are hidden variables within atoms and atoms are generally distinguishable we can apply the M axwell Boltzmann statistics probability of magnetic moment being aligned e UikBT eygBkBT along B 171 eel y 641BT eyBBkBT 8w33kgr einBkT d p2 7 an eyBBkT 6 uBBkT Thus the mean magnetic dipole is 171i1 172th 18171 p2 2 BBkBT BBkBT e e A ltmgt uB BkT 3 k z 645 5 e 53 5T 2 13 tanh IuBBkBT 39 2 Simple picture of two possible energy states Fig 3 High energy state U 4711 LLBB Bgt Low energy state U 4711 uBB 4 ml Fig 3 ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 Hightemperature limit uBBkBT ltlt 1 m B2BBkBT m BzBkBT lt r71 gt 2 lt r71 gtl am 7 BZkBT 2 not n C Iquot A N M E Curie Law C gt Curie constant 1 nambyo kBT T For more general total angular momentum J and according to quantum mechanics mj has 2Jl equally spaced levels U r gyBjE mjgyBB For example if the total angular momentum quantum number 7 32 mj 32 l2 12 and 32 so that 2J1 U k T 2J1 Bk T 39 B quot1143 B z 2 game A lt m gt 2JI 2J1 Z Z eUJkBT Z emJyBBkBT j 139 It can be shown that this becomes llt 1 gtl g 7BBJx where B is called the Brillouin function and g is the Lande g factor and x gyyBBkBT ama cmh w icothi 2J 21 J 2 ltMgtnlt 1gtngJyBBJx2 ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 am E gJuBBJ x Blocal In hightemperature limit we use the Taylor expansion cothx m lx to get n1J12 2 n2 2 C Zmznamuoz guBIHOE Plug 0 3kBT 3kBT T p gJJ J 1 npzzugzluo C gtCurie constant 3kB In an advanced course on Quantum Mechanics it can be shown 177lssl l g 277 1 Example atom with single unpaired spin m nszyOkBT 8 093 X 10 23A m2 n D 5 X 1028m3 T 300K 3 m l3lgtlt10 3 300K This is substantially stronger in magnitude than the diamagnetic for such an atom For partially lled inner shell atoms e g transition elements thejis large and Xm tends to be even larger up to values of 5x10393 F erromagn etism We derived the relation for the ferroelectric effect from relationships between microscopic and macroscopic fields This also works for ferromagnetic materials S 7 S1 3 expect ferromagnetism when nam b u0 m l ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 Take for example spin system whereM m N mmglwd nyBZkBT So am 82 kBT And ferromagnetic condition is n 2uob 3 21 or nZkBTbszyO kBT So at T 300 K n gt 038gtlt 10321 Recall b has max value of 10 in classical magnetostatics So the minimum value is n gt 38 gtlt1031m3 38 X10250m3 Clearly this is much larger than occurs commonly in solids So Pierre Weiss theorized and Heisenberg proved that there is a big contribution to b from spins This can be quanti ed by B 10ml Bm FobM with the possibility that b gtgt 1 Take for example Nickel fcc lattice a 352A n m 0092x1030 92X1028m3 So I gt 038 X103292 gtlt1028 413 to satisfy ferromagnetic condition Simple Model for F erromagnetism Recall temperature dependent M for spin 12 system M r418 tanh LlBBlmalkBT paramagnetic But ferromagnetic spontaneous response requires 31ml guild even when B0 0 Emmi0 Md d gt 0 5 ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 R 30 WWW N01 71 1m1 3 MOM MN o 13E MM So we have nyB unmw kBT This is an implicit equation in M so can be solved by plotting the left and right sides separately and then looking for intersections Increasing Temperature Fig 4 As expected M1T1 gt M2Tz gt M3T3 for T1 lt T2 lt T3 And eventually the intersection point goes to T Tc where M gt 0 The full curve looks as shown below M Ferromagnetic gt Paramagnetic phase 2ndorder phase transition often well fit by MMsl TTEquot TC T As in ferroelecm39cs susceptibilities often are singular m C Often well f1tb YI TTE ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 Transport Theory 4 Boltzmann F armalism We have seen how ionizedimpurity scattering of charged particles is associated with a strong dependence of scattering time 139 on the particle kinetic energy UK In addition we expect that 139 and perhaps other aspects of the transport process such as the number of particles involved will depend on particle identity ie classical MaxwellBoltzmann Fermionic etc This leads us to the Boltzmann transport formalism arguably the most important advancement of transport theory ever made because of its inherent ability to handle multiple physical effects automatically and selfconsistently1 At the foundation of Boltzmann are a set of three axioms each far more plausible than those of kinetic theory 1 First each particle in the transport process can be described by a point in mechanical phase space spanned by position and momentum 17 amp f7 This phase space is six dimensional and is shared by all N particles of the population In other words at any moment in time there are N points in this space each designated by amp I31 where i is the particle index 2 The evolution of each particle point in time is given by the equations of motion which in turn depend on the external forces through the appropriate mechanical laws of motion When Boltzmann developed the formalism the only known laws where those of Newton for which dt 1 where F is the external force and m is the particle mass amp is me NE II 2 a 1 while we apply the Boltzmann transport only to solids it is equally well suited to uid transport plasma transport and even radiative ie photonic transport ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 t t t Ifth 1 V Fig l 3 The entire population of particles is described by a probability distribution function f that is the mean number of particles at each point in phase space at any given time Mathematically we can write f f I 131 t So by the chain rule of elementary calculus 1 i i 3 dt bulimic afl dt a i dt The first term is called the dz39f tsion term since it is proportional to the gradient of the particle number in real space ie with respect to r The second term is called the drift term since from 2 it is proportional to the force on the particle The subscript ballistic reminds us that there is nothing in 3 to account for the collisions or scattering of the particles necessary to maintain an equilibrium state in the solid The negative signs in 3 are important here and represent the fact that the particle transport always tends to go in the opposite direction to the gradient vs V or P To see this we examine a speci c example of 3 shown diagrammatically in Fig l of onedim diffusive along x axis transport in 17 space with vx uniform independent of x and positive and no applied forces This implies 3 gt0 E x constant ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 Hence 6f6x6x6t in 3 is positive and the second drift term in 3 is zero Under these conditions and with evolving time the distribution function will shift to the right as shown in df Fig l for three consecutive times t1 to t3 This means that at any given value of x E must be x negative as shown in Fig 1 Clearly the negative sign in 3 is needed to make makes this happen A similar argument can be constructed for a nonzero drift term in 3 but without diffusion Assuming onedim drift along the x axis with more particles having high velocities than low values would always be opposite in sign to Bf6px6px6t Hence the 1 dt V negative sign in the second term of 3 is needed A second advancement by Boltzmann was to add the effect of collisions into the same probabilityfunctional formalism He was guided by the following two physical considerations regarding the collisions or scattering 1 they have the fundamental role of returning the population of particles to a state of thermodynamic equilibrium when there are no external forces applied and 2 they establish a steadystate in the nonequilibrium when external forces are applied He thus added the following term to 3 ffo dt Cullzszmt 4 where f is the nonequilibrium distribution function f0 is the equilibrium function The quantity 1p is the momentum dependent relaxation time which has a broader meaning than in kinetic theory where it was defined as the mean time between collisions for all particles in the population Now it is the relaxation time for the entire distribution function averaged over all of particle phase space The subscript collision reminds us that this term should vanish in the ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 limit of no scattering which is clear from 4 by taking the limit Ep gtOO Historically 4 has become known as the I la W time 4111 39 39 and is in nearly all applications 1 of transport theory When added to 3 we get the full Boltzmann transport equation 1Ef f dt 6 dt 6 dt 1f7 5 To solve 5 under even the simplest conditions boundary and initial conditions are always required Perhaps the most useful condition often true in solids is that the total number of particles is conserved j jfp dfad7 N If the particles are uniformly distributed in real space then the integral over space just yields V the volume of the sample Hence jfVdVNVn 6 Note that even when 6 it is not true such as when injection of charge carriers is occurring in a semiconductor device it can be included as one term in a carrierbalance equation or rate equation that conserves space charge neutrality We will address this subject starting with the generationrecombination equation later on In the most common applications of 5 and 6 in solids the particles are Fermions electrons or holes so can usually be described statistically by the FermiDirac function 1 f 0 expUF 7UkBT1 where Up is the Fermi energy and U is the total particle energy It is important to realize here that U is the total particle energy per particle kinetic plus potential not just the kinetic energy as in kinetic transport theory The potential energy is usually caused by external forces but it might also arise from inhomogeneity of the solid In statistical mechanics or transporttheory courses general solutions to 5 are studied Here we will look only at specific examples especially electronic devices where the transport is ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 usually just one or twodimensional This solutions are then most easily obtained by a Cartesian decomposition of both 17 and f7 so that 5 becomes z v EV v anf Fyaf anf f fo dz axquot aw 622 mavx mavy mavz m 7 and we have used the Newtonian relations dy V 3V 1V 2 1 mV mV mV x dt y dt 2 dt px xpy ypz Z Boltzman Transport for DC Electrical Current Perhaps the simplest application of the Boltzmann transport equation is onedimensional current ow of charged particles in a uniform electric eld E E02 The electric eld is assumed to have been applied for a long enough time that steadystate exists in the solid so that dfdt 0 Furthermore the electric eld magnitude is assumed to be small enough that the particle density remains homogeneous at its equilibrium value so that 0 6x 6y 62 So we get from 7 m a V Tom qu or f fo Tmv m 6V2 8 Next comes the most important assumption of all 7 that the applied force creates a signi cant difference between f and f0 but an insigni cant difference between Bf vz and fo vz Mathematically we can then write ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 g r m i dv dV dU dV 2 dU z z 2 For F ermi Dirac statistics we get are eU U kBTltkBT 1 1 f1 f 2 0 0 dU 6U7UFk3T 1 kBT mv ZfoUJZ Th f N ere ore k8 Z N qand na nx And 8 becomes f f6 k T f6 f 9 B The next step which is applicable to many but certainly not all types of scattering is to assume that the scattering is isotropic2 meaning that 13967 10 7 This encourages us to write and solve 9 in spherical coordinates VZ V cos 9 leading to 0 I qE0V000s97V0f01 f0 39 k T 10 B fefo To proceed further we need a boundary condition starting with the conservation of particles 6 a N a IfVdV7 En which leads to the important condition gtIf VdV0 11 To check this we calculate from 10 27 7t Vmax qEV GOSH7 V f l f stin dvd dgb Jrltvgtdvt L L 0 0 32 0 B 2 This tends to be true for point scattering such as ionized impurities but must be applied very cautiously to scattering from extended defects such as crystal dislocations ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 7239 sin 26 7239 a The 9 integral is J0 00565111 65m I0 2 d6 0 So If VW 0 as expected For the electrical current we have J qltan gtqnltVgt where lt gt denotes and an ensemble average over the nonequilibrium distribution lnction f JvzfdVJvzf a39v ltV gt MN MW But IszodV I me cosQ sin 9V2d9d 0 j j j qEOkBTV IV0 f01 f0v cos26sin8dvod8d gt2 WWW 7239 7239 For 9 integrals we have I0 0052 6511166116 0053 630 23 in numerator SO lt V2 7239 a 7239 IO f0 Vd9 f0 V0 cos60 2f0 V0 Similarly I integrals yield factor of 211 in numerator and denominator Thus qu 3kBT LOOV31V0f0 1 f0dV0 lt Vz gt 00 2 IO f0 V0V0dV0 This can be regarded as a relationship between ltVZgt and E0 where lt gt ltV gtltygtELE Z In ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 foVSTWoMO mm lt1quot gt 12 00 2 IO f0 v0v0av0 Note that the extra factor of v2 beyond the normal Jacobian factor in spherical coordinates in the numerator comes from Boltzmann equation solution Botlzmann Transport in Semiconductors and other Nondegenerate Solids In solids having a low carrier concentration compared to the atomic density the Fermi Dirac function reduces to the Maxwell distribution of velocities As discussed earlier in our treatment of statistical mechanics the MaxwellBoltzmann MB distribution is a continuum form of the discrete Boltzmann pdf C exp U1 kBT where C is the normalization 71 7U kBT constant C 2 e l In pr1nc1ple U 1s the total energy of the part1cle Accordlng i to the MB approach it is just the kinetic energy l2mvi2 which is consistent with the assumption that the density of particles is homogeneous Hence we can write the MB density 1mVZk T A a 3 3 5 3 3 function as the rindependent function Po 7 Vd rd V C6 2 d rd V and determine C by the conservation of particles f7d3rd3V N 3 1 2 or CVJwexp EmvxkBT alva N This is one of a wellknown set of Gaussian integrals evaluated by the formulae oo 1 7239 3 I0 eXpax2 xndx for n0 2a 1 for nl J4a for n2 Thus we can write ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 Ifwexp mvkBTdvx 1l27rkBTm 32 m and C n Thus the MB distribution can be written3 27rkT 32 A a m P0 rv 214 eXp mV2kBT 13 Abenefit of using the MB distribution is the simpli cation it brings to the calculation of ltTgt in 12 This occurs in part through the equipartition theorem given one degreeof freedom of each particle such as VX the mean value under 13 is clearly zero but V 1 32 1 2 00 a 00 m 7 mv k T ltv2 gt j vimvmz j Vin e 2 B dv dv dv x N oo n oo ZzszT x y z k T 3 since v2 v vi v m l 1 Therefore lt U gt mltvgt kBT 2 2 A second simpli cation is that f0 in 12 becomes ltlt 1 so that l f0 m l and gt m Jun V21vf0 l f0v2dv N m Joo Vzrfovzdv 3kBT 0 Iow v vzdv 3kBT Jow vzdv 0 Since we have done integral in spherical coordinates we can write 2 ltTgt m ltv2239vgtEltVT EVgt 3kBT ltv gt 3 Note the dimensions of P0 are particles per unit volume per velocity cubed ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 Aside on relationship between P0 and a true probability density function P0 is really a distribution function for the density of particles in real space and velocity space In the Boltzmann equation we need a distribution function for particles that like the FermiDirac function is unitless In mathematical terms 32 m 6Xp mV2kBTdfd7 27rkBT P0 Hymn n Integrating over we get assuming that P0 is not a function of r 32 In eX imV2 k T dy d 27mg plt 2 Bl 32 CXp mV2kBTdV Po7d7n N m 27rkBT With U mv2 we have dV V2 sin 6dth9d and dU mvdV So that by integrating 32 lmV2k T m B overe and we get Po VdV e 2 47IV2dV B Or in terms of the energy variable 27Z39N 7U T P U dUz xU BdU 0 kBT3Z e In general this can be written in product form PoUfoUgU ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 where gU is the density of states The density of states allows us to connect P0 to quantum statistics for the particles For example if we assume the particles are free so k is a good Flsz 2m quantum number we have k nnL L being the width of the solid sample U V sz 32 N U 2 372 hz includes a factor of 2 for spin degeneracy dNU V 2m 32 m32 U 2 U1 U 87rV 2 U12 and g dU 2 2h2 g J W Therefore 130U zzNJE eXp UkBTh3 g U szT32 sawE mm f0 g h3 eXp UkBT or fo V 227rkaT3z Note For the purpose of doing problems it is often acceptable to ignore the prefactor and write this as f0 C eXp UkBT since C ends up canceling out To compare directly to the FermiDirac function we should drop the spin factor quoth3eiUkT 1 gt Z kaT32 elU UFV BT 1 0 where f0 now means the mean number of particles in a state of given U and given spin ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 Transport Theory1 Kinetic Theory Kinetic theory is a formalism for describing the motion of a population of free particles in a solid in response to an externally applied force such as an electric eld KeyAssumptions o All energy and momentum is transferred by the particles in motion 0 In a statistical sense each particle is its own subsystem ie the particles are fully distinguishable from each other and are in quasiequilibrium with temperature bath 0 Collisions are instantaneous events that randomize particle motion Features of Kinetic Theory 1 Can define uxes each one being defined as the amount of some physical quantity crossing a unit area per unit time 0 Particle ux in 7139 393 where n is the particle density and V is the velocity Both of these quantities are possibly functions of time and position 0 Charge ux jq 7139 Q39 393 also called the electrical current 2 9 Energy ux JU n 1 2mv JD since all the energy is kinetic 2 Clearly the particle velocity vector is very important in kinetic theory In a first analysis it is usually determined by classical mechanics For example in one dimension we can solve for u from Newton s law dvx Fx 2 max m 7 where ax is the x component of acceleration For an electron in a uniform electric field the mechanical force is Fx eEx de x so 6Ex 2 m 7 3 Ux 7 U0 where v0 is the initial velocity 3 Clearly a velocity increase with time cannot persist forever It will be interrupted by collisions that can be modeled as a damping scattering term in Newton 3 equation mdux mux dt 139 EEx 1 ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 This is a linear lSt order inhomogeneous differential equation with constant coefficients The solution is the sum of a homogeneous solution setting eEx 0 and a particular solution Taking the initial conditions once again as vt0 v0 we get 7 er E U 1106 h l e 11 Yl m Homogeneous Particular o The homogeneous solution is transient and vanishes for long time scales t gtgt 1 o The particular solution has a steady state term for long time scales of 21E 0 gt 5 ME m where u is the mobility 0 Using the kinetic ux for electric current we get ne 21E Jq nqu E 0E m Ohm s Law Where 6 is called the electrical conductivity Transport conductivities usually connect a given transport ux to the spatial gradient of some macroscopic potential that arises from the nonequilibrium condition The gradients vanish in equilibrium J oE o v For example the charge ux is g gt electrostatlc potentlal Similarly the kinetic energy ux can be written JU 7KVT where T is the temperature and K is the thermal conductivity This makes sense because heat is just the macroscopic representation of kinetic energy at the microscopic level Physical Interpretation of 1 0 From Newton s equation 11 is the rate at which the initial velocity changes to the steady state value This certainly makes sense for a continuum jellium model of the particles ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 under transport and has its historic roots in uid mechanics 0 But in the corpuscular view we need to think about an ensemble of particles each of which occasionally undergoes a randomizing collision If we focus on a subset of the ensemble having a common velocity 00 then 11 is the rate at which electrons scatter out of this subset If the number of particles in the subset is n we can write nt 3n n0e7 Kinetic Theory and Collision Statistics The introduction of the relaxation term in the equationof motion l is also consistent with the collision process being inherently random ie stochastic Speci cally the probability that a particle undergoes a collision in any in nitesimal time interval of length dt between t and t dt is given by dtI independent of the time interval chosen This implies that the probability ofa particle not having a collision between t and t dt is simply l dtI Thus if we de ne f t as the probability of the particle not having a collision between 0 amp ti we can write fa ma ifm multiplicative probabilities fUHm 0 1 T or afdtamp 239 or Zft0r ft0 dt 1 dt T The solution becomes ff W 0 i This is synonymous with the probability of the particle going collisionless or ballistic 3 ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 So if we apply the initial conditoin ft 0 1 we get 717 t e We can now calculate the probability Pt t that an electron suffers a single collision between time t and t 5t P I t exp tr tr where the first term on the right side is the probability that the particle goes without collision up to time t and the second term is the probability that a collision occurs between tand t 6t This Pt 5t has all the properties of a bonaflde probability density function pdf For example it is normalized co 0 0 exptr L39 m Pt tdt Ptdt dt 4 1 5le l 0 0 T IexpErll Using this pdf and integration by parts we can calculate the mean time of survival or equivalently the average time between collisions lt t gt Loot Ptdt KM ltTgt T This justifies our labeling of 1 as the mean time between collisions Of greater interest is the probability PNt 5t that an ensemble of electrons suffers N collisions in a time between t and t 5t In the same spirit as the previous derivation this time interval can be subdivided into intervals 0 to t and t to t 5t If St is small enough then only two different outcomes can occur during this interval Either N electrons scatter during t and none during t St or N l electrons scatter during t and l scatters during t 5t ECE215BMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 Mathematically this can be written as PNt5t PN 1 5m PN15tt where l 75tt is the probability that no collisions occur during 5t and 5t t is the probability for a single collision in this time This equation can be rearranged to yield PNt t PNt M 2i 6t 1 T which can be thought of as a recursion relation relating PN to PN1 In the limit that St gt 0 the farleft time becomes a derivative and we get the inhomogeneous rstorder ordinary differential equation dPNlttgt PNO i dt T r 3 This is an important equation of probability theory and leads to the Poisson density function as shown next Derivation of Poisson statistics in kinetic collisions Using the technique of an integrating factor and the initial condition PNt 0 0 one nds the following solution to the differential Eqn 3ii eXp t r V PNt f j eXpt r PN1t dt 4 0 where t is a dummy variable of integration Fortunately we already know the solution to this for N 0 ie from Eqn 2 P0t E ft eXpt39c Substitution of this into Eqn 4 yields ii WE Boyce and RC DiPrima Elementary Differential Equations and Boundary Value Problems 2nd Ed Wiley New York 1969 Sec 21 ECE215BMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 131a exp t1 Substitution of this back into 4 and iterating yields Pza e exp tr 239 By logical deduction the solution for an arbitrary number N has the form t ZquotN PNI N exp t r This is the famous Poisson density function of probability theory Like all bonafrde pdfs it approaches a Gaussian in the limit of large samples in this case the limit of large N This limiting behavior is well known in probability theory through the central limit theorem It leads to the prediction that in a solid sample having a large number of charge carriers one expects the collision rate to uctuate about some mean value with Gaussian statistics This can usually be associated with a Gaussian uctuation in the electrical conductivity of the sample a fundamental result of uctuation theory known as the JohnsonNyquist theorem The resulting Gaussian uctuations in the opencircuit voltage or shortcircuit current through electrical contacts on the sample is a phenomena of paramount importance in solidstate electronics known as JohnsonNyquist noise AC Behavior of Carriers in Kinetic Theory To understand the response of charged particles in motion to timevarying external electric fields we go back and rewrite the Newton equation of motion in terms of the instantaneous position of each particle x We do this because it allows us to develop expressions for the electrodynamic response in the same physical terms as for electrostatics ECE215BMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 namely in terms of a dipole moment or an electrical susceptibility Since V E dXdt we can re write 1 as m5 39c reE 5 We solve this for the special case of sinusoidal timevarying eld for which we can apply the phasor form E E0971 and write the position variable as x xoe 71m Substitution into 5 yields 8E x m a2 jwz39 From the general electrostatic de nition the dipole moment is p qX and 92E p W Again from electrostatics the electric susceptibility is de ned by Xe E PSoE where P is the macroscopic polarization given by P np Hence we can write 2 ne X gt 00 e mgo 032 jwz39 6 This has two interesting properties First it diverges to in nity as n gt 0 electrostatic limit which makes sense physically Since the carriers are free they undergo large displacement in the electric eld limited only by the size of the solid sample In the present analysis this size was not constrained so the susceptibility should diverge Second it is always negative meaning that the freeelectron response is truly dia electric When D E ltlt 1 Eqn 6 can be rewritten in a useful approximate form ECE215BMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 2 n62 ne 1 e mgo 032 jar mgooa l jmr 7 ne 5 Using kinetic de nition of conductivity 60 7 we can write m jco jcac 003 l jmt 003 Us 6 l jan39 When D E gtgt 1 6 can be rewritten in the useful form and then approximated as 2 Ta2 cor 39 2 2 ne co jr p J ap up z e mwso 032 112 00322392 1 02 031 8 where up is the plasma frequency up nezmso1z Now since 8 E l Xe we can write 2 2 2 a a a 8 E l j ml a a T a 9 This leads to the oftmade statement that the dielectric constant of metals is large and negative up to the plasma frequency where it rapidly drops to about unity Of course this statement requires D E gt 1 and completely ignores the atomic and ionic polarizability contributions which can often be added independently of the freeelectron response Eqns 7 8 and 9 are extremely useful in the analysis of of metals semimetals and semiconductors To determine when each is applicable we need to know that for bulk gold 16 1 r E 3gtlt103914 sec and n 59x 1022 cm393 This leads to cop l37gtlt10 s gold and ECE215BMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 g 218gtlt1015 s 1 or Al cfp 0138um which is well into the ultraviolet portion of the spectrum Similarly we can de ne a scattering circular frequency by ms l 139 or ms 33XlO13 s39l or fS 53XlO12 s39l So in the nearinfrared and visible regions we have wt gtgt 1 The overall behavior of Eqns 6 7 and 8 is represented by the qualitative plot 700212 02139 Reze 1mm P 1 01272 P 2 2 given below with 1 0 T Fig 1 Real and imaginary parts of Xe vs frequency Note at a 11quot lRexel lImxel Motion in a Magnetic Field We go back and generalize Newton 3 equation for particle velocity in the presence of a Lorentz force in a magnetic eld We must keep track of vectors and signs d5 m5 a a m dt 139 q gt sign electron sign hole 9 ECE215BMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 q13 X B gt Lorenzt force e13 X B gt for electron In general the Lorentz term complicates the solution greatly unless E is uniform Let s assume 3 302 and write general form 3 or Dy 02 So I3gtltB uxBoAuyBofc Matching cartesian components on both sides we get A dv mu xm d qu quyB0 A dvy muy ymE qu quxB0 2 m dvz m Uz qEZ dt 1 Again each equation is a 1st order linear inhomogeneous differential equation having both a homogeneous and a particular solution er u owe 1 E UyBO1 e e m i T q i 139 my Joye Ey uBO1 e e m I 7 er u 00 r1 E1 e e m The key result of the Lorentz force is that it couples components of velocity lying in the plane perpendicular to In the steady state and for t gtgt I q u gt LLEX a25wy e u gt LLE a w 1 y y c r e v gt uEZ 1 e 10 ECE215BMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 where aC E eBgm is called the cyclotron frequency a ubiquitous constant in magneto transport problems much like the Larmor frequency and Bohr magneton occur in magnetostatic problems Suppose the lateral extent of a sample is nite along y and z axes and an electric eld EX is applied along 3c In steady state Dy and Dz must be zero if charge carriers are con ned to sample This reasoning leads to U2 2 0 3 E 0 Dy 2 0 3 uEy 2 wow 6 and U 1 E x e I x as expected Eqn 6 implies E 01 Ux z wcrEx qEEX nye Tm Linear in vx Linear in B The linear dependence on oxis consistent with experimental observation of linear dependence on J K electrical current density By noting that J nqux kinetic theory we get BO e m JX BOJX Ey q r a famous result called the Hall effect m q er nq nq E The ratio B Ly i is called the Hall Coef cienth electrons ERH quotq 0 x RH gt negative for electrons or any negatively charged particles and positive for holes or any positively charged particles Note RH does not depend on mass or effective mass The Hall effect is very important historically and in modern technology as well ECE215BMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 Hall Effect Example Hall Bar b I Dec Fig 2 Hall bar geometry and interesting physical quantities We know for the geometry in Fig 2 that if the external electric eld is applied along the x axis and external magnetic eld is applied along the z axis then there will be a response B0 JX electric eld along y axis given by Ey where J K is the current density and n is the IX Andif Eyis I mobile charge dens1ty But 1f current 1s unlform J K AX b c Vy unlform Ey so we get b E 301x b nqbc or V B IX y nqc ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 Kinetic Transpon Theory2 Clarification of kinetic theory A gradient of temperature in a solid generally creates a ux of kinetic energy often called thermal current or heat transfer Often this heat transfer occurs by charged particles so there can also be electrical current from the temperature gradient depending on the boundary conditions As we will see shortly thermal current is considerably more complicated in the kinetic formalism than electrical current because particles having different velocity directions contribute differently to the nal result So thermal current forces us to take a deeper look at the foundation of kinetic theory particularly the nature of the particles We have stated previously that kinetic theory treats all the particles as statistically independent and interacting with the temperature bath through randomizing collisions that occur at a rate 1391 where 139 is the relaxation time If the collision rate is fast enough that sampledependent boundary or shape effects do not matter then the distribution of particles velocity vectors in space must be purely random Furthermore if we neglect the fact that some collisions are elastic ie conserve kinetic energy and others are inelastic we can adopt an average kinetic velocity v0 for every particle in the solid This velocity magnitude is assumed to be constant until an external force is applied These points can be expressed by writing the velocity of any given particle as 17v0fv0sin6cos sin6sin 2cos6 1 where f39 is the radial unit vector in spherical coordinates Geometrically the velocity vectors of all particles in a population would terminate on a sphere of radius v0 all directions of the velocity vector being equally probable ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 Energy Flux and Charge F lnx One Dimensional Analysis Besides the charge ux or electrical current the second important particle ux in kinetic theory and transport theories in general is the energy ux jU n 39 1 2 mv2 393 Since all energy is assumed to be kinetic and kinetic energy comprises heat this is also the heat ux and is often designated as JK For the simple geometry shown in Fig 1a of transport of charged particles in a uniform electric eld and homogeneous temperature T there is a simple relationship between the heat ux and the charge ux The electric eld is assumed uniform along the z axis so that JW nqvz nq Ez But because the particles carry both charge and heat the kinetic velocities are the same so that JW 2 JKZ n12mv2vz n12mv2UEZ Thus the ratio of heat ux to electrical ux is given by J 12qu2 U W EH 2 q q q k l where UK is the kinetic energy per particle and H is the Peltier coef cient 7 an important quantity in solids whenever electrical and thermal transport are occurring simultaneously Motion in a Temperature Gradient OneDimensional Analysis Given these clari cations we can also proceed to analyze the electrical and thermal effects together for the special geometry of a parallelapiped in which heat and charge are restricted to owing in one end and out the other We assume that the long axis of the parallelapiped is the z axis as shown in the crosssectional views of Fig lb and c The ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 v1 4 J5 v2 Metal Wire short circuitquot I a 4 2 T1 gt T2 Metal Wire J R short circuitquot Q 390 20 I 4 z O I T1 VR bid VL T2 l O No Interconnect open circuit 0 Fig l temperature at one end is assumed to be T2 and the opposite end T1 where T2 gt T1 We examine the relevant physical quantities in any plane perpendicular to the z axis The analysis is carried out in two extreme cases 1 with a metal wire short circuit connecting both ends so that there is no difference in electrostatic potential between them and 2 no metal wire open circuit connecting the two ends so that an electrostatic potential difference can build up in response to the temperature difference ShortCircuit Conditions In the shortcircuit case represented by Fig lb the heat ux along the z axis is given by the scalar expression J E n 0 U K 5 where nZ and vZ are the 2 components of the density and velocity respectively We assume that the temperature difference T2 7 T1 AT is small compared to T1 and T2 so that n remains ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 approximately constant throughout the sample and the kinetic behavior remains isotropic about any point Intuitively we then expect that the leftgoing heat ux in Fig 1a J g is greater than the right going ux J S because J g is associated with particles coming from a hotter region T2 gt T1 and therefore having higher kinetic energy This guides us to calculate the difference AJKYZ also called the net heat ux by a judicious application of the chain rule of differential calculus 6U dlW dT AJ EJL JR z Q K 939 Q39z 939 dlI dT dz 6 where the minus sign is added to account for the fact that kinetic energy ows from highT regions to lowT regions From 5 we calculate Furthermore all kinetic energy contributes to heat but U 2 is only onethird of the total since the fundamental kinetic assumption of l is an isotropically directed velocity In other words dT n where C V is the specific heat capacity Thus 6 becomes dT AJKYZ E VZCVEAZ 7 The next and most subtle step is to estimate AZ the seemingly arbitrary spatial differential Presumably AZ should be much less than the length of the parallelapiped but larger than the atomic dimension so that it makes sense to be using macroscopic ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 thermodynamic quantities like temperature and heat capacity In most cases a reasonable distance is the meanfreepath AZ vZ 139 8 Substitution of this into 7 yields dT dT AJKZ m VZCV ver K 9 39 dz dz where the last step de nes the thermal conductivity K vzerIV 10 This is a very useful expression for the thermal conductivity of many different types of independent particles quasiparticles or quantized collective excitations Good examples are electrons phonons and magnons The last step is to spatially average over the isotropic kinetic distribution represented by l The average is taken over just the hemisphere for which the z axis is the polar axis and all directions being equally weighted 2rr2 Ir2 a A 2 2 2 39 Ir J J vz s1n 6d6d J v0 cos 6s1n 6d6 A 0033 63 2 1 2 o o 0 0 2 ltVz gt 2mm zz 6M2 EVO 11 j j sin6d6d j sin6d6 cos lo 0 0 0 So 10 become K 13 GC 12 OpenCircuit Conditions We now consider the opencircuit case of Fig 1c whereby a temperature gradient leads to the motion of charged particles and the creation of an internal electric field Historically this is called a thermoelectric effect Electric current ows along with the thermal ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 current until enough charge builds up at the ends of the sample to stop the ow This condition is de ned in general by an internal electric eld Ezzvr where E is called the thermopower or Seebeck coefficient1 Naturally the direction of the electric field in the bar geometry of Fig lb is along the z axis EZ 2dTdz To calculate Z we can inspect in any plane perpendicular to the z axis First we write the net velocity difference at a point zu RzAgz ii ZAz dT dz where again the length scale is the meanfreepath AZ vZ 139 and the negative sign is added to account for the fact that kinetic energy transfers from highT to lowT regions Therefore dUz dT r dvz2 dT AUZ Uzr dT dz 2 dT dz 1 dimuz2 sz 2 m dT dz and Now we spatially average designated by an overbar with respect to the isotropic kinetic distribution of l But since 12m1 U we can write 1 after Thomas Seebeck an earlyl9d1 century German physicist ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 EzidUEdlz TC39VLT 2 3m dT dz 3mn dz 13 In steady state and for the opencircuit conditions the thermal AV will spatially separate the particles creating an internal electric eld when they have charge This internal eld will induce the drift velocity effect derived earlier for purely electrical transport 0 EUE Jib 14 A steady state will be reached when the thermallyinduced velocity difference from 13 is equal and opposite to the drift velocity 12 That is 51 22 le m 3m n dz C V dT or E2 V 3n q dz E C39 C39 3 2 E Z V Z V f l t 15 dT dz 3qn 3en or e ec mus Speci cation of kinetic theory results with Maxwell Boltzmann distribution Transport Laws With relatively little work kinetic theory predicts several important quantities 6 1T Kand E 2 All off them represent linear response to nonequilibrium conditions gradient in temperature electrostatic potential or both But to relate the thermallyrelated quantities to experiment we need to know the values of the kinetic velocity v0 or the related kinetic energy per particle U 2 The simplest and most popular approach is to use the MaxwellBoltzmann 2 This is the big advantage of kinetic theory over all other transport formalisms 7 capturing multiple physical effects with the greatest possible simplicity ECE215BMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 distribution of velocities derived much earlier in the coverage of statistical mechanics A key result of the MaxwellBoltzmann distribution was the equz39pam39tion theorem for which the 1 mean energy of the partlcle 1s EkBT per degreeoffreedom In k1net1c theory there are three degrees of freedom vx vy and vz in total each with equal value v0 and weighting Thus lt 12mv gtElt U gtlt12mvfvjvjgt32kBT dltltlfi gt CV EHTKZGZW B 16 where the lt gt brackets denote statistical averaging For the Peltier coef cient of 2 16 leads to II 32kBT 3k T2e 17 q B where the last step applies to electrons For the thermal conductivity of 12 16 yields 2 2 3kg 7 ink 3nkBT7 3m 2 2m 18 I For the Seebeck coefficient 15 16 yields C Y nk k E V B 2 B 19 3qn 2qn 28 where the last step applies to electrons It is interesting and historic to take the ratio of K to 6 using 16 2 2 2 Kl3CVUOT3 kB T3 kB T a nqzr n 2 q 2 e 20 ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 where the last step pertains to the particles being electrons This linear dependence of 106 on T was discovered in metals during the midl9Lh century and called the Wiedemann Franz Law The ratio of the Peltier to Seebeck coefficients 1518 is simply given H 32kBTq 21 z l2kBq a result known as Kelvin s law and a cornerstone of the rather esoteric eld of irreversible thermodynamics Comparison of Kinetic Theory and Transport Laws with Experiment The linear relation of Kc to T of the WiedemannFranz law forms an important comparison with experiment The proportionality constant KoT is called the Lorenz number and according to 20 has the value L 32 kBe2 111x10 8 MKSA 22 A few experimental values for common metals are listed in Table 1 below Note that the experimental values are just over twotimes larger than the kinetictheory prediction of 14 Not bad agreement for 14 considering the low level of effort expended at deriving it Most metals have Lorenz numbers in the range shown and display the linear dependence 106 LT over a wide temperature range But semiconductors do not generally display this behavior Table 1 at 273 K Material L x108 Cu 223 Au 235 Pb 247 ECE215BMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 The reason as might be expected is that the thermal current and electrical current are not carried by the same particle type in semiconductors The electrical current is carried by free carriers as in metals but the thermal current is carried primarily by phonons To understand the discrepancy between 22 and Table l we recall that in metals semimetals or degenerate semiconductors the Pauli exclusion principle allows only those electrons near or above the Fermi energy to change their kinetic energy signi cantly in response to a signi cant change of temperature This was quanti ed in the coverage of the heat capacity for the Fermi model of free electrons C 2 7239 E nk V 2 T B TF which is generally much smaller than the kinetic theory result 3 2nkB To illustrate this point we recall that the typical Fermi temperature in good metals is TF 5 X 104K for which C1 2 at T 5 300K In addition lmu2 at 2k T in the Fermi model but instead 0012nk8 2 2 B mu2 E kTF Adding these corrections to 13 we get 2 Cvuz inkBEZkBTF 2 2 2 2 K 3 V 3 2 TF m 7 kg 7 k3 72 2 239T 239T 23 039 nq nq 3 q 3 e where the last step pertains to the particles being electrons The difference between 23 and the kinetic theory prediction is stated by K0corrected 23 2 Ko 32 39 kinetic So the corrected Lorenz number becomes ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 2 2 25 k 8 245x10 8 W Q UT 3 q K2 This is now in good agreement with the experimental values in Table I And indeed this agreement was one of the early triumphs of the Fermi model of the free electrons in metals For the Seebeck coefficient of 19 we get the evaluation C V k Zz Vz Bz 43x10 5VK 24 3qn 2q where the last step pertains to electrons This is to be compared to the experimental values listed in Table 11 Clearly the magnitude deviates from experiment for all materials except the semimetals antimony and bismuth To explain the discrepancy common we resort to the Fermi model once again noting that 72392 T 3 CV 7nkBEltlt nkB 7r22nk TT 7r2 k T Z B F B SO 3ne 6 e TF 25 For example gold has T E 64 X104K T273K so that 3 E H 06gtlt10 6 VK still in poor agreement with the experimental value of Q 65x10396 ECEZl SBMatenalsZEl B Fundamentals uf Suhds fur Electxumcs E R BmWnSpnng 2m Trampan 7112on as gt k2 I I I I k1sinel2 I Tgt k1 Hg 1 CanuniczlExamples uf Semiclassiczl SememgThemy 11mized Impurity Summing an L ed mumnm or m man I r anIgn mough screen y I I ponenual for Impurity charge qs Iomzauon number z VrqS4nssn7exprrLD 1 At low camer dmsmes LD 15 the Debye length LD kBTBIEunez m Z and aLnIgna dmsmes It becomes the Thmaera mI screamg laugh gwen by ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 szykl 2IVFCXPJ39II2d3r 3 where qp is the charge of the incident particle VI exp LbgtegtltpjltaJame Z l l lwexpp i I fr2sin6drd6d 5 where the last step is just reeXpression in spherical coordinates This is a classic integral of science and can be solved simply by first de ning the direction of the vector 1 1 which is fixed during the integration arbitrarily along the z axis of a same spherical coordinate system Hence 1 12 4 l 1 12 Fl 0056 E Ak r 0056 and there is no dependence on azimuthal angle so that 5 becomes 61qu eXprLD 2 H eX Aler cos r s1n6drd6 1 2amp8on l r PM l 6 The 9 integral is straightforward and yields 7 H 61qu J exprLD rzdr eXpjAC39VCOS9 28 r J39WH lo 00 61qu Jexp r LD SrSOV 0 s1nAk r dr Ak From standard integral tables or a good symbolic integration tool this can be evaluated as lLD sinAkr AkcosAkr w q q P 5Akexp rLD 880 1LD2 102 H k 7 2 SrSOVKlLD NC a remarkably simple result To turn this into a differential scattering cross section for use in transport calculations we note from the trigonometry of Fig 1 that Ak 2k1 sin92 so that Ak 22k12 sin292 2k12lcos9 where the last step uses a trigonometric identity Hence from the central result of the quantum mechanical scattering theory V2m2 272h4 06H H k12k22 111 qPqS 8 M Zzsrsoff kaLD2 2k12 2kf cos ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 One remarkable effect is already evident in 8 7 the cross section does not depend on the relative signs between incident particle and scatterer To turn 8 into a momentum relaxation time we need to carry out the transport integral lrm 27z39n1vIa6l cos6sin6d6 9 o where n1 is the concentration of ionized impurities The first term in the integrand on the RHS of 9 that goes as sine is an odd function with respect to the center of the integration domain 9 n2 whereas the 9dependence of 8 is even So the first term of 9 vanishes The second term through a good integral table or symbolic match tool yields 2 lrm fw s 1n1 232132 l r 0 where i E 2k1LD Using the sphericalband relation hk122m 2 U1 we get 12 32 2 T 2 quot 6 gr50 1nlt1 32 gt 32 lt1 32 1 1 10 n1 qpqs To do transport evaluation 1 should be averaged over the particle distribution function to get ltTmgt This can be done just over the MaxwellBoltzmann function since the particle concentration has been assumed low from the beginning But the expression in square brackets clearly depends on energy through the definition of 5 But its dependence is much weaker than the dominant term U132 since in the sphericalvalley approximation U hk2 2m or k VZmU h So we treat this expression as a constant C during the integration 2 U32 1n1 2 21 2ac 1UZM CNI qpqx This allows us to integrate 10 using a previous result for the generic form 1U AU39s 2 4A1quot52S Al67r2m ergo lt rm gt where wakgf CNI qpq Since S 32 F52S F4 3 6 and we get lZSVZ m ergo 32 T kBT 11 1 qp 2 lt1quot gt2 This expression is credited to H Brooks and C Herring and was not published until 1951 because of the rather tricky calculus needed at the end ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 The most common way to handle the constant C is to evaluate it at the point U 3kBT since that is where the integrand behind ltTmgt reaches its maximum the socalled Brooks Herring approximation So since k l2m k U h we can write kBH 5 6m kBT 7l lt1quot 128V27Im ergo 2 k T32 B m gt2 N1 1n1 1 H H 1 Hhqpqr 12 where 53H a 2km LD 6m kBT Hi kBTsrsMnezn 2 And of course ltpgt eltrmgtm which is a very handy equation to put into a spreadsheet computation Before applying these expressions we note the following physicallyintuitive but nonobvious aspects of 12 l as the temperature increases ltTmgt and ltpgt both increase consistent with the fact that the increasing temperature increases the mean kinetic energy of the particles and the fact that increasing kinetic energy makes the charged scattering centers look progressively more transparent 1 2 as m decreases ltugt increases slowly consistent with the fact that lower m means higher mean velocity for a given temperature and therefore greater transparency of the scatterers 3 as 8r increases ltTmgt and ltugt both increase rapidly consistent with the fact that higher 8r reduces the electric field and electrostatic potential in the solid per unit ionized charge Note that all three of these effects are favorable to the use of semiconductor devices at room temperature particularly since semiconductors tend to have low m and high 8r as we found out in our studies of band structure and electric atomic polarizability respectively Speci c example of GaAs at 300 and 77K in m 0067meer 130n1 21x1016cm3n 1x1016 cm3 1 First observed by E Rutherford and his group in the early part of the 203911 century on the experimental studies of scattering of charged particle beams from charged stationary target 4 ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 2 LD 428 Ang kBH 37x108 m391 13H 32 and C 59 at 300 K so that ltTmgt 70 ps and u 183 mzVs 18x105 cmzVs which is much higher than the experimental value of 6000 cmzVs in this grade of GaAs But at 77 K we get LD 217 Ang kBH 19x108 m 1 BH 81 and C 32 so that ltTmgt 17 ps and u 44 mzVs 44x104 cmzVs which is quite close to the 77 K experimental value Speci c example of Si at 300 and 77K m m 026me8r 119n1 21x1016cm3 n 21x1016cm3 At 300 K 2 LD 412 Ang kBH 73x108 m391 BH 60 and C 72 at 300 K so that ltTmgt 97 ps and u 66 mzVs 66x104 cmzVs which is much higher than the experimental value of 1300 cmzVs in this grade of Si But at 77 K we get LD 209 Ang kBH 37x108 m391 BH 154 and C 45 so that ltTmgt 20 ps and u 14 mzVs l4x104 cmzVs which is close to the 77K experimental value Important practical points 1 The mobility is usually stated in practica units of cmzVs obviously not MKSA 2 The great discrepancy between the 300K ltTmgt and ltugt values for GaAs and Si is caused by the fact that around room temperature and higher ionized impurities play relatively little role in the scattering compared to phonons which we now start to address ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 2 Acoustic Phonon Scattering The mobility calculated above for the ionized impurity scattering is generally very optimistic in semiconductors around room temperature because it ignores the effect of the effect of lattice vibrations Until about 1950 this effect remained rather mysterious Then in a very elegant theorem by Bardeen and Shockley it was shown how the lattice waves create a uniform perturbation on the electrons because of their effect on the lattice constant and therefore the potential energy By this time it was known that the carriers in semiconductors generally lie in k space near band edges and that the bandedge energy represented the potential energy Up of the electrons relative to the conventional zero of infinity The simplest form of the BardeenShockley proof utilizes elasticity theory to relate the bandedge energy to the strain 11 through the relation 8U E 11 13 where E is the deformationpotential constant generally a known parameter for most semiconductors and usually a surprisingly big number N 10 eV The large size of E can be traced back to the large change of cohesive energy with small change of interatomic separation away from the equilibrium point or similarly the very small thermodynamic compressibility of most semiconductors and solids in general To relate 13 to the temperature and statisticalmechanics of phonons we recall that each phonon represents the quantized amplitude of a specific lattice wave in phasor form a Acosl P 7I P A ReeXpijl P 7I P 14 where u is the deformation at each lattice away from the equilibrium position A is the amplitude kp is the phonon wave vector to not confuse with k for the carriers and up is the circular frequency associated with the dispersion curve 03 vs kp Elasticity theory taught us that the strain can and should be defined in terms of the divergence of the deformation so that from 14 Finnam ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 clearly a phasor representation So if we restrict the analysis temporarily to longitudinal modes so that u and kin are parallel 11 ijkpu and 13 becomes 5UP j E kp u j E kp Aexp kpr 15 The reason 15 is so elegant is that it can be put directly into Fermi s Golden rule with the understanding that Hklykz 5UP This leads to the perturbation Hamiltonian 1395 kPA a a a 3 IHM Tjexpmk kz krd r where V is the sample volume needed for normalization of the wave functions Since there are no external forces at work whatever happens between the electrons and phonons should conserve total crystalmomentum conservation assuming of course that the electrons stay within the same band and that no photons are generated Hence we can write l k ikp 0 jEk A and lszJ 392 Tpl m Zd kpA 16 The nal preparation step for the transport calculations is to relate A to the phonon statistical mechanics We made this correspondence in the coverage of lattice waves and phonons where we showed that the amplitude of lattice waves could be related to the mean number of phonons corresponding to that wave see also Kittel Chap 4 Eqn 29 4lt nk gt12h coppV Bl 17 where ltnkgt is the occupancy for the phonons Planck function up is the phonon dispersion relation p is the density and V is the volume of the sample However in that derivation the lattice wave was a standing wave composed of two equalamplitude but oppositelygoing 7 ECE215BMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 traveling waves u Bcoskxcosoat B2coskxoat B2coskxoat where the last step follows by trigonometric identity So by comparing l4 and 17 we deduce that A B2 or lt nk gt12h coppV A2 18 In our acoustical longwavelength approximation and at room temperature we have in 1 kBT gtgt 1 0 lt 1 gt z most SOlldS k exphmkBT l 71 And 03 m Cmp 2 k where Cmm is the longitudinal stiffness coefficient along the same direction as the phonon propagation Hence we get 12 HM l H kBTVCWH 19 Now we are ready to apply Fermi s golden rule to get the transport parameters We just need to be careful to account for phonon absorption and phonon emission as separate processes since both are tantamount to scattering of an electron Emission of a phonon at wave vector kp results in carrier energy after scattering of Uk1 kp and absorption of such a phonon results in an energy U 13p In most solids at room temperature we can assume that the energy of the carrier is much greater than the energy of the acoustic phonons th so that U031 Ep H U031 Ep U032 Inserting this relation and 19 into Fermi s Golden rule then yields CB T E 2 2 7r 6UltI 2gt UltI gt1 20 R 1392 VCmm h Given this and the fact that 20 does not depend on angle or even on k1 the absorption and emission rates are practically equivalent and the easiest route to the momentum relaxation time is just the integral ECE215BMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 V Ti mw j j Rlyzkzzdkz1 cos 92sin62d62 21 i z kB Tazjk an13 10391 dk 1m hCmm where the 9 integral was carried out using the fact that the term in the integrand cosGsinGdG vanishes in the integral over the range from 0 to 11 but the first term sinGdG yields a factor of 2 As before we must be careful using the Dirac delta function and convert from dkl in terms of dU1 taking advantage of the sphericalband approximation that pervades scattering theory l kBT 2 2 a a m z k 5Uk Uk dU Tm hcm I 2 lt2 mm 2 LNm kkBTizk 1 7th3C H 1 from the sifting property of the Dirac delta function Finally using the spherical band approximation yet again so that k1 2m U 12 h and we get L m mm m 2 lt2gtmltngt kBT 1m 7th Cmm 7th Cmm 7th4Cm 139 quot 212m32kBT32 01 UilZ Ezit77S where the last step defines the energydependent exponent S 12 Now applying our MaxwellBoltzmann energy averaging expression once again we get ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 47rh4C mmr 52 12 lt7 4AF52 S 212m32kBTEz 2327Ih4Cmm m gt 35kBTs 35kBTS quot r232 kBT32 using the fact that F2 l 1 Finally we get a trivial step for the mobility e lt 1m gt e23x27rh4CmmT 32 mgtxlt quot y232 kB32 lt gt 23 This is the famous BardeenShockley T3932 law for the mobility of electrons or holes by acoustic phonons It is the most basic of a long list of phononscattering formulae for semiconductors so deserves attention Speci cally it has some physical properties that deserve mentioning l ltugt and ltTmgt both go down with increasing temperature consistent with the fact that the amplitude of all lattice waves and therefore population of the corresponding phonon modes increases with temperature so there is more deformation of the lattice to scatter the electrons 2 ltugt and ltTmgt both increase with crystal stiffness consistent with the fact that a stiffer crystal suffers less deformation of the lattice for a unit amount of latticewave energy 3 ltugt and ltTmgt both decrease rapidly with increasing deformation potential consistent with the fact that this potential is the perturbative coupling coefficient in the problem and 4 ltugt and ltTmgt increase even more rapidly with decreasing m consistent with the fact that smaller m means higher acceleration and therefore a greater distance traversed by the carrier before a deformation can disturb it To use 23 it is important to have the values of Cmm m and E at hand These values are tabulated below for two important compound semiconductors a narrowbandgap InSb and a normal band gap GaAs as well as Si ECE215BMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 E eV 93 64 77 026 92 Speci c example of GaAs at 300 and 77K m m 0067me8r 129C11 119Jc109Nm2 E 93eV So at 300 K ltTmgt 28 ps and u 72 mzVs 72x104 cmzVs which is much higher than the experimental value of 6000 cmzVs in this grade of GaAs At 77 K we get ltTmgt 21 ps and u 56 mzVs 56x105 cmzVs which is way above the experimental value Speci c example of Si at 300 and 77K m m 026me8r 119C11166x109Nm2E 92eV At 300 K ltTmgt 05 ps and u 036 mzVs 36x103 cmzVs which is about a factor of 25 higher than the experimental value of 1300 cmzVs in this grade of Si At 77 K we get ltTmgt 40 ps and u 27 mzVs 27x104 cmzVs which is higher than the 77K experimental value Clearly the BardeenShockley model for scattering by acoustic phonons overestimates lt39cgt and ltugt in both GaAs and in Si As we shall see shortly the reason for this is that it ignores inelastic scattering which is not so important with acoustical phonons but becomes very important with optical phonons This is particularly true at room temperature where the optical phonon modes are beginning to become signi cantly populated and where the thermal kinetic energy of the carriers NkBT becomes great enough to start emitting optical phononsi a mechanism that is profoundly important in most modern electronic devices because it is ultimately the braking mechanism that causes carriers to saturate their drift velocity in moderate to high bias electric fields 11 ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 Transport Theory 5 Boltzmann Transport in Uniform Electric Field cont We have seen how the solution to the classical Boltzmann transport equation in a uniform electric field has the form jv4rvfo1 f0dv lt 1 gt 3k T 00 1 B Ivzfo vdv 0 where f0 is the equilibrium distribution function the FermiDirac function in the most general case The product f0lf0 has deeper meaning than might first appear From statistical mechanics we know f0 is the mean number of charge carriers assumed to be fermions or mean occupancy of any spacespin state quantum state So lfo is the mean deoccupancy of that same state This re ects an important principle in transport theory at all levels including fully quantum mechanical Which is transport requires the presence of a particle occupying a state and it requires an available state for that particle to transfer into In the limit of low carrier concentration ie nondegenerate population covered in the previous section f0 approaches zero for most particles in the population so the deoccupancy factor can be assumed to be unity This is primarily what makes the calculation of l tractable with the MaxwellBoltzmann function 7 a very common and useful exercise with semiconductors In the intermediate case of moderate carrier concentration 0 lt f0 lt l and 0 lt l f0 lt 1 so that neither factor can be ignored and the calculation of 1 becomes much more complicated than the lowconcentration case In the limit of high carrier concentration f0 is approaching l for most particles in the population and the deoccupancy factor approaches zero but clearly can not be ignored Fortunately the calculation of 1 gets simple again in this degenerate limit because of the behavior of the behavior of the FermiDirac function f U eXpU U F kBT I Plotted in Fig l are f0 lfo and the product f0 1 f0 The only place where f0 and lfo are not nonzero is in the region around UF which makes the product display a narrow symmetric peak ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 II39 R 1 39f0 fo1 39fo Fig l To a good approximation that is best in metals the product looks like a Dirac delta function f0 1 w A5ltU UF We nd the coef cient A by de nition of f0 and its derivative with respect to U 00 00 by k T l0 foU1 JUd l0 kBTEOdUWBUFkg0 EkBT 3AkBT SO f0 1 f0 E kBT5U UF lt Dirac Deltafunction 2 Aside on Dirac Delta function 00 Normalization property L00 5 x x dx l Sifting property Ifwfx5x x dx fx d5x x dxi Derivative sifting property I f x dx dx xx39 Symmetry and factor properties 5 x 5x 5 tax 2 1516 agt0 a 2 ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 1 To ut1l1ze these propert1es we go back and rewr1te 1 1n terms of energy us1ng U Emv2 2 dU v xZUm 2U dUmvdv2Umdv vzdvzv dU dU 32 dU mV m m m J m V4dv 2 U32dU va4dvZIuv2dvZIU ifde and J J m I1de IUlZdU 2JU321Uf0 U1 f0 UdU This leads to lt T gt 12 3kBTjU f0 UdU Given the expression forlt L39U gt we can now apply the delta function approximation of 2 in the numerator and the following Heaviside unit step function approximation in the denominator 1 NJ expU UF kBT 1 w 6W 39 Up U This leads to ltTgt 2 ISOU321UkBT6U UF dU 3kBT 51F UlZdU 2 U21UF EUZTUF ltIgtz T U 3 gUsonF 3 U2 F This is a very important result the nonequilibrium ensemble average lt39Ugt can be taken as value at Fermi energy This leads to the adage often use in the transport theory of metals all the action is a the Fermi energy or more accurately the Fermi surface ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 Boltzmann Equation with Concentration Gradient Going back to the timedependent Boltzmann equation we can establish another solution that is essentially opposite to transport in a uniform electric eld We suppose that 6f6v is zero for all velocity components but 6f6z is nonzero This represents the case of a concentration gradient that as in the case of kinetic theory can be created by injection of carriers at one point of an otherwise homogeneous semiconductor To simplify the analysis we assume the concentration gradient occurs only along one direction of space z Then the timedependent Boltzmann equation in the relaxation approximation can be written im 3 dt dt 62 1v 6f 6f0 As 1n the case of electrical dr1ft we assume a And then we expand by the cha1n rule 2 2 6f 6f dn 6 so that in the steady state 3 becomes af dn 3ff VZTV ff39 4 0 6n dz 0 The particle current Jn nvZ which has a transport ensemble average nJvzfdv nIvzf dv ltJn gtn ltVz gt j fdv j fodv Substitution of 4 yields 6 w 6 d IV4T cosz sm 9dvd9d J v47 ltJ gt2 n an ndno an D n H 2 oo dz j fov sm 8dvd8d 3 dz 10 WOW dz w 6 J v47 v A dv D n 0 an where the diffusivity is defined by g Jowvz dv ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 For special case of the Maxwell Boltzmann distribution 2 f0 2 quotC T eiUkBT quotC T eimv 2kBT Ema 6n n wv4rv dv l fquot lltvzrvgt 5 D21 0 2 3 10Vde 3 and Now we recall that for the MaxwellBoltzmann statistics lt1 gtltv21 gtlt v2 gtltv21gtm3kBT 6 Substitution of 6 into 5 results in the expression 13k Tltrgt kTltrgt kT DB B Llt ugt 3 m m e which is Einstein s relation yet again It turns out that there is a similar Einstein s relation valid in the opposite limit of high particle concentration and everywhere in between This makes the Einstein relation one of the most universal results in all of transport theory Importance of Diffusion in Semiconductors o Diffusion is a process that tends to drive the solid back to equilibrium after excitation by non uniform external forces By contrast drift in an electric eld is a process that tends to drive the solid away from equilibrium So naturally the two are often cooperating in semiconductor devices leading to the driftdiffusion formalism developed after semiclassical transport ECE215BMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 Boltzmann Equation with Temperature Gradient Optional Material for E CE21 SB 2008 The last topic we address on the classical Boltzmann equation is transport in a temperature gradient assumed to occur along the z axis And as in kinetic theory we must consider two difference electrical conditions open circuit and short circuit to fully describe the subtle coupling between temperature and electrical effects We will limit the analysis to thermal transport by charge carriers assumed to be Fermions having welldefined Fermi energy UF T OpenCircuit Conditions We allow for a nonzero electric eld along the same axis to accommodate the Seebeck or other possible thermoelectric effects The Boltzmann equation can be written i iffo dt dzaz dzavz 7v i i 62 62 avz 6Vz and we assume as before So we get in steady state were anfo ffo a 3 f fo 7VVz 8 As with electrical drift by operating on the FermiDirac function for f0 we get 6 sz m 30 9 We expand the spatiallydependent term as ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 la Tl az 6T 02 ell kgr iU UF0Tf1fL6UFU UF 0r 6T ewimkgr 12 6T kBT 02 0 0 kBT 6T kBT2 afo 6T fg1 fgaUF U UF6T Therefm 6T 02 kBT 6T T 02 9 Finally we recall that 61 WV fo 1 fo z E avz kBT q and VZ vcose Substitution of these into 8 along with 9 yields TVVcos6 0UF U UFj T l E f f kBT f miq 6T T 02 10 The most useful quantity to average over this nonequilibrium distribution function is the Ivzfdv 2 component of the velocity ltVzgt J fc By substitution of 10 this becomes 1 mrmvz cos 6f01 f0qE jaa v2 sin 6dvd6d lt vz gt kBT Mao f v2 sin 6dvd6d 11 It It is easy to see that the denominator integral over f vanishes since 0056 Sin 6016 0 o ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 It 3 I 2 cos 6 We can also integrate the numerator over 9 and I to get I cos 6 Sm 66m 3 2 3 0 0 2 and I d 277 and do the same integrals in the denominator to get J sm 66m 2 and o o 271 J d 2 So in total 11 becomes 0 6U U UF ar 4 1 jrvfg1 fgqE 6T T EVdv ltV gt2 z 3kBT Ifgvzdv The only terms that depend on velocity are 139 and U to first order Up is independent of V So if we assume a MaxwellBoltzmann distribution ltVz gt2 qE amp 6 lt7gtlaTltUTgt 12 m 6T T 62 T 62 v4 V dv 2 Where gt m w m ltV22V gt2 ltV TEVgt 3kBT J vzfgvdv 3kg W gt Eqn 12 is very useful in describing various thermal effects such as the thermopower Z ie Seebeck coefficient This is obtained by setting ltvzgt equal to zero in 12 consistent with zero electrical current under opencircuit conditions Then solving for E we get E atLg ltUrgt 612261 2 q 6T T Tlt7gt 62 6162 13 z 0 14 GT T Tlt7gt which means that ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 Boltzmann Transport with Band Structure and Quantum Mechanical Scattering Semiclassical Equations of Motion The last essential upgrade to the Boltzmann transport formalism is to account for the everpresent interaction between the transport particles and the background atoms or lattice in the case of a crystalline solid We have previously dealt speci cally with electrons in the crystalline case and invoked Bloch s theorem as a means of classifying the wavefunctions and categorizing the possible electronic energy states into bands In the process we learned that the good quantum mechanical independent variable in the particle dynamics was the crystal wave vector 1 The classical momentum became the crystal momentum h and the mechanical velocity became the group velocity vg lh Un where is the nth band and the subscript on the gradient operator reminds us that the operation is made in k space In terms of these quantities the particle position is given by d a a Vg h 1VUnk 1 dt And the reaction of an electron to an external force can then be written at A h F 2 dt In electronic solids the two most common external forces are the electric and magnetic F qE E ka x B 3 These are the same as the classical forces except that the velocity in 3 is replaced by the quantummechanical group velocity In the general case of nonzero electric and magnetic fields the combination of l 2 and 3 lead to the semiclassical singleparticle equations of motion with 2 expressable in cartesian coordinates as 2 1de dt 413 vUsz vUBy 1 a a hdky dt qEy VUZBX VUXB 4 ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 hde dt qE vUBy vUyB The combination of l and 2 form a coupled pair similar to Newton s equations but in terms of I and 1 instead of I and E Another difference is that7 in 1 can not be a point in space as in Newton s law but rather the expectation value for the position value of the particle We will see below more precisely how to de ne this expectation value Yet another difference is that l is linked intimately to the band structure of the particle if it is moving within a crystal lattice More speci cally 1 couples the motion in real space to the motion in k space through a map 7 the U41 function for a given band indexed by n Semiclassical Equations of Motion Behavior of particles in real space Before embarking on the development of semiclassical Boltzmann transport formalism it is helpful to better understand what the position vector really means in the semiclassical model This can be accomplished by transforming from I space tospace using Fourier techniques We start with a truly free particle in vacuum for which the solution to the Schrodinger equation has the mathematical form of a plane wave IEHam 17 t wW W w woe 5 As in electromagnetics the plane wave is a physically inadmissible solution when allowed to extend to infinity along any direction To fix this problem we formulate a wavepacket w it 2 go expur f jUUE W k W Fourier amplitude ban5 fum oquot In the presence of a crystal lattice we must generalize 5 to include a cellperiodic portion which is just the Bloch function a 11 a W F e ur 6 To make this physically admissible we again form a wave packet confined to band n ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 We 7 elenWhl basis mction 11quot KI 2 where g is the Fourier amplitude function and the sum extends only up to I the Nyquist wave vector within the lst Brillouin zone BZ The meaning of in the semiclassical theory is now addressed by considering the size of the wave packet using the translation property of Bloch functions jIEf39 VHV39Fe W This should apply to the wavepacket as well meaning that for a particular f IT 11 1717 t g 1 V1 expjl 7 Un I th 7 0 mo We can think of this as a function of F for fixed And because the entire sum extends only over the 1st BZ we expect that the product g Vnyk extends over a Ak much narrower than the Brillouin zone ie Ak ltlt kN na 8 By Fouriertransform theory the extent of the wavepacket Ar in 7 should abide by the mutual uncertainty relation Ar Ak m 1 9 Substitution of 8 into 9 leads to Ar gtgt 1 10 7239 In other words the spatial extent of the Block wave packet in real space must be significantly greater than a lattice constant which means even greater than the interatomic separation The analysis of the Bloch wave packet also guides us to better understand the time evolution associated with 1 We should think of 17 as the position of the center of the Bloch wave packet moving with velocity 7g Through l and 2 any external force thereby changes the location of the particle in real space and k space as well External forces are treated ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 classically but consistent with 10 must vary even move slowly in space than the wavepackets For example the wavelength of an ac electric eld must satisfy 7 gtgt Ar The only remaining subtlety of the semiclassical equations is the implicit requirement that the motion be con ned to one band This is much more dif cult to analyze but can be tested with respect to the external forces by the following general conditions a For dc electric elds eEa ltlt Us 2 UF where UG gt gap to nearest other band and eEa gt electrostatic potential energy shift over a unit cell b For dc and ac magnetic elds ha ltlt Us2 UF where a qB is the cyclotron m frequency and ha is the kinetic energy shift due to magnetic quantization c For ac electric elds ha ltlt Us a 2723901 for ac eld Eoe jw39 Example of semiclassical theory Bloch oscillations We recall from the analysis of band structure that the rstorder effect of the background lattice in crystals is the strong Bragg scattering at the Nyquist wave vector kN na The other key result was quadratic behavior around localminimum points freeelectronlike behavior So for a band having a minimum at k 0 a good approximate onedimensional model of band structure along 2 direction would be Ukx l2UB lcoskza 11 where U3 is the band width a is the lattice constant and the absolute minimum energy of the band is set arbitrarily to zero The group velocity is then given by v U361 sinkza g 2 dz 1 6U UBa v E s1n k a dt 3 hak 2 Z 12 The second semiclassical equation is dk h d F E qE0 eE0 eEot for which the solution is simply kz Tko l3 ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 The initial value of k0 can be set to zero with no loss of generality for the present argument Substitution of 13 back into 12 then yields dz l 6U U Ba V dt flak 2h sineE0at h for which the solution by inspection has the oscillating form UB z ZeEO coseE0at h Az cosaBt 14 where the last step de nes the amplitude AZ and circular frequency DB of the oscillation and again the initial value has been set arbitrarily to zero This is arguably one of the most interesting yet simple predictions of the semiclassical theory predicted first by F Bloch and hence called the Bloch oscillation Clearly the solution for k 13 is progressive in k space meaning that k changes linearly with time across the lst Brillouin zone until it reaches kN after which it suddenly reappears at kx kN and continues the linear progression once again But the solution for the wavepacket center 14 is more interesting displaying oscillations of frequency 013 and amplitude AX Example of Bloch oscillations in a direct bandgap semiconductor Suppose the semiconductor is crystalline and has a conduction band centered at kx 0 with a bandwidth UB 30 eV and a lattice constant a 5 Ang The resulting model band function of l l and the associated group velocity are shown in Figs 1a and b respectively The applied dc electric eld is assumed to be 5XlO4 Vcm 51106 Vm a rather large value but easily sustained in highpurity high resistivity semiconductors Under these conditions the linear Bloch frequency will be fB DB2n eanh 604x1011 Hz 604 GHz ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 V9 mls 0 05 1 k2 xnla k2 XEa Fig l And the amplitude will be AZ UBZeEo 3x10397 m 03 pm In many ways the Bloch oscillation predicted above has been a holy grail of solid state and semiconductor physics since it was predicted And as of the time of composition the author has not heard of such oscillations being observed in any bulk semiconductor at any temperature of operation The reasons for this is rather simple but frustrating to solidstate engineers trying to develop Bloch oscillators They can be seen through the following two requirements In order for the oscillation to occur the crystal wavevector must be progressive over the lSt Brillouin zone meaning that transport must be ballistic ie collisionfree or nearly ballistic Stated differently the oscillation of the wave packet in real space must be sustained over a distance of typically 100to 1000 lattice constants It turns out that these requirements are very difficult if not impossible to achieve in a bulk semiconductor for one fundamental reason that is clear from Fig 1a and b Typically bulk semiconductors have conduction band widths in excess of U3 1 eV As the electron wave packet moves in k space within such a band it will necessarily have to experience a very high group velocity such as the peak values in excess of 106 ms as shown in Fig lb Although still far below the speed of light and thus nonrelativistic such velocities create a very strong interaction with lattice waves or phonons via acoustic or optical phonon absorption in all semiconductors around room temperature or optical phonon emission in polar semiconductors at low temperatures ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 Semiclassical Generalization of Boltzmann Equation The importance of particle collisions and scattering in all solids even in very pure semiconductors forces us to generalize the semiclassical equations in the same fashion as done previously for the classical Newton equation Not surprisingly the first and most common way to generalize 2 is through a phenomenological relaxation term gt d k gt 1 gt gt gt gt gt h q E VkUnkgtltB hkTk 15 dt h where 11 is written explicitly as a function of I to account for energydependent scattering No such relaxation term is added to the realspace semiclassical Eqn 1 since 139 is assumed not to depend on spatial location of the wavepacket Just as in the classical case we expect that certain types of particle scattering will have a strong dependence on kinetic energy and therefore even a stronger dependence on k This compels us to immediately consider a statistical approach to the semiclassical transport A key issue in the development of a statistical semiclassical approach is that the Boltzmannian concept of a phase space in which to place each and every particle and to define a distribution function is still valid We just have to replace of I and E by I and 1 and stay cognizant of the fact that the location of particles in this space is made fuzzy by the inherently probabilistic nature of the quantum mechanics To remind ourselves of this fact we replace the variable I by 1 WP where wp stands for wave packet Fortunately the space is still six dimensional so that we can define the distribution function in the same way mathematically as in the classical Boltzmann case and expand it by the chain rule of partial differential calculus as df 6f dpr 6f dk af T T l6 dt 6er dt ak dt 6t And guided also by the classical case we approximate the scattering term by the small perturbation semiclassical relaxationtime approximation 1 2 f fo 6t 239k scattering scattering so that 16 becomes ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 0f We are ffo dt 6pr dt E ca Given this semiclassical Boltzmann equation if f is independent of 17 we can write the electrical current formally as 17 qnjmdv IEfUn 13d132L2a3 LandfUnlEdl 2L27r3 where the factor LZn is the volume per state in k space So 18 becomes j N bmdvj 1 fUnEdE2L27r3 jbmd hfUn dE 1 7 V N q 473 Similarly the thermal current can be written JK IbmdgoaL mE dIEMf 19 As in the classical case the nonequilibrium distribution functionfUn is found as J an q 8 18 the solution to Boltzmann s equation If we assume f is uniform in real space so that ift3er 0 and analyze the steadystate 17 becomes 0 1 f h 4613a T 0r fag Tl g qiaqdk 6k dt 6U 61 dt 39 Where we show a dot product in the last step between two intrinsically vector derivatives But from the semiclassical equations 1 and 2 we have dk F 6U a an T th dt h 0k Furthermore if the perturbation created by F is small then as in the classical analysis izfolfo 6U 6U kBT Thus the solution to 17 becomes 1 a fsmvagEmf39 B In the special and very useful case ofa uniform electric field this becomes rf0l fOqET f f0 g kBT 20 Example of semiclassical Boltzmann equation uniform electric eld in a solid having a single filled band The general solution for the electrical current is ECE215BMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 a a o ltJq gt 13 jvgfodk L3 I VngO 1 folqE ngk 4 band 4 band kBT But for a lled band the second integral has no available states within band f0 l for all values of k so this term is approximately zero Hence ltJqgt 43vag dE L U d1 21 To evaluate this integral we utilize yet another important consequence of band structure its symmetry in k space Speci cally we recall that for a given band index U41 Uquot 42 22 which in mathematics is called an even function And since Unk is even so is f0Unk Eqn 6Uk 6U I 22 also 1mp11es that wh1ch 1n mathematlcs 1s called an odd functlon 6k 6k So in total 21 is an integral over the product of an odd function times an even function But such a product is also an odd function And from calculus we know that the integral of an odd function over any zerocentered domain of the independent variable must be zero So in the end we get for the electrical current from a full band lt jq gt 0 Similarly for the heat ux we can derive a solution to 17 analogous to 20 and use the nonequilibrium function to formulate the thermal current via 19 13 if IEUn 1 fat ltj gt Q 47 b 1 a v Unfodk 4723 biz1d g Without even going through the work we can immediately derive the solution for free carriers in a filled band based on the same symmetry arguments as above As with the electrical current the nonequilibrium term f in the second integral will be proportional to f0 lfo so will go to zero for a filled band since there are no available states Furthermore the lSt integral also vanishes since the integrand is now a product of two even functions Un and f0 and an odd function vg And from mathematics the product of two evens and one odd is odd This leads to the following profound theorem of semiclassical transport A lled banal cannot conduct electrical or thermal current ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 Corollary Partially lled bands make good conductors In other words it is not the availability of electrons that makes a solid a conductor or insulator Rather it is how the electrons are distributed in bands We came to this same conclusion when covering band structure Hole Theorem Yet another profound consequence of the semiclassical theory is the transport behavior of a band in which most of the states are occupied with electrons and just a few are empty We start with the band picture shown below with just one missing electron at wave vector k and we analyze the effect of this on the crystal momentum and energy of the carriers In terms of crystal momentum the missing electron at 1 means that the net wave crystal momentum of the sample is hko since all other electrons occupied states cancel in pairs In terms of energy we can write from the assumed known electronic band structure and symmetry properties U k0 U ko U ko E U kh 23 where Uh is a fictitious hole band that is a mirrorimage of the real electronic band about the U 0 axis This mirror imaging is shown in the sketch below To complete the hole theorem we need to analyze the semiclassical equations of motion starting with evaluation of the group velocity Since the momentum of all electrons cancel in ECE215BMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 pairs except for the state 1 E then from 2 we can write one semiclassical equation for the entire band dz 61E A 1 A a A dz 2 E VU k X 2 h h dt dt e h k dt 24 Then using 23 along with symmetrical properties of the gradient operator we can write VUk kVUk k0 VUh k0VUh kh 25 where the last equality is justi ed graphically in the sketch below Substitution of the outside equality of 25 into 24 leads to the A Ae vkuh W3 dt or by negation Given this clari cation we can treat the transport of empty states in an otherwise full band simply by the following two tricks which constitute the hole theorem With respect to transport in a uniform electric field by the semiclassical model the effect of an empty hole state in an otherwise full band can be accounted for by l changing the sign of the charge from q e to q e 2 inverting the energy band about a mirror plane of U 0 the highest energy point in the band This is a remarkably simple result and profoundly important for bipolar semiconductors ie semiconductors with some occupied electronics states in the highest conduction band band and an equal number of unoccupied electronic hole states in the next lowest valence energy band We will be examining those shortly The hole theorem can be added to the embodiment of the semiclassical Boltzmann transport equation by adding a counterpart to 15 for holes amp 8ElkaeEXBh k 26 dl h 12 k2 ECE215BMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 dk7 14 a 21E F dt eEEVkUh khgtltBW 27 where 111 is the hole scattering or relaxation time Magnetic Field Conservation Theorem Yet another profound consequence of semiclassical transport comes about when there is a large magnetic eld applied to a solid with an insigni cant electric eld In the special case of E gtgt0 and E 0 we can write the semiclassical equation 1 as 6 261112 kUh When coupled to 26 and 27 above we see the following two important facts 1 the component of kg or E along the B eld is conserved in motion 3 kspace motion is entirely in the perpendicular plane 3 cyclic motion in many crystals 2 the total energy U2 kg or Uh is conserved The proof of the energy conservation is a simple but elegant exercise in semiclassical transport We assume the band structure is known at all points in k space and can be expanded as 6U a a 6U a 519 hvg 519 28 eh where the subscript eh mean either electron or hole But from the semiclassical equation 26 and 27 we can expand 5kg1 as a function of time 5 h ke39h 511FXB51 29 239 dt h g Combining 28 and 29 we get 5Uqgggtltl 5t0 30 for any uniform 5 since Vg X3 is always perpendicular to Vg The practical implications of the magnetic conservation theorem are very important in materials science and for semiconductors metals and semimetals The cyclic motion and its ECE215BMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 associated frequency are known as cyclotron resonance The motion in k space becomes particularly simple 7 circular or elliptical in spherical or spheroidal bands as usually occurs in the conduction band of semiconductors1 This means that the band curvature or equivalently the reciprocal effectivemass tensor components can be determined through measurement of the cyclotron frequency see HW Problem as a function of crystal orientation Even when the solid does not have a simple constantenergy ie Fermi surface the magnetic field theorem is still very useful in understanding the band structure This is because the cyclic motion in k space creates oscillatory behavior in other transport parameters which in turn can be related to the Fermi energy effective mass etc Oscillations created in the electrical resistivity or conductivity are called the Subnikovde Haas effect and oscillations in the macroscopic magnetic susceptibility xm are called the de Haasvan Alphen effect Conductivity Effective Mass We have seen that in the presence of a crystal lattice we must use the semiclassical picture of transport to properly account for energy band structure For example for individual electrons we have d a 1 a a a hi 15 3 e E4 VUkegtltB 9 dt h re Now suppose we have an E eld only and a steady state situation such that I ere hE 31 We suppose further that the electrons are confined to a conduction band consisting of NS valleys having spheroidal constantenergy surfaces In the limit of low carrier concentration we can thus write for the electrical current density a a a e a a a lt Jq gt eZ I vgfsalc 7 Z IVUkefsdk S S where the sum is carried out over all spheroidal valleys indexed by the integer s the integral is over a single spheroid and fS is the nonequilibrium distribution function for each spheroid We assume further that the E field is small enough in magnitude that the carriers remain approximately equally distributed over all the valleys as in the equilibrium state so that fs m fNs Then we can write 1 The effective mass parameters of the conductionbands in both silicon and germanium were first figured out using cyclotron resonance measurements in the mid 19505 by competing groups from UC Berkeley led by C Kittel and MIT led by B Lax These experiments also led to the realization that the constant energy surface of the conduction bands was spheroidal rather than ellipsoidal as was earlier believed l3 ECE215BMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 ltjq gt amphZSJ Ul efdIE 32 1 39Jl nn 39 1 Example of through r39 m J To make further progress on analyzing 32 is helpful to look at conduction in a speci c example ntype silicon In this case the constantenergy spheroids have six different forms when expressed in cartesian coordinates 2 2 2 2 31 mils ggkww 2 or k U k 1n space ml ml ml 2 mt ml M 2 2 2 h kxkykzik022 2 mt mt ml 33 Without loss of generality we can restrict our analysis to transport with a uniform E eld magnitude E0 along an arbitrary axis say x Then we need calculate only the x component of 32 which requires calculating the x component of the group velocity of all six ellipsoids in 33 1 t 2th 2th 2th 2th 4th ZVxUke x x x x x s ml mt mt ml mt But from 3lkx T hE0 so that EE 2 2 0 er 2 4 a a ltJ gt 2 k ak 6h 1 f Lx h ml mi 2 l 2 4 ne lt r gt 6 ml mt where the last step defines the conductivity effective mass ms 2 eEoiiJrgfk6 6 ml mt n62ltTgt E E E 0 0 m0 11241111 mg 67 mt 3m mt mt Numerically we know for silicon that m 098 m0 and mt 019 m0 so that m 026 mo 34 It is important to note that the conductivity mass is a fundamentally different quantity than the densityofstates mass md defined earlier in the statistical mechanics of semiconductors Eqn 34 is an arithmetic average over the ellipsoids whereas the densityof states mass was a geometric average We recall for silicon that m 3 mfml which when evaluated yields md 033 m0 It is not hard to see that the evaluation for silicon above is independent of the direction of the applied electric field provided it is uniform In other words m 026 m0 is a fundamental electrical transport property of silicon It is a bit more difficult to prove that the last step of 34 is valid for all spheroidalvalley conduction bands provided that the semiconductor has cubic symmetry We can think of it as a peculiar arithmetic sum that is independent of where the 14 ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 NOTES 1 Electrostatic Behavior of Solids1 Solids are good at storing mechanical potential energy associated with the microscopic forces between atom ionic or covalent bonds A small compression of expansion of the solid entails a big change of energy Solids are also good at storing other forms of potential energy especially in electric and magnetic elds We start with a look at electrostatic energy and the forces associated with it This is one of the oldest branches of physics and engineering and fundamental to the understanding of solidstate electronics Energy and electric fields In presence of externally applied electric eld 130 there are two terms that contribute to electrical potential energy inside a solid a Work done by external eld alone SW dUE 61570 j E0 lde where UE is the energy stored per unit volume b Potential energy from alignment of microscopic dipoles 6W Elquot d13 V where 132 gt polarization per unit volume Em gt electric eld inside solid For the purpose of solidstate analysis we can subtract out the rst term since it is there even without the solid present The lSt law of thermodynamics then becomes dU m9 PdV Einda3 V By de nition Em i aq intensive variable V a l P l T derivative taken along 13 direction and we have a susceptibility thermodynamic V613 I f electrlc suscept1b111ty 5 l E0 l The more common electrical susceptibility is given by l ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 V6132 I 2 806 l E0 l c 0 so gt permittivity of vacuum 0 Em and Pg are to be thought of as macroscopic averages over a volume much larger than the atoms but smaller than the solid itself Solids generally form dipoles in 3 different ways a Microscopic dipoles are induced by applied eld b Permanent internal dipoles are alig by applied field c Free carriers as in metals or semimetals displace to create a large induced dipole extending over the entire solid sample Independent of the type of atomic dipole the response mechanism is the same and can be drawn graphically as shown below Net positive 9 charge n gt gt Eresponse E E1 So by linear superposition Elquot 2 E1 110 Note Kittel defines E EM 0 The physics of dielectrics is largely the study of 13g and how it depends on a geometrical ie shape effects of the solid sample b microscopic nature of the solid Before getting into these effects we need one important result from electrostatics 2 ECE215BMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 Polarization Theorem The distribution of microscopic dipoles shown in the above gure leads to a simple relation between the polarization vector and the microscopic dipole moment P n I It is assumed thatPe is averaged over enough dipoles to get a macroscopic result and that the 13 do not change appreciably over this volume It is also clear from the figure that the applied electric eld will induce a net surface charge as the microscopic dipoles are induced If P in the solid sample is uniform we will see the sample conditions that create this shortly every negatively charged end of a microscopic dipole must be effectively neutralized by being very close to the positive end of a neighboring dipole And every positively charge end of a dipole must be effectively neutralized by being very close to the negative end of a neighboring dipole So there can be no net charge Q even though P is distinctly nonzero But at the surfaces there is dangling charge ie negative or positive ends of dipoles that are not neutralized by opposite charges The negative ones are enclosed by the dotted ellipse in the above gure It turns out that the surface charge density is given exactly by 0391 P 19 1 where 13 is the normal unit vector to the surface at each point P is the polarization vector at that point and the subscript b stands for bound meaning that charge can move over distances only on a microscopic scale If the P in a solid is nonuniform then an additional contribution to the net charge can build up owing to a imperfect neutralization of each microscopic dipole Electrostatic theory then shows that Pb V Pe 2 where pb is the bulk charge density In most isotropic solids pb is small because PE is uniform But in highly anisotropic solids particularly noncubic crystals spatial variation of 13 can result in signi cant pb The polarization theorem states that no matter what the shape of a solid sample or its microscopic composition that the electric eld outside the sample can be calcualted just simply from 6b and pb by 1 See any good book on Electrostatics such as P Lorrain and D Corson Electromagnetic Fields and Waves 2nd Ed Freeman and Company San Francisco 1970 ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 E 1 650de prdV 47 r V r 0 S where dS and dV are the surface and volume differentials respectively Geometrical Effects 0 These arise from the existence in all solids of surface charges and the dependence of the electric eld on these charges Exam le from electrostatics the slab O gt surface charge density in surface layer a i E 0 5X top or bottom 8 g 81 P dEp in From Poisson equation or E 80 g dx out 0 Em E0 5xiE 2E 8 response 1 0 80 U Em E0 g 6 lt 0 for E0 as shown negative charge on top 0 Z IEml lt lEol 0 We call E1 the depolarization field i a o a From 1 above lEll E g where Pg gt macroscopic internal polarization vector 0 0 Pe 1 So we have E1 g E 3 N E depolarlzatlon factor l for slab 0 0 ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 Other important geometries o sphere below left and cylinder below right Sphere 3 N 13 for sphere Cylinder E1 S J to z axis 3 N 12 for cylinder 0 Important theorem from electrostatics states that ellipsoids in general have the property that a uniform 13 yields a uniform Em Hence the p P P ElxNx X ElyNy y ElzNz z 50 go 50 subject to the constraint that NX NY NZ l The values of each N map onetoone into the principal axes of the ellipsoid This is an amazing result that requires some higher level math solving Laplaces eqn in ellipsoidal coordinates By so doing another result is found the depolarization field E in an ellipsoid is always uniform when the external eld E0 is uniform However E1 is not necessarily parallel to E0 The same statement can be made about spheroids But E1 becomes parallel to E0 when the ellipsoid degenerates into a sphere Electric susceptibility displacement vector and dielectric constant By definition 13 E ZESOE e m s a a a N13 sothat B21280E0E121280E0 82 0 5 Fundamentals of Solids for Electronics ER Brown Spring 2008 ECEZlSBMaterials206B 132 1 XeN Xe 0E0 13 Xe 0E0 01 e I NXe 0 Another useful macroscopic eld vector particularly when there are free charges present is the displacement vectorD This is well de ned in the following way we consider the general case of a solid that can have signi cant bound charge density pb and sign cant free charge represented by density pf a good example is a heavily doped semiconductor The generalized Gauss equation in this case is given by in V E M pf 0 0 0 where the last step follows from 2 Rearrangement leads to vgog pf 565 Hence we see thatD is related to the spatial variation of the free charge density only 0 The dielectric constant also called the relative permittivity is used to gauge the strength of D relative to Em D 80E 13 era a I f Ie SUEin SUEin SUEin This is de ned so that 8r increases as the response of the solid increases If P is parallel to Em as is usually the case then Sr gt 1 And since E1 is generally opposed to Pg and therefore opposed to E0 this is sometimes used to justify the label dielectric since quotdiaquot means across or opposite in Latin but note more generally the dia in dielectric means that the material opposes or blocks the ow of current Microscopic Effects At the atomic level the electric eld is a superposition of the external eld E0 the response eldE1 due to geometry and surface charges and the eld due to other ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 atomic dipoles in the vicinity While complicated it is the interaction between neighboring dipoles that makes the electrostatics of solids so interesting Elocal Em Edipoles 2 E0 E1 Edipoles Recall that Maxwell s equations including the Poisson equation as a special case are scalable to all dimensions including subatomic The exact expression for the electric eld from a neighboring atomic dipole is thus the same as from a macroscopic dipole a i pry g E V e 5 47580 Unfortunately this is very dif cult to evaluate so it is common in solidstate theory to make approximations The simplest and most common is the spherical atom approximation by which all of the charge comprising the atomic or molecular dipole moment j is assumed to be located inside a sphere for which we already know that the internal electric eld will be uniform if the external eld is uniform We then now think of Elam as the eld at the atomic or molecular site caused by charge from everywhere else in the solid other than the chosen atom or molecule This can be depicted as in Fig l by a virtual elimination of the sphere representing the atom and then calculating the eld note the virtual elimination is done in such a way that the charge distribution on the rest of the solid is not perturbed The calculation of Elam now proceeds by the principle of linear superposition 7 arguably one of the most important principles in all of electromagnetism and all of solidstate theory for that matter By this principle the effect of the charges in the solid without the spherical atom is equivalent to the effect of all the charges in the homogeneous solid minus the effect of the charges inside the spherical atom The rst effect is just Em The second effect represented by Esphere is one of the most important results from electrostatic theory which can be stated two ways depending on the spatial variation of the charge in the spherical atom ECE215BMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 Q G9 gt gt p Eu Ein Q E45 Esphere Fig l 1 If the charge is homogeneous and the atom or molecule is big enough then Esphere 38 identical to the depolarization eld for a macroscopic sphere 0 2 If the charge is inhomogeneous as one would certainly expect in any real atom or 1 molecule we can take a spatial average over the sphere and find Esphere Eave W 0 where R is the radius of the sphere 2 Note that the dipole moment here is generalized from the simple case oftwo displaced and opposite charges It is now given by I p7 F dV where p is the charge distribution and 17 de nes the position where the charge occurs relative to some origin within the sphere But because Vsphm 4311R3 we nd 439 p P Eave m 380V 380 The last step is an approximation because spheres can not ll space This important consequence of the spherical atom approximation is called the Lorentz condition the supersmart guy that Einstein once said he admired most amongst all his colleagues So in total we get the very useful result that connects the microscopic to the macroscopic l E local EinE sphere zEE e 0 I 38 0 2 See practically any undergradute book on Electricity and Magnetism such as P Lorrain and D Corson Electromagnetic Fields and Waves 2nd ed WH Freeman New York 1970 Sec 213 8 ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 Important comments 1 In general El opposes E0 but 136380 supports E Hence Elam is generally larger in magnitude than Em 2 Because atoms or molecules rarely if ever have spherical charge distributions a more general expression is often used a a a a 4 b13 ElocalEmExphm E0E1Se where0ltbltl u Electrostatic Feedback Loop 0 To complete the microscopic formulation we need two more relations P p mm p gt density of atomic or molecular dipoles womm 052E 10ml 0 electric polarizability which assumes a linear response 0 We can now show the macroscopic and microscopic relationships graphically through the quotdielectric feedback loopquot with E0 being the input L L E0 7 rEin macroscopic gt m 4 a patomic local mlcroscoplc 39 39 t quot 39 Formulae ClausiusMosotti Relation We know Elam Em 136380 So we can relate the atomic polarizabilities to the macroscopic susceptibility Pg ij j ijajElocal j j j gt dipole type pj gt density of that dipole type ECE215BMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 Thus Pe ijaj EinIje380 j a 1 4 Solving we get Pg I39EZpaj ijajEin or e 0 j j i ij 180 X E a 3 I 80Equot I 39 Z pjaj 380 This is a very useful connection equation But what most books fail to mention is that this is G exactly the form as the ga1n of a twoport feedback system generally wr1tten as GF m where G XomXin is the openloop ie without feedback gain X is the signal could be current or voltage and H E XFXom is the unitless feedback factor3 In our case G corresponds to 21 80 and H 13 139 Back to solidstate theory we have by definition 8 l Z So 05 S 9 12 ij 10 g1ZpJaJ80 l Fail3106 PEEP0 1 2 8 Algebraic rearrangement yields ZZPJaj g g 21 5 0 p105 5 1 or The famous Claus1sMosott1 relatlon j 380 g 2 Example 1 Dielectric Sphere uniform applied field one dipole type at From depolarization factor for sphere N 13 Elquot E0 e 3 See any good book on Electronic Circuit Theory eg Electronics Circuits and Devices by R Smith Wiley New York 1973 Chap 14 This was Prof Brown s undergraduate text 10 ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 NOTES 2 Electrostatic Behavior of Solids2 M39 39 Models of Polari abilitv Like many other physical models developed for the solid state the ClausiusMosotti formula is to be used with caution because of its simplifying assumptions But it demonstrates certain behavior that are important qualitatively and justify our inspecting the microscopic electrostatic behavior in more detail via the polarizability 0c By including only one dipole type of density n we find a 33 8 1 1 n e 2 This is this can be normalized vs 380n to yield the plot in Fig 1 Clearly 0c goes to zero as 8r goes to unity as expected physically But also note the saturation behavior at large Sr above a certain value of at which is roughly 06 380n in Fig 1 Sr rises very quickly with any further increase in at In other words 8 becomes very sensitive to a In analogy with our feedback model developed for Xe previously and since Sr l Xe we can say that the closed loop macroscopic gain of the system is becoming is very sensitive to the open loop microscopic gain In electronics this is a sign that the system is close to instability As mentioned earlier there are two lndamentally different mechanisms for at at the microscopic level 1 induced dipoles of the individual atoms and 2 reorientation of I Baan ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 existing dipoles The rst is most important in covalently bonded solids The second is most important in ionic solids So not surprisingly the microscopic electrostatic behavior is strongly correlated to the type of molecular bonding going on in the solid Atomic Polarizability A fundamental part of atomic structure is the vastly different spatial distributions for the positive and negative charge as shown in the sketch of Fig 2 The nucleus harbors all the positivecharge protons in a radius of between l and 5 x103915 m depending on the atomic weight The electron orbitals harbor all the negativecharge in a radius of 5x1011 m Bohr radius This big difference in radii allows us to treat the nucleus like a point charge Ze where Z is the atomic number Furthermore if there are enough electrons to fill several different shells in a rst approximation we can treat them all as a cloud of radius R and uniform charge density Ze4311R3 The resulting atomic model is as shown in Fig la Now if an electric eld is applied as in Fig lb the nucleus will be repelled slightly and the electron cloud attracted in such a way that the nucleus will no longer be located at the geometric center of the cloud If we further assume that the uniformity and sphericity of the electron cloud are not perturbed signi cantly then we can show the effect of the eld simply as an offset of the nucleus by a distance r from the center of the spherical cloud Next we invoke yet another powerful result from electrostatic theory whereby the electric eld at a test charge Q1 located on the surface of a sphere containing a uniform charge distribution can be described by only the total charge contained within the sphere Q2 Furthermore the same electric eld is obtained by replacing Q2 with a point charge at the center Associating Q1 with the nucleus and calculating the electronic charge Q inside the inner sphere of radius r in Fig lb we get a local electric eld of Elm Q 41180r2 17 where 17 is the unit vector connecting the Q charge to Q1 Since the cloud density remains uniform and spherical this can be rewritten E local Ze4311r34311R341180r2 ac Zerf4nso R3 2 Now the dipole moment associated with Q1 and Q isjust f7 Q Zer E OLEle Substitution for Zer from 2 leads to the interesting result at 411780 R3 3 ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 Nucleus 39 b a Electron cloud Fig 2 The interesting part is that the atomic polarizability is so strongly dependent on the size of the electronic cloud According to 3 it scales with the atomic volume We should then expect that solids with large atoms might have high polarizability and therefore high 8r A good type of solid to test this thinking is one in which the atoms spread their electronic charge in space for some reason As discussed previously in the section on bonding this happens naturally in stronglycovalent solids since each covalent bond will draw at least one electron away from the atom of reference roughly halfway to the neighboring atom ie half of the nearestneighbor separation In most solids this means that the bonding electron will end up much further away from the reference atom than it would be for the same neutral atom in vacuum This is what we mean by spread In the special case of columnIV elemental semiconductors viz diamond silicon and germanium we have four such covalent bonds between each atom and its nearest neighbors neighbor lying at the vertex of a tetrahedron This fourfold spreading of atomic electrons should create a very large effective radius R of the atomic clouds a large atomic polarizability and a large dielectric constant Table I lists the results for the elemental semiconductors Indeed Si and Ge have surprisingly high dielectric constants 7 a fact that has plagued semiconductor devices and integrated circuit technology forever since high Sr usually means high specific capacitance be it in a transistor or in a transmission line T I 1 Note the interesting MKSA units for on which can also be expressed as Fmz 3 ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 Ionic Polarizability A fundamental aspect of compound materials2 7 the vast majority of solids 7 is some degree of ionicity between nearest neighbor atoms This was discussed earlier in the text and arises naturally from the differences in atomic electron affinity Builtin ions are pervasive in solids but interestingly do not create a builtin nonzero macroscopic polarization 15 unless other conditions are satisfied more on this in the section on ferroelectricity The usual condition is a net cancellation of all the atomic dipoles such that 0 This is true even in highly ionic solids such as the alkali halides NaCl KCl etc Although 0 in the absence of applied eld the builtin ions can create 15 0 when E0 0 through the process of dipole reorientation We need to define reorientation broadly since by definition 13 qd f and 15 n 13 where d is the physical separation between the minus and plus charges of the dipole and f is the unit vector connecting them Hence we can get a change in f5 and thus a nonzero polarizability at from an applied E0 in two ways 1 a change of through angular reorientation of the dipole and 2 a change in d through compression or expansion of molecular bond Many books on dielectrics discuss only the first effect largely because of its predominance in liquids But in solids both effects are important and indeed the second one becomes the predominant one and the simplest to understand in many crystals such as the alkali halides To derive an expression for the ionic microscopic polarizability we imagine a simple twodim square lattice as shown in Fig 3 in which without external forces all ions are separated by the same distance d This can be considered as a cut through one lattice plane of a NaCl crystal one of the cubic facets for example In the absence of external electric field the builtin dipoles along any line of atoms parallel to the applied eld cancel by pairs as shown through the opposing 251 and Z72 atomic dipoles Similarly any chosen dipole in a line of atoms along the applied field will be cancelled by a dipole in the closest parallel line The combination of these two reasons is why ionic crystals have no builtin polarizationl6 no matter how the surfaces are terminated In other words the unit cell primitive and 2 Here compound means that the solid contains at two or more atomic species 4 ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 conventional of an ionic crystal is electrically neutral both in a unipolar ie thal 0 and dipolar F 0 sense In the presence of the external eld E0 all the positive ions are displaced by u in the direction of the eld and all the negative ions are displaced oppositely by the same amount In other words if we choose any atomic ion and consider only its nearestneighbor interactions along the direction of the applied eld each contributes to two equal and opposite dipoles The induced dipole moment from the parallel dipole pair is 1339 171 3392 qd 21456 qd2u5 4 1 39 I 4 where d is the interatomic separation with zero eld If the electric eld is small enough we can assume u ltlt d and then u can be approximated in the linear elastic limit By choosing any ion in the lattice in Fig 3 inspection of the elastic response yields a restoring force Fm Cd2u Cd 2u4Cu 5 where C is the interplanar atomic spring constant Note that the factor of 4 arises because the electrostatic displacement creates cooperative elastic response from the two planes nearest to the plane containing the chosen atom ie if the spring connecting to one nearestneighbor plane is in compression then the spring to the opposite plane is in tension and vice versa In equilibrium this elastic force must balance the electrostatic force quocal ion by ion so that we can write u quocal4C Substitution back into 4 yields 2 13 4qdx q E C low E aEloca 6 Fig 3 Sketch of the electrostatic response of two neighboring rows of atoms in an ionic crystal to an electric eld The vertical dashed lines denote parallel planes which collectively account for all atoms of the crystal see ECE215A Notes XX ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 So we end up with the remarkably simple result at qz C which is intuitively correct in that an increase in C is tantamount to an increase in stiffness and thus a reduced displacement to any applied force Intuitively the ionic charge should occur quadratically once for the dipolar magnitude and once for the unipolar reaction to the applied eld To test this result against experiment we harken back to the analysis of lattice waves for a crystal having a basis of two atoms In this case the interplanar spring constant was essential to the dispersion curves 0 vs k for acoustic and optical lattice waves the acoustical waves being described in the small k limit as 02 m mez 2m1 m2 where a 2d is the crystal period This leads to C m 2m1 m2 tnka2 l2m1mzvsd2 where vs nk is the speed of sound in the smallk limit This leads to the interesting result 2 2d2 a m quot 1 Zmz sz showing once again that the atomic size scale matters As an example we take NaCl along the 100 direction for which q e d a2 563 Ang2 2815 Ang vs CHp 2 4820 ms where the stiffness coef cient C11 4 87XlO10 Nm2 and the density p 2096 KGm3 3 m1 22mp atomic weight of Na and m 34mp atomic weight of Cl with mp 167X1039 27 KG the proton mass The result show for HW problem is ac 187X103939 Cbmz V We focus only on nearestneighbor builtin dipoles The clue to understanding the result harkens back to our analysis of lattice waves and phonons If we assume the crystal is perfect the displacement shown in Fig 3 will be uniform over the entire solid sample Since different atoms in a crystal form all or part of the basis of a nonBravais lattice the displacement shown in the gure must be occurring within a primitive cell of the lattice I Therefore because the eld is static this J39 J 39 is to a 39 quot optical LO lattice wave of in nite wavelength ie k 0 3 See Kittel Introduction to Solid State Physics 73911 Ed Chap 3 Eqn 6 6 ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 Classi cation of Dielectric Types 0 The quot of r39 to r39 also allows us to classify macroscopic effects based on microscopic quantities In addition to classical dielectric response of induced dipoles already described we also have paraelectric and ferroelectric To separate out shape depolarization effects the classification is based on the relationship of Pg to En ie on Xe mammal relation of P6 to En Xe Normal dielectric Parallel Em lt E0 gt 0 induced dipoles in electrical insulator Paraelectric Parallel Em lt E0 gt 0 permanent dipoles Ferroelectric Independent singular 13 gtspontaneous Metallic Parallel Unde ned4 since Xe E good electrical conductor PesoEin and E gt 0 0 To quantify these relationships better we assume the Lorentz relation again t t 13 a N a a N a 8 melm aEm Pe3go 3 Pe anpj2njaj Em 3g J J 0 zna a a j J J m P PE m and solv1ng for e we get 1 njaj 380 Now the definitions becomes more obvious Zn 1380 Metallic Effectively gtgtl owing to the macroscopic 4 A metallic sample will tend to screen out the internal electric field We will show later in transport theory how 8 is large and negative in metals so Xe must do that too ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 Important comments 0 One must be careful with this simple picture because solids usually have several dipole types each having different 09 Because ferroelectric state is singular ie Xe diverges it is very sensitive to other macroscopic variables such as temperature and stress Normal dielectric induced dipoles and paraelectric permanent dipoles are similar in that for both types the solid responds in such a way that the electric polarization Pg increases with applied field 0 Types of ferroelectrics to be J in lecture I order disorder eg KDP KHzPO4 TgtTg T ltTs e ca ii i 99 e G G 2 displacement eg Perovskites BaTi03 LiTi03 most important to engineering C 6 69 6 TgtTc TltTc 669 666 999 999 eg Perovskites BaTi03 LlTlO3 0 Because of the critical dependence of PC on the density and polarizability of atomic dipoles many of the thermodynamic variables are also critical For example PC is observed to depend critically on T ECE215BMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 P6 E0 0 lst order phase transition characteristic of orderdisorder ferroelectrics Tc T Pe 2nd order hase transition P characteristic of displacement ferroelectrics Tc T Charateristics of phase transitions from Thermodynamic theory 1st order 2nd order 1 Entropy discontinuous across TC because of l Entropy continuous acrossTc latent heat no latent heat 2 Derivatives of entropy are continuous 2 Derivatives of entropy are eg heat capacity singular 3 Other thermodynamic derivatives are 3 Other thermodynamic continuous derivatives are singular 1st order derivatives 2 Dielectric susceptibility C Heat capacity 0 Many of the common ferroelectrics such as LiNbO3 and other Perovskites display these characteristics The sensitive nature of Thermodynamic variables leads to many useful effects in ferroelectric solids ECE215BMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 g 6 K gt pyroelectric coef cient Pyroelectric Effect Room temperature infrared detectors Tc T e I P Piezoelectric constant 1 Piezoelectric effect acoustic transducers electromechanical transducers Important comments about ferroelectrics At T gt Tc the ferroelectric goes into a paraelecm39c state whereby there are still permanent dipoles in the solid but they all cancel in the absence of an external electric eld But the electric susceptibility can still be very high in magnitude because we still have 71ij 38 0 1 But note that in this state Xe is still positive J Clari cation of Energy Density Earlier in discussions of dielectrics we had stated we Em d13 V 80Ein2du potential energy from dipoles potential energy of eld alone In the simple case of a parallel plate capacitor lled with dielectric material this is easy to evaluate for energy UEin since for dielectric P XZSOE we nd 2 m ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 Magnetic Effects in Solids 1 Diamagnetism Magnetic effects in solids go back in history far before electric effects back to the ancient civilizations The reason for this is simple ferromagnetism the magnetic analogue of ferroelectricity 7 is naturally occurring particularly in a iron ore called magnetite and is very strong Whereas two ferroelectric samples placed sidebyside will have a very weak interactive force two ferromagnetic samples can interact so strongly as to y together or apart depending on their orientation One of our goals in covering ferromagnetic effects is to explain this fascinating phenomena 7 a phenomena that was the basis for the first recorded electromagnetic device in history 7 the magnetic compass We will see that ferromagnetism is so strong because of a hidden nonclassical force 7 the quantummechanical interaction between particle spins Unfortunately for ferroelectrics no such interaction occurs so that they are stuck with the relatively weak classical interaction between electric dipoles Another important distinction between magnetic and electric effects is the nature of the fundamental particles In electric effects there exists monopoles ie negative electrons and positive protons and dipoles positivenegative pair This creates the important distinction between conductors dominated by monopoles and insulators dominated by dipoles in response to externally applied electric fields In magnetic effects there are only dipoles1 But unlike electric effects the dipoles originate from two fundamentallydifferent sources 1 electrons in orbital motion and 2 particle spin While both are similar forms of angular momentum they have radically different interactive strengths and therefore lead to different macroscopic effects Specifically electronic orbital motion leads to true diamagnetism unpaired spins usually lead to paramagnetism and under special conditions to be defined later create ferromagnetism As in insulating electric materials the fundamental quantity in magnetic quantities is the atomic magnetic dipole m The definition of m and the other important quantities in magnetic solids follow by analogy with electric quantities i5 6139 d r71 139 21 charge separation current area E gt E magnetic induction 132 gt A71 magnetization WhereAj n 71 71 l Fig 1 Classical magnetic dipole with area A and current i number per unit volume 21 gt area vector points along perpendicular direction to current loop according to righthand rule 1 The search for magnetic monopoles has proceeded offandon for over a century One of the more recent endeavors was conducted by Prof L Alvarez of of the Physics Dept at UC Berkeley while the author was an undergraduate there in the early 19705 Unfortunately all such endeavors have failed 1 ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 Z Z t Hm I dM M M ZMB M e m ma ne IC susce 1 1 1 I m a a 01 g p y 6H 0 dB m m in most materials 0 the exception being ferromagnets m gt 0 with permanent magnetic dipoles 3 paramagnetic m gtgt 1 with ordered permanent dipoles 3 ferromagnetic m lt 0 diamagnetic Recall that in electrostatic phenomenology there was no true diaelectric response dielectric meant simply that the solid opposed the ow of electric current But in magnetic phenomena there is a true dia response meaning that the internal magnetization is opposite in direction to the internal magnetic eld Hence there is a distinct possibility of negative magnetic susceptibility M Magnetic P6 Em M I Bin macroscopic l lt l l gt I I P p 06E local m Blow mlcroscoplc m 053 And as in electrostatics we have a builtin feedback loop see diagrams above between the microscopic and the macroscopic levels In other words there is a local magnetic induction that the atomic magnetic dipoles sense and that can be different than the macroscopic induction Bin inside the solid 1 Sample Geometrical Effects As in the depolarization effect of electrostatics there is a screening of the internal magnetic induction owing to dipoles current loops that build up on the surface of the sample These surface current always tend to reduce the internal induction to a degree that ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 depends on the shape of the sample If we write 31 as the response induction created by the surface currents then the mascroscopic induction inside the solid is given by BinZBOBIZB0NOM a B 0 t Fig 2 Relative orientation of response induction B1 and external induction B0 for a simple ellipsoidal geometric shape N for sphere N 1 for slab where N is the depolarization factor 2 Microscopic Effects Magnetostatic analysis shows that M Note39 is in numerator Blow Bin 0 This is the magnetic analog of the Lorentz condition for dielectric and again only approximate In general Blow Bin l lt I lt1 3 The 39 r39 quot to the r39 is again just the sum over atomic dipoles M nrn namglwd nam 31quot byOAZ analogous to 13 n naZElml where 0cm is the magnetic polarizability So we can write M l nambu0 namgln not 3 not M m m and Zm E mILlO 1 mmbtuo 1 mmbtuo As in the calculation of the electric polarizability for atomic ionic and ferroelectric responses the calculation of 0cm is a great exercise in microscopic physics We will do it separately for the diamagnetic paramagnetic and ferromagnetic responses ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 Classical model of atomic 139 The much greater mobility of electrons compared to protons creates a generic magnetic response analogous to the atomic polarizability derived for electrostatics We start by assuming that the electrons execute circular orbits around a fixed nucleus To maintain the orbits there must be a centripetal inward radial force F mgv2 mg R R 27m fm2 mngZR where frm is the rotational frequency For such an orbit without there is a builtin magnetic moment along the z axis of a iA2e 7zR22 27139 Fig 3 This assumes that the orbit is occurring in the equatorial plane as shown in Fig 3 moving with negative helicity with respect to the velocity vector anal the z axis2 But as in the case of electrostatics an analysis of real atoms must start with a three dimensional distribution of charge not just an orbit The simplest model assumes a spherical charge distribution that can be built up from circular orbits by a taking a vector sum of a very large number of such orbits in random orientation This can be expressed mathematically by pointing each area vector along the radial vector Zl A r so that r71 i A r recall that in spherical coordinates r by itself can be pointing in any direction until 9 and I are speci ed We get the total magnetic moment by summing over 9 in spherical coordinates at an arbitrary value of I 10 with equal weighting at all angles 2 Ir 2 in JiArd dqp JiAchin cosqpo J3sin6sin 0 2cos6a 6 0 3 0 tot 0 2 Helicity pertains to the direction of rotation of a physical quantity with respect to a chosen axis Righthand circular RHC means that when the thumb of the right hand points along the chosen axis the fingers wrap along the rotation of the chosen variable So while Fig 3 above is RHC with respect to the z axis and electrical current it is lefthand circular LHC with respect to the z axis and electron velocity 4 ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 Important points to achieve a spherical distribution starting with Z one orbit we integrate over 9 from 0 to 211 we do not need to 9 B integrate over I because this is taken care of by the rotating 3 current in the orbit F1g 4 From the zero result there is no builtin magnetic dipole from a rotating spherical charge distribution This is intuitively obvious With an external magnetic eld applied the situation changes There is now aLorentz force on the electron that is always given by F e3 X E Without loss of generality we can start with a current loop that lies in the equitorial plane of a spherical coordinate system such that Zl is along the z axis as shown in Fig 4 The applied B must be along some direction E BO 94 such that it can be always be decomposed into a z component B2 and an equitorial component Bp Clearly the equitorial component will generate no net Lorentz force around an orbit since at any point on the circle the Lorentz force is exactly cancelled in amplitude and direction by the Lorentz force on the diametricallyopposed side of the circle But the B2 component will generate an orbitconstant force a A a A A F evsz e27rR2 BZp eRa Bcost9p 7239 where lt gt again denotes angular averaging and B l B l and is the radial vector lying in the equitorial plane and unspeci ed in direction same as radial vector in cylindrical coordinates Note that this force is centrifugal ie directed outward from the center of rotation if 0 lt 9 lt n2 or 311 2 lt 9 lt 211 but is centripetal otherwise 3 Using the useful identity from vector spherical coordinates f fcsin 9 cos 0 9 sin 9 sin 0 2 cos 9 5 ECE215BMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 To proceed further we make the educated assumption that the Lorentz force does not change the orbit radius4 And since the mass must stay xed then the Lorentz force can be counterbalanced by a change of the always centrifugal force of orbital motion only if there is a change in orbital frequency We denote this by the change in orbital frequency from no to 030 A03 for which the equitorialplane force counterbalance condition becomes me 00 Am2R memozR eRmOB cos6 If A03 ltlt 030 we can drop terms of order A002 and obtain eaOB cos 9 m 2m2Aa a20 B 9 or M m 2m 63 where 2 1s called the Larmor frequency 7 a pervas1ve quant1ty 1n magnetlc calculatlons and m magneticbased systems and derived before the development of quantum theory For example it is a common quantity in nuclear magnetic resonance imaging MRI Of great interest is the new magnetic moment if any in the presence of B To calculate this we must as with zero B form a sphericallysymmetric charge distribution by rotating the orbit in Fig 4 But common sense dictates that this is equivalent to keeping the orbit xed and rotating B over all possible orientations via the random expression 3 BO This is a convenient trick because we already know that the Lorentz force only depends on B2 as Bocose where again 9 is the angle betweenlgS andgl So the total magnetic dipole moment is It ram J12eaR2f sin 6d6 0 4 An assumption justified by the shell model of electronic orbitals via the Pauli exclusion principle 6 ECE215BMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 This is slightly different than the case for B 0 because the orbit is staying xed and the magnetic induction is being rotated down at an arbitrary value of I 10 Hence we need to multiply by sine the Jacobian in spherical coordinates This leads to rhtot Jl2ea0 Am R20 sin 6 cos 110 3 sin 6 sin 00 2 cos 6 sin 6d6 0 Ir 0 Jl2e eBO 2mecos6R2 sin Gcosqpo fsin 6sin 00 Ecos sin 6d6 0 7r 7r 3 I from integral tables we have Isin cosz 95m co3 6 and Ismz 69059616 0 0 0 2 2 a B 2 A B A So we get ltmgte0RZZe oRzz 4m 3 6mg The fact that this is nonzero and negative means that the response of the spherical current orbit is truly dia magnetic Note that the direction of the magnetic moment is exactly along the z axis This is because of the trick we invoked of xing the orbit and rotating B By deduction if we had kept the magnetic induction xed and rotated the orbit to all possible orientations then the magnetic moment would have ended up pointing along B This an important bit of deduction for all magnetic eld effects with atoms Since there is an arbitrariness in the orientation of the electron orbits in any isolated atom related to the pointlike nature of the nucleus then the external magnetic induction removes this arbitrariness and de nes a preferred direction in space By convention the direction of E is usually set along the 2 axis in spherical coordinates This allows us to determine the electronic magnetic polarizability as ECE215BMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 71 e2R2 am E 6mg an expression that makes it clear that the MKSA units for 0cm are Cbzmz KG It should also be compared to the classical expression we previously derived for the atomic electric polarizability Xe 41180R3 As in the electrostatic case size matters although not as much for diamagnetism as for the electric case The most important difference is the sign Our analysis predicts that all atoms contribute a diamagnetic effect ie xm negative in contrast to the paraelectric effect ie Xe positive they display from electrostatics Re nements to Diamagnetic Model 1 It is easy to deduce that we should get one unit of this polarizability for each electron orbit in an atom of radius R So we can generalize to get the total atomic polarizabilty cm 11 N ltrhgt N w b am 0 E y t Z 4mg Knowing the polarizability we can find go back to the macroscopic level and find the magnetic susceptibility N 2 2 nu e E 6m Wm o 0 E111 l na b N 39 0 lnbyoezz Rf 6m il Zm where n is the density of atoms in the solid Example Silicon l4 electrons per atom 71 8613 8543 gtlt103910 m3 5 X IONm3 0 47239 gtlt10397MKSA From the section on atomic polarizability we have for silicon an average radius of the electron cloud of ltpgt N 15 A 3 ocmm
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