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by: Miss Alberto Prohaska
Miss Alberto Prohaska
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R. Seshadri

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R. Seshadri
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This 22 page Class Notes was uploaded by Miss Alberto Prohaska on Thursday October 22, 2015. The Class Notes belongs to MATRL 218 at University of California Santa Barbara taught by R. Seshadri in Fall. Since its upload, it has received 36 views. For similar materials see /class/226926/matrl-218-university-of-california-santa-barbara in Material Science and Engineering at University of California Santa Barbara.

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Date Created: 10/22/15
Materials 218UCSB Class VIII More crystal structures A303 perovskite A3204 spinel A2304 K2NIF4 Ram Seshadri seshadrimrlucsbedu AB03 perovskite BaZr03 A Ba7 B Zr SG ngm Not 221 a 4194 A Atom Wyckoff z y 2 W Zr 1a M 3d is 500 050 007 ABe BZr 0 20 2312 octehedre Most perovskite structures are not cubicl BaZrOg7 BaSnOg7 BaHng are a few examples of the cubic ones In the cubic perovskite7 we have TA To Vi2M TB To IBIO 20 2 12 m Wm M m mAEO yew51m 5 dmd x Farmxdadmmymsln gl mymmmcmwmmmmm Amwmsmmmm 112103 Wm mmxsgmhm Caho sazmmu za53lkb Am WyM t 3 x a 44 um new w 39n u m u m Ab mm 7mm w 02 34 um mm mm musmmammmwmuw mmmmmdnmdz vgn Tmmmya mrsmaa m Ammuwly an Pemslnte mm mm seshadrimrlucsbedu MATERIALS 218CHEMISTRY 277 ELECTRONIC STRUCTURE II MATRL 218CHEM 277 Class 9 More on electronic structure Ram Seshadri seshadriCerlucsbedu o In traditional solid state physics treatments Kittel we are taught that band gaps arise due to translational periodicity that there is Bragg re ection of free electrons at the edges of the Brillouin zone and this open up gaps In 1D the free electron wavefunction is was expwe I These are plane wave solutions to the free electron Schrodinger equation The energy and momentum are given by 2 6k k2 phk Instead of a free electron in 1D consider a 1D lattice with lattice constant a The Bragg condition for diffraction by waves of wavevector k is k G2 k2 where G is the reciprocal lattice vector and 27rna for the 1D lattice where n is some integer Therefore the Bragg condition solves to k i G inTra The first re ections and therefore the first energy gaps occur at k iTra B B k 4113 113 k On the left is a sketch of the free electron wave function and on the right is a sketch of the effect of imposing a lattice on the free electron wave function with lattice parameter a1 At the k points k iTra an energy gap A opens up The second allowed band starts after the first one gives up The second band is in the second Brillouin zone 0 Such a description is limited in its applicability What about a defect a surface or an amorphous material that would not have translational periodicity and therefore would not have Bragg re ection of electrons 392 We know that such materials do have band gaps window glass The solution is to look at realespace pictures and return to tightebinding models For the real space description consider two energy levels on two different orbitals A and B When the orbitals are far apart the energy levels are at the atomic limit When they approach bonding and an tibonding combinations form splitting the two levels The bonding energy level is the bottom of that particular DOS and the antibonding level the top 1This is referred to as the nearly free electron model Note its resemblance to the kinds of dispersion relations we derived for s orbital bands seshadrimrlucsbedu MATERIALS 218CHEMISTRY 277 ELECTRONIC STRUCTURE II energy 03 7 V antibonding mu m L bonding inverse distance Notice how the levels broaden Somewhere inibetween there is some dispersion of the individual states but there is still a gap When the atoms approach very close the gap vanishes Such a picture is physically quite realistic Many semiconductors which have a gap become metallic on being subject to hydrostatic pressure 0 When states in a crystal are lled up the rules are the same as what is required for lling up atomic orbital states Start with the lowest energies and pay heed to Pauli s exclusion principle The exclusion principle says that no band can have more than two electrons insulator semi metal metal Electrons in lled bands do not carry a current because they cannot move without violating Pauli s exclui sion principle If they do move their motion must be compensated by the motion of a hole in the same direction or of an electron in the opposite direction 0 The Peierls distortion 1D Lattices with 1 Zi lled and indeed ln lled bands susceptible to distort in a special way that permits opening of a gap at the Fermi energy This is called the Peierls distortion Polyacetylene is a typical 1D system that undergoes such a distortion as rst suggested by Salem and LonguetiHiggins The simple tightibinding picture of such a gap opening up can be obtained by considering crystal orbitals formed from a 1D lattice of s orbitals At the center of the band the number of bonds equals the number of antibonds Two kinds of crystal orbitals can be envisioned which are degenerate in energy seshadrimrlucsbedu MATERIALS 218CHEMISTRY 277 ELECTRONIC STRUCTURE II If the lattice is alternately contracted and expanded as shown using the blue and red arrows the cell parameter becomes 2a instead of a This means that the X point is now at k 7r2a The doubling of the cell in real space corresponds to halving in k space Also the bands are no longer disperse The scheme below is the opening of a gap as a result fold back halve distort The dotted line is the Fermi energy The Peierls distortion is an example of symmetryebreaking lifting a degeneracy The JahneTeller distortion in d orbital solids is another such example 0 Some examples of real electronic structures Cu metal 10 Energy eV 10 l l 2 4 DOS states eVquot cellquot seshadrimrlucsbedu MATERIALS 218CHEMISTRY 277 ELECTRONIC STRUCTURE II Energy eV 0 T 7ltgt 10 L F XW L K o C graphite W TR 01 T i E e 6 0 a a E LJ LJ 5 S f T T T 10 F K M FA L H A F K H A 0 Screening in metals In a good metal the density of electrons is very high 7 of the order of 1022 cm g These electrons should repel strongly We recollect that hard spheres crystallize and that repulsive interactions are sufficient for this It is therefore surprising that electrons do not crystallize in metals If they did crystallize they would stop moving and the metal would no longer conduct The answer to this is that paradoxically when the concentration of electrons is high the electrons form a sea of negative charge that actually prevent one electron from seeing another The formula for such screening in a crystal is the soicalled ThomasiFermi formula for the screened Coulomb potential KittelZ Tom gawk where q is the charge and r is the distance The wavevector k5 defines the screening length lks k5 is a function of the DOS at the Fermi energy k 47reQDEF 2Note the similarity with the repulsive part of the DLVO potential for colloids 39 in h edu MATERIALS ZlSCHEMISTRY 227 COLOSSAL MAGNETORESISTANCE MATRL 218CHEM 227 Class Xlll Colossal Magnetoresistance Ram Seshadri seshadrimrlucsbedu o The simplest example of magnetoresistance is transverse magnetoresistance associated with the Hall ef fect Ex jx When an electrical conductor is subject to an electric field Ex along x and simultaneously a magnetic field H along 2 a new transverse field arises due to the Lorentz force of H on the electrons moving along x This field Ey acts along y because the Lorentz force is a cross product giving rise to the Hall voltage There is a magnetoresistance associated with the transverse field Such magnetoresistance measurements allow the number of free carriers to be obtained experimentally as well as allow mapping of the Fermi surface etc 0 Negative magnetoresistance is the term given to the large decrease in the electrical resistance when cer tain systems are exposed to a magnetic field The negative magnetoresistance is usually defined as a percentage ratio MR 7 x100 where pH is the resistivity in the presence of a magnetic field of strength H and 30 is the resistivity in the absence of a magnetic field 39 in h edu MATERIALS ZlSCHEMISTRY 227 COLOSSAL MAGNETORESISTANCE o The term Giant Magnetoresistance GMR has come to be associated with certain thin metallic multilayer devices that are commonly used in the readheads of magnetic hard disks 0 1 O 1 O N ferromagnetic free spacer ferromagnetic pinned antiferromagnetic pinning layer unmagnetized magnetized O 1 O 1 O N ferromagnetic free spacer ferromagnetic pinned antiferromagnetic pinning layer When the head the free ferromagnetic layer runs over a magnetized region on the hard disk the mag netization of the ma netic layer is aligned with the magnetization of the pinned magnetic layer This reduces the electrical resistivity of the assembly 0 Many ferromagnetic elements display an intrinsic negative magnetoresistance in the vicinity of their ferromagnetic transitions This is because in the vicinity of the ferromagnetic transition conduction elec trons are scattered by magnetic uctuations Switching on a magnetic field supresses such uctuations and this results in a reduction of such scattering and consequently a reduction in the electrical resistivity p lt9 cmgt 39 in h edu MATERIALS ZlSCHEMISTRY 227 COLOSSAL MAGNETORESISTANCE o In 1950 Ionker and van Santen in the Netherlands found that the perovskite LaMnOs which is an an tiferromagnetic insulator becomes metallic when La is substituted by Sr in La1xerMnOg when x is around 03 the system becomes displays an insulatormetal transition on cooling Concurrently at the same temperature as the metalinsulator transition the system becomes ferromagnetic 3 S2 cm Tc Memu g 1 T T K At TC the system becomes metallic indicated by the change of temperature coefficient of resistance TCR from negative to positive and ferromagnetic indicated by a sharp rise in the magnetization The reson why this happens is called Zener Double Exchange DEX o The electronic configuration of d4 Mn in LaMnOs is octahedral and high spin with a Iahn Teller distor f J ahane11er distorted lt 2 yz 7L 52 dzl dwdyz 5quot elongated o The substitution of 30 Sr in the La site results in a change in the oxidation state according to 39 in h edu MATERIALS Z18CHEMISTRY 227 COLOSSAL MAGNETORESISTANCE Lafor Mn IIMn IVOs with x 03 The removal of electrons from trivalent d4 Mn results in a loss of most of the Iahn Teller nature the elongation of 2 Mn O bonds in the MnO6 octahedra and the possibility that the 88 electron on one MnaH can hop to its neighboring MnaV l l Mn II Mnav A 7L7 if The important ingredient in the Zener DEX mechanism is that the electron retains a memory of its spin when it hops from one site to the next Such hopping is therefore favored in the ferromagnetic state This is why La07Sr03Mn03 displays metallic behavior and ferromagnetism at the same time What is the mechanism for the colossal magnetoresistance Near the magnetic transition when the spins are tending to line up switching on a magnetic field helps align neighboring spins Hopping from MnaH to neighboring Mnav is therefore facilitated In certain materials the application of a 7 T magnetic field can result in a 13 order of magnitude decrease in the electrical resistivity turning wood into silver 0 Issues in colossal magnetoresistance CMR manganese oxides The effect of oxidation state in Ln1xAan03 where Ln is a trivalent rareearth Ln La Pr Nd and A is an alkaline earth metal A Ca Sr Ba The most studied compositions are x N 03 when about onethird of the Mn are in the 4 oxidation state The effect of the perovskite tolerance factor When the average size on the A site the weighted average of Ln1xAx is small t is small and the Mn OMn bond angle deviates greatly from 180 This results in the electrical resistivity being greater and the transition from insulator to metal taking place at lower temperatures l wols 3r MnOMn near 180 MnOMn less than 180 in h edu MATERIALS ZlSCHEMISTRY 227 COLOSSAL MAGNETORESISTANCE The reduction of the Mn OMn bond angle because of tilting of octahedra a consequence of small t results in poorer overlap between orbitals the figure shows dxziyz on Mn and px on O This makes the system a poorer metal TA 13 1 20 115 1 130 400 l I I A07A3903Mn03 300 l La 7ca i Lamsro s Lag 7Bao a 35 PM FMM v 200 B warming Pro vcaon OPEHJFUM 100 closed 391 J A Laxyh 7caoa u LaPro7caos 0 Lau7CaSrua 0 Lao7SFrB u3 0 I 0 l I 89 090 091 092 093 094 09 Tolerance factor From H Y Huang et ll Phys Rev Lett 75 1995 914 This plot shows the temperature at which the different compounds become ferromagnetic as a function of the average size of the A cation The tolerance factor t and the average 9coordinate ShannonPrewitt radii form the different ordinate x axes Note how the highest transition temperature is found for a specific tolerance factor PMI paramagnetic insulator FMM ferromagnetic metal FMI ferromagnetic insulator Charge ordering when x 05 In compositions such as La05Ca05Mn03 when the amounts of Mn111 and MnIV are equal there is the possibility of forming a crystalline ordering of two different kinds of octahedra The material becomes an antiferromagnetic insulator at a certain temperature called the Chargeeordering temperature Such a phase transition is also called Wigner crystallization or the Verweij transition 39 uc h edu MATERIALS ZlSCHEMISTRY 227 CRYSTALS GLASSES ETC MATRL 218CHEM 227 Class II Classification of materials as amor phous and crystalline and the structural hierarchy in a polycrystalline ma terial Ram Seshadri seshadrimrlucsbedu o Molecules vs extended solids Molecules are finite and have a distinct identity whether solid liquid or gas eg the methane molecule Extended solids are defined by the phase When they melt or vaporize they are no longer the same material eg graphite A The methane molecule left and a portion of a graphite crystal right In the graphite crys tal two sheets are displayed The actual crystal has effectively an infinite number of sheets stacked one upon another and the sheets in addition extend infinitely in the plane 0 Extended solids whether crystalline or amorphous can be classsified according to their bonding 7 whether it is directional as in graphite or diamond or not as in most inorganic materials that we shall discuss in this course A caveat Many solids for example silicate glasses and minerals are somewhere inbetween Solids with directional bonding typically have atoms with low coordination numbers CN number of neighbors Polymers are extended directional solids in 1D 0 Examples of nondirectional solids iNaCl left and CsCl right Under pressure NaCl in the NaCl structure transforms to the CsCl structure going from CN 6 to CN 8 7 nondirectional solids are not terribly particular about CN 39 in h edu MATERIALS ZlSCHEMISTRY 227 CRYSTALS GLASSES ETC o The formation of a crystalline solid 7 One can either cool across vaporsolid or liquidsolid coexistence curves solid arrows or one can crush dotted arrows across them T In all cases the phase transition is strongly rsteorder and is characterized by symmetry breaking Re member that gases and liquids have higher symmetry than do crystals characterized by the ability of the atoms to translate continuously This is lost in the crystal 0 In crystals atoms are ordered at the microscopic level This microscopic order manifests at a macro scopic level i as sharp edges and regular and constant angles between faces of a crystal The external morphology of crystals allowed 19th century crystallographers to guess the nature of atomic ordering in simple crystals in NaCl for example well before the advent of Xrays o The work of Alder and Wainright on hard sphere computer simulations demonstrated that purely re pulsive interactions are sufficient for crystals to form the Alder transition The formation of crystals is entropy stabilized See the note Insights into phase transition kinetics from colloid science V I Ander son and H N W Lekkerkerker Nature 416 2002 811815 DOI o The formation of a glass cannot be described using an equilibrium phase diagram Glasses form when a liquid is cooled sufficiently rapidly that crystallization is avoided gas molar volume crystal glass formation Please also see Allen and Thomas The Structure of Materials MIT Press 39 in h edu MATERIALS ZlSCHEMISTRY 227 CRYSTALS GLASSES ETC o A liquid can therefore solidify in two ways discontinuously into a crystal continuously into a glass Schematic arrangements of atoms in a planar crystal left and glass right Note that the coordination numbers of the atoms CN 3 remains the same 0 The tendency to form glasses see D R Nelson and F Spaepen Polytetruhedrul Order in Condensed Matter an article in the series Solid State Physics Academic Press San Diego 42 1989 190 In 2D close packing of 3 discs results in triangles Triangles can tile the plane 7 two dimensions are not frustrated In 3D close packing of 3 spheres gives a triangle The fourth sphere gives a tetrahedron Tetrahedra cannot be used to tile 3D space1 7 three dimensional space is frustrated and glasses form easily 0 Glass formers are of two kinds Systems with low coordination numbers such as planar borates and tetrahedral silicates that form network glasses 7 this is the continuous random network model Look at the handout Systems with high coordination number such as metals that are unable to find the potential energy minumum corresponding to the crystal Such glasses were first found in AuSi alloys See Spaepen and Nelson to find out more 0 The hierarchy of structure in a polycrystalline material Atoms come together in a crystal the basic builiding block being the unit cell 1 Many unit cells make the crystal 2 Depending on how the unit cells are put together the crystal s habit can change 3 The polycrystalline material comprises of many crystals separated by grain boundaries 4 Crystals can themselves have defects 5 turn 1Just as pentagons cannot le a at surface Matenals 21sucsa Class v Packings CCP and HCP voids radlus ratio rules the structures 039 elements mp0 re u Mg 5 diamond c graphite mmsesm seshadrl ml unshadu mmylme 9mm mmmmmae mnemdammmem sweeten mwmemymmmWWmmmmmmmtm mmmmh mmmm messes mspkmrmyaudmy mmmt see Wt mmmmmm ee Wt The Meta aammmmue 1M WWWEWsmaww Mahcuah Materials 218UCSB Class XV Polar Materials Ram Seshadri seshadrimrlucsbedu o Crystals comprising cations and anions can be classi ed into four types7 according to their polar behaV1or cation Oanion applied eld 4 T gt TC T lt Tc T lt Tc roelectric py Q Q O O O I O I O O 0 O no on on I O O O O O O 0 O antipolar m an ferroelecuiciiiii m 60 O O I o Piezoelectric materials There is coupling between electrical and mechanical energies For example7 an applied stress results in the generation of polarization 0 Pyroelectric materials A material With a temperature dependent polarization This requires a unique polar axis 0 Ferroelectrics A subgroup of pyroelectric materials in Which the spontaneous polarization can be reori ented between equilibrium states by applying an electric eld All ferroelectrics are both pyroelectric and piezoelectric o The possibility of inorganic crystals being polar pyroelectric or piezoelectric is strictly a function of their point group symmetry 32 crystallographic point groups 11 c emmsymmemc mm mm nu o The ferroelectric phase transition for example7 in Ple03 is characterized by the development of a spontaneous zero eld polarization7 changes in the dielectric constant7 and crystal structural changes P Cm2 T K a T K cell parameters T K 0 At the origins of such a phase transition are developments of a dipole Within the unit cell7 due to the centers of positive and negative charges not coinciding PbTiO3 Pmim PbTiO3 P4mm ME 0 In the particular case of PbTiOg7 the phase transition from the cubic7 paraelectric phase to the tetrago nal ferroelectric phase a displacive phase transition is associated With the freezing of a phonon mode that is referred to as the soft mode h Phonon modes are speci c vibrational modes of the crystal lattice In the case of PbTiOg7 this phonon mode is associated With the Ti atom in the center of the octahedron hm meV frozen Ti vibrating Ti lt UK The frequency energy of the soft mode goes to zero as the phase transition is approached o Ferroelectric materials are also characterized by hysteresis of the polarization below the ferroelectric To just as are ferromagnets are characterized by a hysteresis of the magnetization P 0 As is true for ferromagnets7 the hysteretic behavior is a consequence of the presence of domains in the materia i 0 Some materials undergo an orderdisorder phase transition from the paraelectric phase to the ferro electric phasei An example is NaNOg sodium nitrite Room temperature ferroelectric structure of NaNOg projected on 001 The rigid NO groups have been shown as little cheVrons7 and the NaJr ions as circles Hatching indicates that the atoms are at a height of 12 Above the phase transition at 438 K7 the structure is nonpolar and has the Immm space group Below this temperature7 the material is ferroelectric and has the space group ImZm the ferroelectric structure is displayed The dipole moment is lost in the high temperature structure because of disorder Half the cheVrons point to the left and half to the right7 and correspondingly the Na ions also occupy two different sites at randomi o Antiferroelectrics These are usually characterized by antiparallel dipole moments in the unit cellr The formation of antiparallel moments as in antiferromagnetic systems results in the formation of larger unit cel sr cation Q anion 4 a parameter 0 O O O Q paraelectric Q 0 O Q Q 4 aparameter O Q 0 G O ferroelectric Q 0 D O O a parameter Q Q Q Q antiferroelectric 0 O Q G 0 An example of an antiferroelectric is PbZrOg7 Which is cubic ngm7 paraelectric above 503 Kr Below this temperature7 a combination of two effects7 the tendency of the ZrOg octahedra to tilt as a consequence of the tolerance factor being less than 17 and the tendency of the Pb2 ions to go offcenter7 result in the antiferroelectric7 orthorhombic crystal structure In this depiction of the crystal structure of PbZrOg7 the lone pairs77 on the Pb atoms are Visualized using DFT calculations of the electronic structure Note that the lone pairs77 creates a structural distortion that cancels itself o What are lone pairs 7 The three ions Tl Pb2 and Bi3 have the electronic con guration Xel4f145d10632l The pair of s electrons in the valence shell the lone pair should have spherical symmetry if they retain pure 8 character However they like to mix With anion oxygen p orbitals and this mixing results in their going off center h Such offcentering is a good recipe for obtaining polar behaviorl Sketch displaying the tendency ofa lone pair ion blue such as Pb2 to go offcenter in its coordination polyhedron leaving in the center of the polyhedron the 32 lone pair Such a sketch is quite accurate for describing the crystal structure of PbO in the litharge modi cationl Antiferroelectrics do not display hysteresis except at high elds Where they develop separate loops at positive and negative elds A similar dependence of magnetization on the magnetic elds is seen in metamagnetsl o Ordered double perovskites elpasolites and relaxor ferroelectrics Ordered double perovskites have the general formula AQBBOgl One kind of octahedra in the structure is B05 and the other is B Ogl For the octahedra to be distinct meaning that there is no mixing of B and B in the lattice sites of the structure there must usually be a large difference in size and charge between B and BC For example in the ordered double perovskite BagMgWOg there is no mixing of Mg2 and W6 on a single lattice site Instead of the ngm space group of simple ideal perovskites Which have cell parameters of the order of 4 A ideal double perovskites crystallize in the Fmgm space group With a cell parameter around 8 Indeed the two kinds of octahedra in the elpasolite structure are arranged as in the rock salt structure The yellow squares indicate the unit cell 0 Many elpasolitelike compositions With Pb in the Asite display disorder of some sort because the ordering between the B and B sites is not complete Many of these disordered materials display relaxm behaviori For example PbQMgQng4SOB PMN PngchOg PSN etc In PMN and PSN there are two different kinds of octahedra in the structure one that is Nb rich and the other that is Nb poori Such disorder seems to be important for observing relaxer behavior Which is characterized by 7 Broad phase transitions 7 Dispersion characteristics of the dielectric constant The e 7 T behavior is frequency dependent Er UK Relaxors are useful since they have a high dielectric constance even at radio frequencies and they saturate at low voltagesi Because of the broad phase transition the T dependence of 67 does not create problems in devices In addition the hysteresis is small the energy densities are high and the materials are eld tunable


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