Solid State Physics - Problems and Solutions, Provas de Física

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SOLID STATE PHYSICS Problems and Solutions Lászió Mihály Michael C. Martin State University of New York at Stony Brook A WILEY-INTERSCIENCE PUBLICATION JOHN WILEY & SONS, INC. New'York e Chichester * Brisbane * Toronto + Singapore This texr is printed on acid-free paper. Copyright O 1996 by John Wiley & Sons, Inc. All rights reserved. Published simultaneously in Canada. Reproduction or translation of any part of this work beyond that permitted by Section 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. Requests for permission or further information should he addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012 Library of Congress Cataloging in Publication Dara: Mihály, Lászió, 1949- Solid state physics : problems and solutions / László Mihály, Michael C. Martin p. cm. includes bibliographical references and index. ISBN 0-471-15287-0 (paper : alk. paper) |. Sold state physics—Problems, exercises, etc. 1. Martin, Michael €. [1 Title. QCI76.515.M54 1996 530.4'1'076-—de20 96-6184 Printed in the United States of America 1098765432 vi Contents 2.11 Problem: Grineisen Parameter 22 2.12 Problem: Diatomic Chain 22 2.13 Problem: Damped Oscillation 22 2.14 Problem: Two-Dimensional Debye 23 2.15 Problem: Soft Optical Phonons 23 2.16 Problem: Soft Phonons Again 23 3 Electronic Band Structure 25 3.1 Problem: Nearly Free Electrons in One Dimension 30 3.2 Problem: Nearly Free Electrons in Dirac-Delta Potentials 30 3.3 Problem: Tight-Binding in Dirac-Delta Potentials 30 34 Problem: Dirac-Delta Potentials 31 3.5 Problem: Band Overlap 31 3.6 Problem: Nearly Free Electrons in Two Dimensions 31 3.7 Problem: Nearly Free Electron Bands 32 3.8 Problem: Instability at the Fermi Wavenumber 32 3.9 Problem: Electrons in 2D Nearly Free Electron Band 32 3.10 Problem: Square Lattice 32 3.11 Problem: Tight-Binding Band in Two Dimensions 33 3.12 Problem: Electrons in 2D Tight-Binding Band 33 3.13 Problem: Dirac-Delta Potentials in Two Dimensions 33 3.15 Problem: Cyclotron Frequency 34 3.16 Problem: deHaas-Van Alphen 34 3.17-Problem: Fermi Energy —34 4 Density of States 35 4.1 Problem: Density of States 57 42 Problem: Two-Dimensional Density of States 38 4.2 Problem: Two-Dimensional Tight Binding 38 4.4 Problem: Quasi-One-Dimensional Metal 38 4.5 Problem: Crossover to Quasi-One-Dimensional Metal 39 4.6 Problem: Phonon Mode of Two-Dimensional System 39 4.7 Problem: Saddle Point 40 4.8 Problem: Density of States in Superconductors 40 4.9 Problem: Energy Gap 41 4.10 Problem: Density of States for Hybridized Bands 41 4.1] Problem: Infinite-Dimensional DOS 41 4.12 Problem: Two-Dimensional Electron Gas 42 5 Elementary Excitations 43 5.1 Problem: Tight-Binding Model 46 5.2 Problem: Hybridization of Energy Bands 46 5.3 Problem: Polarons 47 Contents vii 5.4 Problem: Polaritons 47 5.5 Problem: Excitons 48 5.6 Problem: Holstein-Primakoff Transformation 48 5.7 Problem: Dyson-Maleev Representation 48 5.8 Problem: Spin Waves 49 5.9 Problem: Spin Waves Again 49 5.10 Problem: Anisotropic Heisenberg Model 49 5.11 Problem: Solitons 50 6 Thermodynamies of Noninteracting Quasiparticles 53 6.1 Problem: Specific Heat of Metals and Insulators 60 62 Problem: NumberofPhonons 60 6.3 Problem: Energy ofthe Phonon Gas 60 6.4 Problem: Bulk Modulus of Phonon Gas 60 6.5 Problem: Phonons in One Dimension 60 6.6 Problem: Electron-Hole Symmetry 61 6.7 Problem: Entropy of the Noninteracting Electron Gas 61 6.8 Problem: Free Energy with Gap at the Fermi Energy 61 6.9 Problem: Bulk Modulus at T=0 62 6.10 Problem: Temperature Dependence of the Bulk Modulus 62 6.11 Problem: Chemical Potential of the Free-Electron Gas 62 6.12 Problem: Ful Specific Heat 62 6.13 Problem: Magnetization at Low Temperatures 62 6.14 Problem: Electronic Specific Heat 63 6:15 Problem: Quantum Hall Effect 63 7 Transport Properties 6s 7.1 Problem: Temperature Dependent Resistance 72 7.2 Problem: Conductivity Tensor 72 7.3 Problem: Montgomery Method 73 74 Problem: Anisotropic Layer 73 7.5 Problem: Two-Charge-Carrier Drude Model 74 7.6 Problem: Thermal Conductivity 74 7.7 Problem: Residual Resistivity 74 7.8 Problem: Electric and Heat Transport 75 7.9 Problem: Conductivity of Tight-Binding Band 75 7.10 Problem: Hall Effect in Two-Dimensional Metals 75 7.11 Problem: Free-Electron Results from the Boltzmann Equations 76 7.12 Problem: p-n Junctions 76 8 Optical Properties 79 8.1 Problem: Fourier Transform Infrared Spectroscopy 86 8.2 Problem: Optical Mode of KBr 87 Contenis ix 1.16 Hint: Vacancies 121 1.17 Hint: Integrated Scattering Intensity 122 2 Interatomic Forces, Lattice Vibrations 123 2.1 Solution: Madelung Constant 123 2.2 Solution: NaCl Bulk Modulus 125 2.3 Hint: Madelung with Screened Potential 127 2.4 Solution: Triple-axis Spectrometer 127 2.5 Solution: Phonons in Silicon 127 2.6 Solution: Linear Array of Emitters, Phonons 128 27 Hint: Long Range Interaction 128 2.8 Solution: Mass Defect 129 29 Solution: Debye Frequency 130 2.10 Solution: Vibrations ofa Square Lattice 131 2.11 Solution: Griineisen Parameter 133 212 Solution: Diatomic Chain 134 2.13 Hint: Damped Oscillation 135 2.14 Solution: Two-Dimensional Debye 136 2.15 Solution: Soft Optical Phonons 13 2.16 Hint: Soft Phonons Again 139 3-EI icBandS 3.1 Solution: Nearly Free Electrons in One Dimension 14] 3.2 Solution: Nearly Free Electrons in Dirac-Delta Potentials 141 3.3 Solution: Tight-Binding in Dirac-Delta Potentials 142 3.4 Solution: Dirac-Delta Potentials 143 3.5 Solution: Band Overlap 147 3.6 Solution: Nearly Free Electrons in Two Dimensions 147 3.7 Solution: Nearly Free Electron Bands 149 3.8 Solution: Instability at the Fermi Wavenumber 150 3.9 Solution: Electrons in 2D Nearly Free Electron Band 152 3.10 Solution: Square Lattice 153 3.11 Solution: Tight-Binding Band in Two Dimensions 154 3.12 Solution: Electrons in 2D Tight-Binding Band 156 3.13 Solution: Dirac-Delta Potentials in Two Dimensions 156 3.14 Hint: Effective Mass 157 3.15 Hint: Cyclotron Frequency 158 3.16 Solution: deHaas-Van Alphen 159 3.17 Hint: Fermi Energy 159 4 Density of States 161 4.1 Solution: Density of States 161 42 Solution: Two-Dimensional Density of States 161 x Contents 4.3 Solution: Two-Dimensional Tight-Binding 162 44 Solution: Quasi-One-Dimensional Metal 163 4.5 Solution: Crossover to Quasi-One-Dimensional Metal 165 4.6 Solution: Phonon Mode of Two-Dimensional System 167 4.7 Solution: Saddle Point 168 4.8 Solution: Density of States in Superconductors 170 4.9 Solution: Energy Gap 171 4.10 Solution: Density of States for Hybridized Bands 17] 4.11 Solution: Infinite-Dimensional DOS 172 4.12 Solution: Two-Dimensional Electron Gas 173 5 Elementary Excitations 175 5.1 Solution: Tight-Binding Model 175 5.2 Solution: Hybridization of Energy Bands 177 5.3 Solution: Polarons 179 5.4 Solution: Polaritons 181 5.5 Hint: Excitons 182 5.6 Solution: Holstein-Primakov Transformation 182 5.7 Hint: Dyson-Maleev Representation 185 5.8 Solutions: Spin Waves 183 5.9 Solution: Spin Waves Again 184 5.11 Hint: Solitons 186 Th jynamíes of Noni ing Quasiparticl 187 6.1 Solution: Specific Heat of Metals and Insulators 187 6.2 Solution: Number of Phonons 188 6.3 Solution: Energy of the Phonon Gas [89 6.4 Solution: Bulk Modulus of Phonon Gas 190 6.5 Hint: Phonons in One Dimension 191 6.6 Solution: Electron-Hole Symmetry 191 6.7 Solution: Entropy of the Noninteracting Electron Gas 193 6.8 Solution: Free Energy with Gap at the Fermi Energy 194 6.9 Solution: Bulk Modulusat T=0 195 6.10 Solution: Temperature Dependence of the Bulk Modulus 195 6.11 Solution: Chemical Potential of the Free-Electron Gas 197 6.12 Solution: EuO Specific Heat 198 6.13 Hint: Magnetization at Low Temperatures 199 6.14 Solution: Electronic Specific Heat 200 6.15 Solution: Quantum Hall Effect 201 7 Transport Properties 203 7] Solution: Temperature Dependent Resistance 203 7.2 Solution: Conductivity Tensor 204 Preface A number of years ago, one of us had the opportunity to attend lectures by a partic- ularly talented and dedicated high-school science teacher, Miklós Vermes. The lec- tures were full of interesting and entertaining demonstrations, and he told the audi- ence à lot about how the basic laws of physics fit together into a logical structure. Yet, when Professor Vermes was asked about what is the really important part in learning physics, he answered: “Solve homework problems. That way, you will get used to the stu , In this spirit, we hand this book to advanced undergraduate and introductory graduate students in Condensed Matter Physics. “Get used to it” When you are used for the trees more clearly. We wrote this book to satisfy another need as well: to help measure the progress must have a a set tof carefully guarded problems, appropriate to the final exam of the course she/he is teaching (better yet to have several sets, if you are teaching in sev- — eral consecutive semesters). One would like to share and discuss these—presum- ably interesting —physics questions with the students, but practical considerations do not allow for this; 1f a problem is given out as a homework assignment, its value as a final exam problem is greatly and understandably reduced. Our goal here was to provide a volume of problems greater in number than the “critical mass” number which can be memorized. If a student remembers al! of the solutions for a suffi- ciently large set of problems, he or she pretty much knows the subject (for proof, see the argument in the first paragraph). The problems and solutions presented in this book stem from several years of teaching advanced undergraduate and introductory graduate solid state physics courses in the Physics Department of SUNY at Stony Brook. During these courses we used several of the excellent textbooks available; some of them are listed as ref- erences to the present collection of problems. Naturally, the problems were devel- oped and organized each year, more or less in accord with the textbook used in that particular course. As we began to assemble the present collection, we planned to di- vide the problems into chapters corresponding to one of the standard organizations of the subject matter. However, we discovered that we do not need to be tied by the same constraints as the typical introductory textbook: A particular aspect of super- conductivity may be effectively used to illustrate the concept of density of states; magnetism-and charge density waves fit reasonably well under the umbrella of in- teracting electron systems. PARTI Problems 4 1. Crystal Structures even if there is only one type of atom in the crystal (see the example in Figure 11.9). The Bravais lattice, or space lattice, is an infinite array of points, deter- mined by the lattice vectors R, where R = ma + n242 + n343 such that every n; is an integer. The q's are the three primitive vectors of the Bravais lattice; in three dimensions they must have a nonzero q, - (42 x «3) product. There are an infinite number of different choices for the primitive vectors of a given lattice. For example, a, = q) +02;0)=G/— q2; and q; = «3 will describe the same lattice. The lattice spacings are the lengths of the shortest possible set of primitive vectors. All three crystals in Figure 1.1.1 have the same Bravais lattice. Note that 11.2 shows a honeycomb lattice and a choice for its Bravais lattice and basis. o... “e... s hd X X x XxX e... Basis: o & 6 é é “0 0“ “e. .. X X x X . ..- Ss. . .... ese. o... Bravais | & ss. q e... Lattice: . ss. . X XxX XxX X .... +. 6 Fig. 1.1.2. Example of a regular array of points that is not a Bravais lattice (hon- eycomb lattice). In addition to the translational Symmetry, most crystals also have other symmetries, including reflection, rotation, or inversion symmetry, or more complicated symmetry operations, like the combination of rotation and trans- lation by a fraction of the lattice vector. In Figure 1.1.1 the three-fold rota- tional symmetry around the point P is common to all three crystals. The honeycomb lattice (Figure 1.1.1c) also has a “mirror line” m, while the other two crystals do not have this symmetry. À less trivial symmetry operation is mirroring the honeycomb lattice with respect of the line m”, and then shifting it parallel to the line, until it overlaps with itself. The collection of symmetry operations forms a symmetry group. The im- portant property that defines a symmetry group is the relationship between the symmetry elements — i.e., what happens if two symmetry operations are applied subsequently. In the language of group theory, this relationship is described by the multiplication (direct product) table.? The symmetry group can be represented in many ways (collections of matrices, symmetry opera- 2 An introduction to basic group theory is given by Yu and Cardona [6] pp. 21-43 or Harrison [5] pp. 16-20. 1. Crystal Structures 5 tions of a simple geometric object, and so on). As long as the multiplication table is the same, we are dealing with the same group. The crystals in Fig- ures Ll.1a and 1.1.1b have equivalent symmetry groups, while some of the symmetries of the honeycomb lattice are different. When all possible symmetry operations are taken into account we talk about crystallographic space groups. Any given three-dimensional crystal belongs to one of the 230 possible crystallographic space groups. (Two- dimensional crystals are much simpler; there are only 17 inequivalent “crystal- lographic plane groups”.) The symmetries are often identified by the name of a representative material, like “sodium chloride structure”, “diamond struc- ture”, “wurzite (or zincblende, zinc sulfide) structure”, and so on. More so- complex structure the identification of the symmetry group may be a rather nontrivial task. A subset of symmetry operations that leaves at least one point invariant makes up the crystallographic point group. There are 32 different crystallo- graphie point groups in three dimensions, and 10 in two dimensions. Consid- ering the examples in Figure 1.1.1, the rotations around P and the mirror line m are point group symmetries, but the combination of mirroring around m! and the subsequent shift is not a point group operation. Sorting out the symmetries of the Bravais lattices is much simpler. There are 14 different space groups for three-dimensional Bravais lattices, including the simple cubic (sc), face centered cubic (fcc), body centered cubic (bcc), simple tetragonal, body centered tetragonal, and others. Figure 1.1.3 shows all i vais ices in two di ions. E isi i the symmetries of the Bravais lattice are intimately related to the symmetries of the original lattice. For example, the three-fold rotational symmetry of the have three-fold rotational invariance (which leaves the hexagonal lattice as the only choice, see Figure 1.1.2). Finally, when the point group symmetries of the Bravais lattices are con- sidered, the choices are further limited, and in three dimensions only seven distinct groups are left. These define the seven crystal systems: Cubic, tetrag- onal, orthorhombic, monoclinie, triclinie, trigonal, and hexagonal. (In two di- mensions there are four crystal systems. The rectangular and centered rect- angular Bravais lattices shown in Figure 1.1.3 make up the “rectangular” system.) The primitive unit cell or primitive cellis a volume which will fill space completely, without overlap, if shifted by each of the lattice vectors. The primitive unit cell contains exactly one Bravais lattice point and the atoms in it can be used as the basis to construct the crystal. The volume made up by the primitive vectors is a possible primitive unit cell, but there are 3 See Ashcroft and Mermin [1] pp. 122-126. 1. Crystal Structures 7 wavelength, d is the spacing between subsequent lattice planes (that is, planes containing a high density of lattice points) and 9 is the angle between the incident beam and the lattice planes. The scattered beam will have the same angle with respect to the planes as the incident beam, so the total scattering angle is 20. Instead of wavelength, the concept of the wavevector is often used to characterize the a plane wave. The wavevector k points in the direction of the propagation of the wave, and the magnitude of the vector is |k| = 27/A. The condition for constructive interference can be expressed in terms of wave vectors as (k — k')-d = 2mn, where n is an integer, k and k' are the incident and scattered wavevectors, and d is the vector pointing from one scattering center to another. The reciprocal lattice is a very useful tool to handle the diffraction of waves; it is generally used to describe all things of “wavy nature” (like elec- trons and lattice vibrations). Definitions of the reciprocal lattice are as fol- lows: eriod of — The collection of all wave vectors that yield plane waves with a p lattice). — A collection of vectors G satisfying G-R=27n,or ciSR =1. — There is al ical definition in tl % a ga= mta ae , (11.1) and eyclic permutations of 12 3, where V = a, - (42 x a3) is the volume of the unit cell. 3 The crystal system of the reciprocal lattice is the same as the direct lattice (for example, cubic remains cubic), but the Bravais lattice may be different (e.g., fcc becomes bcc). The Brillouin. zone is the WS cell in the reciprocal lattice. Using the reciprocal lattice, the condition for constructive interference becomes quite simple: If the difference between the incident (k) and scattered (k') wave vectors is equal to a reciprocal lattice vector, the diffracted intensity may be nonzero. This is the Lave condition. With K = k' — k, this leads to the simple equation K = G,or K-G = 1/2]G|?. The Ewald construction is a geometric representation of these equations. The Miller indices, A, k, and I, are obtained from the “coordinates” of a reciprocal lattice vector G = hg, + kg» +lg3. By definition, the Miller indices are integers. For a simple cubic lattice these numbers are real coordinates in a Cartesian coordinate system. There is an interesting relationship between Miller indices and lattice planes. For any plane there is an infinite number of other, parallel lattice planes, separated by a distance d. It is easy to see that the ratio 7:y: 2 is the 8 1. Crystal Structures same for all parallel planes, where x, y, and z are the intercepts of a given Plane with the coordinate axis defined by the primitive vectors a1, q2, Ga. Sometimes the need arises to classify these planes. There is a conve- nient mapping between a given class of lattice planes and a lattice vector in reciprocal space: For any family of lattice planes separated by a dis- tance d, there is a reciprocal vector with length |G| = 2m/d, and this vec- tor is perpendicular to the lattice planes. One can show (nontrivially) that hikil= (1/2): (1/9): (1/2). The Laue condition is based solely on the Bravais lattice, so the positions of the diffraction peaks are independent of the atomie basis. However, the intensities of the peaks are strongly influenced by the basis. The structure actor, 5 and th t depend on the atoms making up the crystals. These quantities are calculated as a sum (or integral) within the unit cell; therefore they may be totally different for two different crystals, even if the crystals have the same Bravais lattice. In the simplest approximation the scattering depends on the atomic charge distribution pa(r), and the intensity is proportional to the absolute value squared of HG =), fas To (11.2) and falG) = 5 | polrje Star, (113) where e is the electron charge, the sum is over the atoms in the unit cell, for electron and neutron scattering, except the form factor integral is differ- ent depending on the microscopic interaction at Play. Even for X- “Tay, the only for the simplest cases. . For a realistic calculation of scattered radiation intensities one has to include factors representing the directional dependence of the scattering by a single point charge, the absorption of the radiation, and other effects. For powder samples this process is called Rietveld analysis. The expression becomes much more complicated for example, if there is a match between the energy of atomic transitions and the X-ray quanta. When the atomic positions are time-dependent (for example, if lattice waves are excited in the crystal), the crystal scatters radiation at a frequency different from the incident frequency. In this case energy is either absorbed or emitted by the crystal, The process can be described by the dynamic structure factor, which depends also on the frequency difference w between the incident and scattered radiation: S = S(k,w). The general expression Su) = | SS, O)p(-k, 0), (11.4) relates the structure factor to (o(k,0)p(—k, t)), the density-density correla- tion function. (Here N is the number of primitive unit cells and p(k,t) is