Lecture Notes On Condensed Matter Physics (A Work In
Lecture Notes on Condensed Matter Physics(A Work in Progress)Daniel ArovasDepartment of PhysicsUniversity of California, San DiegoMarch 14, 2010
Contents0.1Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .0 Introductory Information0.1vi1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 Boltzmann Transport251.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .51.2Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .51.3Boltzmann Equation in Solids . . . . . . . . . . . . . . . . . . . . . . . . .61.3.1Semiclassical Dynamics and Distribution Functions . . . . . . . . .61.3.2Local Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . .10Conductivity of Normal Metals . . . . . . . . . . . . . . . . . . . . . . . . .111.4.1Relaxation Time Approximation . . . . . . . . . . . . . . . . . . . .111.4.2Optical Reflectivity of Metals and Semiconductors . . . . . . . . . .141.4.3Optical Conductivity of Semiconductors . . . . . . . . . . . . . . .161.4.4Optical Conductivity and the Fermi Surface . . . . . . . . . . . . .18Calculation of the Scattering Time . . . . . . . . . . . . . . . . . . . . . . .191.5.1Potential Scattering and Fermi’s Golden Rule . . . . . . . . . . . .191.5.2Screening and the Transport Lifetime . . . . . . . . . . . . . . . . .23Boltzmann Equation for Holes . . . . . . . . . . . . . . . . . . . . . . . . .251.6.1Properties of Holes . . . . . . . . . . . . . . . . . . . . . . . . . . .25Magnetoresistance and Hall Effect . . . . . . . . . . . . . . . . . . . . . . .281.41.51.61.7i
iiCONTENTS1.81.91.7.1Boltzmann Theory for ραβ (ω, B) . . . . . . . . . . . . . . . . . . .281.7.2Cyclotron Resonance in Semiconductors . . . . . . . . . . . . . . .311.7.3Magnetoresistance: Two-Band Model . . . . . . . . . . . . . . . . .321.7.4Hall Effect in High Fields . . . . . . . . . . . . . . . . . . . . . . . .34Thermal Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .361.8.1Boltzmann Theory . . . . . . . . . . . . . . . . . . . . . . . . . . .361.8.2The Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . . .401.8.3Calculation of Transport Coefficients . . . . . . . . . . . . . . . . .411.8.4Onsager Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . .43Electron-Phonon Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . .451.9.1Introductory Remarks. . . . . . . . . . . . . . . . . . . . . . . . .451.9.2Electron-Phonon Interaction . . . . . . . . . . . . . . . . . . . . . .451.9.3Boltzmann Equation for Electron-Phonon Scattering . . . . . . . .482 Mesoscopia512.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .512.2Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .512.3The Landauer Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . .512.3.1Example: Potential Step . . . . . . . . . . . . . . . . . . . . . . . .54Multichannel Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .562.4.1Transfer Matrices: The Pichard Formula . . . . . . . . . . . . . . .602.4.2Discussion of the Pichard Formula . . . . . . . . . . . . . . . . . . .622.4.3Two Quantum Resistors in Series . . . . . . . . . . . . . . . . . . .642.4.4Two Quantum Resistors in Parallel . . . . . . . . . . . . . . . . . .67Universal Conductance Fluctuations in Dirty Metals . . . . . . . . . . . . .752.5.1Weak Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . .78Anderson Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .802.6.1Characterization of Localized and Extended States . . . . . . . . .822.6.2Numerical Studies of the Localization Transition . . . . . . . . . . .832.42.52.6
CONTENTSiii2.6.3Scaling Theory of Localization . . . . . . . . . . . . . . . . . . . . .852.6.4Finite Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . .893 Linear Response Theory3.191Response and Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . .913.1.1Energy Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . .933.2Kramers-Kronig Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . .933.3Quantum Mechanical Response Functions . . . . . . . . . . . . . . . . . . .953.3.1Spectral Representation . . . . . . . . . . . . . . . . . . . . . . . . .973.3.2Energy Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . .993.3.3Correlation Functions . . . . . . . . . . . . . . . . . . . . . . . . . .1003.3.4Continuous Systems . . . . . . . . . . . . . . . . . . . . . . . . . . .1013.4Example: S Object in a Magnetic Field . . . . . . . . . . . . . . . . .101Bloch Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .102Electromagnetic Response . . . . . . . . . . . . . . . . . . . . . . . . . . . .1043.5.1Gauge Invariance and Charge Conservation . . . . . . . . . . . . . .1063.5.2A Sum Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1063.5.3Longitudinal and Transverse Response . . . . . . . . . . . . . . . .1073.5.4Neutral Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1073.5.5The Meissner Effect and Superfluid Density . . . . . . . . . . . . .108Density-Density Correlations . . . . . . . . . . . . . . . . . . . . . . . . . .1103.6.1Sum Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .112Dynamic Structure Factor for the Electron Gas . . . . . . . . . . . . . . . .1133.7.1Explicit T 0 Calculation . . . . . . . . . . . . . . . . . . . . . . .114Charged Systems: Screening and Dielectric Response . . . . . . . . . . . .1193.8.1Definition of the Charge Response Functions . . . . . . . . . . . . .1193.8.2Static Screening: Thomas-Fermi Approximation . . . . . . . . . . .1203.8.3High Frequency Behavior of (q, ω) . . . . . . . . . . . . . . . . . .1213.8.4Random Phase Approximation (RPA) . . . . . . . . . . . . . . . . .1223.4.13.53.63.73.812
ivCONTENTS3.8.5Plasmons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4 Magnetism1251274.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1274.2Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1274.2.1Absence of Orbital Magnetism within Classical Physics . . . . . . .129Basic Atomic Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1294.3.1Single electron Hamiltonian . . . . . . . . . . . . . . . . . . . . . .1294.3.2The Darwin Term . . . . . . . . . . . . . . . . . . . . . . . . . . . .1304.3.3Many electron Hamiltonian . . . . . . . . . . . . . . . . . . . . . . .1304.3.4The Periodic Table . . . . . . . . . . . . . . . . . . . . . . . . . . .1324.3.5Splitting of Configurations: Hund’s Rules . . . . . . . . . . . . . . .1334.3.6Spin-Orbit Interaction . . . . . . . . . . . . . . . . . . . . . . . . .1354.3.7Crystal Field Splittings . . . . . . . . . . . . . . . . . . . . . . . . .136Magnetic Susceptibility of Atomic and Ionic Systems . . . . . . . . . . . . .1374.4.1Filled Shells: Larmor Diamagnetism . . . . . . . . . . . . . . . . . .1384.4.2Partially Filled Shells: van Vleck Paramagnetism . . . . . . . . . .139Itinerant Magnetism of Noninteracting Systems . . . . . . . . . . . . . . . .1414.5.1Pauli Paramagnetism . . . . . . . . . . . . . . . . . . . . . . . . . .1414.5.2Landau Diamagnetism . . . . . . . . . . . . . . . . . . . . . . . . .143Moment Formation in Interacting Itinerant Systems . . . . . . . . . . . . .1454.6.1The Hubbard Model . . . . . . . . . . . . . . . . . . . . . . . . . .1454.6.2Stoner Mean Field Theory . . . . . . . . . . . . . . . . . . . . . . .1464.6.3Antiferromagnetic Solution . . . . . . . . . . . . . . . . . . . . . . .1504.6.4Mean Field Phase Diagram of the Hubbard Model . . . . . . . . . .1514.34.44.54.64.7Interaction of Local Moments: the Heisenberg Model. . . . . . . . . . . .1534.7.1Ferromagnetic Exchange of Orthogonal Orbitals . . . . . . . . . . .1534.7.2Heitler-London Theory of the H2 Molecule . . . . . . . . . . . . . .1554.7.3Failure of Heitler-London Theory . . . . . . . . . . . . . . . . . . .157
CONTENTS4.7.4vHerring’s approach . . . . . . . . . . . . . . . . . . . . . . . . . . .157Mean Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1584.8.1Ferromagnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1614.8.2Antiferromagnets . . . . . . . . . . . . . . . . . . . . . . . . . . . .1614.8.3Susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1624.8.4Variational Probability Distribution . . . . . . . . . . . . . . . . . .163Magnetic Ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1664.9.1Mean Field Theory of Anisotropic Magnetic Systems . . . . . . . .1684.9.2Quasi-1D Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . .1694.10 Spin Wave Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1704.84.94.10.1Ferromagnetic Spin Waves . . . . . . . . . . . . . . . . . . . . . . .1714.10.2Static Correlations in the Ferromagnet . . . . . . . . . . . . . . . .1734.10.3Antiferromagnetic Spin Waves . . . . . . . . . . . . . . . . . . . . .1734.10.4Specific Heat due to Spin Waves . . . . . . . . . . . . . . . . . . . .178
vi0.1CONTENTSPrefaceThis is a proto-preface. A more complete preface will be written after these notes arecompleted.These lecture notes are intended to supplement a graduate level course in condensed matterphysics.
Chapter 0Introductory InformationInstructor:Contact :Lectures:Office Hours:Daniel ArovasMayer Hall 5671 / 534-6323 / darovas@ucsd.eduTu Th / 9:30 am - 10:50 am / Mayer Hall 5301W 2:00 pm - 3:30 pm / Mayer Hall 5671A strong emphasis of this class will be on learning how to calculate. I plan to cover thefollowing topics this quarter:Transport: Boltzmann equation, transport coefficients, cyclotron resonance, magnetoresistance, thermal transport, electron-phonon scatteringMesoscopic Physics: Landauer formula, conductance fluctuations, Aharonov-Bohm effect, disorder, weak localization, Anderson localizationMagnetism: Weak vs. strong, local vs. itinerant, Hubbard and Heisenberg models, spinwave theory, magnetic ordering, Kondo effectOther: Linear response theory, Fermi liquid theory (time permitting)There will be about four assignments and a take-home final examination. I will be followingmy own notes, which are available from the course web site.1
2CHAPTER 0. INTRODUCTORY INFORMATION0.1References D. Feng and G. Jin, Introduction to Condensed Matter Physics (I)(World Scientific, Singapore, 2005)New and with a distinctly modern flavor and set of topics. Looks good. N, Ashcroft and N. D. Mermin, Solid State Physics(Saunders College Press, Philadelphia, 1976)Beautifully written, this classic text is still one of the best comprehensive guides. M. Marder, Condensed Matter Physics(John Wiley & Sons, New York, 2000)A thorough and advanced level treatment of transport theory in gases, metals, semiconductors, insulators, and superconductors. D. Pines, Elementary Excitations in Solids(Perseus, New York, 1999)An advanced level text on the quantum theory of solids, treating phonons, electrons,plasmons, and photons. P. L. Taylor and O. Heinonen, A Quantum Approach to Condensed Matter Physics(Cambridge University Press, New York, 2002)A modern, intermediate level treatment of the quantum theory of solids. J. M. Ziman, Principles of the Theory of Solids(Cambridge University Press, New York, 1979).A classic text on solid state physics. Very readable.
0.1. REFERENCES3 C. Kittel, Quantum Theory of Solids(John Wiley & Sons, New York, 1963)A graduate level text with several detailed derivations. H. Smith and H. H. Jensen, Transport Phenomena(Oxford University Press, New York, 1989).A detailed and lucid account of transport theory in gases, liquids, and solids, bothclassical and quantum. J. Imry, Introduction to Mesoscopic Physics(Oxford University Press, New York, 1997) D. Ferry and S. M. Goodnick, Transport in Nanostructures(Cambdridge University Press, New York, 1999) S. Datta, Electronic Transport in Mesoscopic Systems(Cambridge University Press, New York, 1997) M. Janssen, Fluctuations and Localization(World Scientific, Singapore, 2001) A. Auerbach, Interacting Electrons and Quantum Magnetism(Springer-Verlag, New York, 1994) N. Spaldin, Magnetic Materials(Cambridge University Press, New York, 2003) A. C. Hewson, The Kondo Problem to Heavy Fermions(Springer-Verlag, New York, 2001)
4CHAPTER 0. INTRODUCTORY INFORMATION
Chapter 1Boltzmann Transport1.1References H. Smith and H. H. Jensen, Transport Phenomena N. W. Ashcroft and N. D. Mermin, Solid State Physics, chapter 13. P. L. Taylor and O. Heinonen, Condensed Matter Physics, chapter 8. J. M. Ziman, Principles of the Theory of Solids, chapter 7.1.2IntroductionTransport is the phenomenon of currents flowing in response to applied fields. By ‘current’we generally mean an electrical current j, or thermal current jq . By ‘applied field’ wegenerally mean an electric field E or a temperature gradient T . The currents and fieldsare linearly related, and it will be our goal to calculate the coefficients (known as transportcoefficients) of these linear relations. Implicit in our discussion is the assumption that weare always dealing with systems near equilibrium.5
6CHAPTER 1. BOLTZMANN TRANSPORT1.31.3.1Boltzmann Equation in SolidsSemiclassical Dynamics and Distribution FunctionsThe semiclassical dynamics of a wavepacket in a solid are described by the equationsdrdt vn (k) 1 εn (k) k(1.1)dkee E(r, t) vn (k) B(r, t) .(1.2)dt cHere, n is the band index and εn (k) is the dispersion relation for band n. The wavevectoris k ( k is the ‘crystal momentum’), and εn (k) is periodic under k k G, where G isany reciprocal lattice vector. These formulae are valid only at sufficiently weak fields. Theyneglect, for example, Zener tunneling processes in which an electron may change its bandindex as it traverses the Brillouin zone. We also neglect the spin-orbit interaction in ourdiscussion.We are of course interested in more than just a single electron, hence to that end let usconsider the distribution function fn (r, k, t), defined such that1fnσ (r, k, t)# of electrons of spin σ in band n with positions withind3r d3k 3d3r of r and wavevectors within d3k of k at time t.(2π)(1.3)Note that the distribution function is dimensionless. By performing integrals over thedistribution function, we can obtain various physical quantities. For example, the currentdensity at r is given byX Z d3kj(r, t) efnσ (r, k, t) vn (k) .(1.4)(2π)3n,σΩ̂The symbol Ω̂ in the above formula is to remind us that the wavevector integral is performedonly over the first Brillouin zone.We now ask how the distribution functions fnσ (r, k, t) evolve in time. To simplify matters,we will consider a single band and drop the indices nσ. It is clear that in the absence ofcollisions, the distribution function must satisfy the continuity equation, f · (uf ) 0 .(1.5) tThis is just the condition of number conservation for electrons. Take care to note that and u are six -dimensional phase space vectors:u ( ẋ , ẏ , ż , k̇x , k̇y , k̇z ) , , ,,,. x y z kx ky kz1(1.6)(1.7)We will assume three space dimensions. The discussion may be generalized to quasi-two dimensionaland quasi-one dimensional systems as well.
1.3. BOLTZMANN EQUATION IN SOLIDS7Now note that as a consequence of the dynamics (1.1,1.2) that · u 0, i.e. phase spaceflow is incompressible, provided that ε(k) is a function of k alone, and not of r. Thus, inthe absence of collisions, we have f u · f 0 . t(1.8)The differential operator Dt t u · is sometimes called the ‘convective derivative’.EXERCISE: Show that · u 0.Next we must consider the effect of collisions, which are not accounted for by the semiclassical dynamics. In a collision process, an electron with wavevector k and one withwavevector k0 can instantaneously convert into a pair with wavevectors k q and k0 q(modulo a reciprocal lattice vector G), where q is the wavevector transfer. Note that thetotal wavevector is preserved (mod G). This means that Dt f 6 0. Rather, we should write f f f ṙ · k̇ · t r k f t Ik {f }(1.9)collwhere the right side is known as the collision integral . The collision integral is in generala function of r, k, and t and a functional of the distribution f . As the k-dependence isthe most important for our concerns, we will write Ik in order to make this dependenceexplicit. Some examples should help clarify the situation.First, let’s consider a very simple model of the collision integral,Ik {f } f (r, k, t) f 0 (r, k).τ (ε(k))(1.10)This model is known as the relaxation time approximation. Here, f 0 (r, k) is a static distribution function which describes a local equilibrium at r. The quantity τ (ε(k)) is therelaxation time, which we allow to be energy-dependent. Note that the collision integral indeed depends on the variables (r, k, t), and has a particularly simple functional dependenceon the distribution f .A more sophisticated model might invoke Fermi’s golden rule, Consider elastic scatteringfrom a static potential U(r) which induces transitions between different momentum states.We can then writeIk {f } 2π X k0 U k 2 (fk0 fk ) δ(εk εk0 ) k0 Ω̂Z 3 02πdk Û(k k0 ) 2 (fk0 fk ) δ(εk εk0 ) V (2π)3(1.11)(1.12)Ω̂where we abbreviate fk f (r, k, t). In deriving the last line we’ve used plane wave wave-
8CHAPTER 1. BOLTZMANN TRANSPORT functions2 ψk (r) exp(ik · r)/ V , as well as the resultZ 3XdkA(k) VA(k)(2π)3k Ω̂(1.13)Ω̂for smooth functions A(k). Note the factor of V 1 in front of the integral in eqn. 1.12.What this tells us is that for a bounded localized potential U(r), the contribution to thecollision integral is inversely proportional to the size of the system. This makes sensebecause the number of electrons scales as V but the potential is only appreciable over aregion of volume V 0 . Later on, we shall consider a finite density of scatterers, writingPNimpU(r) i 1U (r Ri ), where the impurity density nimp Nimp /V is finite, scaling as0V . In this case Û(k k0 ) apparently scales as V , which would mean Ik {f } scales as V ,which is unphysical. As we shall see, the random positioning of the impurities means thatthe O(V 2 ) contribution to Û(k k0 ) 2 is incoherent and averages out to zero. The coherentpiece scales as V , canceling the V in the denominator of eqn. 1.12, resulting in a finite valuefor the collision integral in the thermodynamic limit (i.e. neither infinite nor infinitesimal).Later on we will discuss electron-phonon scattering, which is inelastic. An electron withwavevector k0 can scatter into a state with wavevector k k0 q mod G by absorption ofa phonon of wavevector q or emission of a phonon of wavevector q. Similarly, an electronof wavevector k can scatter into the state k0 by emission of a phonon of wavevector q orabsorption of a phonon of wavevector q. The matrix element for these processes dependson k, k0 , and the polarization index of the phonon. Overall, energy is conserved. Theseconsiderations lead us to the following collision integral:n2π XIk {f, n} gλ (k, k0 ) 2 (1 fk ) fk0 (1 nq,λ ) δ(εk ωqλ εk0 ) V 0k ,λ (1 fk ) fk0 n qλ δ(εk ω qλ εk0 ) fk (1 fk0 ) (1 n qλ ) δ(εk ω qλ εk0 )o fk (1 fk0 ) nqλ δ(εk ωqλ εk0 ) δq,k0 kmod G, (1.14)which is a functional of both the electron distribution fk as well as the phonon distributionnqλ . The four terms inside the curly brackets correspond, respectively, to cases (a) through(d) in fig. 1.1.While collisions will violate crystal momentum conservation, they do not violate conservation of particle number. Hence we should have3Z Z 3dkIk {f } 0 .(1.15)d3r(2π)3Ω̂2
An advanced level text on the quantum theory of solids, treating phonons, electrons, plasmons, and photons. P. L. Taylor and O. Heinonen, A Quantum Approach to Condensed Matter Physics (Cambridge University Press, New York, 2002) A modern, intermediate level treatment of the quantum theory of solids. J. M. Z
Introduction of Chemical Reaction Engineering Introduction about Chemical Engineering 0:31:15 0:31:09. Lecture 14 Lecture 15 Lecture 16 Lecture 17 Lecture 18 Lecture 19 Lecture 20 Lecture 21 Lecture 22 Lecture 23 Lecture 24 Lecture 25 Lecture 26 Lecture 27 Lecture 28 Lecture
in condensed matter physics. [6] P. W. Anderson, Basic Notions in Condensed Matter Physics, Benjamin (1984). Inspirational but not tutorial — a highly subjective view of modern condensed matter physics from one of the most influential figures. [7] S. Coleman, Aspects in Symmetry — Selected Erice Lectures (CUP), 1985. Chapter
GEOMETRY NOTES Lecture 1 Notes GEO001-01 GEO001-02 . 2 Lecture 2 Notes GEO002-01 GEO002-02 GEO002-03 GEO002-04 . 3 Lecture 3 Notes GEO003-01 GEO003-02 GEO003-03 GEO003-04 . 4 Lecture 4 Notes GEO004-01 GEO004-02 GEO004-03 GEO004-04 . 5 Lecture 4 Notes, Continued GEO004-05 . 6
Condensed consolidated statement of cash flows . 16 Notes to the condensed consolidated interim financial statements . 17 Independent auditor’s review report on the condensed . Prosus condensed consolidated interim financial statements for the six months ended 30 September 2020 1 Commentary
These condensed interim financial statements were approved for issue on 29 August 2020. The financial statements have been reviewed, not audited. Condensed consolidated statement of profit or loss 5 Condensed consolidated statement of comprehensive income 6 Condensed consolidated balance sheet 7
07 Condensed consolidated statement of financial position 08 Condensed consolidated statement of changes in equity 09 Condensed consolidated statement of cash flows . MultiChoice Group / Condensed consolidated interim financial statements for the period ended 30 September 2020 / 01. slightly more than the prior year due to higher
Lecture 1: A Beginner's Guide Lecture 2: Introduction to Programming Lecture 3: Introduction to C, structure of C programming Lecture 4: Elements of C Lecture 5: Variables, Statements, Expressions Lecture 6: Input-Output in C Lecture 7: Formatted Input-Output Lecture 8: Operators Lecture 9: Operators continued
Quantum Condensed Matter Physics - Lecture Notes Chetan Nayak November 5, 2004
