Lecture Notes on Condensed Matter Physics (A Work in …

1 Lecture Notes on Condensed Matter Physics (A Work in Progress)Daniel ArovasDepartment of PhysicsUniversity of California, San DiegoMarch 14, ..vi0 Introductory ..21 Boltzmann .. Equation in Solids .. Dynamics and Distribution Functions .. Equilibrium .. of Normal Metals .. Time Approximation .. Reflectivity of Metals and Semiconductors .. Conductivity of Semiconductors .. Conductivity and the Fermi Surface .. of the Scattering Time .. Scattering and Fermi s Golden Rule .. and the Transport Lifetime .. Equation for Holes .. of Holes .. and Hall Effect .. Theory for ( ,B) .. Resonance in Semiconductors .. : Two-Band Model .. Effect in High Fields .. Transport .. Theory .. Heat Equation .. of Transport Coefficients .. Relations .. Scattering .. Remarks .. Interaction .. Equation for electron -Phonon Scattering ..482 .. Landauer Formula .. : Potential Step .. Systems .. Matrices: The Pichard Formula.

2 Of the Pichard Formula .. Quantum Resistors in Series .. Quantum Resistors in Parallel .. Conductance Fluctuations in Dirty Metals .. Localization .. Localization .. of Localized and Extended States .. Studies of the Localization Transition .. Theory of Localization .. Temperature ..893 Linear Response and Resonance .. Dissipation .. Relations .. Mechanical Response Functions .. Representation .. Dissipation .. Functions .. Systems .. :S=12 Object in a Magnetic Field .. Equations .. Response .. Invariance and Charge Conservation .. Sum Rule .. and Transverse Response .. Systems .. Meissner Effect and Superfluid Density .. Correlations .. Rules .. Structure Factor for the electron Gas .. 0 Calculation .. Systems: Screening and Dielectric Response .. of the Charge Response Functions .. Screening: Thomas-Fermi Approximation .. Frequency Behavior of (q, ).

3 Phase Approximation (RPA) .. 1254 .. of Orbital Magnetism within Classical Physics .. Atomic Physics .. electron Hamiltonian .. Darwin Term .. electron Hamiltonian .. Periodic Table .. of Configurations: Hund s Rules .. Interaction .. Field Splittings .. Susceptibility of Atomic and Ionic Systems .. Shells: Larmor Diamagnetism .. Filled Shells: van Vleck Paramagnetism .. Magnetism of Noninteracting Systems .. Paramagnetism .. Diamagnetism .. Formation in Interacting Itinerant Systems .. Hubbard Model .. Mean Field Theory .. Solution .. Field Phase Diagram of the Hubbard Model .. of Local Moments: the Heisenberg Model .. Exchange of Orthogonal Orbitals .. Theory of the H2 Molecule .. of Heitler-London Theory .. s approach .. Field Theory .. Probability Distribution .. Ordering .. Field Theory of Anisotropic Magnetic Systems.

4 Chains .. Spin Wave Theory .. Ferromagnetic Spin Waves .. Static Correlations in the Ferromagnet .. Antiferromagnetic Spin Waves .. Specific Heat due to Spin Waves .. PrefaceThis is a proto-preface. A more complete preface will be written after these Notes Lecture Notes are intended to supplement a graduate level course in Condensed 0 Introductory InformationInstructor:Daniel ArovasContact :Mayer Hall 5671 / 534-6323 / Th / 9:30 am - 10:50 am / Mayer Hall 5301 Office Hours:W 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, magnetore-sistance, thermal transport, electron -phonon scatteringMesoscopic Physics : Landauer formula, conductance fluctuations, Aharonov-Bohm ef-fect, disorder, weak localization, Anderson localizationMagnetism: Weak vs.

5 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 0. INTRODUCTORY References 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, semi-conductors, 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.

6 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 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.

7 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)4 CHAPTER 0. INTRODUCTORY INFORMATIONC hapter 1 Boltzmann References 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 IntroductionTransport is the phenomenon of currents flowing in response to applied fields. By current we generally mean an electrical currentj, or thermal currentjq. By applied field wegenerally mean an electric fieldEor 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 1.

8 BOLTZMANN Boltzmann Equation in Semiclassical Dynamics and Distribution FunctionsThe semiclassical dynamics of a wavepacket in a solid are described by the equationsdrdt=vn(k) =1~ n(k) k( )dkdt= e~E(r,t) e~cvn(k) B(r,t).( )Here,nis the band index and n(k) is the dispersion relation for bandn. The wavevectorisk(~kis the crystal momentum ), and n(k) is periodic underk k+G, whereGisany 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 are of course interested in more than just a single electron , hence to that end let usconsider the distribution functionfn(r,k,t), defined such that1fn (r,k,t)d3rd3k(2 )3 # of electrons of spin in bandnwith positions withind3rofrand wavevectors withind3kofkat timet.( )Note that the distribution function is dimensionless.

9 By performing integrals over thedistribution function, we can obtain various physical quantities. For example, the currentdensity atris given byj(r,t) = e n, d3k(2 )3fn (r,k,t)vn(k).( )The symbol in the above formula is to remind us that the wavevector integral is performedonly over the first Brillouin now ask how the distribution functionsfn (r,k,t) evolve in time. To simplify matters,we will consider a single band and drop the indicesn . It is clear that in the absence ofcollisions, the distribution function must satisfy the continuity equation, f t+ (uf) = 0.( )This is just the condition of number conservation for electrons. Take care to note that anduaresix-dimensionalphase spacevectors:u= ( x , y , z , kx, ky, kz)( ) =( x, y, z, kx, ky, kz).( )1We will assume three space dimensions. The discussion may be generalized to quasi-two dimensionaland quasi-one dimensional systems as BOLTZMANN EQUATION IN SOLIDS7 Now note that as a consequence of the dynamics ( , ) that u= 0, spaceflow isincompressible, provided that (k) is a function ofkalone, and not ofr.

10 Thus, inthe absence of collisions, we have f t+u f= 0.( )The differential operatorDt t+u is sometimes called the convective derivative .EXERCISE: Show that u= we must consider the effect of collisions, which are not accounted for by the semi-classical dynamics. In a collision process, an electron with wavevectorkand one withwavevectork can instantaneously convert into a pair with wavevectorsk+qandk q(modulo a reciprocal lattice vectorG), whereqis the wavevector transfer. Note that thetotal wavevector is preserved (modG). This means thatDtf6= 0. Rather, we should write f t+ r f r+ k f k=( f t)coll Ik{f}( )where the right side is known as thecollision integral. The collision integral is in generalafunctionofr,k, andtand afunctionalof the distributionf. As thek-dependence isthe most important for our concerns, we will writeIkin order to make this dependenceexplicit. Some examples should help clarify the , let s consider a very simple model of the collision integral,Ik{f}= f(r,k,t) f0(r,k) ( (k)).