Lecture Notes on Solid State Physics

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Lecture Notes on Solid State Physics (A Work in Progress) Daniel Arovas Department of Physics University of California, San Diego April 13, 2014Chapter 0 Introductory Information Instructor: Daniel Arovas Contact : Mayer Hall 5438 / 534-6323 / darovasucsd.edu Lectures: Tu Th / 11:00 am - 12:20 pm / York Hall 3050B Office Hours: by appointment A strong emphasis of this class will be on learning how to calculate. I plan to cover the following topics this quarter: Transport: Boltzmann equation, transport coecients, cyclotron resonance, magnetore- sistance, thermal transport, electron-phonon scattering Mesoscopic Physics: Landauer formula, conductance uctuations, Aharonov-Bohm ef- fect, disorder, weak localization, Anderson localization Magnetism: Weak vs. strong, local vs. itinerant, Hubbard and Heisenberg models, spin wave theory, magnetic ordering, Kondo e ect Other: Linear response theory, Fermi liquid theory (time permitting) There will be about six assignments. In lieu of a nal examination you will write nal paper on a topic you will select from a list I will provide sometime in the middle of the quarter. I will be following my own notes, which are available from the course web site. 12 CHAPTER 0. INTRODUCTORY INFORMATION 0.1 References  D. I. Khomskii, Basic Aspects of the Quantum Theory of Solids (Cambridge University Press, 2010) This is listed as the ocial course text. It will have little overlap with much of the course, but it will be useful when we get to the topic of magnetism. I chose it because it is an excellent text and I may assign some reading and problems from it.  D. Feng and G. Jin, Introduction to Condensed Matter Physics (I) (World Scienti c, Singapore, 2005) A recent text with a nice modern avor and good set of topics.  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.  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. REFERENCES 3  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, both classical 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 Scienti c, 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)Chapter 1 Boltzmann Transport 1.1 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 7. 1.2 Introduction Transport is the phenomenon of currents owing in response to applied elds. By `current' we generally mean an electrical current j, or thermal current j . By `applied eld' we q generally mean an electric eld E or a temperature gradientrT . The currents and elds are linearly related, and it will be our goal to calculate the coecients (known as transport coecients) of these linear relations. Implicit in our discussion is the assumption that we are always dealing with systems near equilibrium. 56 CHAPTER 1. BOLTZMANN TRANSPORT 1.3 Boltzmann Equation in Solids 1.3.1 Semiclassical Dynamics and Distribution Functions The semiclassical dynamics of a wavepacket in a solid are described by the equations dr 1" (k) n = v (k) = (1.1) n dt k dk e e = E(r;t) v (k)B(r;t) : (1.2) n dt c Here, n is the band index and " (k) is the dispersion relation for band n. The wavevector n isk (k is the `crystal momentum'), and " (k) is periodic underkk +G, whereG is n any reciprocal lattice vector. These formulae are valid only at suciently weak elds. They neglect, for example, Zener tunneling processes in which an electron may change its band index as it traverses the Brillouin zone. We also neglect the spin-orbit interaction in our discussion. We are of course interested in more than just a single electron, hence to that end let us 1 consider the distribution function f (r;k;t), de ned such that n 3 3 drdk of electrons of spin  in band n with positions within f (r;k;t)  (1.3) n 3 3 3 (2) dr ofr and wavevectors within dk ofk at time t. Note that the distribution function is dimensionless. By performing integrals over the distribution function, we can obtain various physical quantities. For example, the current density atr is given by Z 3 X dk j(r;t) =e f (r;k;t)v (k) : (1.4) n n 3 (2) n; The symbol in the above formula is to remind us that the wavevector integral is performed only over the rst Brillouin zone. We now ask how the distribution functions f (r;k;t) evolve in time. To simplify matters, n we will consider a single band and drop the indices n. It is clear that in the absence of collisions, the distribution function must satisfy the continuity equation, f +r (uf) = 0 : (1.5) t This is just the condition of number conservation for electrons. Take care to note that r andu are six-dimensional phase space vectors: _ _ _ u = ( x _ ; y _ ; z _ ; k ; k ; k ) (1.6) x y z   r = ; ; ; ; ; : (1.7) x y z k k k x y z 1 We will assume three space dimensions. The discussion may be generalized to quasi-two dimensional and quasi-one dimensional systems as well.1.3. BOLTZMANN EQUATION IN SOLIDS 7 Now note that as a consequence of the dynamics (1.1,1.2) that ru = 0, i.e. phase space ow is incompressible, provided that "(k) is a function ofk alone, and not ofr. Thus, in the absence of collisions, we have f +urf = 0 : (1.8) t The di erential operator D  +ur is sometimes called the `convective derivative'. t t EXERCISE: Show thatru = 0. Next we must consider the e ect of collisions, which are not accounted for by the semi- classical dynamics. In a collision process, an electron with wavevector k and one with 0 0 wavevector k can instantaneously convert into a pair with wavevectors k +q and k q (modulo a reciprocal lattice vector G), whereq is the wavevector transfer. Note that the total wavevector is preserved (modG). This means thatD f =6 0. Rather, we should write t   f f f f _ +r _ +k = I ffg (1.9) k t r k t coll where the right side is known as the collision integral. The collision integral is in general a function of r, k, and t and a functional of the distribution f. As the k-dependence is the most important for our concerns, we will writeI in order to make this dependence k explicit. Some examples should help clarify the situation. First, let's consider a very simple model of the collision integral, 0 f(r;k;t)f (r;k) I ffg = : (1.10) k ("(k)) 0 This model is known as the relaxation time approximation. Here, f (r;k) is a static dis- tribution function which describes a local equilibrium at r. The quantity ("(k)) is the relaxation time, which we allow to be energy-dependent. Note that the collision integral in- deed depends on the variables (r;k;t), and has a particularly simple functional dependence on the distribution f. A more sophisticated model might invoke Fermi's golden rule, Consider elastic scattering from a static potentialU(r) which induces transitions between di erent momentum states. We can then write X 2 0 2 I ffg = j k U k j (f 0f )(" " 0) (1.11) k k k k k 0 k2 Z 3 0 2 dk 0 2 0 0 = jU(kk )j (f f )(" " ) (1.12) k k k k 3 V (2) where we abbreviate f f(r;k;t). In deriving the last line we've used plane wave wave- k8 CHAPTER 1. BOLTZMANN TRANSPORT p 2 functions (r) = exp(ikr)= V , as well as the result k Z 3 X dk A(k) =V A(k) (1.13) 3 (2) k2 1 for smooth functions A(k). Note the factor of V in front of the integral in eqn. 1.12. What this tells us is that for a bounded localized potentialU(r), the contribution to the collision integral is inversely proportional to the size of the system. This makes sense because the number of electrons scales as V but the potential is only appreciable over a 0 region of volume/ V . Later on, we shall consider a nite density of scatterers, writing P N imp U(r) = U(rR ), where the impurity density n = N =V is nite, scaling as i i=1 imp imp 0 0 V . In this caseU(kk ) apparently scales as V , which would meanI ffg scales as V , k which is unphysical. As we shall see, the random positioning of the impurities means that 2 0 2 theO(V ) contribution tojU(kk )j is incoherent and averages out to zero. The coherent piece scales asV , canceling theV in the denominator of eqn. 1.12, resulting in a nite value for the collision integral in the thermodynamic limit (i.e. neither in nite nor in nitesimal). Later on we will discuss electron-phonon scattering, which is inelastic. An electron with 0 0 wavevectork can scatter into a state with wavevectork =kq modG by absorption of a phonon of wavevectorq or emission of a phonon of wavevectorq. Similarly, an electron 0 of wavevectork can scatter into the statek by emission of a phonon of wavevectorq or absorption of a phonon of wavevector q. The matrix element for these processes depends 0 on k, k , and the polarization index of the phonon. Overall, energy is conserved. These considerations lead us to the following collision integral: n X 2 0 2 I ff;ng = jg (k;k )j (1f )f 0 (1 +n )(" + " 0) k k k q; k q k  V 0 k ; +(1f )f 0n (" " 0) k k q k q k 0 0 f (1f ) (1 +n )(" " ) k k q k q k o 0 0 0 f (1f )n (" + " )  ; (1.14) k k q k q k q;kk mod G which is a functional of both the electron distribution f as well as the phonon distribution k n . The four terms inside the curly brackets correspond, respectively, to cases (a) through q (d) in g. 1.1. While collisions will violate crystal momentum conservation, they do not violate conserva- 3 tion of particle number. Hence we should have Z Z 3 dk 3 dr I ffg = 0 : (1.15) k 3 (2) 2 Rather than plane waves, we should use Bloch waves (r) = exp(ikr)u (r), where cell function nk nk u (r) satis es u (r +R) = u (r), whereR is any direct lattice vector. Plane waves do not contain nk nk nk the cell functions, although they do exhibit Bloch periodicity (r +R) = exp(ikR) (r). nk nk R 3 3 d k If collisions are purely local, then I ffg = 0 at every pointr in space. 3 (2) k 1.3. BOLTZMANN EQUATION IN SOLIDS 9 Figure 1.1: Electron-phonon vertices. The total particle number, Z Z 3 dk 3 N = dr f(r;k;t) (1.16) 3 (2) is a collisional invariant - a quantity which is preserved in the collision process. Other collisional invariants include energy (when all sources are accounted for), spin (total spin), 4 and crystal momentum (if there is no breaking of lattice translation symmetry) . Consider a function F (r;k) of position and wavevector. Its average value is Z Z 3 dk 3  F (t) = d r F (r;k)f(r;k;t) : (1.17) 3 (2) Taking the time derivative, Z Z   3   dF F dk 3 _ = = d r F (r;k)  (r _f)  (kf) +I ffg k 3 dt t (2) r k    Z Z 3 dk F dr F dk 3 = d r  +  f +FI ffg : (1.18) k 3 (2) r dt k dt 4 Note that the relaxation time approximation violates all such conservation laws. Within the relaxation time approximation, there are no collisional invariants.10 CHAPTER 1. BOLTZMANN TRANSPORT Hence, if F is preserved by the dynamics between collisions, then Z Z 3  dF dk 3 = d r FI ffg ; (1.19) k 3 dt (2)  which says that F (t) changes only as a result of collisions. If F is a collisional invariant, _   then F = 0. This is the case when F = 1, in which case F is the total number of particles,  or when F ="(k), in which case F is the total energy. 1.3.2 Local Equilibrium The equilibrium Fermi distribution,     1 "(k) 0 f (k) = exp + 1 (1.20) k T B is a space-independent and time-independent solution to the Boltzmann equation. Since collisions act locally in space, they act on short time scales to establish a local equilibrium described by a distribution function     1 "(k)(r;t) 0 f (r;k;t) = exp + 1 (1.21) k T (r;t) B This is, however, not a solution to the full Boltzmann equation due to the `streaming terms' _ r _ +k in the convective derivative. These, though, act on longer time scales than r k those responsible for the establishment of local equilibrium. To obtain a solution, we write 0 f(r;k;t) =f (r;k;t) +f(r;k;t) (1.22) and solve for the deviation f(r;k;t). We will assume  = (r) and T = T (r) are time- 0 independent. We rst compute the di erential of f ,   0 f " 0 df = k T d B " k T B ( ) 0 f d (")dT d" = k T + B 2 " k T k T k T B B B ( ) 0 f  " T " = dr + dr dk ; (1.23) " r T r k from which we read o ( )   0 0 f  " T f = + (1.24) r r T r " 0 0 f f = v : (1.25) k "1.4. CONDUCTIVITY OF NORMAL METALS 11 We thereby obtain      0 f e 1 f " f 0 +vrf E + vB  +v eE + rT =I ff +fg k t c k T " (1.26) whereE =r(=e) is the gradient of the `electrochemical potential'; we'll henceforth refer toE as the electric eld. Eqn (1.26) is a nonlinear integrodi erential equation in f, with the nonlinearity coming from the collision integral. (In some cases, such as impurity scattering, the collision integral may be a linear functional.) We will solve a linearized version of this equation, assuming the system is always close to a state of local equilibrium. Note that the inhomogeneous term in (1.26) involves the electric eld and the temperature gradientrT . This means that f is proportional to these quantities, and if they are small then f is small. The gradient of f is then of second order in smallness, since the external elds=e andT are assumed to be slowly varying in space. To lowest order in smallness, then, we obtain the following linearized Boltzmann equation:    0 f e f " f vB +v eE + rT =Lf (1.27) t c k T " whereLf is the linearized collision integral;L is a linear operator acting onf (we suppress denoting thek dependence ofL). Note that we have not assumed thatB is small. Indeed later on we will derive expressions for high B transport coecients. 1.4 Conductivity of Normal Metals 1.4.1 Relaxation Time Approximation Consider a normal metal in the presence of an electric eldE. We'll assumeB = 0,rT = 0, and also thatE is spatially uniform as well. This in turn guarantees thatf itself is spatially uniform. The Boltzmann equation then reduces to 0 f f 0 evE =I ff +fg : (1.28) k t " We'll solve this by adopting the relaxation time approximation forI ffg: k 0 ff f I ffg = = ; (1.29) k   where , which may be k-dependent, is the relaxation time. In the absence of any elds or temperature and electrochemical potential gradients, the Boltzmann equation becomes _ f =f=, with the solution f(t) =f(0) exp(t=). The distribution thereby relaxes to the equilibrium one on the scale of .12 CHAPTER 1. BOLTZMANN TRANSPORT it WritingE(t) =Ee , we solve 0 f(k;t) f f(k;t) it ev(k)Ee = (1.30) t " ("(k)) and obtain 0 eEv(k)("(k)) f it f(k;t) = e : (1.31) 1i("(k)) " 0 0 0 The equilibrium distribution f (k) results in zero current, since f (k) = f (k). Thus, the current density is given by the expression Z 3 dk j (r;t) = 2e fv 3 (2)   Z 3 0 dk ("(k))v (k)v (k) f 2 it = 2e E e : (1.32) 3 (2) 1i("(k)) " In the above calculation, the factor of two arises from summing over spin polarizations. The conductivity tensor is de ned by the linear relation j () =  ()E (). We have thus derived an expression for the conductivity tensor,   Z 3 0 dk ("(k))v (k)v (k) f 2  () = 2e (1.33) 3 (2) 1i("(k)) " Note that the conductivity is a property of the Fermi surface. For k T  " , we have F B 0 f ="(" "(k)) and the above integral is over the Fermi surface alone. Explicitly, F we change variables to energy " and coordinates along a constant energy surface, writing d"dS d"dS " " 3 dk = = ; (1.34) j"=kj jvj where dS is the di erential area on the constant energy surface "(k) = ", and v(k) = " 1 r "(k) is the velocity. For TT , then, F k Z 2 e (" ) v (k)v (k) F  () = dS : (1.35) F 3 4 jv(k)j 1i(" ) F 2 2   For free electrons in a parabolic band, we write "(k) = k =2m , so v (k) =k =m . To further simplify matters, let us assume that  is constant, or at least very slowly varying in the vicinity of the Fermi surface. We nd Z   2 0 2 e  f  () = d"g(")" ; (1.36)  3m 1i " where g(") is the density of states, Z 3 dk g(") = 2  (""(k)) : (1.37) 3 (2) 1.4. CONDUCTIVITY OF NORMAL METALS 13 The (three-dimensional) parabolic band density of states is found to be  3=2 p (2m ) g(") = " (") ; (1.38) 2 3 2 where (x) is the step function. In fact, integrating (1.36) by parts, we only need to know p about the " dependence in g("), and not the details of its prefactor:   Z Z 0 f 0 d""g(") = d"f (") ("g(")) " " Z 0 3 3 = d"g(")f (") = n ; (1.39) 2 2 where n = N=V is the electron number density for the conduction band. The nal result for the conductivity tensor is 2 ne    () = (1.40)  m 1i This is called the Drude model of electrical conduction in metals. The dissipative part of the conductivity is Re. Writing = and separating into real and imaginary parts 0 00  = +i , we have 2 ne  1 0  () = : (1.41)  2 2 m 1 +  The peak at = 0 is known as the Drude peak. Here's an elementary derivation of this result. Let p(t) be the momentum of an electron, and solve the equation of motion dp p it = eEe (1.42) dt  to obtain   eE eE it t= p(t) = e + p(0) + e : (1.43) 1i 1i The second term above is a transient solution to the homogeneous equation p _ +p= = 0.  At long times, then, the currentj =nep=m is 2 ne  it j(t) = Ee : (1.44)  m (1i) In the Boltzmann equation approach, however, we understand that n is the conduction electron density, which does not include contributions from lled bands.   In solids the e ective mass m typically varies over a small range: m  (0:1 1)m . The e two factors which principally determine the conductivity are then the carrier density n and the scattering time . The mobility , de ned as the ratio ( = 0)=ne, is thus (roughly) 5 independent of carrier density . Since j =nev = E, where v is an average carrier  velocity, we havev =E, and the mobility  =e=m measures the ratio of the carrier velocity to the applied electric eld. 5  Inasmuch as both andm can depend on the Fermi energy, is not completely independent of carrier density.14 CHAPTER 1. BOLTZMANN TRANSPORT Figure 1.2: Frequency-dependent conductivity of liquid sodium by T. Inagaki et al, Phys. Rev. B 13, 5610 (1976). 1.4.2 Optical Re ectivity of Metals and Semiconductors What happens when an electromagnetic wave is incident on a metal? Inside the metal we have Maxwell's equations,   4 1D 4 i rH = j + =) ikB = E (1.45) c c t c c 1B i rE = =) ikE = B (1.46) c t c rE =rB = 0 =) ikE =ikB = 0 ; (1.47) where we've assumed  =  = 1 inside the metal, ignoring polarization due to virtual interband transitions (i.e. from core electrons). Hence, 2 4i 2 k = + () (1.48) 2 2 c c 2 2 2 i p = + () ; (1.49) 2 2 2 c c 1i c p 2  where = 4ne =m is the plasma frequency for the conduction band. The dielectric p function, 2 4i() i p () = 1 + = 1 + (1.50) 2 1i1.4. CONDUCTIVITY OF NORMAL METALS 15 p determines the complex refractive index, N() = (), leading to the electromagnetic dispersion relation k =N()=c. Consider a wave normally incident upon a metallic surface normal to z. In the vacuum (z 0), we write iz=c it iz=c it E(r;t) =E x e e +E x e e (1.51) 1 2 c iz=c it iz=c it B(r;t) = rE =E y e e E y e e (1.52) 1 2 i while in the metal (z 0), iNz=c it E(r;t) =E x e e (1.53) 3 c iNz=c it B(r;t) = rE =NE y e e (1.54) 3 i Continuity ofEn givesE +E =E . Continuity ofHn givesE E =NE . Thus, 1 2 3 1 2 3 E 1N E 2 2 3 = ; = (1.55) E 1 +N E 1 +N 1 1 and the re ection and transmission coecients are 2 2 E 1N() 2 R() = = (1.56) E 1 +N() 1 2 E 4 3 T () = = : (1.57) 2 E 1 1 +N() We've now solved the electromagnetic boundary value problem. 22 3  Typical values For a metal with n = 10 cm and m = m , the plasma frequency is e 15 1 = 5:710 s . The scattering time varies considerably as a function of temperature. In p 9 7 14 high purity copper atT = 4 K, 210 s and  10 . AtT = 300 K, 210 s p and  100. In either case,  1. There are then three regimes to consider. p p   1 : p We may approximate 1i 1, hence 2 2 i  i  p p 2 N () = 1 +  (1i) p 1=2 2  1 +i 2 2 p N() p =) R 1 : (1.58)  2 p Hence R 1 and the metal re ects.  1 : p In this regime, 2 2 i p p 2 N () 1 + (1.59) 2 3 16 CHAPTER 1. BOLTZMANN TRANSPORT Figure 1.3: Frequency-dependent absorption of hcp cobalt by J. Weaver et al., Phys. Rev. B 19, 3850 (1979). which is almost purely real and negative. Hence N is almost purely imaginary and R 1. (To lowest nontrivial order, R = 1 2= .) Still high re ectivity. p  1 : p Here we have 2 p p 2 N () 1 =) R = (1.60) 2 2 and R 1 the metal is transparent at frequencies large compared to . p 1.4.3 Optical Conductivity of Semiconductors In our analysis of the electrodynamics of metals, we assumed that the dielectric constant due to all the lled bands was simply  = 1. This is not quite right. We should instead have written 2 4i() 2 k = + (1.61) 1 2 2 c c ( ) 2 i p () = 1 + ; (1.62) 1 2 1i1.4. CONDUCTIVITY OF NORMAL METALS 17 Figure 1.4: Frequency-dependent conductivity of hcp cobalt by J. Weaver et al., Phys. Rev. B 19, 3850 (1979). This curve is derived from the data of g. 1.3 using a Kramers- Kr onig transformation. A Drude peak is observed at low frequencies. At higher frequencies, interband e ects dominate. where  is the dielectric constant due to virtual transitions to fully occupied (i.e. core) 1 and fully unoccupied bands, at a frequency small compared to the interband frequency. The plasma frequency is now de ned as   1=2 2 4ne = (1.63) p  m  1 wheren is the conduction electron density. Note that (1) = , although again this 1 is only true for smaller than the gap to neighboring bands. It turns out that for insulators one can write 2 pv  ' 1 + (1.64) 1 2 g p 2 where = 4n e =m , with n the number density of valence electrons, and is pv v e v g the energy gap between valence and conduction bands. In semiconductors such as Si and Ge,  4 eV, while  16 eV, hence   17, which is in rough agreement with the 1 g pv experimental values of 12 for Si and 16 for Ge. In metals, the band gaps generally are considerably larger. There are some important di erences to consider in comparing semiconductors and metals:18 CHAPTER 1. BOLTZMANN TRANSPORT  The carrier density n typically is much smaller in semiconductors than in metals, 16 3 ranging from n 10 cm in intrinsic (i.e. undoped, thermally excited at room 19 3 temperature) materials to n 10 cm in doped materials.      10 20 andm =m  0:1. The product m thus di ers only slightly from its 1 e 1 free electron value. 4 Since n 10 n , one has semi metal semi 2 metal 14  10  10 s : (1.65) p p  5 2 In high purity semiconductors the mobility  = e=m 10 cm =vs the low temperature  11 15 1 scattering time is typically  10 s. Thus, for 3 10 s in the optical range, we  p have   1, in which case N()  and the re ectivity is p 1 p 2 1  1 R = : (1.66) p 1 +  1 Taking  = 10, one obtains R = 0:27, which is high enough so that polished Si wafers 1 appear shiny. 1.4.4 Optical Conductivity and the Fermi Surface At high frequencies, when  1, our expression for the conductivity, eqn. (1.33), yields Z  Z 2 0 ie f () = d" dS v(k) ; (1.67) " 3 12 " where we have presumed sucient crystalline symmetry to guarantee that  =  is diagonal. In the isotropic case, and at temperatures low compared with T , the integral F 2 3  2  over the Fermi surface gives 4k v = 12 n=m , whence  = ine =m , which is the F F large frequency limit of our previous result. For a general Fermi surface, we can de ne 2 ine 1 ( ) (1.68) m opt where the optical mass m is given by opt Z  Z 0 1 1 f = d" dS v(k) : (1.69) " 3 12 n " m opt Note that at high frequencies () is purely imaginary. What does this mean? If  1 it +it E(t) =E cos(t) = E e +e (1.70) 21.5. CALCULATION OF THE SCATTERING TIME 19 then  1 it +it j(t) = E ()e +()e 2 2 ne = E sin(t) ; (1.71) m opt   where we have invoked () = (). The current is therefore 90 out of phase with the voltage, and the average over a cyclehj(t)E(t)i = 0. Recall that we found metals to be 1 transparent for   . p At zero temperature, the optical mass is given by Z 1 1 = dS v(k) : (1.72) F 3 12 n m opt The density of states, g(" ), is F Z 1 1 g(" ) = dS v(k) ; (1.73) F F 3 4  from which one can de ne the thermodynamic e ective mass m , appealing to the low th temperature form of the speci c heat, 2   m 2 th 0 c = k Tg(" ) c ; (1.74) F B V V 3 m e where 2 m k T e 0 B 2 1=3 c  (3 n) (1.75) V 2 3 is the speci c heat for a free electron gas of density n. Thus, Z 1  m = dS v(k) (1.76) F th 2 1=3 4(3 n) 1.5 Calculation of the Scattering Time 1.5.1 Potential Scattering and Fermi's Golden Rule Let us go beyond the relaxation time approximation and calculate the scattering time  from rst principles. We will concern ourselves with scattering of electrons from crystalline 6 impurities. We begin with Fermi's Golden Rule , X 2 2 0 0 I ffg = k U k (f 0f )("(k)"(k )) ; (1.77) k k k 0 k 6 We'll treat the scattering of each spin species separately. We assume no spin- ip scattering takes place.20 CHAPTER 1. BOLTZMANN TRANSPORT   m =m m =m e e opt th Metal thy expt thy expt Li 1.45 1.57 1.64 2.23 Na 1.00 1.13 1.00 1.27 K 1.02 1.16 1.07 1.26 Rb 1.08 1.16 1.18 1.36 Cs 1.29 1.19 1.75 1.79 Cu - - 1.46 1.38 Ag - - 1.00 1.00 Au - - 1.09 1.08 Table 1.1: Optical and thermodynamic e ective masses of monovalent metals. (Taken from Smith and Jensen). whereU(r) is a sum over individual impurity ion potentials, N imp X U(r) = U(rR ) (1.78) j j=1 N imp 2 X 0 2 0 2 0 2 i(kk )R j k U k =V jU(kk )j  e ; (1.79) j=1 where V is the volume of the solid and Z 3 iqr U(q) = drU(r)e (1.80) is the Fourier transform of the impurity potential. Note that we are assuming a single species of impurities; the method can be generalized to account for di erent impurity species. To make progress, we assume the impurity positions are random and uncorrelated, and we average over them. Using N imp 2 X iqR j e =N +N (N 1) ; (1.81) imp imp imp q;0 j=1 we obtain N N (N 1) 2 imp imp imp 0 2 2 0 k U k = jU(kk )j + jU(0)j  0 : (1.82) kk 2 2 V V EXERCISE: Verify eqn. (1.81). We will neglect the second term in eqn. 1.82 arising from the spatial average (q = 0 Fourier component) of the potential. As we will see, in the end it will cancel out. Writing

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