Lecture Notes on Solid State Physics (A Work in Progress), Lecture notes of 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

May 10, 2018

Contents

vi CONTENTS

0.1 Preface

This is a proto-preface. A more complete preface will be written after these notes are completed. These lecture notes are intended to supplement a graduate level course in condensed matter physics.

Chapter 0

Introductory Information

Instructor: Daniel Arovas Contact : Mayer Hall 5438 / 534-6323 / [email protected] Lectures: Tu Th / 11:00 am - 12:20 pm / location TBA 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 coefficients, cyclotron resonance, magnetore- sistance, thermal transport, electron-phonon scattering Mesoscopic Physics: Landauer formula, conductance fluctuations, 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 effect Other: Linear response theory, Fermi liquid theory (time permitting) There will be about six assignments. In lieu of a final examination you will write final 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.

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 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)

4 CHAPTER 0. INTRODUCTORY INFORMATION

6 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^1

dr dt =^

∂εn(k) ∂k −^

dk dt ×^ Ωn(k)^ (1.1) dk dt =^ −^

e ℏ E(r, t)^ −^

e ℏc

dr dt ×^ B(r, t)^.^ (1.2) Here n is the band index and εn(k) is the dispersion relation for band n. The wavevector is k (ℏk is the ‘crystal momentum’), and εn(k) is periodic under k → k + G, where G is any reciprocal lattice vector. The second term on the RHS of Eqn. 1.1 is the so-called Karplus-Luttinger term, defined by

Aμn(k) = −i 〈^ un(k) ∣∣^ ∂k∂μ^ ∣∣^ un(k) 〉^ (1.3)

Ωnμ (k) = μνλ^ ∂A λn(k) ∂kν^ ,^ (1.4) arising from the Berry phases generated by the one-particle Bloch cell functions |un(k)〉. These formulae are valid only at sufficiently weak fields. They neglect, for example, Zener tunneling processes in which an electron may change its band index as it traverses the Brillouin zone. We assume Ωn(k) = 0 in our discussion, i.e. we assume the Bloch bands are non topological. Finally, we neglect the orbital magnetization of the Bloch wavepacket and contributions from the spin-orbit interaction. When the orbital moment of the Bloch electrons is included, we must substitute εn(k) → εn(k) − Mn(k) · B(r, t) (1.5) where M (^) nμ (k) = e μνλ^ Im

〈 (^) ∂un ∂kν

∣∣εn(k) − H 0 (k)

∣∣^ ∂un ∂kλ

where Hˆ 0 (k) = eik·r^ Hˆ 0 e−ik·r^ and Hˆ 0 = 2 pm^2 + V (r) is the one-electron Hamiltonian in the crystalline potential V (r) = V (r + R), where R is any direct lattice vector. Note Hˆ 0 (k) |un(k)〉 = εn(k) |un(k)〉 and that un(k, r + R) = un(k, r) is periodic in the direct lattice. We are of course interested in more than just a single electron, hence to that end let us consider the distribution function fn(r, k, t), defined such that^2

fnσ(r, k, t) d (^3) r d (^3) k (2π)^3 ≡^

of electrons of spin σ in band n with positions within

d^3 r of r and wavevectors within d^3 k of k at time t. (1.7) (^1) See G. Sundaram and Q. Niu, Phys. Rev. B 59 , 14915 (1999). (^2) 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

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 at r is given by

j(r, t) = −e

n,σ

Ω^ ˆ

d^3 k (2π)^3 fnσ(r,^ k, t)^ vn(k)^.^ (1.8)

The symbol Ω in the above formula is to remind us that the wavevector integral is performedˆ only 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 of collisions, the distribution function must satisfy the continuity equation, ∂f ∂t +^ ∇^ ·^ (uf^ ) = 0^.^ (1.9) This is just the condition of number conservation for electrons. Take care to note that ∇ and u are six -dimensional phase space vectors:

u = ( ˙x , y ,˙ z ,˙ k˙x , k˙y , k˙z ) (1.10) ∇ =

∂x ,

∂y ,

∂z ,^

∂kx^ ,^

∂ky^ ,^

∂kz

Now note that as a consequence of the dynamics (1.1,1.2) that ∇ · u = 0, i.e. phase space flow is incompressible, provided that ε(k) is a function of k alone, and not of r. Thus, in the absence of collisions, we have ∂f ∂t +^ u^ ·^ ∇f^ = 0^.^ (1.12) 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 semi- classical dynamics. In a collision process, an electron with wavevector k and one with wavevector k′^ can instantaneously convert into a pair with wavevectors k + q and k′^ − q (modulo a reciprocal lattice vector G), where q is the wavevector transfer. Note that the total wavevector is preserved (mod G). This means that Dtf 6 = 0. Rather, we should write

∂f ∂t + ˙r^ ·^

∂f ∂r + k˙ · ∂f ∂k =

( (^) ∂f ∂t

coll ≡ Ik{f } (1.13)

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 write Ik in order to make this dependence explicit. Some examples should help clarify the situation.

1.3. BOLTZMANN EQUATION IN SOLIDS 9

Figure 1.1: Electron-phonon vertices.

absorption of a phonon of wavevector q. The matrix element for these processes depends on k, k′, and the polarization index of the phonon. Overall, energy is conserved. These considerations lead us to the following collision integral:

Ik{f, n} = (^) ℏ^2 Vπ

k′,λ

|gλ(k, k′)|^2

(1 − fk) fk′^ (1 + nq,λ) δ(εk + ℏωqλ − εk′^ ) +(1 − fk) fk′ n−qλ δ(εk − ℏω−qλ − εk′ ) −fk (1 − fk′^ ) (1 + n−qλ) δ(εk − ℏω−qλ − εk′^ ) −fk (1 − fk′ ) nqλ δ(εk + ℏωqλ − εk′ )

δq,k′−k mod G , (1.18)

which is a functional of both the electron distribution fk as well as the phonon distribution nqλ. 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 conserva- tion of particle number. Hence we should have^4 ∫ d^3 r

Ω^ ˆ

d^3 k (2π)^3 Ik{f^ }^ = 0^.^ (1.19)

(^4) If collisions are purely local, then ∫ Ω^ ˆ (2^ dπ^3 k)^3 Ik{f^ }^ = 0 at every point^ r^ in space.

10 CHAPTER 1. BOLTZMANN TRANSPORT

The total particle number, N =

d^3 r

Ω^ ˆ

d^3 k (2π)^3 f^ (r,^ k, t)^ (1.20)

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), and crystal momentum (if there is no breaking of lattice translation symmetry)^5. Consider a function F (r, k) of position and wavevector. Its average value is

F¯ (t) =

d^3 r

Ω^ ˆ

d^3 k (2π)^3 F^ (r,^ k)^ f^ (r,^ k, t)^.^ (1.21)

Taking the time derivative, d F¯ dt =^

∂ F¯

∂t =

d^3 r

Ω^ ˆ

d^3 k (2π)^3 F^ (r,^ k)

− (^) ∂∂r · ( ˙rf ) − (^) ∂∂k · ( k˙f ) + Ik{f }

d^3 r

Ω^ ˆ

d^3 k (2π)^3

{[ ∂F

∂r ·^

dr dt +^

∂F

∂k ·^

dk dt

]

f + F Ik{f }

Hence, if F is preserved by the dynamics between collisions, then d F¯ dt =

d^3 r

Ω^ ˆ

d^3 k (2π)^3 F^ Ik{f^ }^ ,^ (1.23)

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 = ε(k), in which case F¯ is the total energy.

1.3.2 Local Equilibrium

The equilibrium Fermi distribution,

f 0 (k) =

exp

( (^) ε(k) − μ kBT

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

f 0 (r, k, t) =

exp

( (^) ε(k) − μ(r, t) kBT (r, t)

(^5) Note that the relaxation time approximation violates all such conservation laws. Within the relaxation time approximation, there are no collisional invariants.

12 CHAPTER 1. BOLTZMANN TRANSPORT

1.4 Conductivity of Normal Metals

1.4.1 Relaxation Time Approximation

Consider a normal metal in the presence of an electric field E. We’ll assume B = 0, ∇ T = 0, and also that E is spatially uniform as well. This in turn guarantees that δf itself is spatially uniform. The Boltzmann equation then reduces to

∂ δf ∂t −^

∂f 0 ∂ε ev^ ·^ E^ =^ Ik{f^

(^0) + δf }. (1.32)

We’ll solve this by adopting the relaxation time approximation for Ik{f }:

Ik{f } = − f^ −^ f^

0 τ =^ −^

δf τ ,^ (1.33) where τ , which may be k-dependent, is the relaxation time. In the absence of any fields or temperature and electrochemical potential gradients, the Boltzmann equation becomes δf˙ = −δf /τ , with the solution δf (t) = δf (0) exp(−t/τ ). The distribution thereby relaxes to the equilibrium one on the scale of τ. Writing E(t) = E e−iωt, we solve

∂ δf (k, t) ∂t −^ e^ v(k)^ ·^ E^ e

−iωt ∂f^0 ∂ε =^ −^

δf (k, t) τ (ε(k)) (1.34)

and obtain δf (k, t) = e^ E 1 −·^ v iωτ(k) (^ τε^ ((εk())k))^ ∂f^

0 ∂ε e

−iωt (^). (1.35)

The equilibrium distribution f 0 (k) results in zero current, since f 0 (−k) = f 0 (k). Thus, the current density is given by the expression

jα(r, t) = − 2 e

Ω^ ˆ

d^3 k (2π)^3 δf v

α

= 2 e^2 Eβ^ e−iωt

ˆΩ

d^3 k (2π)^3

τ (ε(k)) vα(k) vβ^ (k) 1 − iωτ (ε(k))

− ∂f^ 0 ∂ε

In the above calculation, the factor of two arises from summing over spin polarizations. The conductivity tensor is defined by the linear relation jα(ω) = σαβ (ω) Eβ^ (ω). We have thus derived an expression for the conductivity tensor,

σαβ (ω) = 2e^2

Ω^ ˆ

d^3 k (2π)^3

τ (ε(k)) vα(k) vβ^ (k) 1 − iωτ (ε(k))

− ∂f^ 0 ∂ε

1.4. CONDUCTIVITY OF NORMAL METALS 13

Note that the conductivity is a property of the Fermi surface. For kBT  εF, we have −∂f 0 /∂ε ≈ δ(εF − ε(k)) and the above integral is over the Fermi surface alone. Explicitly, we change variables to energy ε and coordinates along a constant energy surface, writing

d^3 k = (^) |dε dS∂ε/∂kε| = dε dS ℏ|v| ε, (1.38)

where dSε is the differential area on the constant energy surface ε(k) = ε, and v(k) = ℏ−^1 ∇k ε(k) is the velocity. For T  TF, then,

σαβ (ω) = e 2 4 π^3 ℏ

τ (εF) 1 − iωτ (εF)

dSF^ v α(k) vβ (^) (k) |v(k)|.^ (1.39)

For free electrons in a parabolic band, we write ε(k) = ℏ^2 k^2 / 2 m∗, so vα(k) = ℏkα/m∗. To further simplify matters, let us assume that τ is constant, or at least very slowly varying in the vicinity of the Fermi surface. We find

σαβ (ω) = δαβ 3 m^2 ∗^ e

(^2) τ 1 − iωτ

dε g(ε) ε

− ∂f^

0 ∂ε

where g(ε) is the density of states,

g(ε) = 2

Ω^ ˆ

d^3 k (2π)^3 δ^ (ε^ −^ ε(k))^.^ (1.41)

The (three-dimensional) parabolic band density of states is found to be

g(ε) = (2m

2 π^2 ℏ^3

√ε Θ(ε) , (1.42)

where Θ(x) is the step function. In fact, integrating (1.40) by parts, we only need to know about the √ε dependence in g(ε), and not the details of its prefactor: ∫ dε ε g(ε)

− ∂f^ 0 ∂ε

dε f 0 (ε) (^) ∂ε∂ (ε g(ε)) = (^32)

dε g(ε) f 0 (ε) = 32 n , (1.43)

where n = N/V is the electron number density for the conduction band. The final result for the conductivity tensor is

σαβ (ω) = ne (^2) τ m∗

δαβ 1 − iωτ (1.44) This is called the Drude model of electrical conduction in metals. The dissipative part of the conductivity is Re σ. Writing σαβ = σδαβ and separating into real and imaginary parts σ = σ′^ + iσ′′, we have σ′(ω) = ne (^2) τ m∗

1 + ω^2 τ 2.^ (1.45)