Marder M.P. Condensed Matter Physics

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The Kubo-Greenwood

Formula

623

The frequency w

is the plasma frequency, and it generalizes the quantity de-

fined by Eqs. (20.31) and (20.32). The optical mass m

opt

is defined by Eq. (20.52).

Thus general lessons to learn from Eq. (20.49) are as follows:

A collection of absorbing modes whose frequency lies well below the probe

frequency w

acts like a collection of unbound electrons, and its influence

falls off as l/io

A collection of absorbing modes whose frequency lies well above the probe

frequency

acts like a high-frequency dielectric constant e°°.

20.4 The Kubo-Greenwood Formula

One of the most important quantum-mechanical calculations in the theory of solids

concerns the linear response of a collection of electrons to an externally applied

electromagnetic field. Not only does this calculation permit one to find the electri-

cal conductivity of solids, it describes their interaction with light and also permits

a self-consistent description of the interaction of electrons with one another. The

strength of the calculation is that it starts from quantum-mechanical principles,

introducing few phenomenological assumptions. This feature is also its greatest

weakness, and for the purposes of studying thermoelectric properties of solids the

Boltzmann equation is more powerful.

The setting of this calculation is quite general. Consider a collection of elec-

trons occupying states of energy

£.l=hcüi The calculation is within the independent elec- (20.53)

tron approximation.

with probability //. Usually // will be a Fermi factor, which means essentially

that states up to the Fermi energy are occupied, while those above are unoccupied.

However, the calculation continues to function when // has other forms, a fact of

particular importance for lasers. Apply an external electromagnetic field, and ask

for the expectation value of the current that results in response. The calculation

has two steps. First, one has to work out the response to first order of an arbitrary

quantum-mechanical state to a potential U(t) with time dependence cos uit. Next,

one finds the expectation value of the current in this state.

20.4.1 Born Approximation

Begin with the Born approximation, which states that an eigenstate |/) of "K that

comes in contact with a weak time-dependent potential U(t) evolves into

/>)) « ttfe-'**

|/>

+ / dt'e-^^r

Wile-™!;

\l)

(20.54)

The eigenstates of 'K are denoted by ]/), while the time-dependent state evolving from |/)

under the combined action of 3-fand Û is called

|/(/)).

Nis a normalization constant.

624 Chapter 20. Phenomenological Theory

l0J

\l) +

f dt'liy-^^'-^^^-e-^'-^"'] (20.55)

Take U(t') to have time dependence exp(—iuit'); to must have a small positive imaginary

part

for the time integral converge, or else take üj

i to have an imaginary part —y

f\l\<\^\l'\

V I^IO

e lUJ

' \ -iuj.t Excluding / from the sum takes

= » t/J' /w ; 7 (

■ care of the normalization. (2U.56)

,Y/ Δ(w,-o;//+a;)J

If, on the other hand, the time dependent potential were to have the form

exp[ico*t],

then one would have instead

«-^ (l'\Û*\l)e

iu}

*' 1

i'('>Hi'>

gi^-!^.)R

°-

Imaginary

Frequencies.

For purely formal reasons, so that the integrals in (20.56)

converge, it is necessary to take the frequencies

UJ/I

to have imaginary parts;

needs a small positive imaginary part

r],

while

U[i

needs an imaginary part

—7//.

The constant

does not have much physical interest. It corresponds to turning on

the interaction potential U very slowly, a long time in the past, so that the interact-

ing system can adiabatically adjust itself to the new potential and reach a steady

state.

The constants 7// are more important. By making them nonzero, one is able

to describe transitions into metastable states. The initial state of a system is usu-

ally very stable; the system has been there for a long time, and it would remain if

not perturbed by U. However, the intermediate states indexed by /' are usually not

true eigenstates. In a formally exact description of a quantum system, the eigen-

values and frequencies should all be real. The common situation is, however, that

the quantum states /' used in describing excited states of a system are not exact,

both because one has only an approximate solution of the Hamiltonian and be-

cause the Hamiltonian actually has many more degrees of freedom than one is able

to describe. Not being exact eigenstates, these states decay, and the decay can be

modeled by giving them complex energies. All Bloch states

should really be

described in this way; they decay because electron-electron interactions have not

been treated exactly, because the real wave functions involve many electrons not

one,

because the crystal in which they live has impurities, or because of interac-

tions with phonons. Localized states, described for example by Wannier functions,

should be taken to decay for the same sorts of

reasons.

The important point is that

many different types of ignorance and error can be accommodated with a few con-

stants 7/', and only in this way does detailed comparison with experiment become

possible. The initial states |/) usually do not decay. They are, after all, states the

system chose to go to itself in the absence of outside interference.

Interaction with Electromagnetic Field. The next step is to use Eq. (20.56) to

describe a situation in which a collection of electrons interacts with an electromag-

netic field. As in Section 16.3 the sensible way for an electrical field to coexist with

periodic boundary conditions is to introduce it through a vector potential. Special-

izing to the case described by Eq. (20.24) where the wavelength of light is much

greater than unit cell dimensions or electron mean free paths, one can treat the light

The Kubo-Greenwood

Formula

625

spatially uniform oscillating field with

cÈ

A =

,iU>

-\-C.C.. c.c.

means "complex conjugate." (20.58)

The next task

is to

find

the

expectation value

the current

such

state

When

electromagnetic field

described classically

vector potential

must be understood as the canonical momentum, where classically

p =

—

eAjc.

So the current operator

—

— [p+

-A], See Goldstein (1980), pp. 341-342. (20.59)

and the Hamiltonian changes also because

P-*P+-A,

(20.60)

so the kinetic energy becomes

{P+-AY

e ^

(20.61)

[A-P + P-A]+ . . .

Terms

order/I

are irrelevant in a theory

(20.62)

2m 2mc

linear response.

- ~

= 1 \A

■

-\-

. . . . "

conventional

specialize

the

trans- (20.63)

2m mc

verse gauge

■

= 0 so as to

obtain this

simplification.

To linear order in applied fields, the Hamiltonian

changed by addition

a term

Û(t) =

—\È-P]e-

iu}

'--^[Ë-P]e^'.

(20.64)

;

micu

miu*

The contribution

the current

state

|/) is

therefore

~t Using Eq. (20.58)

for

the

vector potential

J = Vj =

(I\P-\

\!)

and using Eq. (20.56)

for

the state

|/~).

(20.65)

2p This term comes from

the

expectation value

(/|P|/)

—

lwt

+C.C.1 ClilO-

The

two

lines result from

the

m imtü terms proportional

E in |/)

acting

left

and right-hand sides.

*-, -, - r e~

iu)

Y(l\P\l'){l'\E-P\l){-r-^

T r-

ihm

ff/

M !

' '

^ÜJ(UI-U,>+UJ) UJ*(UJI-LÜI>-U;*)}

(/lE-Pl/'X/'l/

!/)!—

—

-1.(20.66)

The conductivity tensor a{oj)

defined

as the

coefficient

e~'

' that relates

the

current

to the

applied electric field. Assuming that state

|/) is

occupied with

626 Chapter 20. Phenomenological Theory

probability //, assuming that the ground state had no net current, and summing

over all states / gives the Kubo-Greenwood formula

°aß(u)

imuV

// f{i\p

\i')(i'\p

\i)

{

{i\p

\i')(i'\p

\i)

*-rf nm i-

u>i —

u>['

uij

—

u)J,

— ui

. (20.67)

This expression simplifies if all the w/ are real. Then exchanging / and /' in the

second term gives

ß(uj)

imcoV

»ß +

fl-fy(l\P

\l')(l'\Pß\l)

Hm LÜI—LÜI'+(JJ + irj

i v

Adding

a small imaginary part

i-q

explicitly.

(20.68)

Absorption of incoming energy is given by the real part of the conductivity

tensor, and is worth writing out separately. When all the decay rates 7/' = —Im uif

are zero, the real part is entirely induced by the small imaginary term r\ in the

denominator of Eq. (20.68), and is

Re [a

aß

{oj)} = -Im

musV

ft-fv{i\p

\i'){i'\h\i)'

Hm WI—U)[I+UJ + ir\

TÏ

£(//-//'WI

—1/')</'|^|/><5(£,'

w mm

£;

—Hui).

Recall

Im[l/(jc-

ir))]

=-K5(X).

(20.69)

(20.70)

More generally, if decay rates 7/ = — Im

LUI

are not zero,

Re [(J

ß(uj)}

e IT

hu)m

E(/'

-fi>)(l\P

\l')(l'\Pß\l)Fu>(u), (20.71)

where

FU>(UJ)

= (7/'

—

ji)/(TV[(LJ

—

UJI>

+w;)

])

is the Hneshape, a Lorentzian

of width 7//

—

7/ centered at

CJ;

—

uii>.

Expression (20.71) can be simplified slightly with some additional assump-

tions.

First assume that the occupation numbers // are either 1 for occupied states

or 0 for unoccupied states. Then the only terms contributing to the conductivity are

those where / or /' is occupied, but not both. Since the occupied states are stable,

take 7/ = 0 for occupied states. Finally, choose to > 0 and suppose that Fut is neg-

ligibly small whenever /' is an occupied state and / is not. Under these conditions

one can write

Re [a

ß(u)}

htum

! occupied

/' unoccupied

(l\P

\l')(l'\P

\l)

[u;-(uj

-uji)}

+ 'yj,

(20.72)

The Kubo-Greenwood

Formula

627

Equation (20.67) has a dual interpretation. For sufficiently small values of

reduces to Eq. (16.29), and it describes the motion of electrons within a single band

with an effective mass. However, as

rises toward optical frequencies, the effec-

tive mass picture breaks down, and Eq. (20.67) now describes transitions between

levels that become possible whenever there is a resonance such that

w/<

—UJI=UJ.

For a material in equilibrium, only low-lying states are occupied, higher-lying

states are empty, and the real part of the conductivity tensor (20.71) describes the

absorption of energy from incoming waves. However, if a material can be pre-

pared out of equilibrium, with higher-lying states occupied and low-lying states

empty, the sign of energy absorption reverses, and Eq. (20.71) describes stimulated

emission. Therefore, this equation describes the basic operation of the laser.

20.4.2 Susceptibility

A slight variant on the preceding calculation is valuable in order to produce es-

timates of the interactions of electrons with one another. The calculation differs

from the previous one in several respects. First and most important, electrical po-

tentials created by electrons in solids have wavelengths on the order of interatomic

spacings. The q

—>

0 limit employed until now is not legitimate, and the perturbing

potential U must be taken to behave as

U{q,uj)e^

iwt

+c.c. (20.73)

In compensation for the difficulties introduced by the extra spatial dependence of

U, it is customary to take the unperturbed electron states |/) simply to be plane

waves, not so much because the approximation is carefully justified as because

otherwise the expressions become unwieldy. The calculation is also simplified by

focusing upon the expectation value of the charge density rather than upon the

current, making expressions somewhat more compact.

The electron density n is defined to be

n(r, 0 = X) fl(Kt)\r)(m))- (20.74)

If one uses the simplification that \l) is a plane wave, takes the potential U to be an

electrical potential

U(q,

to) = ~eV(q, u), (20.75)

and proceeds in just the fashion that leads to Eq. (20.68), Problem 5 shows that

-en(q,

u>)

= xdq, v)V(q, u) (20.76)

with

xM")

Y.^ ^"^ ■

(20.77)

rr" nv

toy . ^ —

to-;

— u)

628

Chapter 20. Phenomenological Theory

Xc is not the dielectric susceptibility. It relates charge density to applied field,

rather than dipole moments to field. Its relation to the dielectric constant

is deter-

mined by writing

V = V

t +

47re«

= 47ren

ext

+ 47ren SeeEq.(20.i0). (20.78)

V = -V-D +

47T(?n

(20.79)

=>•

-q

V(q, to) = -iq-D +

47ren(q,

u) (20.80)

=* ~q

V{q, u) = -iq-b-4TT

{q, w)V(q, w) (20.81)

=►

(4vrXc-^

)^(^, w) = -iq-D (20.82)

=>•

ATTXC)E

= q(q

■

D) Because £ = -iV^V. (20.83)

For longitudinal vibrations, where q-D

—

qD, one therefore has

e(& w) =

- ^^ (20.84)

In this form, e(q,

LO)

is called the dynamic Lindhard dielectric function.

20.4.3 Many-Body Green Functions

The Born approximation is only the first step in formal schemes that track the

effects of perturbations. The most powerful formal methods employ Green's func-

tions similar to those in Section 18.4.2, but generalized to describe many interact-

ing electrons. These methods make it possible to obtain a variety of

results,

includ-

ing expressions for electrons interacting with light and with each other. Many of

the results can also be obtained by more elementary means, although the elemen-

tary derivations are not as systematic and are sometimes less convincing. Some of

the many books describing many-body Green functions are Kadanoff and Baym

(1962),

Abrikosov et al. (1965), and Fetter and Walecka (1971).

Problems

Numerical evaluation of Kramers-Kronig relations: While the Kramers-

Kronig relations in the form (20.39) are valuable for analytical work, they are

almost useless for practical evaluation of response functions. The integrals

are very singular and are hard to evaluate accurately.

One practical way to employ these relations is by returning to the arguments

from which they originally were derived. To illustrate the process, suppose

that the imaginary part of a dielectric constant has been measured and that the

experimental data give

ImeH = e

M « -^-«"A")

)

/* +

10e

-((-M)

-0

/5]

(20

.85)

with

u}\

= 4 Hz and w

= 13 Hz. (20.86)

The goal is to find Re[e(a;)] =

(a;).

Problems

629

(a) Recall that e(t)

real. Show that

(UJ)

an even function

and that

62(0;) is an odd function of

(b) Show that the inverse Fourier transform of e\{u) is an even function of

and

that the inverse Fourier transform of 62(0;) is an odd function of

(t) =

j^e-^e

(20.87)

how can

{t) =

^e-^'zxiu) (20.88)

be determined, using the fact that e(t)

0 for

< 0?

(d) Making use of these suggestions and employing numerical fast Fourier trans-

forms,

find the real part of the dielectric constant given by Eq. (20.85)

Connection between Kubo formula and Boltzmann equation:

It is

reas-

suring to verify that the Boltzmann equation does indeed emerge as the low-

frequency limit of the Kubo formula for the conductivity. Expression (20.67)

becomes somewhat problematic

this limit, because from Eq. (20.58)

the

vector potential and hence the current diverge as

—>

0. The dissipative pro-

cesses preventing this growth

can be

represented

taking

—> ui

i/r.

Making this substitution and sending the real part,

u, to

zero, verify that

Eq. (20.67) coincides with a result of the Boltzmann approach.

Cubic symmetry: Show that

a conductivity tensor

ß is

invariant under

the cubic symmetry operations, then it must be a multiple of the unit tensor.

Sums of

poles:

Consider the function

f(cv) =

J2 -•

(20-89)

LO[

- IT]

(a) Plot this function for

G [-1, 2] when

LO,

1/30 for

10~

and

0.1.

(b) What integral does the sum correspond to for the larger values of

77?

What is

the value of the integral, and does it compare well with the sum?

makes

qualitative transition between one type of behavior

for small

and another type

behavior

for

large

77.

What determines the

value of

at which the transition takes place?

(d) What does this problem reveal about the relation between eigenvalues

of a

quantum mechanical Hamiltonian and the idea of a density of states? What is

the significance of

77?

630 Chapter 20. Phenomenological Theory

Susceptibility

the electron gas:

(a) Writing the electron density

defined

Eq. (20.74)

in k

space, show that

n(q,

t) = j

dre-

^n{7,

t) = £

fi(ï\qi)(q + qi\ï) (20.90)

(b) Consider

potential

the form

U(r,

t) =

U(q,

Lü)e^

iu;t

/V.

(20.91)

Take

the

unperturbed states

|/) to be

plane waves. Show that

Eq.

(20.56)

produces

for n

eU(q,

u) y^ 1

**ft

k+q

-^k+q

+UJ

^k-^l+q-^

(20.92)

To obtain this result, drop

term proportional

8$$

order

focus upon

spatially nonuniform conditions.

and

(20.77).

6. Static Lindhard dielectric function:

(a) By relabeling the first sum

Eq. (20.77), show that

for

= 0,

—

y /_^

fk~

)

q-k

(20.93)

(b) Transform

Eq.

(20.93) into

integral over

k in

polar coordinates. First

integrate over

cos 6, and

next integrate over

from

0 to kf. The

answer

should

me2(

44-*

)log(f^f)

= -^ö ^-ql _

(m94)

—>

and show that

me kf

(20.95)

Screening

bare nuclei:

The

Lindhard dielectric function,

Eq.

(20.84),

can

used

in a

wide variety

physical contexts, both dynamic

and

static,

to solve problems involving

the

interaction

many electrons. Consider,

for

example,

nucleus

charge

surrounded

by a

cloud

Z electrons, which

produces

potential

U for

the purposes

Eq.

(20.73) equal

ext

(q)

4-irZe

This

is the

Fourier transform

the Coulomb

potential —Ze

/r.

(20.96)

References 631

(a) Show that Eq. (20.83) can be rewritten as

jj/- \

e]i

t(q,w)

/inn-7\

w =

"i—A

/~2' (20.97)

where U is the total potential produced by the nucleus together with the cloud

of surrounding electrons.

(b) Using the result of Problem 6, show that

Lu{q = 0) = ~E

, (20.98)

where 0 is the volume of the unit cell, and tip is the Fermi energy h

/2m

appropriate for Z free electrons per unit cell.

References

A. A. Abrikosov, L. P. Gor'kov, and I. Y. Dzyaloshinskii (1965), Quantum Field Theoretical Methods

in Statistical Physics, 2nd ed., Pergamon Press, Oxford.

A. Einstein (1905), The production and transformation of light: A heuristic point of view, Annalen

der Physik (Leipzig), 17, 132-148. In German.

A. Fetter and J. D. Walecka (1971), Quantum Theory of Many-Particle Systems, McGraw-Hill, San

Franci

so.

H. Goldstein (1980), Classical Mechanics, 2nd ed., Addison-Wesley, Reading, MA.

O. Heaviside (1892), Electrical

Papers,

London, Macmillan.

H. Hertz (1887), The effect of ultraviolet light on an electrical discharge, Annalen der Physik

(Leipzig),

31, 983-1000. in German.

Y. Il'inskii and L. V. Keldysh (1994), Electromagnetic Response of Material Media, Plenum Press,

New York.

J. D. Jackson

(

1999),

Classical Electrodynamics, 3rd ed., John Wiley and Sons, New York.

L. P. Kadanoff and G. Bay m (1962), Quantum Statistical Mechanics; Green's Function Methods in

Equilibrium and Nonequilibrium Problems, W. A. Benjamin, New York.

J. J. Thomson (1897), Cathode rays, Philosophical Magazine, 44, 293-316.

Y. Yu and M. Cardona (1996), Fundamentals of Semiconductors, Springer-Verlag, Berlin.