is anisotropic, falling to zero in the four azimuthal directions where cos 2

vanishes. These are the directions in which the impurity wavefunction, and

the tunneling currents associated with it, extend farthest.

While the standard BCS analysis can be readily modi¬ed to handle d-wave

pairing, there are many other pieces of evidence to suggest that the BCS

formalism cannot be applied to high-temperature superconductors without

more drastic modi¬cation. Examples include the temperature dependence of

the speci¬c heat, the very small coherence length, and the large value of the

ratio 2Á°0Þ=kTc . In Section 7.5 the BCS prediction for this ratio was shown

to be 3.50, but the experimental result for HgBa2 Ca2 Cu3 O8 is 12.8 in the

direction that maximizes Á. It is clear that superconductivity is a phenom-

enon that can appear in a number of forms, and whose complete explanation

requires a correspondingly wide range of theoretical approaches.

Problems

7.1 Would you characterize the BCS theory as a mean ¬eld theory in the

sense discussed at the beginning of Section 3.11? If so, at what stage is

the mean ¬eld approximation introduced?

7.2 An alternative approach to the ¬nite-temperature theory of Section 7.5

involves minimizing the Helmholtz energy, hHBCS i À TS, with respect

to both xk and fk , with the entropy given by

X

S ¼ À2k ½ fk ln fk þ °1 À fk Þ ln°1 À fk Þ:

k

Show that this approach leads to the same expression for Á°TÞ as given

in Eq. (7.5.8).

7.3 Anderson™s pseudospin formulation of the BCS theory starts by trans-

y y

forming from the pair creation operators b k ¼ c k" cy to operators de-

Àk#

y y

¬ned by 2sx °kÞ ¼ b k þ bk ; 2sy °kÞ ¼ i°b k À bk Þ; 2sz °kÞ ¼ 1 À nk" À nÀk# .

Verify that in units where 0 ¼ 1 these operators have the commutation

properties of spins as de¬ned by Eq. (3.10.11).

7.4 Verify that when the transformation of Problem 7.3 is substituted in the

BCS Hamiltonian (7.2.4) one ¬nds a result of the form

X

HBCS ¼ À H°kÞ Á s°kÞ

k

283

Problems

where H is an ˜˜e¬ective pseudomagnetic ¬eld™™ given by

X

X

^

Vkk 0 sx °k 0 Þ; Vkk 0 sy °k 0 Þ; 2E k :

H°kÞ ¼

k0 k0

[The energy of this system can now be minimized by arguing in analogy

with the theory of domain walls in ferromagnets.]

7.5 Prove that no material can have a macroscopic magnetic susceptibility

more negative than À1=4. [Hint: consider a long cylinder held parallel

to an applied ¬eld, and plot Bz as a function of position.]

7.6 Show that the fourth-order o¬-diagonal terms omitted from Eq. (7.3.8)

have a negligible expectation value in the BCS ground state.

7.7 How does the electronic speci¬c heat of a superconductor vary with

temperature T as T tends to zero?

7.8 Provide the missing steps in the calculation that leads from Eq. (7.7.4)

to Eq. (7.7.5).

7.9 The operator that describes the spin magnetic moment of the electron

gas is

X

M ¼ B °nk" À nk# Þ:

k

In the ground state of the BCS superconductor, the magnetization

vanishes. What is the minimum energy needed to create a state for

which the expectation value of M is 2B ?

7.10 An exception to our general statement that an electron annihilation

y

operator ck only acts in partnership with a creation operator ck arises

if we introduce a positron to the system. Consider a positron at rest in a

BCS superconductor at zero temperature annihilating with an electron

to produce two photons of total momentum 0k1 . What is the di¬erence

between the total energy they would have in the superconductor and

that in a normal metal?

7.11 Now raise the temperature in problem 7.10 to T1 , so that Á is reduced

to Á°T1 Þ. There are now two possible answers to the problem. What are

284 Superconductivity

they, and what is the ratio of the probabilities of occurrence of these

two answers?

7.12 Calculate the electronic speci¬c heat C°TÞ for a BCS superconductor in

the limit as T ! Tc , and hence ¬nd the ratio of C°Tc À Þ to C°Tc þ Þ

as ! 0.

7.13 How does the penetration depth vary with temperature in a BCS

superconductor as T ! Tc ? Express your answer in terms of T; Tc ,

and °T ¼ 0Þ.

Chapter 8

Semiclassical theory of conductivity in metals

8.1 The Boltzmann equation

We now have at our command many of the ingredients of the theory of the

conduction of heat and electricity. In Section 3.9 we considered the heat

current operator for phonons in a lattice, and in Section 4.6 we calculated

the velocity of Bloch electrons and their dynamics in applied ¬elds. The

missing ingredients of the theory of the transport of heat or electricity, how-

ever, are the statistical concepts necessary to understand such irreversible

processes. In this chapter we shall adopt the simplest attitude to these statis-

tical problems, and begin with a discussion of the probable occupation number

of a given phonon mode or Bloch state.

Let us start by considering a system of independent phonons or electrons.

We know that we can de¬ne operators nq and nk whose eigenvalues are

integers. If, for instance, there are three phonons of wavenumber q 0 present

then the expectation value hnq 0 i of the operator nq 0 will be equal to three. If the

crystal is not in an eigenstate of the Hamiltonian, however, then hnq 0 i may

take on some nonintegral value. If we wish to discuss the thermal conductiv-

ity of the lattice we should have to interpret the idea of a temperature gra-

dient, and this must certainly involve some departure from the eigenstates of

the lattice. We are thus obliged to consider linear combinations of di¬erent

eigenstates as describing, for example, a lattice with a temperature gradient.

Because there will be many di¬erent combinations of eigenstates that all give

the appearance of a crystal with a temperature gradient we shall only have a

very incomplete knowledge of the state of any particular crystal. We can,

however, discuss the average results fq and fk that we expect to ¬nd when

we make measurements of nq and nk , respectively, and can proceed to use

semi-intuitive methods to derive equations that will govern their variation in

time.

285

286 Semiclassical theory of conductivity in metals

If we are to include the possibility of temperature gradients we shall have to

allow fq and fk to be functions of position as well as of wavenumber. This

appears self-contradictory, in that phonon and Bloch states are not localized,

and so one cannot attempt to specify even the approximate location of a

particle if one knows its wavenumber exactly. We must thus sacri¬ce some

precision in determining the wavenumber if we allow fq and fk to vary with

position. This need not be a serious limitation provided we restrict ourselves

to slowly varying functions in r-space. If for instance we consider a tempera-

ture that varies as

T°rÞ ¼ T0 þ T1 cos °qT Á rÞ

then we must make qT small enough that the concept of a local temperature

is valid. We must certainly have a high probability that all the excitations

have many collisions in traveling a distance qÀ1 so that they are properly

T

˜˜thermalized.™™

We now consider the equation of continuity for the function f , which can be

the probable occupation number of either electrons or phonons. This relates

the rate of change of f in the absence of collisions to the number of particles

leaving an element of volume of six-dimensional k-r space. By a simple

generalization of the usual three-dimensional version we ¬nd

@f @ @

dr dk

¼À Á f À Áf :

@t @r @k

dt dt

Upon di¬erentiation this becomes

@f @f dk @f @ @ dk

¼ Àv Á À Á Àf Áv þ Á : °8:1:1Þ

@t @r dt @k @r @k dt

The term in brackets vanishes both for phonons and for Bloch electrons in

applied ¬elds. For phonons this is obvious, since v ¼ @!=@q, and does not

depend on position while k ¼ q, the wavenumber, and is constant. For elec-

trons

dk e

¼ °Ec þ v ‚ HÞ;

dt 0c

and the fact that 0v ¼ @E=@k ensures this result. What remains of Eq. (8.1.1)

constitutes the Liouville equation. We now specialize to consider only the

steady state, in which @f =@t vanishes, but we add to the right-hand side a

287

8.1 The Boltzmann equation

term to allow for changes in f due to collisions. We then have a form of the

Boltzmann equation, which states

@f @f dk @f

À vÁ À Á ¼ 0:

@t collisions @r dt @k

The spatial variation in f may be attributed solely to a variation in tempera-

ture, giving

@f @f

¼ rT:

@r @T

We then have the following equations for the phonon and electron systems:

@fq @fq

@!

¼ Á rT °8:1:2Þ

@t collisions @q @T

@fk 1 @E @fk @fk

e

¼ Á rT þ °Ec þ v ‚ HÞ Á : °8:1:3Þ

@t collisions 0 @k @T 0c @k

In general these two equations are coupled, since in the presence of the

electron“phonon interaction the scattering probability for the electrons

depends on the phonon occupation function, and conversely.

Before proceeding to an investigation of the solution of these equations in

particular circumstances, we remind ourselves of the fact that applied electric

¬elds and temperature gradients are usually rather small. If one is measuring

thermal conductivity, for example, one usually wishes to determine this quan-

tity as a function of temperature, and no accurate result could be obtained if

the two ends of the sample were at widely di¬erent temperatures. The result-

ing currents of heat and electricity are consequently linear in E and rT. This

leads us to a linearization of Eqs. (8.1.2) and (8.1.3). We expand f in powers

of E and rT, and keep only the ¬rst two terms so that

f ™ f 0 þ f 1:

Here f 0 will represent the distribution in the equilibrium situation, and will

be given by the Bose“Einstein function for phonons and the Fermi“Dirac

function for electrons. By de¬nition °@f 0 =@tÞcollisions vanishes. We also note

that for electrons f 0 depends on k only through the function E k , and so

@fk

0

@f 0 @E

¼ :

@k @E @k

288 Semiclassical theory of conductivity in metals

It thus follows that the term v ‚ H Á °@f 0 =@kÞ also vanishes. On neglect of

terms of second order of smallness we are then left with

@fq1 @fq0

¼ vq Á rT °8:1:4Þ

@t @T

collisions

@fk1

@fk

1

@f 0 @f 0

e

¼ v ‚ HÁ þ vk Á rT þ eE : °8:1:5Þ

@t 0c k @k @T @E

collisions

This is as far as we can conveniently go in simplifying the Boltzmann equa-

tions without specializing to a consideration of speci¬c models and situa-

tions. We shall now proceed to investigate a few of these and attempt to

see how some of the simple geometrical ideas such as the mean free path can

be rescued from the complexity of the Boltzmann equation.

8.2 Calculating the conductivity of metals

We now specialize to consider the electrical conductivity of a metal in which

the electrons are scattered elastically by a random array of n impurities. We

argue that if the positions of the scattering centers are not correlated in any

way, then we can neglect coherent scattering by the array of impurities as a

whole, and assume that the scattering probability between Bloch states is just

the scattering probability due to a single impurity multiplied by n. This

argument will be valid if the density of the impurities is low, as we can

then consider our calculation as ¬nding the ¬rst term of an expansion of

the resistivity in powers of the impurity density n. While the function °nÞ

turns out not to be analytic at n ¼ 0, the derivative d=dn does exist at this

point, and so for small enough n we can write the scattering probability

following Eq. (4.7.3) as

2

Q°k; k 0 Þ ¼ n jTkk 0 j2 °E k À E k 0 Þ:

0

The rate of change of fk will be the average net number of electrons entering

state k from all other states k 0 . Making allowances for the Exclusion

Principle, which will prevent an electron from entering the state k if it is

already occupied, we ¬nd

X

@fk

½ fk 0 Q°k; k 0 Þ°1 À fk Þ À fk Q°k 0 ; kÞ°1 À fk 0 Þ:

¼

@t k0

collisions

289

8.2 Calculating the conductivity of metals

It was shown in Section 4.7 that Q°k; k 0 ) was equal to Q°k 0 ; k), and so

X

@fk

Q°k; k 0 Þ° fk 0 À fk Þ:

¼

@t k0

collisions

Because energy is conserved Q°k; k 0 ) vanishes unless E°kÞ ¼ E°k 0 Þ, and as fk

0

depends only on E°kÞ we immediately verify that

X

Q°k; k 0 Þ° fk 0 À fk Þ ¼ 0:

0 0

k0

We may thus write

1

@fk @fk

¼

@t @t collisions

collisions

X

Q°k; k 0 Þ° fk 0 À fk Þ:

¼ 1 1

k0

In the absence of a temperature gradient the linearized Boltzmann equation

(8.1.5) then becomes

X @fk

1

@fk

0

e

0

Þ° fk 0

1

À fk Þ

1

À vk ‚ H Á ¼ eE Á vk : °8:2:1Þ

Q°k; k

0c @k @E

k0

1

The changes, fk , that occur in f due to the action of the electric ¬eld are, of

course, con¬ned to the region of k-space in the vicinity of the Fermi surface.

This is evident from the presence of the factor @fk =@E on the right-hand side

0

of (8.2.1). For Fermi“Dirac statistics

@fk

0

@ 1

¼

@E @E exp ½°E k À Þ=kT þ 1

exp ½°E k À Þ=kT

¼À ;

kT°exp ½°E k À Þ=kT þ 1Þ2

which may be rewritten in the form

@fk

0

fk °1 À fk Þ

0 0

¼À :

@E kT

290 Semiclassical theory of conductivity in metals

In the limit of low temperatures

@fk

0

! À°E k À E F Þ; °8:2:2Þ

@E

the Fermi energy E F being equal to the chemical potential at zero tempera-

1

ture. It is then clear that fk is too rapidly varying a function for convenience

1

in computation. We also note that the statement that fk is linear in E means

that the total change in fk due to the ¬eld (Ex , Ey , Ez ) is the sum of the

1 1 1

changes fk;x , fk;y , and fk;z that would be produced by the three components of

E acting separately. It is then convenient to de¬ne quantities Ãk;x , Ãk;y , and

Ãk;z that satisfy

@fk

0

1

¼ ÀeEx Ãk;x

fk;x

@E

1 1

and similarly for fk;y and fk;z . More brie¬‚y we state that a vector ,k exists

such that

@fk

0

1

¼ ÀeE Á ,k °8:2:3Þ

fk

@E

and which is independent of E. Equation (8.2.1) then becomes

0

X @fk 0

0

@fk

0

À eE Á Q°k; k Þ ,k 0 À ,k

@E @E

0

k

@ @fk

0

@fk

0

e

þ v ‚ HÁ eE Á ,k ¼ eE Á vk :

0c k @k @E @E

Now since fk 0 ¼ fk for elastic scattering we may take the derivatives of fk

0 0 0

outside the summation. We also note that

0

@ @fk

vk ‚ H Á ¼ 0;

@k @E

which allows us to cancel terms in eE and @fk =@E when we substitute back

0

into (8.2.1). We are left with the equation

X @

e

0

Q°k; k Þ°,k À ,k 0 Þ þ v ‚ HÁ ,k ¼ vk : °8:2:4Þ

0c k @k

0

k

291

8.2 Calculating the conductivity of metals

This bears no reference to the Fermi energy or to the temperature, and thus

may be expected to lead to a smoothly varying function as its solution for the

quantity ,k .

Our object in solving the Boltzmann equation in this instance is the evalua-

tion of the electrical conductivity tensor p in terms of the band structure and

the scattering probabilities. We have already noted that the current density

due to a single Bloch electron is equal to evk =, and we may then write the

total current density j as

X

À1

j¼ evk fk

k

X

¼ À1 evk fk ;

1

k

since in equilibrium no net current will ¬‚ow. With use of (8.2.3) this becomes

e2 X @fk

0

j¼À v °, Á EÞ :

k kk @E

As p is de¬ned by

j ¼ pÁE

we see that

e2 X @fk

0

p¼À : °8:2:5Þ

v,

k k k @E

This expression includes a summation over the two spin directions.

At this point we can make a connection with some pictorial concepts of

transport theory by considering the conductivity of the simplest possible

system “ the free-electron gas in the absence of a magnetic ¬eld. In this

case the conductivity tensor is by symmetry diagonal and

xx ¼ yy ¼ zz ¼ 1 Tr p

3

e2 X @fk

0

¼À v Á, :

3 k k k @E

Also by symmetry v and , must always be parallel to k and of uniform

magnitude over the Fermi surface. Because thermal energies are small com-

pared with the typical Fermi energy we adopt (8.2.2) and replace @fk =@E

0

292 Semiclassical theory of conductivity in metals

by À°E k À E F Þ to ¬nd

e2 vÃ X

xx ¼ °E k À E F Þ

3 k

e2 vÃ

¼ D°E F Þ:

3

p¬¬¬

We recall that the energy density of states D°EÞ is proportional to E in the

free electron gas, and that N, the total number of electrons, is given by

°EF

N¼ D°EÞ dE:

0

Thus

3N 3N

D°E F Þ ¼ ¼

2E F mv2

and

Ne2 Ã

xx ¼ :

mv

This is similar to a formula of elementary kinetic theory which expresses the

conductivity in terms of a mean free path or a relaxation time , de¬ned

such that À1 is the probability per unit time of an electron having a collision

in which it loses any momentum gained from the electric ¬eld. In such a

theory one argues that at any instant an average electron has been traveling

for a time since its last collision, and hence has an average extra velocity of

eE=m. The system thus has a conductivity

Ne2

¼ ;

m

which is equivalent to our previous expression when Ã is identi¬ed with v.

Looking back to Eq. (8.2.5) we see that in contrast to these simple ideas the

solution of the Boltzmann equation in the more general circumstances of a

metal with a nonspherical Fermi surface cannot, in general, be expressed in

terms of a relaxation time. It is, however, an extremely useful approximation

to make when the detailed nature of the scattering is not thought to be

central to the problem under investigation. This may, for example, be the

293

8.2 Calculating the conductivity of metals

case in multilayered systems, such as metallic or semiconducting superlat-

tices. These consist of a sequence of layers of two or more di¬erent materials,

with each layer perhaps a few nanometers thick. At the interface between two

di¬erent layers, there is in general a mismatch in the electronic band struc-

ture. Disorder due to interdi¬usion of atoms across the interface may also

lead to signi¬cant scattering. A lack of precise information about the form of

the interface scattering can then make it unnecessary to worry about the

lesser errors introduced by the relaxation-time approximation, which corre-

sponds to writing

@fk 1

fk

¼À

@t

collisions

in the Boltzmann equation. When this approximation is not valid one must

return to a calculation of ,k , which is known as the vector mean free path of

the Bloch electrons, it being a natural generalization of the of kinetic

theory.

When the system is spatially inhomogeneous on a macroscopic scale, there

will be spatial derivatives in the Boltzmann equation. As a consequence, the

distribution function will depend on r as well as on k, in mild violation of the

Uncertainty Principle as discussed at the beginning of this chapter. The

simplest such system is a thin ¬lm, which also happens to be an important

subject in such practical applications as the technology of integrated circuits.

The conducting paths connecting di¬erent circuit elements are ¬lms of Cu or

Al with thicknesses ranging from a few to a few tens of nanometers. In order

to model the transport properties of such thin ¬lms correctly, we must

include scattering at the boundaries.

With the ¬lm thickness d along the z-axis and with an electric ¬eld along

the x-axis, the linearized Boltzmann equation is

@fk °zÞ

1

@fk

0

fk °zÞ

1

þ eEvx ¼À :

vz

@z @E

For simplicity, we are here using the relaxation-time approximation and are

also assuming that the ¬lm is homogeneous in the xy plane. We again remove

1

the inconveniently rapidly varying part of fk by de¬ning a new smoother

function hk °zÞ, just as we did in Eq. (8.2.3), and writing

@fk

0

fk °zÞ

1

¼ hk °zÞ :

@E

294 Semiclassical theory of conductivity in metals

The Boltzmann equation in terms of hk °zÞ is then

@hk °zÞ hk °zÞ v eE

þ ¼À x :

@z vz vz

In order to solve this equation we shall need to stipulate the boundary con-

ditions for hk °zÞ. A simple set of such conditions, which seems to work rather

well in most practical applications, was given by Fuchs and Sondheimer.

Their idea was that an electron incident on a boundary has a probability S

of being specularly re¬‚ected. In such a re¬‚ection, the momentum of the

electron parallel to the surface is conserved, and the momentum perpendi-

cular to the boundary changes sign but preserves its magnitude. The energy

of the electron is conserved. In addition, there will be nonspecular scattering

due to imperfections at the surface, and so we assume that the electron has a

¬nite probability D ¼ 1 À S of being di¬usively re¬‚ected in a process in

which only energy is conserved as the electron moves away from the surface

after the re¬‚ection. In general, we may not know precisely how the electron is

scattered di¬usively. Usually one assumes that the distribution after a di¬use

re¬‚ection is the equilibrium distribution for electrons moving away from the

boundary. This is convenient for subsequent calculations of the total current,

since the electrons that have scattered di¬usively do not contribute to the total

current. With these Fuchs“Sondheimer boundary conditions, the solutions

will be di¬erent for electrons traveling in the positive and negative z-direc-

tions. Let us denote by hþ °zÞ and hÀ °zÞ the distribution functions of electrons

k

k

of wavevector k and position z with vz > 0 and vz < 0, respectively. We can

then write

@hÆ °zÞ hÆ °zÞ v eE

k

Æk ¼Ç x :

@z jvz j jvz j

The general solution for hÆ °zÞ is

k

hÆ °zÞ ¼ Àevx E½1 À Fk eÇz=°jvz jÞ :

Æ

k

Æ

The coe¬cients Fk are to be determined using the boundary conditions.

These are

hþ °z ¼ 0Þ ¼ ShÀ °z ¼ 0Þ

k

k

hÀ °z ¼ dÞ ¼ Shþ °z ¼ dÞ:

k k

295

8.3 Effects in magnetic fields

We can then calculate the current density and conductance of the system.

Here we state only the qualitative result without going through the details. If

the specularity coe¬cients at the boundaries are not unity, there is current

lost near the boundaries due to di¬use re¬‚ections. If the thickness d is less

than or of the order of the elastic mean free path, the e¬ect is an apparent

increase in the resistivity of the material. As the system thickness d becomes

greater than the elastic mean free path, the increase in resistivity becomes less

and less important, and the resistivity approaches that of bulk material. As

an example, in Cu the mean free path at room temperature is of the order of

20 nm. Therefore, the increase in resistivity can be quite appreciable for ¬lms

of thickness 10 nm or less. We note that if S ¼ 1, so that the electrons are

re¬‚ected perfectly at the boundaries, a ¬lm of any thickness will have the

same e¬ective resistivity as bulk material since in this case there is no reduc-

tion in current density due to di¬use re¬‚ection at the boundaries. In real

applications, the specularity coe¬cient S is generally quite low (close to

zero) at the interface between a metal and either vacuum, metal, or insulator

because of roughness and di¬usion across the interface.

8.3 Effects in magnetic ¬elds

The presence of the magnetic ¬eld term in the Boltzmann equation is a great

complication, and makes the evaluation of the conductivity tensor a di¬cult

task. The k-vectors of the Bloch electrons now follow orbits around the

Fermi surface as described in Section 4.6, until they are scattered to some

new k-state to start their journey again. At the same time they are accelerated

by the electric ¬eld, but then have their extra velocity reversed as the mag-

netic ¬eld changes their direction of motion. We shall here outline in the

briefest possible manner the formal solution for the conductivity tensor,

and then indicate a few qualitative conclusions that may be drawn from it.

Let us ¬rst abbreviate the Boltzmann equation (8.2.4) by noting that the

left-hand side is a sum of two terms, each of which represents an operator

acting on ,. We could thus rewrite (8.2.4) as

S,k À i!W,k ¼ vk °8:3:1Þ

with S and W operators and ! the cyclotron frequency, eH=mc. In the

absence of a magnetic ¬eld the vector mean free path is then the solution of

S,0 ¼ vk :

k

296 Semiclassical theory of conductivity in metals

If we de¬ne a new operator T equal to S À1 W then we can rewrite (8.3.1) as

°1 À i!TÞ,k ¼ ,0 : °8:3:2Þ

k

The operator T has a complete set of eigenfunctions and real eigenvalues l ,

and so we can solve by expanding ,0 in terms of these. This allows us to

k

invert (8.3.2), substitute the solution for ,k into the expression (8.2.5) for the

conductivity, and ¬nd an expression of the form

X °lÞ

¼ ; °8:3:3Þ

1 À i!l

l

°lÞ

where the

are numbers that depend on the band structure, the direction of

H, and the form of the scattering, but are independent of the magnitude of H.

If we choose axes so that when H ¼ 0 the conductivity tensor is diagonal it

°lÞ °lÞ

happens that the diagonal

are real while the o¬-diagonal

are pure

imaginary. One can then make simpli¬cations of the form

X X i!l

°lÞ

°lÞ

¼ ; ¼ : °8:3:4Þ

1 þ !2 l2 1 þ !2 l2

°6¼Þ

l l

An added complication that has to be considered is that in an experiment the

current is constrained by the boundaries of the sample to lie in a certain

direction. The total electric ¬eld acting on the sample must then be in such

a direction that p Á E conforms with the sample geometry. There are thus

electric ¬elds set up in the sample whose direction is not within the control

of the experimenter, and it is these that must be measured when a given

current is ¬‚owing. One thus measures the resistivity tensor, o, which is the

inverse of the conductivity tensor.

The diagonal elements of p are functions of !2 , and are thus unchanged

when the magnetic ¬eld is reversed. The same is true for the diagonal ele-

ments of o. One de¬nes the magnetoresistance Á as the increase in as a

function of the magnitude and orientation of H. This quantity is propor-

tional to !2 at low ¬elds. When H is in the -direction then °!Þ À °0Þ is

said to be the longitudinal magnetoresistance, while the transverse magne-

toresistance is measured with H ¼ 0. The fact that the magnitude of H

occurs only in the combination !l gives rise to the approximation known

as Kohler™s rule. One argues that if Q°k; k 0 ) were increased in the same

proportion as H the products !l would be unchanged, and the magneto-

resistance would remain the same proportion of the zero-¬eld resistance.

297

8.3 Effects in magnetic fields

Thus one should be able to ¬nd some function F such that for a wide range of

impurity concentration

H

Á ¼ °0ÞF :

°0Þ

In practice deviations occur from this rule, as it is not possible to alter the

scattering without altering various other factors such as the electron veloci-

ties.

The o¬-diagonal elements of p are odd functions of !, and thus are

changed in sign when the magnetic ¬eld is reversed. The o¬-diagonal

elements of o, on the other hand, are neither odd nor even in !. In the

particular case where H is perpendicular to j the presence of these terms

constitutes the Hall e¬ect. Let us now change to a coordinate system in

which H is in the z-direction and j in the y-direction, and expand xy in

powers of Hz , so that

xy °!Þ ¼ xy °0Þ þ RHz þ SHz þ Á Á Á :

2

2

In this expression R is known as the Hall coe¬cient, while the term in Hz is

responsible for the so-called transverse-even voltage, which does not change

sign when the magnetic ¬eld is reversed. One sometimes calls S the trans-

verse-even coe¬cient for low ¬elds.

Some interesting e¬ects occur at high ¬elds, where from (8.3.4) it appears

at ¬rst glance that all elements of p become small. This will be the case unless

one of the l should be equal to zero, in which case the diagonal elements of p

°0Þ

will tend to a constant value . To see when this situation will arise we look

back to the de¬nitions of T and W in (8.3.2), (8.3.1), and (8.2.4). The pre-

sence of a term equivalent to H ‚ @=@k in W (and hence in T) gives the

obvious answer that T acting on a constant always vanishes. However, if

we are to expand ,°0Þ in eigenfunctions of T we do not expect to ¬nd any

constant vector as a component of ,°0Þ , since by symmetry there is no pre-

ferred direction on an orbit of the type in Fig. 4.6.5. However, when we

look at the open orbits

, we see that it would be possible for ,°0Þ to be a

di¬erent constant on each part of the orbit without violating any symmetry

requirements. Physically we could say that the magnetic ¬eld is incapable of

reversing the velocity of the electrons on these orbits, and so they contribute

an anomalously large amount to the conductivity in high magnetic ¬elds. The

transverse magnetoconductivity at high ¬elds can thus give information

about the topology of the Fermi surface. We also note that symmetry does

298 Semiclassical theory of conductivity in metals

not exclude constant components of ,0 in the direction of H, as shown in

k

Fig. 8.3.1, from contributing to the current. This means that when H is in the

z-direction, zz will again tend to a constant at large magnetic ¬elds.

We note, incidentally, that the free-electron metal with an isotropic scatter-

ing probability is a pathological case and is of little use as a model in which to

understand magnetoresistance. This arises from the fact that ,°0Þ is composed

of only two eigenfunctions of T in this case, and they both share the same

eigenvalue , which we can identify with the relaxation time de¬ned as in

Section 8.2. Then, because the conductivity is of the form

0 1

À!

1

B 1 þ !2 2 1 þ !2 2 0 C

B C

B ! C

p ¼ 0 B C;

1

B 0C

@ 1 þ !2 2 1 þ !2 2 A

0 0 1

Figure 8.3.1 The component of the vector mean free path in the direction of the

applied magnetic ¬eld contributes a term to the conductivity tensor that always

causes the longitudinal magnetoresistance to saturate.

299

8.4 Inelastic scattering and the temperature dependence of resistivity

the resistivity tensor is

0 1

!

1 0

À1 B C

o ¼ 0 @ À! 0A

1

0 0 1

and shows no magnetoresistance. One must thus at least generalize to the case

of two parabolic bands to ¬nd a nonvanishing magnetoresistance.

8.4 Inelastic scattering and the temperature dependence of resistivity

The theory of the electrical conductivity of metals presented in Section 8.2

was based on the assumption that only elastic scattering occurred between

Bloch states. At temperatures di¬erent from zero this assumption will not be

valid, since then the electron“phonon interaction will cause electrons to be

scattered between Bloch states with the emission or absorption of phonons.

The matrix elements that appear in the expression for the scattering prob-

ability will be those of the electron“phonon interaction, and are functions of

the occupation of the phonon states as well as of the Bloch states. We shall

write the scattering probability between states to ¬rst order as

2

P°1; 2Þ ¼ jh1jHeÀp j2ij2 °E 1 À E 2 Þ °8:4:1Þ

0

where now j1i and j2i are descriptions of all the nk and all the nq of particular

many-body states, and E 1 and E 2 the corresponding energies. In lowest order

X

Mk 0 k cy 0 ck °aq þ ay Þ

HeÀp ¼ Àq

k

kk 0

with q ¼ k 0 À k, suitably reduced to lie within the ¬rst Brillouin zone. This

interaction changes the state of the metal by scattering the electron from k to

k 0 and either absorbing a phonon of wavenumber q or emitting one of

wavenumber Àq. The corresponding Born approximation for the scattering

probability is found on substitution in (8.4.1) to be

2

P°k; k 0 Þ ¼ jMkk 0 j2 hcy ck 0 cy 0 ck ay aq i°E k 0 À E k À 0!q Þ

0 k k q

2

¼ jMkk 0 j2 hnk °1 À nk 0 Þnq i°E k 0 À E k À 0!q Þ

0

300 Semiclassical theory of conductivity in metals

when a phonon is absorbed and

2

P°k; k 0 Þ ¼ jMkk 0 j2 hnk °1 À nk 0 Þ°1 þ nÀq Þi°E k 0 À E k þ 0!Àq Þ

0

when a phonon is emitted. The collision term in the Boltzmann equation for

the electrons (8.1.5) will then be of the form

X 2

@fk

¼À jMkk 0 j2 f fk °1 À fk 0 Þ½ fq °E k 0 À E k À 0!q Þ

@t collisions 0

k0

þ °1 þ fÀq Þ°E k 0 À E k þ 0!Àq Þ À fk 0 °1 À fk Þ

‚ ½ fÀq °E k 0 À E k þ 0!Àq Þ þ °1 þ fq Þ°E k 0 À E k À 0!q Þg:

This very complicated expression can be simpli¬ed by a number of steps, but

still remains di¬cult to interpret even within the framework of the free-

electron model. One customarily assumes the phonon distribution to be in

equilibrium so that fq may be replaced by the Bose“Einstein distribution fq0 .

A numerical solution of the Boltzmann equation with H ¼ 0 then shows that

the mean free path ,, de¬ned as before, has a ˜˜hump™™ in it at the Fermi

surface (Fig. 8.4.1). This re¬‚ects the fact that it is no longer possible to

eliminate the chemical potential from the Boltzmann equation.

We shall leave the details of such a calculation to the more specialized

texts, and simply examine the qualitative nature of the scattering and the

Figure 8.4.1 The inelastic nature of the scattering of electrons by phonons causes the

mean free path to be a few per cent greater within the thermal thickness of the Fermi

surface than elsewhere.

301

8.4 Inelastic scattering and the temperature dependence of resistivity

Figure 8.4.2. Scattering of an electron by absorption or emission of a phonon takes

the electron onto one of the two surfaces de¬ned by the energy-conservation relation

E k 0 ¼ E k Æ 0!q .

electrical resistance it causes. Firstly we note that an electron in state k is not

scattered onto a surface of constant energy E k 0 ¼ E k , but onto one of

two surfaces slightly displaced in energy from it, so that E k 0 ¼ E k Æ 0!q

(Fig. 8.4.2). If we were to make the approximation that the scattering is elastic,

we should misrepresent the e¬ect of the Exclusion Principle, in that some

phonon emission processes are forbidden because of the low energy of the

¬nal electron state. However, only electrons within about kT of the Fermi

surface can be scattered, and they would have to lose more than kT in energy

to be scattered into a region where fk 0 was close to unity. Since we do not expect

to ¬nd many phonons present in equilibrium with energy more than kT, we can

argue that the approximation only introduces a small error. Secondly we recall

that the scattering matrix element will be proportional to the change in density

in the sound wave, i.e., to qyq , where yq is the amplitude. For a harmonic

oscillator the average potential energy is half the total, and so

2 °nq þ 1Þ0!q $ 1 m!2 y2 ;

1

2 2

or

1=2

nq

qyq / q

!q

/ °qnq Þ1=2

for long waves, for which !q / q. Thus at low temperatures, when long waves

are most important, we can make the approximation

Q°k; k 0 Þ / qfq0 °E k À E k 0 Þ

302 Semiclassical theory of conductivity in metals

where

q ¼ jk 0 À kj:

In calculating ,k from the Boltzmann equation we should then write

X

q fq0 °E k À E k 0 Þ°,k À ,k 0 Þ / vk :

k0

The principal contribution to the di¬erence between ,k and ,k 0 will be

a change in direction, and one may then argue that the contribution of

,k À ,k 0 to the sum will be about the same as ,k °1 À cos kk 0 Þ, where is the

angle between ,k and ,k 0 . For low temperatures it will only be small that

will be important, and then

1 À cos kk 0 ™ 1 kk 0

2

2

/ q2 :

Equation (8.2.1) is thus of the form

X

q3 fq0 °E k À E k 0 Þ / vk :

,k

k0

The delta-function in energy restricts the sum to a surface in k-space.

Changing the sum to an integral we ¬nd

°

q3 fq0 q dq / vk :

,k

Since fq0 is a function of 0!q =kT, which for small q is proportional to q=T, we

¬nd the integral over q to be proportional to T 5 ; then Ã, and hence the

conductivity, is proportional to T À5 . At high temperatures, on the other

hand, the Bose“Einstein function fq0 may be approximated by kT=0!q . The

detailed shape of the scattering probability is then unimportant, as the tem-

perature only enters the Boltzmann equation through the function fq0 . If fq0 is

proportional to the temperature then Ã must vary as T À1 . The resistivity of a

pure metal should thus be proportional to T 5 at low temperatures and to T at

high temperatures. This prediction appears to be veri¬ed experimentally

for the simple metals, but not for transition metals such as nickel, palladium,

platinum, rhenium, and osmium. In these elements electron“electron

303

8.4 Inelastic scattering and the temperature dependence of resistivity

scattering appears to play a major role in limiting the current and in leading

to a resistivity that varies as T 2 rather than T 5 .

When both impurities and phonons are present the total probability of

scattering from a Bloch state will be approximately the sum of the two

scattering probabilities taken separately. This is so because the inelastic scat-

tering by phonons will connect the state k with ¬nal states k 0 which are

di¬erent from those entered by elastic scattering. The two processes must

be considered incoherent, and one adds the scattering probabilities rather

than the scattering amplitudes. This leads to Matthiessen™s rule, which

expresses the idea that the electrical resistivity can be considered as a sum

of two independent parts, one of which is a function of the purity of the metal

and the other a function of temperature characteristic of the pure metal; i.e.,

¼ i þ 0 °TÞ:

The addition of further impurities to a metal is then predicted to displace the

curve of °TÞ and not to alter its shape (Fig 8.4.3).

In practice deviations of a few percent occur from this rule as a conse-

quence of a variety of e¬ects. Adding impurities, for example, may change

the phonon spectrum, the electron“phonon interaction, or even the shape of

the Fermi surface, while raising the temperature introduces the ˜˜hump™™ of

Fig. 8.4.1 in Ã, which, in turn, changes the resistance.

Figure 8.4.3 Matthiessen™s rule predicts that addition of impurities to a metal has the

e¬ect of increasing the resistivity by an amount that does not depend on the

temperature.

304 Semiclassical theory of conductivity in metals

8.5 Thermal conductivity in metals

One recognizes a metal not only by its large electrical conductivity but also

by its large thermal conductivity. This indicates that the electrons must play

an important role in the transport of heat “ a fact which is at ¬rst surprising

when one remembers the small heat capacity of the electron gas. The impor-

tance of the electrons lies in their long mean free paths and in the high

velocities with which they travel, which more than compensate for their

small heat capacity when compared with the phonon system.

Just as the electric current density was calculated from the expression

eX

j¼ v f; °8:5:1Þ

k kk

so one can write the energy current density u 0 as

X

0 À1

u ¼ E k vk fk :

k

This, however, is not the same thing as the heat current density u of the

electrons, as can be seen by picturing the arrival at one end of a piece of

metal of an electron of zero energy. As the only unoccupied k-states would be

those with energy close to E F , all the thermally excited electrons would have

to donate a small amount of their thermal energy to the new arrival, with the

net result that the electron gas would be cooled; a zero-energy electron thus

carries a large amount of coldness! We consequently have to measure E k

relative to some carefully chosen reference energy. This is the same problem

that we encountered in Section 3.3. There we saw that the appropriate zero of

energy is the chemical potential ; adding an electron of energy to the

metal does not change the temperature of the system. This can be stated in

thermodynamic terms by noting that

@F

¼ ;

@N

with F the Helmholtz energy. We accordingly write the heat current density

due to the electrons as

X

u ¼ À1 °E k À Þvk fk : °8:5:2Þ

k

305

8.5 Thermal conductivity in metals

The ¬‚ow of heat resulting from the presence of any combination of ¬elds

and temperature gradients may now be calculated from the linearized

Boltzmann equation (8.1.5). The thermal conductivity, for example, may

be found by putting H and E equal to zero to obtain the equation

0

@fk @fk

1

¼ vk Á rT : °8:5:3Þ

@t @T

collisions

Strictly speaking, the thermal conductivity tensor i is de¬ned by the equation

u ¼ Ài Á rT °8:5:4Þ

under the condition that j rather than E be equal to zero. This only adds a

very small correction, however (Problem 8.5), and so we shall neglect it here.

We then proceed in analogy with the discussion of electrical conductivity

given in Section 8.2. This time we de¬ne a vector mean free path by putting

rT @fk

0

1

¼ °E k À Þ Á ,k : °8:5:5Þ

fk

@E

T

Then since

@fk

0

E À @f 0

¼À ;

@T @E

T

we ¬nd that when we consider only elastic scattering all the arguments used

in Section 8.2 apply and we end up with the identical equation for Ãk . Since

from Eqs. (8.5.2) and (8.5.5) the heat current density is

X rT @fk

0

À1

u¼ °E k À Þ vk 2

Á ,k ; °8:5:6Þ

@E

T

k

then from Eq. (8.5.4)

1X 2 @fk

0

i¼À v , °E À Þ : °8:5:7Þ

T k k k k @E

The fact that for elastic scattering it is identically the same ,k that occurs in

Eq. (8.5.7) as occurred in the Eq. (8.2.5) that de¬ned the electrical conduc-

tivity leads us to the remarkable Wiedemann“Franz law. We recall from

306 Semiclassical theory of conductivity in metals

Section 2.1 that a summation over allowed values of k can be replaced by an

integration over k-space, so that for any well behaved function A°kÞ

°°

X 2 dSk

A°kÞ ¼ ; °8:5:8Þ

A°kÞ dE

0vk

°2Þ3

k

the ¬rst factor of 2 arising from the sum over spin directions, and dSk

representing an element of area of a surface in k-space of constant energy

E. If the functions vk and ,k do not have any unusual kinks at the Fermi

surface, then when Eq. (8.5.7) is changed into a double integral in the manner

of Eq. (8.5.8) the result may be factorized to obtain

° °

@f 0

1 vk ,k

i¼À 3 °E À Þ 2

°8:5:9Þ

dSk dE:

0vk @E

4 T

This is possible because the presence of the term df 0 =@E has the consequence

that the integrand is only appreciable within the thickness kT of the Fermi

surface. Since for a typical metal at room temperature kT=EF has a mag-

nitude of 10À2 or less, then vk and ,k can to a good approximation be

considered independent of energy when the integration over E is performed.

Similar arguments applied to Eq. (8.2.5) yield

° °

e2 @f 0

v k ,k

p¼À 3 °8:5:10Þ

dSk dE:

0vk @E

4

The function À@f 0 =@E has the shape shown schematically in Fig. 8.5.1(a),

while À°E À Þ2 @f 0 =@E is the double-humped curve of Fig. 8.5.1(b). Both

functions may be integrated by extending the limits to Æ1, and one ¬nds

the results 1 and (kTÞ2 =3, respectively. Comparison of Eqs. (8.5.9) and

(8.5.10) then yields the Wiedemann“Franz law, which states that

i ¼ LTp °8:5:11Þ

where L is known as the Lorenz number, and in our simple calculation is

equal to

2 k 2

L0 ¼ ; °8:5:12Þ

3e2

307

8.5 Thermal conductivity in metals

Figure 8.5.1 (a) The expression for the electrical conductivity contains the factor

@f 0 =@E, and thus samples the electron distribution in the immediate vicinity of the

Fermi surface. (b) The thermal conductivity of a metal, on the other hand, involves

the double-humped function (E k À Þ2 @f 0 =@E, and measures , immediately below and

above the Fermi energy. (c) If the mean free path varies appreciably over the energy

range from À kT to þ kT then the Lorenz number deviates from its usual value.

which has the value 2:45 ‚ 10À8 V2 KÀ2 . We deduce from this result that i is

directly proportional to T at low enough temperatures, since the electrical

conductivity then tends to a constant in the absence of superconductivity.

The Wiedemann“Franz law is obeyed to within a few percent by most good

metals at most temperatures. Deviations occur whenever the mean free path

becomes a function of energy, as may be seen by a glance back at Fig. 8.5.1.

The electrical conductivity contains the factor À@f 0 =@E, and thus re¬‚ects the

mean free path at the Fermi energy . The thermal conductivity, on the other

hand, contains the factor shown in Fig. 8.5.1(b), and thus measures ,k at

energies slightly below and slightly above . If ,k had a dip in it at the energy

E ¼ , as indicated in Fig. 8.5.1(c), then the calculation of p would sample a

lower value of j,k j than would the calculation of i, and the Lorenz number L

would show a positive deviation from the value L0 that we previously

derived. We should expect deviations in particular in alloys that exhibit the

308 Semiclassical theory of conductivity in metals

Kondo e¬ect discussed in Chapter 11, as well as in any situation where

inelastic scattering can occur. Because an electron loses or gains energy 0

!q in scattering by phonons, one generally observes deviations of L from L0

as the temperature is increased from zero, and the electron“phonon interac-

tion gradually takes over from the impurity potentials as the dominant scat-

tering mechanism. At temperatures much greater than the Debye

temperature it will be the case that kT ) 0!q for all phonons, and it again

becomes a good approximation to consider the scattering as elastic. Then the

Lorenz number returns to its ideal value, L0 .

8.6 Thermoelectric effects

In the preceding section we calculated the heat current density u that results

in a metal when a temperature gradient exists. It is a simple matter to extend

this calculation to ¬nd the heat current density caused by the application of

an electric ¬eld alone. Substitution of Eq. (8.2.3) in Eq. (8.5.2) yields the

result

eX @f 0

u¼À °E k À Þvk , Á E: °8:6:1Þ

k @E k

The existence of this heat current is known as the Peltier e¬ect. The Peltier

coe¬cient & is de¬ned as the ratio of the heat current to the electric current

induced in a sample by a weak electric ¬eld in the absence of temperature

gradients.

Let us write Eq. (8.6.1) in the form

u ¼ rÁE °8:6:2Þ

and evaluate the components of the tensor r by changing the sum over

k-states into a double integral in the manner of Eq. (8.5.8). We ¬nd

° °

@f 0

e vk ,k

r¼À 3 °E À Þ dSk : °8:6:3Þ

dE

@E 0vk

4

We then immediately see that r is a very small quantity, for if we make the

assumption that the integral over dSk is independent of energy then the whole

expression vanishes, as (E À Þ°@f 0 =@EÞ is an odd function of E À . This is

illustrated schematically in Fig. 8.6.1, which shows the electron distribution

309

8.6 Thermoelectric effects

Figure 8.6.1 Although all the electrons in the region between A and B carry an

electric current of approximately the same magnitude in the same direction, the

heat current carried by those between A and C is almost exactly cancelled by

those between C and B.

shifted in k-space by the action of the electric ¬eld. By symmetry the net

electric and thermal currents are carried by those electrons between the

points marked A and B. Because all these electrons are moving with approxi-

mately the Fermi velocity they all contribute a similar amount to the electric

current. This is not the situation with the heat current, however, for those

electrons between A and C have energies less than , and carry a current of

coldness that almost cancels the positive heat current of the electrons between

C and B. We thus have to take into account the energy dependence of vk and

,k , and so we make a Taylor expansion to ¬rst order of the integral over dSk

in Eq. (8.6.3). We write

° °

°

d

v k ,k vk ,k v k ,k

dSk ™ þ °E À Þ

dSk dSk

0vk 0vk 0vk

dE

E¼ E¼

and substitute this in Eq. (8.6.3). The energy integral that survives is identical

to that in Eq. (8.5.9), and so we ¬nd

°

e °kTÞ2 d vk ,k

r¼ 3 : °8:6:4Þ

dSk

0vk

4 3 dE E¼

310 Semiclassical theory of conductivity in metals

The fact that Eq. (8.5.10) can be written in the form

°

43

v k ,k

dSk ¼ 2 p

0vk e

is sometimes made use of to put Eq. (8.6.4) in the form

°kTÞ2 @p

r¼ : °8:6:5Þ

3e @

The derivative @p=@ is taken to mean the rate of change of conductivity with

Fermi energy when it is assumed that the scattering and band structure

remain constant. It is very important to realize that @p=@ cannot be

found simply by adding more electrons to the metal and measuring the

change in resistance; as we saw in Section 6.4, certain kinks in the band

structure and in the scattering are linked to the position of the Fermi energy,

and would be altered by the addition of extra electrons.

Since u ¼ r Á E and E ¼ pÀ1 Á j, we can use Eq. (8.6.5) to write the Peltier

coe¬cient (de¬ned by u ¼ & Á j) in the form

L0 eT 2 @ ln p

&¼ ; °8:6:6Þ

@ ln

with L0 the ideal Lorenz number given in Eq. (8.5.12). For a material with

a scalar conductivity (as, for example, a cubic crystal) the dimensionless

quantity

@ ln

¼ °8:6:7Þ

@ ln

is a useful measure of the Peltier e¬ect. It is generally of the order of magni-

tude of unity (Problem 8.7) but is extremely sensitive to the type of scattering.

Alloys exhibiting the Kondo e¬ect, for instance, have anomalously large

Peltier coe¬cients as a consequence of the strong energy dependence of the

scattering due to magnetic impurities.

In the same way that we calculated the heat current caused by an applied

electric ¬eld we can investigate the electric current that results from the

presence of a temperature gradient. On substituting Eq. (8.5.5) into Eq.

(8.5.1) we ¬nd

eX @f 0

j¼ °E k À Þvk , Á rT: °8:6:8Þ

T k @E k

311

8.6 Thermoelectric effects

In an isolated piece of metal charge will move to one end of the sample until

an electric ¬eld E is built up that is just su¬cient to induce an equal and

opposite current that cancels that given by Eq. (8.6.8). The presence of

this ¬eld constitutes the Seebeck e¬ect; the ratio of E to rT is known as

the absolute thermoelectric power or thermopower of the metal. If we write

Eq. (8.6.8) in the form

j ¼ Àt Á rT °8:6:9Þ

then the thermopower S is equal to pÀ1 Á t.

We fortunately do not need to spend much time analyzing the Seebeck

e¬ect, for the thermopower S is related to the Peltier coe¬cient & in a very

simple way. Comparison of t [as de¬ned by Eqs. (8.6.8) and (8.6.9)] with r

[as de¬ned by Eqs. (8.6.1) and (8.6.2)] yields the result

r ¼ Tt

or more generally

˜

& ¼ TS °8:6:10Þ

˜

with S the transpose of S. A relationship of this kind was ¬rst derived by

Lord Kelvin by arguments that are still appealing, but alas, no longer

respectable. It is now thought of as an example of one of the Onsager rela-

tions that form the basis of the macroscopic theory of irreversible processes.

From Eqs. (8.6.6) and (8.6.10) one can write

L0 eT @ ln p

S¼ :

@ ln

For a cubic metal this becomes

L0 eT

S¼ : °8:6:11Þ

As is generally a few electron volts and is of the order of magnitude

of unity one ¬nds that thermopowers in metals at room temperature have

magnitudes of a few microvolts per kelvin. In semimetals such as bismuth,

where is small, the thermopower is correspondingly larger. The pro-

portionality of S to the absolute temperature suggested by Eq. (8.6.11) is

dependent on being independent of T. Because impurity scattering and

phonon scattering may lead to completely di¬erent expressions for it is

not uncommon for even the sign of S to change as the temperature of the

312 Semiclassical theory of conductivity in metals

sample is changed. Thus dilute AuMn alloys have thermopowers that at the

lowest temperatures are positive (i.e., a negative value of , the electronic

charge e being considered negative in Eq. (8.6.11)), but which become nega-

tive as the temperature is raised above a few kelvin.

Before leaving the topic of thermoelectric e¬ects we should brie¬‚y consider

an e¬ect that occurs when phonon Umklapp processes are rare. One mechan-

ism we have considered by which an electron can be scattered is the electron“

phonon interaction, a phonon being created that carries o¬ some of the

momentum of the electron. We have implicitly been assuming that the

momentum carried by this phonon is rapidly destroyed, either by a phonon

Umklapp process or by scattering by lattice imperfections and impurities. In

mathematical terms we have been uncoupling the Boltzmann equation for the

phonons (Eq. (8.1.2)) from that for the electrons (Eq. (8.1.3)) by assuming

the phonon relaxation time ph to be very short. If this assumption is not

valid then we must include in our computation of the heat current density the

contribution of the perturbed phonon distribution. This added contribution

to the Peltier coe¬cient (and hence also to the thermopower) is said to be due

to phonon drag, the phonons being thought of as swept along by their inter-

action with the electrons. Such e¬ects are negligible at very low temperatures

(when there are few phonons available for ˜˜dragging™™) and at high tempera-

tures (when phonon Umklapp processes ˜˜anchor™™ the phonon distribution)

and thus cause a ˜˜hump™™ in the thermopower of the form shown in Fig. 8.6.2

at temperatures in the neighborhood of ‚=4.

Figure 8.6.2 In very pure specimens the phenomenon of phonon drag may contribute

appreciably to the thermopower at temperatures well below the Debye temperature.

313

Problems

Problems

8.1 When a certain type of impurity is added to a free-electron gas of Fermi

energy E F it is found that jTkk 0 j2 is approximately constant, so that

Q°k; k 0 Þ ™ constant ‚ °E k À E k 0 Þ:

How does the electrical conductivity, , of this system vary with E F ?

[Hint: In Eq. (8.2.4) the summation over k 0 may be replaced by an

integral over k 0 -space. That is

°

X

! 3 dk:

8

k

Also

° ° °

dS

dk ! dE;

j@E=@kj

where dS is an element of a surface of constant energy.]

8.2 With another type of impurity one ¬nds that the scattering probability of

Problem 8.1 is modi¬ed to the form

Q°k; k 0 Þ ¼ °k À k 0 Þ°E k À E k 0 Þ;

where