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282 Superconductivity
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 ފ:
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 ފ:
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=kTŠ þ 1
exp ½°E k À Þ=kTŠ
¼À ;
kT°exp ½°E k À Þ=kTŠ 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 và 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 à 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 Š:
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 ފ 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 ފg:

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
qq
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 à 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

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