Marder M.P. Condensed Matter Physics

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18.

Microscopic Theories of Conduction

18.1 Introduction

This chapter is devoted to exploring the difference between electrical conductors

and insulators. Electrical resistance arises from the scattering of electrons as elec-

tric fields drag them through a sample. To connect electrical resistivity with the

underlying physics at the atomic scale, it is necessary to grapple with the scat-

tering problem directly. The simplest version of this theory treats impurities and

thermal fluctuations as weak scattering sites, amenable to the same techniques that

solve the problem of inelastic neutron scattering.

As the strength and density of impurities that scatter electrons increases, it is

natural to expect the resistance of the solid to increase as well. What might not be

expected is that past a certain density of impurities, the solid undergoes a qualitative

change. It no longer can support traveling waves, and it turns from a conductor

to an insulator. This mechanism is a subtle quantum-mechanical effect, valid at

low temperatures. It is only one in a series of ways that qualitative shifts from

conductors to insulators can occur. The essential difference between insulators

and conductors is more than a question of filled versus unfilled bands in perfect

crystals, and the most general explanation has probably not yet been formulated.

18.2 Weak Scattering Theory of Conductivity

18.2.1 General Formula for Relaxation Time

Although the calculation of electronic states in disordered materials is an extremely

difficult problem, there is one fairly straightforward case, which is an extension of

the nearly-free electron picture to the case of disordered metals or random alloys.

Imagine that a collection of electrons sits in a weak but no longer periodic potential.

If the effects of the potential may safely be treated to low order in perturbation

theory, then its interactions with electrons may be solved in the same way that X-

ray scattering in amorphous structures can be solved. As first noted by Bhatia and

Krishnan (1948), the same correlation function governs the two cases, because the

electrons, like the X-rays, are simply waves bouncing off weak scatterers.

The starting point is Section

10.2.1,

which showed that in some cases, partic-

ularly the alkali metals, Schrodinger's equation for the conduction electrons in the

presence of a periodic potential can be recast as a problem for plane waves in weak

potentials. Formally, these weak potentials are nonlocal and energy-dependent,

because the fact that they are so weak results from the cooperative screening pro-

vided by all the different electrons. All one needs to know, however, is that there

523

Condensed Matter

Physics,

Second Edition

by Michael P. Marder

524 Chapter 18. Microscopic Theories of Conduction

are cases where regarding a metal as free electrons in weak potentials has hope

of quantitative success. Take U(R) to be an appropriate pseudopotential, centered

at R. In the case of a liquid metal or metallic glass, the locations R label all the

sites inhabited by atoms. In the case of a random alloy, or a metal with impuri-

ties,

the locations R label all the places where the second species or impurity has

been placed, and U(R) describes the difference between the pseudopotential of the

original metal and the pseudopotential of the impurity.

The Collision Term. The collision term dg/dt\

\\ is central to the Boltzmann

equation. It has been treated so far in the relaxation time approximation. The first

step in a microscopic theory of resistance is the attempt to calculate the relaxation

time from an underlying picture of electron collisions.

The relaxation time approximation is based upon the view that there are effec-

tively random processes causing electrons to alter their wave vectors. What form

should these processes have? The question can be answered in a purely formal

manner, using techniques to be presented in Section 20.4. The formal derivation

can obscure the physical meaning of

the

various terms that arise, so the presentation

in this section will emphasize physical arguments over formal ones.

Consider a collection of electrons described by the distribution function gj,.

There is no need for the index r, so it will not be displayed explicitly. If an electron

scatters off nonmagnetic impurities, the probability

for it to make a transition

from state k and spin a to state k! with spin a' must have the form

V(k^V,t)=g

[l-gp]

Ôaa>W

(18.1)

Because the scattering potential is nonmagnetic, the jump can only occur if the

electron spin a does not change. The rate must be proportional to the number

g of electrons occupying state k, proportional to the number

1 —

g of vacancies

at destination k', and finally proportional to a rate W^, that is independent of the

occupation number g. Accounting both for electrons that jump into k by these

processes and for those that jump out gives

J[dk'}

,[\

}

-g

[l-

gll

rkl

(18.2)

dt coll. 2

The leading 1/2 is due the fact that the spin state of the electron cannot change, effectively

cutting the density of states in

half.

\dk'\ defined in Eq. (6.15).

Equation (18.2) is equivalent to the relaxation time approximation when the

following conditions obtain:

Scattering potentials into which electrons collide are spherically symmetrical,

and can be treated as weak.

The energies of occupied electronic states £^ are isotropic, and they depend

only upon the magnitude of k.

Weak

Scattering Theory of Conductivity 525

The first assumption allows one to write the potential seen by electrons as

The functions

are spherically symmetrical scattering

Utnt

= / U(r

—

R).

potentials, centered

sites R. They are best thought

(18.3)

■"—^ as pseudopotentials of impurity atoms added to a perfect

R crystal, or else as atoms of the crystals out of place.

Adopting this form of scattering potential, and retracing steps starting with Fermi's

Golden Rule that led to Eq. (13.104) gives

M>=

{

-n\(*

\Û

\V)\

(18.4)

final states

= 27rS'(q,uj

)\U(q)\

^. (18.5)

Here q = k

—

is the change in wave number of the electron upon scattering, and

huj

8.J.

—

£~

gives its change in energy. The inelastic structure factor

appears

just as

for

neutron scattering from phonons, now depending upon the change

electron energy. The number

scatterers N

may refer to

dilute impurity with

strong scattering properties, or to every atom in a liquid metal.

Because the potential U(r)

spherically symmetrical, U(q)

symmetric

k and

and indeed only depends upon the angle

between them. The inelas-

tic structure factor is not

general insensitive to interchange

of k

and k', but

scattering is nearly elastic and uj

very small, then one sees from Eq. (13.104)

(q,

S'(—q, 0), and assuming one can use this approximation then

(18.6)

Next assume that the solution of the Boltzmann equation has the form

Here C is a vector that either is constant, or is

= fr -\-Q

•

function

E-

;

it cannot otherwise depend

(

18.7)

upon

This form is general enough to encompass spatially uniform electric fields, uniform

temperature gradients, and uniform magnetic fields. Putting Eqs. (18.6) and (18.7)

intoEq. (18.2) gives

coll.

-e-iv

J[dV](k-k?)W

„

(18.8)

and so

j„

—

Because

W-g.,

depends only on the angle between

and

k!, only the component

of V

parallel

to k

survives the (18.9)

' £

integration.

one takes

kz, W depends only on

ki,

and

is easy to see that the integrals of

and

vanish

by symmetry. Note that C-k = g

—

with

—

-yj[S!] W

,{\-k-k').

(18.10)

_ 1

Inserting Eq. (18.5) into Eq. (18.10), noting that

[\-k-k')

2(^-)\

(18.11)

526 Chapter 18. Microscopic Theories of Conduction

Table 18.1. Mean free paths of electrons in liquid metals

Metal: Li Na Cu Ag Au Zn Hg AI Ga Sn Pb Sb Bi Fe

IT (A): 45 157 34 51 27 15 5 20 17 5 6 4 4 3

Values obtained by treating the metals as free electron gases where l

VfT, and the relaxation time r is obtained from the resistivity p at the

melting point through

= m/e

np. Source: Cusack (1987) and Grigoriev

and Meilkhov (1997)

555.

and averaging over initial phonon states as discussed prior to Eq. (13.113) gives

J[dq]S(q,oJ

)\U(q)\

^. (18-12)

T, ~ 2 V

In this way, the relaxation time may be related to an integral over the inelastic

structure factor and the pseudopotential U(q).

Resistivity of Liquid Metals. A liquid metal is as disordered as a metal can

be,

yet somewhat surprisingly the weak scattering limit is often appropriate, and

Eq. (18.12) allows quantitative description of resistivity. As evidence, one can use

the measured resistivity of liquid metals to deduce the characteristic mean free path

h = vpT electrons travel between scattering events. As shown in Table 18.1, this

distance is tens of angstroms in the noble metals, although it shrinks to atomic

dimensions in iron or bismuth.

For liquid metals, the static structure factor is spherically symmetric. Purely

elastic electron scattering from such a disordered array of atoms leads to nonzero

electrical resistivity. So it is enough to consider the limit

= 0 where change in

electron energy can be neglected and write from Eqs. (3.55) and (13.104),

S{q,

Lü

= 0) = S(q)

6(E

- £{\k - q\)). (18.13)

In writing this expression, make use of the fact that transport formulas such as

Eq. (17.54) use k only on the Fermi surface where £^ = £/?. For elastic scattering,

£.p must lie on the Fermi surface as well.

To continue the evaluation of

Eq.

(18.12), insert Eq. (18.13). Doing the angular

integral over q

6{2k

-q)

</(cos 0)5(£

- E(yJ4 +q

- 2k

q cos 6)) = g

£/Q

^ (18-14)

gives finally an expression due to Ziman (1961),

l l

N, r

1 IN

LKF

-, -,

— = y-: ^ dqq

S(q)\U(q)\

is number of (18.15)

4irh

V Jo

HH KHn KHn

scatters.

î-

^2&iW r

dqq'S{q)\U{q)\

(18.16)

Ak%

Weak

Scattering Theory of

Conductivity

527

The final expression uses free-electron relations between the electron density, the

Fermi wave vector, and the Fermi velocity vp. So in the weak scattering limit, the

resistivity is just given by an integral over the static scattering structure factor S

and the pseudopotential U.

In order to evaluate Eq. (18.16), it is necessary to know both the static structure

function S(q) and the pseudopotential U(q). The former can be measured by X-

ray scattering, and the latter can be obtained either from the density functional

calculations described in Section

10.2.1,

from de Haas-van Alphen measurements,

or else from optical data to be described in Section 23.4. An example of assembling

these various ingredients to determine the resistivity of liquid aluminum is provided

as Problem 1. The calculation outlined in the problem should not be expected to be

accurate to more than around 50%, but the more careful evaluation of Ashcroft and

Guild (1965) duplicated experimental values within a few percent. Other properties

of liquid metals are reviewed by Ashcroft and Stroud (1978) and March (1990).

Application to Phonons.

Calculation of

the

resistance of crystals proceeds in a slightly different

way.

absolutely perfect crystal has almost no electrical resistance, as Bloch first realized

[Section 7.1]. Formally this fact emerges without the need for any calculation

from Eq. (18.12). The leading term Eq. (13.129) in the inelastic structure function

is a product of delta functions restricting q to the reciprocal lattice, and a delta

function restricting k and k! to the Fermi surface. These two delta functions are

incompatible: when k and V are on the Fermi surface there is vanishing chance

that q = k

— K

is in the reciprocal lattice.

For metallic crystals resistivity comes from phonons, which means employing

Eq. (13.134) to take into account the creation or destruction of one phonon in each

electron scattering event.

To simplify matters, keep only longitudinal phonons corresponding to acoustic

modes, for which the polarization e^\\q. This type of interaction is called normal

scattering (N-process). The contribution of other phonon bands is not necessarily

negligible, when they enter in this context it is called called Umklapp scattering

[U-process;

Umklapp is German for "flopping over" and was introduced by Peierls

(1929)].

It is hard to see without detailed calculation whether normal or Umklapp

processes should be more important, and both should be included for accurate re-

sults.

In the interests of simplicity, the calculation that follows will keep only the

contribution from acoustic phonons, which is sufficient to give a good idea of how

the general procedure runs, but not to achieve comparison with experiment on the

order of

1 %

that is possible in the most ruthlessly correct calculations, presented

by Bass et al. (1990).

Returning to Eq. (13.134), there are two terms to inspect. In one of them, pro-

portional to

8{u)^

the phonon of energy

%ujq

steals energy from the electron.

This requires the final electron state to have energy lower than the initial state.

Looking now at Eq. (18.1), one sees that since k is on the Fermi surface, both

terms on the right hand side of Eq. (18.1) vanish unless the final state is higher in

528 Chapter 18. Microscopic Theories of Conduction

energy than the initial state; the electron must scatter from occupied to unoccupied

states.

Terms where the phonon takes energy from the electron must be negli-

gible at temperatures low compared with the Debye temperature. In the second

process described by Eq. (13.134), proportional to

S(LO^

— u)

the phonon gives

energy to the electron. This processes is allowed because the electron moves from

occupied to unoccupied states. The main effect of keeping track of the energy as-

sociated with phonons is to decide which scattering process is allowed; otherwise

the phonon energy is negligible in comparison with the electron energies. Thus one

can write

t2 2 Keep only the phonon absorption term of

S(fl, u) « —n

h6(£

-£(\k-q\) Eq.U

3.134);

once this is done it is all

(

18#17

)

IMHLÜ

ght to neglect the phonon energy inside

Q the delta function. n

is a phonon

occupation number.

Thus

Ht/^

M^^-^-^0

(18

18)

3TT

/N \ 1 f

/j£j2

^=^(^44

dqq3

2lt^

USeEq(1814) (J8

19)

3TT

WV,N 1 H ,k

T^5 rWT • i ,

Il(

zTs

^(v)^2Mêy

dzz

F^T

|£/

(-e-)'

(i8

2o)

where c is the longitudinal wave speed, M is ion mass, 0 = hk^c/kß is a De-

bye temperature, and z =

Oq/kpT.

At temperatures T large compared with the

0, Eq. (18.20) is linear in T, while at much lower temperatures the resistivity is

predicted to vanish as T

, the Bloch T

law.

Comparison with Experiment. The most careful comparison of these predictions

with experiment has been carried out for the alkali metals. As shown in Figure

18.1,

the resistivity is linear to high accuracy at temperatures over 100 K, and it

begins crossing over to a decay as T

at temperatures on the order of 20 K. By a

temperature of 2 K, there is yet another crossover to a decay of resistivity as T

The origin of this behavior is electron-electron scattering, as described in Section

17.5.

Computing the coefficient of T

is difficult, and it is discussed by Bass et al.

(1990).

18.2.2 Matthiessen's Rule

Comparison of theory and experiment for the alkali metals relies upon extremely

pure crystalline samples. A piece of wire pulled from the shelf follows predic-

tions of Eq. (18.20) down to around 100 K, but at lower temperatures it begins

to differ. Grain boundaries, dislocations, and various trace elements scatter elec-

trons,

and therefore all contribute additively to resistivity according to Eq. (18.16).

Matthiessen

rule states that separate sources of resistivity, such as phonons and

impurities, sum linearly to produce the total resistivity of a sample just as resistors

Weak

Scattering Theory of

Conductivity

529

Figure 18.1. Resistivity of potassium from 0.1 to 300 K. At the lowest temperatures, the

resistance varies as T

, from 2 to

it varies as T

, and from 100

upwards, it varies

as T

. [Source: Bass et al. (1990).]

in a series sum linearly. In any but the purest samples, resistivity is therefore the

sum of two pieces: a temperature-dependent part resulting from Eq. (18.16), and a

temperature-independent part that depends exclusively upon preparation and purity

of the sample. The size of the temperature-independent resistivity is in fact used

to define the purity of electrical samples. A common figure of merit is the residual

resistivity ratio (RRR) which is the ratio of resistivity at 30Q K to resistivity at 4

K. A "typical" sample of aluminum is 99.5% pure and has RRR « 11. A "pure"

sample of aluminum is six nines (99.9999%) pure and might have RRR = 2500.

Matthiessen's rule is no more than a good rule of

thumb.

It relies upon the lin-

earity inherent in the weak scattering approximation, and when this approximation

fails,

so may Matthiessen's rule as well.

18.2.3 Fluctuations

At any temperature above zero, resistance is never entirely constant, but fluctuates

in time. These fluctuations are conventionally separated into three components

Thermal noise. Thermal noise consists of current fluctuations that are present

whether a current is flowing or not. When the thermal noise signal is Fourier

transformed, its root mean square amplitude is independent of frequency, and

530

Chapter 18. Microscopic Theories of

Conduction

the magnitude of voltage fluctuations in frequency interval

dco

for a wire of

resistance R is

(ÔV

)=4k

TRduj. (18.21)

These noise properties were identified by Johnson (1927) and were explained

byNyquist(1928).

Shot noise. Shot noise is an additional source of noise that is proportional to mean

current. The current fluctuations 5J it produces have strength per frequency

interval duj

(SJ

)=2eJduj. (18.22)

// or flicker noise.

// noise is an additional noise source whose amplitude

grows with decreasing frequency / roughly as l/f. It is fairly ubiquitous,

yet the specific mechanism that produces it varies from one system to another.

Dutta and Horn (1981) provide a general explanation of how it might arise.

18.3 Metal-Insulator Transitions in Disordered Solids

18.3.1 Impurities and Disorder

No crystal produced in the laboratory is ever completely perfect. In the case of sil-

icon that has been prepared for use in electronics, the degree of perfection is very

high. Atoms such as silver and gold whose presence would disrupt electronic cir-

cuitry are excluded to better than one part in 10 '

, and wafers of tens of centimeters

in diameter are regularly prepared without a single dislocation or grain boundary

disrupting the crystalline order. However, silicon binds very easily to oxygen, and

silicon wafers used in electronics typically contain 10

atoms/cm

oxygen, which

is tolerated and rarely mentioned as it does not disrupt electrical properties. Other

materials such as aluminum or iron cannot be made nearly as pure, and in addition

to traces of other metals and oxygen contain many dislocations. Small mixtures

of other elements in a nearly perfect crystals are called impurities, and when one

wants to call attention to the ways a solid differs from a perfect crystal one calls it

disordered.

Disorder destroys crystalline regularity, and the conditions for Bloch's theorem

no longer apply. One must ask whether the corrections to Bloch's picture are mi-

nor, or whether they produce qualitative changes in the nature of the solid. Both

outcomes are possible, depending on how strongly electrons interact with the dis-

order. Sufficiently strong interactions can turn metals into insulators. Much of the

work in this area was instigated by Mott, and is reviewed in Mott (1990).

An important class of scattering sites in solids are point defects produced either

vacancies,

sites where atoms simply are missing, or else by substitutional atoms,

replacing their hosts on randomly selected lattice sites. The effects of point defects

divide into several different classes. Sometimes they have magnetic moments and

can flip the spin of passing electrons. Discussion of this case is deferred to Section

26.6.

The nonmagnetic impurities fall into two additional groups.

Metal-Insulator Transitions in Disordered Solids 531

An impurity is called compensated or isoelectronic when it comes from the

same column of the periodic table as the host crystal. Germanium or tin dissolved

in silicon, or silver dissolved in gold, provide examples. Adding an atom of this

sort amounts to a very localized perturbation on the crystal. The impurity carries a

different number of electrons with it from the host. However, the extra (or missing)

electrons are entirely confined to core states that are tightly bound to the nucleus,

localized in space, and largely inert. Section 18.4 will be dedicated to studying the

properties of electrons in such random potentials.

18.3.2 Non-Compensated Impurities and the Mott Transition

When impurities are not compensated, the electrons they carry cannot all be swept

into a static potential. The effect is as if one has scattered throughout the solid

atomic-sized spots of extra charge. All the conduction electrons in the solid inter-

act strongly with these extra charges. Simple physical considerations based upon

classical electrostatics make it possible to explain some of the main experimental

results in a simple fashion.

Consider adding an atom of phosphorus to silicon. Phosphorus lies just to

the right of silicon in the periodic table, and the perturbation it produces is best

described by adding a single proton to a silicon atom, resulting in a long-range

potential, falling off as \/r. The phosphorus comes accompanied by an extra elec-

tron as well, but this electron must be viewed as a member of the population of

conduction electrons, responding to the potential. Its wave function spreads over

hundreds of adjacent lattice sites.

Precisely because the spread is so large, the nature of the phosphorus impurity

may be captured by a simple semiclassical argument, as shown by Kohn (1957).

The static dielectric constant of silicon e, like that of other semiconductors, is large

and, according to Table 19.1, equals 11.8. The phosphorus nucleus is therefore

strongly screened. In addition, as shown in Figures 10.10 and 23.16, silicon is

technically an insulator. The valence band is filled, and any extra electron must

enter a superposition of states from the conduction band.. The lowest-lying states

behave approximately like free electrons, with the provision that the mass of the

electron be replaced by an effective mass m*, which, according to Table 19.1, is

on the order of 0.2. A proton in empty space binds an electron at radius ao

—

/me

with binding energy £^ = e

/2ao (this is a hydrogen atom). For phosphorus

in silicon, the screening of the proton replaces e

by e

/e, where e is the static

dielectric constant, and the binding energy becomes instead

eh e m* 1

o* = =■ and £* = -— = ~-13.6eV. (18.23)

m*e

2ea* m e

Because

jm ~ 0.1 and e ~ 10, a* is approximately 100 times larger than the

Bohr radius, and the binding energy is approximately 1000 times smaller than the

binding energy of hydrogen. Some comparisons of

Eq.

(18.23) with actual binding

energies appear in Table 19.2, showing that it successfully captures orders of mag-

nitude despite simplifications such as assuming effective masses to be isotropic.

532 Chapter 18. Microscopic Theories of

Conduction

The large size of a* retroactively justifies using concepts such as dielectric con-

stant and effective mass that are only valid on scales much larger than a lattice

spacing. Because of its small binding energy, phosphorus is called a shallow im-

purity, and because it adds an electron to the conduction band, it is called a donor.

Elements to the left of silicon in the periodic table, such as boron or aluminum,

come with one less electron than the host and may be treated as a weakly bound

positronium atom, where a delocalized hole, built from states at the top of the va-

lence band, is attracted to a screened

—

\/r potential. These elements are called

acceptors. Other impurities, such as gold, produce much more violent changes in

the local electronic environment of silicon than do phosphorus or boron and cannot

be described as hydrogen atoms. They are called deep impurities.

Metal-Insulator Transition in

Si:P.

The subtleties of impurity potentials, as well

as the difficulty of precisely defining the difference between metals and insulators,

are both well illustrated by examining the behavior of silicon as the density of

phosphorus impurities increases. Take phosphorus-doped silicon (Si:P) down to a

temperature on the order of

K, so that the chance of thermally exciting a phos-

phorus electron out of

its

bound state becomes negligible. At low densities of

phos-

phorus, the material is nothing but pure silicon, an excellent insulator, and bound

phosphorus electrons, also insulating. As the density of phosphorus increases, the

bound electron wave functions begin to interact, and at a critical density the doped

semiconductor turns into a metal.

This transition cannot be explained by the view that insulators are solids with

filled Bloch bands, while metals have unfilled bands. From such a point of view,

Si:P should always be a metal. Even at low densities, the phosphorus atoms could

be arranged in a periodic way throughout the silicon. Bloch's theorem could then

be brought to bear on the rather large resulting unit cell, which would produce

bands nearly identical to those of silicon, but with the Fermi level moved upwards

to populate the bottom of the conduction band with extra electrons provided by

phosphorus.

However a dilute arrangement of phosphorus in silicon is no more likely to

produce a metal than a lattice of silver atoms, spaced one meter apart. Coulomb re-

pulsion between distant localized electrons produces an insulator that one-electron

theory cannot explain. In fact many of

the

crystalline compounds predicted by one-

electron density functional theory to be metals are in fact insulators. A prototype of

these compounds is CuO, which will be the subject of further discussion in Section

23.6.3.

Experimental realizations of Si:P do not actually array the phosphorus in a

crystalline array, but the basic physics is believed to be similar whether the ar-

rangement of the phosphorus is regular or random, with a simple semiclassical

analysis providing a simple picture of how an insulator gives way to a metal.

Polarization Argument for Mott

Transition.

View Si:P as a cubic lattice of phos-

phorus, treating the silicon as a continuous dielectric medium of dielectric constant

e. When an external electric field is applied to the sample, the phosphorus atoms