Marder M.P. Condensed Matter Physics

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Metal-Insulator Transitions in Disordered Solids

533

polarize slightly in response. Continuing to view the phosphorus as producing a

weakly bound hydrogen-like state, its polarizability is

9 -,

(X =

—

ß See, for example, Landau and Lifshitz

( 1977)

(18 24)

2 ' pp. 286 and 290.

The Clausius-Mossotti relation (Section 22.2.3) relates the polarizability a of the

phosphorus to the dielectric constant of Si:P through

e-1 4TT

—- = --

a (18.25)

e + 2 3

3 + 87T«pa

—

4irnpa '

where np is the density of the phosphorus.

(18.26)

Figure 18.2. Metal-insulator transition in silicon doped with phosphorus. Enough phos-

phorus is added to bring the material to the verge of become metallic at low temperatures,

so that application of slight uni-axial pressure, slightly increasing the density, is enough

to push it over the edge. Dielectric susceptibility is measured on the insulating side, and

metallic conductivity on the conducting side of the transition. [Source: Rosenbaum (1985),

P-3.]

Defining a dielectric constant makes it possible to discuss long-range interac-

tions in a particularly simple way. It is plain from Eq. (18.26) that when the density

np rises to

3 0.053

1/3

= -. = —5- ^»rit

* =

°-

(

47ra a%

there is a polarization catastrophe in which the dielectric constant diverges. Further

increase of the density would seem to send the dielectric constant negative, but in

fact the whole basis of the calculation breaks down beyond this point, and the

534

Chapter 18. Microscopic Theories of

Conduction

Figure 18.3. A host of different systems displays metal-insulator transitions when the

condition

a*n

1//3

= 0.26 obtains, as found by Edwards and Sienko (1982), p. 92. Note the

large range of length scales covered by these data.

system becomes a metal rather than an insulator with local dipole moments. This

metal-insulator

transition

is also known as the Mott transition.

Using a* = 48

ÜQ

for silicon gives a value of a

—

1.02

•

-19

; Eq. (18.27)

predicts a critical density of

= 2.33

•

-3

. A measurement of this transi-

tion by Rosenbaum (1985), displayed in Figure 18.2, finds that the transition actu-

ally occurs at 3.74

•

cm"

, rather good agreement for such a simple argument.

Edwards and Sienko (1982) find empirically that a whole host of metal-insulator

transitions can be described by this simple theory, with the constant in Eq. (18.27)

slightly altered. A summary of their results is presented in Figure 18.3. In none

of these experiments is the impurity arrayed in a crystalline fashion, but the results

seem insensitive to this detail.

18.4 Compensated Impurity Scattering and Green's Functions

18.4.1 Tight-Binding Models of Disordered Solids

Compensated impurities can be treated as static potentials in a single-electron prob-

lem. They do not carry extra charge, and electrostatics is no longer a guide to what

they do. Compensated impurities have a very large effect upon electronic motion

when they are sufficiently numerous, and perturbation theory is frequently not ade-

Compensated Impurity Scattering and Green's Functions 535

quate to understand their effect upon electronic states. In trying to understand such

processes, simple models have proved particularly useful. The most basic of these

is the tight-binding model, introduced in Section 8.4.4.

Tight-Binding Hamiltonian. The tight-binding Hamiltonian may be dismissed

by lovers of accuracy and realism, but it provides a simple starting point for con-

ceptual advances. Following Eq. (8.67), it can be written

ÄTB

= J2

)fi\ + E

t|Ä>(#1

+t|#>(4 (18.28)

R {RR')

where the sum over {RR') is over nearest-neighbor pairs only. Notice that the

diagonal components U$ depend upon R, allowing the possibility that different

sites have different energies. By allowing

to vary randomly from site to site one

models a disordered solid.

It is convenient to ask questions of the tight-binding Hamiltonian that are more

difficult in a general context. For example, one can ask what happens after adding

an impurity to one site in a crystal, altering the energy

at the origin. This

problem can be solved exactly. The impurity Hamiltonian is

;>£] = (/o|0)(0|. Without loss of generality, take the disturbance (18.29)

to be at the origin.

Taking

"KQ

to be the tight-binding Hamiltonian (18.28) with all diagonal energies

Uß = 0, the goal is to solve

£|V>}= (Äo + Äi)

\i>)-

(18.30)

To solve this problem requires a certain amount of formal development, so it is

useful to summarize the main results in advance.

One dimension. Adding an impurity to a one-dimensional chain always produces

precisely one bound state, localized about the impurity.

Two dimensions. Adding an impurity to a two-dimensional net also always pro-

duces precisely one bound state. However, if the impurity potential is weak,

the binding energy is exponentially small.

Three dimensions. Adding an impurity to a three-dimensional crystal produces a

bound state only if the impurity is strong enough. A weak impurity warps the

traveling waves in its vicinity but does not change them qualitatively.

To verify these claims, it is necessary to introduce some formalism concerning

Green's functions.

536

Chapter 18. Microscopic Theories of

Conduction

18.4.2 Green's Functions

Green's functions originate in the question, "Starting with a particle at site |0) at

time 0, what is the amplitude for finding it at site \R) at time ??." The answer to

this question is that the amplitude is

(R\G(t)\0) = {R\e-

iiu/n

\0}. (18.31)

It is more convenient to work with Fourier transform of this operator, so define

1 r°°

G(£) = - / dte

iEt

G{t) (18.32)

in JQ

=>G(£) = (£-Ä)

. (18.33)

Equation (18.32) can only be expected to converge if £ has a positive imaginary

part; this fact signals that £ must be allowed to vary in the complex plane, and

warns that the physical significance of Green's functions may depend in an im-

portant way on the complex part of the energy, even when it is very small. The

operator defined in Eq. (18.32) obtains its information from the future of a particle,

starting at t = 0; if instead one had chosen to look at the past history of the particle,

then one would have defined

G{Z)=

[ dt e

G{t), (18.34)

n J-oo

which is in fact identical to Eq. (18.33) except that now £ must have a negative

imaginary part in order for the integral to converge. So Green's function contains

full information about the time evolution of a particle: When £ has a positive

imaginary part, the information is about the future, and when £ has a negative

imaginary part, the information is about the past. From Eq. (18.33), it should be

clear that Green's function has a pole whenever

has an eigenvalue, because if

|£„) are the eigenstates of 3ï with energies £„, then

|p \/p I

G=(£-J{)~

= ^(£-?{)-

|£„)(£

| = ^y^-^. (18.35)

The poles of G identify all the energy eigenvalues of

"K.

Letting £ =

£,.

irj,

with

and

real, in the neighborhood of one of the poles gives

^ !£„)(£„!(£,-£„)

ir,\£

)(E

h^yje^y-

(£

— £„) +7/ (£

— £„) +

superscript on G tells which

sign for

was chosen.

= |£»>v5»l (^TW^r-2«)) • If

»7.

very small and (18.37)

[ t

— £„ J positive,

nS(x) =

/(x

Compensated Impurity Scattering

and

Green's Functions 537

The delta function resulting from

the

imaginary part

of G

keeps track

of the

density

states.

particular,

T-Im[(

R|G

(£)|

R)]

^5(£

-£„)|(^)|

(18.38)

defines

the

local density

states

position R. This function

can be

employed

find the probability that

electron

localized

some point

space; Problem

in Chapter

provides

example.

Green's Function

One Dimension. Green's functions

can be

determined

ex-

plicitly

for

tight-binding models when they

are

perfectly crystalline,

and all the

diagonal elements

vanish. Denoting Green's function

this case

Go, return

the

solution

the tight-binding Hamiltonian given

Section

8.4.

Making

use

of the exact eigenfunctions

and

eigenvalues from Eq. (8.72) gives

(R\k) (k\R'\

(R\GO\R')

= y^ -— —-

Assume the lattice spacing

unity.

(18 39)

Y N £ - 2t cos (2TT///V) "*

Jo 2^£-2t cos(Jk)'

(18

40)

Without loss

generality,

let R > R'.

Equation (18.40)

can be

transformed into

contour integral about

the

unit circle

defining

~ik

lli

^dk=^

(18.41)

so that

{R\GQ\R')

becomes

2vnz(£-t(z

+ z-

))

27i7 £z

—

(18.42)

(18.43)

The denominator has two roots:

=Z-orz

, (18.44)

where

z~ is

the smaller

the two

absolute value, while

the larger. Observ-

ing that

the

product

and

z+ equals

1, one

sees that

always lies within

the

unit circle, while z+ lies

out of it

(unless both

lie on it, a

singular case).

The

result

is that

<#|Go(£)|tf')=

£ 1

|£|v

-4t

£ /£

2t |£IVV2t

\R-R'\

(18.45)

538

Chapter 18. Microscopic Theories

Conduction

| £ |

> 2t, and

otherwise

\R-R'\

(R\G

(£

±ir,)\R)^

( )±l

- £?

IWHD'

(18.46)

Notice

the

branch

cut

between — 2t

and 2t

This branch

cut

represents

line

poles that have merged together (Problem

4 of

Chapter 20).

The

diagonal element

of this operator

plotted

Figure

18.4.

Two

and

Three

Dimensions. Pictures

(0|Go|0)

for

two-

and

three-dimensional

square crystals appear

Figure

18.4. The

computations involve elliptic integrals

and

are

somewhat cumbersome,

but the

main feature that will

needed

for the

impurity problem

is the

behavior near the band edge, which

easily estimated.

a two-dimensional array with

sites,

for £

near the band edge

—4i,

(0|Go(£)|0)

= - Y \= 7=-

(18.47)

&-2t[cos2Tr^/VN + cos27rk

/VN]

dki

/ dk

- „

: rr

(18.48)

(2-7r)

JO JO £

—

2t [cos k\ +

cos k

]

1 f

dkidk

2T7

,— 7T-r—,

(18.49)

(2vr)

OS.

- 4t - 2t

[cos

ife]

cos k

]

/• 1

/ kdk — ~

Defining

ö£ = £ +

4t, moving

polar (18.50)

(27T)

O£

—

t /t

coordinates,

and

expanding the cosines

around

k\ =

= f to

obtain

the

leading

singularity.

~""-^'>.

(,8.5,)

4vrt

three-dimensional crystal, Green's function goes

to a

constant near

the

band

edge,

shown

Figure

18.4.

Perturbation Theory.

One of

the most important properties

Green's functions

the way

they change when

Hamiltonian

perturbed.

3~C

!Ko

3~Ci,

(18.52)

then defining

(£-Äor' (18.53)

gives

G=(£-Ä)"

(£-Äo-Äi)~

(18.54)

((£

-Ä

)(l

- (£

-Äor'Ä,))-

(1 -GoÄiT'Go (18.55)

= 2^(GoÄi)

;

+ GQ"K\GQ + GQ'KIGQ'KXGQ +. . .

(18.56)

GQO~CIG

+ G3{,\GQ.

(18.57)

The

matrix

defined

to be the

operator satisfying the equality

GofG

. (18.58)

Compensated Impurity Scattering and Green's Functions

539

One dimension Two dimensions Three dimensions

-2

-4

-2 0 2 _4 0 4-6 0 6

£/t £/t £/t

Figure 18.4. Green's functions for perfect square tight-binding lattice in one, two and

three dimensions. The band of energies where propagating extended states are possible is

shaded in each case. Because the imaginary part of Green's function gives the density of

states,

the singularities appearing in it are the van Hove singularities described in Section

7.2.5.

The curves were determined from equations of Economou (1983), p. 82.

18.4.3 Single Impurity

All the formalism now is in place to find the effect of an impurity placed at site 0.

Equation (18.29) states that "K\ = L/n|0)(0|, so Green's function corresponding to

the Hamiltonian of Eq. (18.30) is

G = G

+ Go|0)C/o(0|Go + Go|0)f/o(0|Go|0)[/o(0|Go + . . . See Eq.(l8.56). (18.59)

= Go + Go|0)£/

(0|Goy^ fî/o(0|G

|0>y Only one matrix element ever (18.60)

—' V / appears in the sum.

p=0

= Go -I °^ ' °' J——. Recognizing the infinite sum to (18.61)

1 -f/o(0|Go|0)' be a Taylor expansion.

The simplicity of Eq. (18.61) illustrates the usefulness of Green's functions. By

carrying out perturbation theory infinitely far, the effect of a single impurity is

determined exactly.

Recall that the poles of (18.61) give the energy eigenvalues of "K. Do any of

the original poles of Go remain poles of G once the perturbation is applied? No:

Consider what happens when £ approaches £„n, some pole of Go. Then Go begins

to diverge, and

J£*o)(£,,o|

(1862)

—

£„o

|£«o)(£«o| |£/io) (£no|0) (0|£„o) (£«o|

£-£„0 (£-£

)(0|£„o) (£n0|0) ' Eq. (18.61).

Put Eq. (18.62) into (18.63)

540

Chapter 18. Microscopic Theories of Conduction

so that the pole cancels. The only poles of (18.61) remaining are those due to the

vanishing of

l-£/o(0|Go(£)|0>=0. (18.64)

Equation (18.64) determines the energy eigenvalues of the impurity Hamilto-

nian. With the energy eigenvalues £„ in hand, the eigenfunctions can be extracted

from (18.35) by choosing £ to be near to some energy eigenvalue £„ satisfying

(18.64).

Return to Eq. (18.61) and write

1 - t/

(0|Go(£)|0) « -t/

(0|G

(£„)|0)(£ - £„) (18.65)

with

9£ '

!£„)(£„! Go(£„)|0)<0|Go(£„) 1

= ^, (18.66)

(18.67)

£-£„ -(0|G

(£„)|0) £-£„

C (F )\0)

|£ \ = ——-— This is an explicit expression for (18 68)

/_(0|^(£„)|0>"

theei

enfunction

The term inside the square root is positive for bound states.

Impurities in One Dimension. The effects of impurities depend upon whether

they are added to one-, two-, or three-dimensional systems. Begin with one di-

mension. In one dimension the unperturbed Hamiltonian has a continuous band

of states stretching in energy from —2t to 2t. Green's function Go in Eq. (18.40)

can be viewed as a tightly grouped collection of N poles (see Problem 4 in Chapter

20),

spaced at distance on the order of

/N. Around each of these

poles the

Green function varies from —oo to oo, and somewhere in this range, for each pole,

Eq. (18.64) must be satisfied. Therefore, after the impurity is added there is still a

continuous band of states stretching in energy from — 2t to 2t. Although somewhat

distorted near the impurity, these states look like plane waves far from it. They are

extended and can carry electrical current from one end of the system to the other.

However, a qualitatively different type of state exists as well. FromEq. (18.45),

outside the band (0|G

|0) = l/V£

-4t

, and (18.64) can be satisfied when

£ = ±

J4t

+ UZ.

The

lies for u

o > 0 and the - sign ( 18.69)

V applies for

< 0; see Figure 18.4.

So,

in one dimension, adding an impurity to a crystal always produces an en-

ergy eigenstate outside the band. If the impurity potential is attractive, there is

a bound state below the band; even if it is repulsive, there is still a bound state,

but now above the band. Because the energy of the bound state lies outside the

conduction band, one finds by combining Eqs. (18.68) and (18.45) that the bound

state is localized. It falls off in amplitude exponentially away from the location of

the impurity at |0), because the matrix elements of Go fall off in this way. Local-

ized electrons cannot easily contribute to electrical conduction because they do not

extend across the sample.

Compensated Impurity Scattering and Green's Functions 541

Single Impurity in

Two

Dimensions. The addition of a single impurity does not

remove the band of extended states stretching from —4t to 4L As in one dimen-

sion the addition of an impurity always leads to the creation of a localized state.

However, the binding is extraordinarily weak, because Green's function goes as a

logarithm near the band edge. For

ÖE-

< 0, (18.50) is real and diverges logarithmi-

cally. Using Eq. (18.51) to solve (18.64) with t/

< 0 gives

E = -4t-te~

47ri/m

. (18.70)

The exponentially small binding energy appearing in Eq. (18.70) ends up appear-

ing in unexpected places, such as the binding energy of Cooper pairs in supercon-

ductivity or the binding energies of magnetic impurities in the Kondo problem.

These problems become effectively two-dimensional because they concern the in-

teractions of a collection of interactions restricted to the Fermi surface, a nearly

two-dimensional subset of the original collection of electrons.

Three Dimensions. In three dimensions, the unperturbed Green function is finite

at the band edge, as shown in Figure 18.4. A sufficiently small impurity does not

lead to the creation of a bound state because

1 —

UO(0\GQ(E)

|0)

does not have any

zeros.

For a simple cubic lattice, the smallest value of

leading to a bound state

is approximately

= 1.5 x 6t. If a weaker impurity is added, all the extended

plane-wave eigenstates deform slightly, but they are not affected qualitatively. This

special feature of three-dimensional space is one of the reasons that it is so often

legitimate to treat scattering potentials as weak.

Why then does a noncompensated impurity always create a bound state in three

dimensions, no matter how weak it may be? To model a single noncompensated

potential in the context of

the

tight-binding model, one would need to set an infinite

number of energies U$ nonzero and to have them fall off in amplitude as \/R

moving away from the origin. The long range of this potential always produces a

bound state.

18.4.4 Coherent Potential Approximation

The coherent potential approximation is an elaborate approximation scheme that

describes disordered systems in which the impurities are too numerous or too

strong for weak scattering theory to be satisfactory, but not strong enough, in three

dimensions, to produce large numbers of localized states. The starting point is

the idea of an effective medium surrounding each site that is spatially uniform but

represents the average effect of disorder. To model this situation, write

aCo

+ ^(f/

-E)|/n)(/n| + E, (18.71)

which for the moment is just an identity, but will be chosen later to make approxi-

mations as good as possible. The energies U

will be viewed as random variables,

characterized by a probability distribution 7{U

). Define

!K£

:KO

+ £, Äf = ^(£/

-£)|m)(m| (18.72)

542

Chapter 18. Microscopic Theories of

Conduction

so that

G^(£) = G

(£-E). (18-73)

In general, £ will turn out to be a function of

the

energy £ and will also be complex.

An equation for the full Green function is now

G =

+ G$T

'G%, (18.74)

where T^ is the T-matrix appropriate to the perturbation 'Kf, as defined in

Eq. (18.58).

So far everything is exact. The only approximation is to put

m, (18.75)

where T^ is the T matrix that would result if the only nonzero part of Hf were at

The problem of a single impurity has already been solved, and from Eq. (18.61)

one sees immediately that the T matrix in this case is given by

T-'ZJ

\m)(U

-T,)(m\

T£ = '

^ (18.76)

l-(f/

-S)(m|Gf|m)

There is still the task of choosing £. In order to do it effectively, write

G = G$ +

G%f

G%.

(18.77)

and average both sides over disorder potentials U

. Choose E by requiring that the

difference between G^ and G vanish on average:

G = Gn(£.) = Gn(£ — E) The bar means to average all the potentials (18 78)

vy J v n

with probability distribution 7{U

with the consequence from Eq. (18.77) that

= 0 (18.79)

dU7{U) -—

; ^

^ = 0. (18.80)

(t/-s)

l-(£/-£)(0|G

^|0)

Equation (18.80) is the basic equation of the coherent potential approximation.

From it one calculates E(£), and then the average of

from Eq. (18.78). This for-

malism replaces Eq. (18.16) when scattering is not weak; Ziman (1979) discusses

results obtained from this approximation.

18.5 Localization

The study of electronic states in random lattices was initiated by Anderson (1958).

Adding a single impurity at a single site can lead to the creation of

single localized

state.

Life would be simple, and the ultimate conclusions correct, if one could