R?ssler U. Solid State Theory: An Introduction

Подождите немного. Документ загружается.

270 9 Defects, Disorder, and Localization

anti-

bonding

deep trap

hyper – deep

trap

Fig. 9.3. Schematic level diagram with valence and conduction band formed from

bonding and anti-bonding states of host atoms Ga and P (left) and of deep impurity

states formed with s states of a substitutional N impurity (right)

and anti-bonding states of the s orbitals of Ga and of the impurity X, one

may write the simpli ﬁed secular problem

− EU

− E

= 0 (9.16)

with E

Ga(X)

being the energy of the atomic s orbital of the Ga(X) atom, and

U the coupling between these orbitals. The eigenvalues of (

9.16)

+ E

− E



4|U|

− E

)



1/2

(9.17)

are depicted as a function of E

− E

in Fig.

9.4. Depending on the sign of

−E

, the anti-bonding or bonding states are shown: they evolve from the

valence or conduction b and edge, r espectively, and converge with increasing

− E

| toward the energy E

of the atomic Ga-s orbital which acts as

a pinning level. The symmetry of these impurity states is A

due to the s

orbital [

269]. Similar considerations for the p orbitals lead to deep impurity

states with T

symmetry.

The other concept to describe deep impurities is based on scattering theory

and employs the Green function of the impurity problem [

270]

G(E) = lim

δ→0

E − H + i δ

. (9.18)

We make use of the separation of the system Hamiltonian H = H

+U into the

Hamiltonian H

for the unperturbed periodic solid and the impurity potential

U(r), now understood as the diﬀerence of the self-consistent DFT-LDA single-

particle potentials with and without impurity (see Sect.

5.1)

U(r)=V

eﬀ

[n(r), r] − V

eﬀ

[n(r), r]. (9.19)

9.1 Point Defects 271

Fig. 9.4. Chemical shift of deep traps evolving from p-bonding and s-anti-bonding

states

The Green function G(E) of the full problem can be expanded in the Born

series

G(E)=G

(E)+G

(E)UG

(E)+...=(1− G

(E)U)

−1

(E). (9.20)

The ﬁrst factor in the last expression gives rise to additional poles of G(E)

caused by the impurity potential U . They exist besides those of the second

factor, the Green function G

(E) of the unperturbed band structure. For the

density of states

D(E)=−

Tr Im G(E) (9.21)

one ﬁnds with G(E)=G

(E)+(G(E) − G

(E))

D(E)=−

Tr Im G

(E) −

Tr Im (G(E) − G

(E))

= D

(E)+∆D(E)

(9.22)

where D

(E) is the density of states of the unperturbed soli d and ∆D(E)

its change due to the impurity both in the gap and in the continuum of the

valence an d conduction bands.

In a representation with lo calized functions, e.g., Wannier functions

(r − R)=

√



ik·R

(r) (9.23)

with



∗

(r − R)φ

(r − R

′

r = δ

′

R,R

′

(9.24)

272 9 Defects, Disorder, and Localization

the matrix elements (take for simplicity n = n

′

)

′



∗

(r − R)U(r)φ

(r − R

′

r (9.25)

are diﬀerent from zero only for a small number of R, R

′

out of a set of lattice

points {R

} around the impurity site. This deﬁnes a short range defect matrix

′

}

(9.26)

for which the additional poles of G(E)in(

9.20) can be calculated from

(Problem 9.1)

1 − G

(E)U = 1 − G

0,{R

}

(E)U

}

 =0. (9.27)

Here G

0,{R

}

(E) is part of the matrix representation of the Green function

connected with the set {R

A point defect of particular interest is the anti-site defect, which has

been studied intensively in GaAs and found to be responsible fo r realizing

semi-insulating material [

271]. Anti-site defects exist also in other compound

semiconductors. The schematic level diagram of the As

anti-site defect is

showninFig.

9.5. Besides resonances, deriving from p-anti-bonding and bond-

ing states in the valence and conduction bands, there is a trap in the middle

of the gap connected with the s-anti-bonding states of A

symmetry. Note

the degeneracy of the respective states, it is sixfold (with spin) for T

deriving

from p-states and twofold for A

deriving from s-states.

Transition metal atoms in semiconductors form deep impurity states as a

consequence of their tightly bound d electrons.[

272] Their orbital multiplets

are split by the crystal ﬁeld into states characterized by the point symmetry

(e.g., in zinc blende material A

,E,T

and T

, see Appendix).

Fig. 9.5. Level diagram of the As

anti-site defect (center) and its composition

out of the Ga vacancy (left)andp states of the As atom (right). Fu ll (open) dots

indicate occupied (empty) states

9.2 Disorder 273

9.2 Disorder

Following the systematics of dimensionality in the deviations from the crys-

talline order, we mention brieﬂy the dislocations as one-dimensional defects

and may address interfaces (or surfaces) as two-dimensional defects. A par-

ticular deviation from crystalline conﬁgurations is a disorder by composition

or structure, which can exist in a ll three spatial dimensions. Compositional

disorder results, e.g., by placing two or more diﬀerent kinds of atoms statisti-

cally on the otherwise unperturbed lattice po ints. This is the case in so-called

mixed crystals or alloys, in which the long-ran ge o rder is destroyed because

there is no correlation in the chemical nature of the ions occupying the lattice

points. Also statistical occupation of the lattice points with diﬀerent isotopes

of one kind of atoms creates compositional disorder. We mentioned already

the broadening of phonon resonances due to isotope disorder in Chap.

A more dramatic thr ee-dimensional disord er is the structural disorder, typical

for amorphous solids, which do not exhibit any crystalline order.

Adopting a single-particle description in the sense of Chap.

5, we write the

electron Hamiltonian for a system with disor der as

H = −

¯h

∆ +



(r − R

) (9.28)

with the ion conﬁguration {R

} and the eﬀective single-particle potentials

(r − R

) provided by the speciﬁc atom (or ion) at R

. Disorder can

be accounted for by deviations from the crystalline ion conﬁguration {R

}

and/or by varying the atomic species with the potential v

from site to site.

We may write the Hamiltonian H also in the form

H =



†



j,l

†

(9.29)

by using fermion operators c

†

in the site representation. Here the single-

particle energies ǫ

(the site energies) account for disorder in the occupatio n

of the sites R

, while the transfer matrix elements t

do it for the hopping

between sites R

and R

. The actual conﬁguration of a disordered material

is not k nown but is not essential for the physical prope rties. Instead, disorder

is considered by means of probability distributions of the site energies ǫ

and/or of the transfer matrix elements t

. For obvious reasons, the former is

called diagonal disorder, the latter oﬀ-diagonal disorder. Both are related to

probability distributions of the conﬁgurations P ({R

}).

A simple case of diagonal disorder is that the single-particle energies

scatter around a mean value ǫ

,andwemaywrite

H =



†



j,l

†



(ǫ

− ǫ

†

, (9.30)

274 9 Defects, Disorder, and Localization

where the ﬁrst two terms are the tight-binding Hamiltonian for a single band

(see Sect.

5.4), while the last term represents the diagonal disorder. We may

switch to the Bloch representation and write

H =



†



k,q

u(q)c

†

k+q

= H

+ H

disorder

(9.31)

which in a natural way splits into the unperturbed energy band and the per-

turbation due to disorder. The potential matrix element u(q) is the Fourier

transform of the impurity potential U (r − R

) and can be s eparated accord-

ing to u(q)=



exp (iq · R

)U(q)=ρ(q)U(q) into structure factor ρ(q)and

form factor U(q).

For a systematic treatment of the disorder, we study the single-particle

Green function (see Sect.

7.1)

G(k, k

′

,t− t

′

)=−

¯h

θ(t − t

′

){c

(t),c

†

′

)}. (9.32)

The equation of motion for G(k, k

′

,t − t

′

) is easily obtained with H from

(

9.31)andreads



i¯h

∂

∂t

− ǫ



G(k, k

′

,t− t

′

)=δ

′

δ(t − t

′

)



′′

u(k − k

′′

)

−i

¯h

θ(t − t

′

){c

′′

(t),c

†

′

)}



 

G(k

′′

′

,t−t

′

)

(9.33)

After Fourier transformation with respect to t − t

′

with E

=¯hω +iδ,the

retarded Green function of the disorder problem is given by

G(k, k

′

; E

− ǫ



′



′′

u(k − k

′′

)G(k

′′

, k

′

; E

)



. (9.34)

By identifying the ﬁrst factor on the rhs as the unperturbed retarded Green

function G

(k,E

) for the Bloch band, this can be written in the form of the

Dyson equation

G(k, k

′

; E

)=G

(k,E

)



′



′′

u(k − k

′′

)G(k

′′

, k

′

; E

)



(9.35)

By iteration, it leads to the Born series

G = G

+ G

+ ... (9.36)

and by introducing the t-matrix

9.2 Disorder 275

t = u (1 + G

u + G

u + ...)=u

∞



n=0





(9.37)

it takes the form

G(k, k

′

; E

)=G

(k; E

)



k,k

′

+ t(k, k

′

; E

′

; E

)



. (9.38)

We note that the full Green function depends on the conﬁguration {R

}

of the impurities, which in principle is not known. On the other hand, we

do not expect, that the physical properti es of a suﬃciently large sample are

determined by its speciﬁc impurity conﬁguration. Since on macroscopic length

scales (large compared e.g. with the mean free path) diﬀerent conﬁgurations

are realized, a statistical average can be taken which depends only on the

impurity concentration.

However, this argument does not apply, in the mesoscopic regime with

sample sizes comparable to the typical transport lengths. In this regime, the

observation of universal conductance ﬂuctuations can be taken as a ﬁngerprint

of the distinct impurity conﬁguration realized in the sample [

159, 160]. Here we

rely on the property of a macroscopic observable A,thatitisself-averaging,

i.e., for the variance VarA = (A −A)

 it holds that

lim

V →∞

VarA

A

=0, (9.39)

where V is the volume of the system. An example can b e treated as Prob-

lem 9.2.

The averaging over impurity conﬁgurations aﬀects only the structure factor

and is written as

f(q)=ρ(q)

conf



}

P ({R

})



j=1

iq·R

, (9.40)

where P ({R

}) is the probability distribution for the conﬁgurations of N

impurities. The t-matrix consists of terms with diﬀerent powers of the poten-

tial u(q). Thus, the conﬁguration average has to be performed in all orders of

ρ(q), and we write for the nth order

f(q

,...,q

)=ρ(q

) ...ρ(q

). (9.41)

There is no preference for the impurities to be on particular sites, and we can

simplify the double sum over conﬁgurations and sites by replacing



}



j=1

···→c



all sites

... (9.42)

with the impurity concentration c. When taking the sum over all sites and not

only over those occupied by an impurity, we have



exp (iq · R

)=Nδ

q,0

and ﬁnd

276 9 Defects, Disorder, and Localization

f(q

,...,q

)=p

(c)δ

...δ

(9.43)

with a polynomial p

(c)ofdegreen in the impurity concentration. The lowest

order terms can explicitly be written

f(q





·R



= Nδ

q,0

(9.44)

f(q

, q





i(q

)







i=j

i(q

)·R



·R



i=j

·R



= Nδ

+ Nδ

(N − 1)δ

. (9.45)

Whenever a su m is to be taken over the wave vectors, it can be converted

into an integral with the consequence that the Kronecker symbo l becomes a

δ function or

f(q

)=cδ(q

)andf(q

, q

)=cδ(q

+ q

)+c

δ(q

)δ(q

). (9.46)

The power of the impurity concentration in these expressions indicates the

number of impurities involved in a scattering process. Scattering with sin-

gle impurities is dominating, while scattering with two or more impurities

becomes less l ikely according to the correspon din g power of c<1.

Turning back to the Dyson equation, we now perform the conﬁguration

average with the Green function by replacing

G(k, k

′

; E

) →

G(k, k

′

; E

)=G(k, k

′

; E

)

conf

(9.47)

and write (suppressing the energy argument E

to simplify notation)

G(k, k

′

)=G

(0)

(k)



k,k

′

(2π)



Σ(k, k

′′

)

G(k

′′

, k

′

′′



(9.48)

with the self-energy Σ(k, k

′′

) due to the disorder, which in general depends

on the energy.

9.3 Approximations for Impurity Scattering

The conﬁguration averaged Green function, likewise the self-energy, can be

evaluated in diﬀerent approximations, which can be classiﬁed by makin g use

of a diagrammatic representation [

9, 206, 273]. Thi s is achieved by replacing

the physical objects of our formulas by graphs:

G :

(0)

, ρU : .

9.3 Approximations for Impurity Scattering 277

The leading terms of the Dyson equation can be arranged according to their

number of interaction lines with the impurities and lead to the following

graphical representation of the averaged Green function

G or the Dyson

equation:

G =

kkkk

kk’kkkk’

k’k

>>>> >>>>

>>>>>>

. (9.49)

We recognize a growing complexity of the individual diagrams. Some of them

can be reduced to simpler ones by cutting a free propagator line, while others

are irreducible in t his sense. This is used in the graphical representation of

the self-energy, which contains only the irreducible diagrams

Σ(k, k

′′

; E

+++

>>>

(9.50)

With reference to the graphical form, we may characterize some standard

approximations [

9, 206, 273].

1. Virtual Crystal Approximation (VCA): This simplest approximation

consists in taking into account the impur i ty interaction only in lowest

order. The self-energy becomes

VCA

(k, k

′′

= ρU = cU(0)δ(k − k

′′

) (9.51)

and does not depend on the energy. For this case, the Dyson equation

278 9 Defects, Disorder, and Localization

VCA

(k, k

′

; E

)=G

(0)

(k; E

)



k,k

′

+ cU(0)G

VCA

(k, k

′

; E

)



(9.52)

can be solved to give

VCA

(k, k

′

; E

(0)

(k; E

)

1 − G

(0)

(k; E

)cU(0)

k,k

′

− ǫ

− cU(0)

(9.53)

and ﬁnally

VCA

(k, k

′

; E

)=G

(0)

(k,E

− cU(0))δ

k,k

′

. (9.54)

In the virtual crystal approximation, the single particle energies are simply

shifted by cU (0), which is the averaged perturbation by the impurities and

does not break the translational symmetry of the solid. Thi s approximation

can be understood also as a r epl acement of the host atoms by virtual

atoms, each of them modiﬁed by an averaged impurity contribution.

2. Averaged t-matrix Approximation (ATA): In this approximation,

multiple scattering with a single impurity is considered in all orders as

expressed by the self-energy

ATA

(k, k

′′

; E

= cδ

k,k

′′

t(k, k

′′

; E

) (9.55)

with the t-matrix

t(k, k

′′

; E

)=U(k − k

′′

)

(2π)



U(k − k

(0)

(k; E

)U(k

− k

′′

+ ... .

(9.56)

3. Born Approximation: Here only the leading terms of the ATA, the single

and double scattering with an impurity are considered, i.e., the self-energy

consists of two terms

(k, k

′′

; E

(9.57)

and the t-matrix takes the form

(k, k; E

)=U(0) +

(2π)



|U(k − k

(0)

; E

. (9.58)

4. Self-Consi stent Born Approximation (SCBA): The Born approxi-

mation can be improved by replacing in the second graph the unperturbed

propagator G

(0)

; E

), describing the free propagation of the elec-

tron between two s cattering events, by the conﬁguration averaged Green

function

G(k

; E

), or in graphical form

9.3 Approximations for Impurity Scattering 279

SCBA

(k, k

′′

; E

(9.59)

At the end of this section, we shall apply the SCBA to a system with

discrete energy levels in order to demonstrate th e mechanism of level

broadening due to impurity scattering.

5. Coherent Potential Approximation (CPA): It combines the SCBA

with the ATA by taking into account all graphs of the latter but with the

replacement of the free Green function by the conﬁguration averaged one

as in the former. Thus, the self-energy is represented by

CPA

(k, k

′′

; E

(9.60)

In all these approximations, the self-energy turns out to be diagonal in

the wave vector, which allows one to solve the Dyson equation to obtain the

conﬁguration averaged Green function

G(k, k

′

; E

k,k

′

− ǫ

− Σ(k; E

)

(9.61)

with the complex self-energy

Σ(k; E

)=cU(0) + c

(2π)



|t(k − k

− ǫ

. (9.62)

Its imaginary part

ImΣ(k; E

)=c

(2π)



|t(k − k

δ(E

− ǫ

¯h

(9.63)

can be expressed by the quasi-particle lifetime τ

and quantiﬁes the rate by

which the Bloch electron is scattered by disorder out oﬀ its state with wave

vector k . Note that the approximations 1–5 consider only scattering from

a single impurity, therefore, the self-energy is proportional to the impurity

concentration c. In the CPA, all diagrams with up to four interaction lines,

except those where these lines cross each other, are included. This point needs

further discussion in Sect.

9.5.

The modiﬁcation of the electron energy by impurity scattering becomes

particularly evident for systems with a discrete level spectrum, such as the

Hubbard bands in the atomic limit (with energies at t

and t

+ U ,see

Sect.

7.2) and the Landau levels of a two-dimensional electron system (with

=¯hω

(n +1/2), see Sect.

5.6). Their density of states has the form

(0)

(E)=



δ(E − ǫ

) (9.64)

with the degeneracy factor g

. Let us assume the impurity potential to be

extremely short-ranged as described by