Spectral limits in the tight-binding approximation

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Spectral limits in the tight-binding approximation

F. Ducastelle

To cite this version:

F. Ducastelle. Spectral limits in the tight-binding approximation. Journal de Physique, 1974, 35 (12),

pp.983-988. �10.1051/jphys:019740035012098300�. �jpa-00208221�

SPECTRAL LIMITS IN THE TIGHT-BINDING APPROXIMATION

F. DUCASTELLE

Office National d’Etudes et de Recherches

Aérospatiales (ONERA)

92320

Châtillon,

^France

(Reçu

^{le 15 mai}

1974)

Résumé. ²⁰¹⁴ On passe ^{en revue} les méthodes et théorèmes

principaux

qui permettent d’estimer les limites spectrales pour des systèmes décrits dans

l’approximation

des liaisons fortes. On donne

quelques applications concernant les métaux de transition, ^les alliages désordonnés et les semi- conducteurs amorphes.

Abstract. ²⁰¹⁴ A brief review is given ^{of some} basic theorems and methods which

give

estimates of the spectral limits in systems described within the

tight-binding approximation. Applications

^to

transition metals, disordered alloys ^and amorphous semiconductors are given.

Classification Physics ^Abstracts

8.120

1. Introduction. ^--- The

tight-binding approxima-

tion is

widely

^{used to describe} systems

involving

electrons which are more or less localized on the atoms. In

spite

^{of the}

approximations

made in the

simplest

^schemes

(the assumptions

that the atomic orbitals are

orthogonal

and that the transfer

integrals

are

only important

between first

neighbours),

^it

often leads to very convenient model Hamiltonians.

In

particular

this kind of model has been used often in recent years ^to describe disordered

systems

^such

as disordered

alloys, liquids

^or

amorphous

^semi-

conductors.

It is well known that within the

tight-binding approximation

the bandwidth is

finite,

^{and that this}

width is

mainly

determined

by

the value of the transfer

integrals.

^When

dealing

with disordered systems, the Bloch theorem cannot be

used,

^{and it is in}

general impossible

^{to calculate}

exactly

^the

density

^of states ; it is therefore _very useful to

dispose

^{of some theorems}

which

give

^the

regions

^{of the} real axis where the

eigenvalues

^{of a}

given Hamiltonian

must lie. As shown

by Cyrot-Lackmann [1]

^and

by

^Khomskii

[2],

two useful theorems in this context are the Perron- Frobenius theorem and the

Hadamard-Gerschgorin theorem,

^{but some} other methods can also be used.

In this

article,

^{we want to}

give

^{a brief review of}

the basic theorems or methods which

gives

^estimates

of the

spectral

^limits and which do ^not

depend

^on

the lattice

periodicity.

^Some

portions

of this work

are not

actually

^{new, but we have}

thought

^{that it}

might

be useful to show that the various results obtained

recently

^{in this field are}

straightforward

consequences of ^a very limited number of

general

theorems.

In section

2,

^we recall ’the Perron-Frobenius theorem. In section

3,

^{we recall the Hadamard-}

Gerschgorin

theorem and we

give

^an extension of it well fitted to the

study

^of

degenerate

^{d bands,}

Section 4 is concemed with a method referred to as the scalar

product

^method and section 5 is devoted to the min-max theorem of Courant and Fischer.

In these two last

sections, particular

attention is

paid

^{to the}

applications

^to disordered

alloys

^and

amorphous

semiconductors.

2. The Perron-Frobenius theorem. ^-- The Perron- Frobenius theorem states that if all the elements

aij

of a matrix are

positive,

^{then the}

eigenvalue Àmax

whose absolute value is maximum is

real, positive

and non

degenerate.

Moreover the

components

^of the

corresponding eigenvector

may be taken to be

positive

^and

Àmax

may be characterized ^as follows

[3], [4] :

^:

where x varies over all non

negative

^{vectors different}

from zero. In

particular

^from

(2. 1)

^{we obtain :}

As shown

by Cyrot-Lackmann [1],

this theorem

applies

^{in a}

quite

natural way ^to

non-degenerate tight-binding

systems in which all the transfer inte-.

grals

^are

negative.

^It

gives

the well known result that the lowest _energy in the spectrum is obtained for

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019740035012098300

984

a linear combination of atomic orbitals with

positive

coefficients

(bonding state) ;

furthermore it tells us that the absolute value of the lower bound is

larger

than

(or equal to)

^{that of the} upper bound. For

example

^{let us consider a}

periodic

lattice in which every ^atom has Z first

neighbours;

^from

(2.2)

^we

have :

where

fl

is the transfer

integral

^{between first}

neighbours.

3. The

Gerschgorin

theorem and its

generalization.

- 3.1 THE GERSCHGORIN THEOREM. - The Gersch-

gorin

theorem is

equivalent

^to the Hadamard criterion which

gives

^a condition for a

determinant

^to be diffe- rent from zero. The

proof

^{is so}

simple

that it is worth

giving

^it

explicitly [5], [6]. Let t/1

^{be an}

eigenvector

^of

the matrix aij

(with

components

t/1i)

^{and  the corres-}

ponding eigenvalue ;

^{then :}

and therefore :

Let us chose i such that

1 t/1i

⁼

max 1 j 1,

it follows

j

that :

Since the

inequality (3.3)

^is satisfied for at least one

value

of i,

^we may ^assert that is contained in the union of the discs centred _{on aii} and with radii

Furthermore, by using continuity

arguments, ^{it can} be shown that if the union of _p discs is a connected part of the total

union,

then it contains

exactly

p

eigenvalues.

^{In the case where an}

off-diagonal

^matrix

element _aik is

larger

in modulus than the sum of the moduli of the other _ones, we have also

[5] :

The

Gerschgorin

^criterion

generalizes

^{a small} part ^of the Perron theorem.,From

(3.3)

^we indeed deduce :

which is to be

compared

^to

(2.2).

Let us notice _{that eq.}

(3.6)

^{may be} considered as a

particular

^{case of a more}

general result, namely :

where Il A II denotes

^{the norm of the matrix A which}

is subordinate to some vector norm. Such a norm is defined

by :

Let us also define the _p-norm of a vector x :

with the

particular

^{cases :}

The

corresponding

subordinate norms are

[6] :

Il A Il 2

^{is of course the usual norm and is}

independent

of the

basis,

^{which is not the case} of the other ones.

, A subordinate norm has the

particular properties :

Let us now write

that §

^{is an}

eigenvector

^{of A with}

eigenvalue :

By taking

^{the norm} of both sides

of (3.13),

^{we obtain :}

which _proves

(3.7).

Useful

applications

^{of the}

Gerschgorin

^theorem

have been

given by Cyrot-Lackmann [1] and

^Khom-

skii

[2].

^{We shall now} show that it is

possible

^to

extend this criterion.

3.2 A GENERALIZATION OF THE GERSCHGORIN THEO- REM. ^- We shall try ^to

generalize

^the

Gerschgorin

theorem in

allowing

the matrix elements aij ^{to be} square matrices. We have in mind a

degenerate tight- binding

system ⁱⁿ

which q

^{atomic states are associated}

with each site. The matrix of the Hamiltonian in the basis

generated by

^{these states is a} block matrix in which each block is a q x q matrix. Let

Pi

^{be the}

projection

operator ^which separates ^out all orbitals

on

site i,

^{and Â}

and Çi

^an

eigenvalue

^of the Hamilto- nian H and its

corresponding eigenvector ;

^{then :}

Using

^the

simplified

^notation

Hij ^{for Pi} HPj,

^and

taking

^the norm, ^we have :

where :

Now we take into account the

specific properties

^of

our

model ;

^{that H is an hermitian} _operator ^and we

choose the usual ^norm in the Hilbert _space defined from the scalar

product. Let pf

^{be the}

eigenvalues

of

Hih

it is clear that :

and therefore :

We may ^now argue ^as in section

3.1 ;

^{we chose i in}

such a way

that Il t/1i Il

⁼

max II ; II,

_j ^which

^implies

that :

Finally

^{we obtain :}

which is the

required generalization

^of

(3.4).

Let us

apply

this criterion to the case of a pure transition metal with coordination number Z. We

keep only

^transfer

integrals

between first

neighbours separated by

^{a distance R. Then} all blocks

Hij

^{are the}

same,

H;;

⁼

^H(R),

^{and we have :}

Il H(R) Il

^is

given by

^the

largest

^transfer

integral (in modulus),

^{that is}

by 1 ddu 1

ⁱⁿ Slater-Koster’s ^nota- tions

[7] ;

^{hence :}

This result is the natural extension of the corres-

ponding inequality

obtained for non

degenerate

^bands

in eq.

(2.3).

^{The main} difference is that because the various transfer

integrals

^{included in}

Hij

have different

signs,

^{we are no}

longer

^sure that the bound

Z [ ddJ 1

is

actually

^{attained. In}

fact,

ⁱⁿ three dimensions the

true bound is much smaller

(for example

^{in a FCC}

lattice, 1 À Imax ~ 5 ) 1 ddu with

realistic values of the transfer

integrals [8]).

^{The one} dimensional case is a

particular

^one because the orbitals of different _sym- metries do not

mix;

^{as a} consequence the Perron theorem still

applies

^{for each} sub-band and the bound

2 1 dda is

attained. This indicates that better bounds cannot be obtained in

general

without further information on the system. ^From

(3.20)

^{we can also}

derive a criterion for the occurrence of a

crystal

^field

splitting

of the bands. Let us assume that

Hu

^{has two}

different

eigenvalues (which

^would

correspond

^{to the}

t2g and eg symmetries

for d electrons in a cubic envi-

ronment),

oc, and _(X2, then a gap will

necessarily

appear

if 1 oc, - (X2 1 > 2 Z 1 dda [ .

Finally

^{it should be}

pointed

^{out that}

(3.21)

^{is still a}

consequence of the

general

^result

(3.7) provided

^we

define a new

nonn Il H Il’ :

^:

which is the norm subordinate to the

following

^vecto-

rial norm :

4. The scalar

product

method. -- 4. 1 FORMULATION

OF THE METHOD. - When

dealing

^with

physical

^sys-

tems, ^we know that the Hamiltonian H is hermitian and therefore the

spectral

^radius

(that

^{is the}

eigenvalue

whose absolute value is

maximum)

^is

equal

^{to the}

usual norm defined

by (3.llb).

^More

generally

^we

may ^use the

properties

of the Hilbert _{space. In}

parti- cular,

considerations based on scalar

products

may be fruitful as

explained

^{below. For}

example,

^{let us}

assume that H can be written as :

where

Ho

^{and V are hermitian}

operators

^whose

spectra

are known. Let

hi and

^{be the}

eigenvalues

^of

Ho and V, and 1 t/J

> an

eigenstate

^{of H} associated with the

eigenvalue

^{À. From :}

we obtain :

where we have assumed

that Il t/J Il

⁼

t/J 1 > 1/2

^{,-- 1.}

But :

and

finally :

In the same way :

986

Equations (4.5)

^and

(4.6)

^are

equivalent

^{to the more}

explicit propositions :

with :

4.2 APPLICATION TO AMORPHOUS SEMICONDUCTORS.

- As an

application

of the criterion

(4.7),

^{let us}

consider the now well-known model hamiltonian for

amorphous

semiconductors used

by Thorpe,

^Weaire

[9]

and others. We shall use here the notations of Friedel and Lannoo

[10] :

. The

state 1 iJ >

^is the atomic

hybridized sp"

^orbital

pointing

^{from site i}

along

^{the bond J. The}

positive promotion

energy

Ep - ^ES

^is

^equal

^{to -}

⁽ⁿ

⁺

¹⁾

^d

and fl

^{is the}

negative

^transfer

integral.

The

spectra of Ho

^{and V are} very

simple.

^The

eigen-

values

hi and vi

^are

given by :

Therefore

min 1 vi 1 = max 1 vi 1 = 1 P 1,

and the inter-

i i

vals

Di and di

^{have the same} bounds. As a conse-

quence ^we realize at once that there is

always

^a gap in the

spectrum.

^{This is} clear if the intervals

Dl

^and

D2

^{do not}

overlap (that ^{is d} 1/1 PI> 2/(n

⁺

1)),

^but

from

(4.7b)

this is also true in the other case because the interior of the intersection of the intervals is forbidden. From

continuity arguments

^{it is then} rather obvious to count the number of ^states in each subband. In the atomic

limit P

^--.

0,

^{the subbands} have s and _p character with the

weights

^{1 and n}

respectively ;

in the other limit 4 -

0,

each subband

contains (n

⁺

1)/2

^states.

Actually,

the Hamiltonian

(4.9)

^{is so}

simple

^{that a}

much more detailed

description

^{of the}

spectrum

^can be obtained rather

easily (see

references

[9], [10], [11]

for

example),

^{but the}

^advantage

^{of our} criterion is that it allows

generalizations

^{to more realistic}

models ;

for

example

^we could take a distribution of transfer

integrals ;

^{it is clear that in this case the} gap will survive in the covalent limit 4 - 0

provided

^that

min 1 Vi 1 =1=

^0.

i

Finally

^{let us} put ^some

emphasis

^{on the}

duality

of the

descriptions

from atomic orbitals and bond orbitals in this

problem.

^In

(4.9)

the basis of atomic orbitals was

chosen,

^{but we}

might

^{have taken a}

basis of bond orbitals from which the

eigenstates

^{of V}

are

easily

^built up. This is

equivalent

^to

exchange

^the parts

played by Ho

^{and V. This}

duality

^is

quite

^clear

if we rewrite H in the

following

canonical form :

Now V and

Jeo

^{have the same}

property :

and we can still

apply

the criterion

(4.7)

^where

H0 plays

^the part ^{of V and}

conversely :

,with :

and

d’1,2

^{are the}

corresponding

open intervals. Of

course from

(4.14)

^{we find the same} gap ^as before.

4.3 APPLICATION ’TO DISORDERED ALLOYS. ^- We shall now consider the usual model hamiltonian for disordered

alloys :

This is a

simple tight-binding

Hamiltonian in which the disorder is introduced

through

^a statistical distri- bution of the atomic levels _El. As we may notice from the discussion of section

4.2,

^the best bounds are

found when the

spectral

limits of V are

symmetric.

We assume that this is the case here. If this was not true, it would be sufficient ^to substract

from

Ho

^{and to add the same}

quantity

^{to V. For}

simplicity,

^{we also assume} that there is no gap in the pure system ^described

by V,

^{so that}

min 1 vi 1

^{= 0.}

i

Therefore we are left with the

single

^criterion

(4. 7a) :

which is the well-known theorem about spectra ^of disordered

alloys ;

^the spectrum ^{of an}

alloy

is contained in the union of the spectra ^{of the}

alloy

components.

4.4 DISCUSSION. ^- The

proofs given

^{in section 4. 2}

and 4.3 are not

actually

^very different from those used

by

^other authors. For

example,

Schwartz and Ehrenreich

[12] (for

^{the case of}

amorphous

^semi-

conductors)

^and

Kirkpatrick, Velickÿ

and Ehren- reich

[13] (for

^{the case} of disordered

alloys)

^have

given

^similar

proofs by using

^the convergence proper- ties of series

expansions

of the resolvent

[14].

^Both

methods are

equivalent

^{but the} present ^{one is}

perhaps simpler.

It should also be

pointed

^{out that from the}

previous

discussion

nothing

^can be said about the

possibility

of the bounds to be

actually

attained. Further infor- mation can

only

be obtained from a detailed

analysis

of the

problem

^under

study.

^For

example

^{it can be}

shown that the bounds in

alloys

^are attained for a

perfectly

disordered

alloy [15], [16].

^The

assumption

of

perfect

disorder is crucial. This is clear when

looking

^{at the}

following example.

^{Let us consider an}

AB

alloy

in which all A atoms have Z first

neighbours

of B

type

^and

conversely,

^{as in a CICs}

type

^structure for

example.

^The energy levels _BA

and EB

^{are measured}

from the centre of the

symmetrical

spectrum :

Let

PA(B)

^{be the}

projection operator

^which

separates

out the orbitals on

A(B)

^{sites. The} Hamiltonian H

can then be written as

(1) :

If we notice that :

then :

so that the

eigenvalues

^{of H are}

given by :

We find therefore that there is

always

a ap and that the extreme bounds are

given by

^±

ô2 ^{+ II Y II2}

which are smaller in modulus than the

corresponding

bounds of the disordered

system.

^±

[ô

⁺

^il V Il].

5. The min-max theorem. ^-- The min-max theorem of Courant and Fischer is a very

elegant

^theorem

which

gives

^a characterization of each

eigenvalue

^of

an hermitian

operator

^A

[18], [3].

^{If these}

eigenvalues

are labelled in

decreasing

^{order :}

then Âk

^is

given by :

where

the y’

^are

^linearly independent

^vectors.

(1) This formulation has been already ^used by Straley [17] ⁱⁿ

the case of compound semiconductors.

If the

increasing order À 1

^...

ÀN

is chosen the theorem becomes a max-min theorem. An

important

consequence of this theorem is that if ^we consider the

projection PiAPi = Ai

^{of the} operator ^A into a

subspace

of dimension

N - i,

^{then :}

Another consequence is that if a

non-negative

ope- rator B is added to

A,

^then

This theorem has

already

^{been used}

by

^Thouless

[16]

to

give

^a

general proof

of the theorem on bounds in disordered

alloys

^mentioned in section 4.3. The basic idea is to notice that we can find a

projection

operator ^which

gives

^{the same}

operator

^when

applied

either to

the

disordered

alloy hamiltonian,

^{or to the}

pure metal one.

(The operators PA

^and

PB

^{defined in}

section 4.4 fulfil this

condition.)

^{We will}

apply

^here

the same method to the

amorphous

semiconductor

problem.

Let

Ps

^and

Pp

^{be the}

projection operators

^which

separate

^out the s and _p orbitals on each site. The Hamiltonian

(4.9)

^can then be written as :

Let us define

H2

^and

H3 by :

The

eigenvalues

^of

Hi, H2

^and

H3

^are

Âp,

lip ^and vp

respectively.

^{Since nN is} the dimension of the _space

generated by the p

^{atomic states}

(N

is the number of atoms, and the dimension of the whole Hilbert space is

(n

⁺

1) N),

^{we obtain froID}

^{(5.2) :}

But since L1

0, H2 - Hi

^{is non}

negative,

^and

(5.3)

leads to :

Finally

^from

(5.7)

^and

(5.8),

^we get :

The

eigenvalues

^of

H2

^{= V are :}

988

and therefore

(5.9) gives :

This result is remarkable because it shows that there

are two ô functions in the spectrum

of Hl

^with

weight

n 2 corresponding

^{to the}

énergies that

^{is to}

- L1

± p

^{for the} spectrum ^{of H. For these states}

Àp =’ Jlp

^{= vp which proves} that these states are

purep ^states. Of course this result is _{not new,}

[9], [10],

but the present

proof

^{is a} clear illustration of the

possibilities

^{of the min-max} theorem. Furthermore there is no

difficulty

ⁱⁿ

applying

^{if to more elaborate}

models.

Let us now

study

^{the 2}

N remaining

^states.

Eq. (5 .11) gives

^some

information,

^{but it can be}

completed

^if

we repeat ^the

previous

arguments

by using

^the pro-

jection

operator ^{into the} s-space. We therefore define

H2

^and

H3 by :

The

comparison

^{of the}

eigenvalues Àp, /Zp

^and

vp

^of

Hl, H2

^and

H3 gives :

hence :

Let us

forget

^the à functions and relabel the 2 N

remaining

^{states. From}

(5 .11 )

^and

(5.15),

^we get :

where :

Therefore we have

always

^a gap whose width is

12 1 fl 1 - (n + 1) Id 11 .

^.

It is rather obvious that we

might repeat

^{the same}

proofs by starting

^{from a} bond orbital

representation.

It is

easily

checked that the same result would be obtained once more.

Although

^more

complete,

^the

argument ^{would be}

actually

^{similar to that}

given by

Heine

[19].

We

might give

^{some other}

applications

^{of the}

min-max theorem. For

example,

^{if we} consider the Slater-Koster

problem

^{of a}

single

^localized

potential,

eq.

(5.2) gives directly

^the familiar result that the

eigenvalues

^{of the} pure system

separate

^the

eigenvalues

of the

perturbed

system,

except

^{for one}

eigenvalue

which _may

correspond

^{to a localized state.}

As a conclusion of this

section,

^{let us}

point

^{out that}

the min-max theorem enables us to recover all the results obtained in section 4. The min-max theorem is a little more subtle to

apply

than the scalar

product method,

^{but it}

provides

^us with results which are more

general

^and

complete

^{as can be seen from the}

previous example (see

also Thouless

[16]

^{for the case}

of disordered

alloys).

6. Conclusion. ^-- The main conclusion which ^can be drawn from the

present study

^is

^that,

^{within a}

tight-binding approximation,

^{we can} in fact obtain a

substantial information on the

spectrum

^without any reference to

periodicity.

^{It can} also be said that the theorems we used are

particularly

well suited to the

tight-binding approximation

because in this case

advantage

is taken of a

representation

^{in real} space.

These theorems have allowed us to recover some well known

results,

^{but due to their}

generality they

^can

lead to

straightforward

extensions as shown in some cases. There are

probably

many other

applications

to be

explored,

and it is

hoped

that this work has

some use in

stimulating

further researches in this field.

Références [1] CYROT-LACKMANN, F., ^J. Phys. ^C. Solid State Phys. ⁵ (1972)

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