Edge singularity in tight binding approximation

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Submitted on 1 Jan 1972

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Edge singularity in tight binding approximation

P. Joyes

To cite this version:

P. Joyes. Edge singularity in tight binding approximation. Journal de Physique, 1972, 33 (11-12),

pp.1081-1088. �10.1051/jphys:019720033011-120108100�. �jpa-00207334�

EDGE SINGULARITY IN TIGHT BINDING APPROXIMATION

P. JOYES

Laboratoire de

Physique

^{des Solides}

(*)

Université

Paris-Sud,

^Centre

d’Orsay,

^Essonne

(Reçu

^{le 22 décembre}

1971,

^{révisé le 23}

juin 1972)

Résumé. ²⁰¹⁴ Les spectres

d’énergie caractéristiques

^des transitions

électroniques

^{dans les métaux}

présentent

^un fort écart _par rapport ^aux spectres ^{à « un} electron » au

voisinage

^du

seuil,

^écart

qui

est dû à la réaction des électrons de conduction

(d’énergies ~ l’énergie

^de

Fermi)

^au

potentiel

transitoire. Nous étendons le traitement de Nozières et De Dominicis aux métaux dont les bandes

d’énergies

peuvent ^être décrites en « liaisons fortes ». Nous considérons le cas de la transition entre

une niveau lié et le continuum et celui de la transition entre deux niveaux liés. Ce calcul devrait permettre ^{une meilleure}

compréhension

^des spectres

expérimentaux.

Abstract. ²⁰¹⁴ The _energy spectra of electronic transitions in metals show ^a strong

divergency

from « one electron ^» spectra ^{in the}

edge vicinity

which is due to the reaction of conduction elec- trons

(with energies ~

^Fermi

energy)

^{to the transient}

potential.

^{We extend here the treatment of}

Nozières and De Dominicis to metals whose _energy bands can be described

by tight binding

^methods.

We consider transitions between the conduction band and a bound level and between two bound levels. This calculation should allow a better

understanding

^{of some}

experimental

^data.

Classification Physics ^Abstracts

18.30

The

positive potential

^{due to a hole on a bound}

level

severely

^{disturbs the} conduction electrons of

a metal. This effect has ^a

large

^{influence on} X ray transitions

[1] ]

^to

[6]. Studying

ⁱⁿ

particular

^{X ray}

spectra

^{near the} upper

edge,

Nozières and De Domi- ° nicis

[5]

^have

given

^an exact treatment of the

dyna-

mical

screening

^{of the transient}

potential.

^{Later on}

other electronic transitions have been studied with similar methods

(Photoemission [7], Auger

^effect

[8]).

Our aim is to

adjust

these models to metals whose energy bands can be described in the

tight binding approximation.

^The discussion will include electro- nic transitions between a bound level and the conduc- tion band

(especially

^the

absorption

^{case but the}

results can be extended without any

difhculty

^to

emission)

and between two bound levels. The main difference between our

study

^and the forementionned is that the

perturbation potential

^{is not}

supposed

^to

be

separable.

This makes the calculation more intri-

cate,

^however the final results can be

expressed

ⁱⁿ

simple

^{formulae. The}

presentation

of the calculations will follow the same lines as in the Nozières and De Dominicis _paper

[5].

1.

Tight binding

^model. Notations and

intégral

^equa-

tions. ^-

a)

MODEL. - The

deep

^hole

potential Vo

alters the electron

hopping

^{between sites. The}

pheno-

menon can be described

by

^the

following

hamiltonian

(as

ⁱⁿ

^(5),

the Coulomb interaction between electrons is

neglected),

^where

the

cio

and ci,, ^are

operators

for creation and destruc- tion of an electron with

spin 6

^{on an atomic state}

03A6i

i

which we suppose ^non

degenerate

^for

simplification.

b, ^{b +}

^and

Eo

^{are the}

operators

^and the energy for the bound

state, V

is the lattice

potential.

For a

given frequency

^{o, the} interaction hamilto- nian with the X _ray field is

and, using

^{the same notation as in}

[5],

the transition

probability

^is

proportional

^{to the real}

part

^{of the} Fourier transform of

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019720033011-120108100

1082

which can also be written

with and

As we shall see

further, only

^{the terms in W which}

are related to electric and

magnetic dipole

transitions will be considered.

b) ^EQUATION

^{OF MOTION. -} The function

Fij(t - ^t’)

are calculated with the

help

^{of new}

propagators adapted

^to the transient nature of the

problem :

Following

^the

diagramatic

^method

[5]

^or the equa- tion of motion method of

Langreth [9]

^{it can be shown}

that,

with the definition of the

deep

^hole

propagator :

and a new notation :

the ({Jij ^{functions are, in the}

absorption

^case

(where

t

t’),

solutions of the

system

^of

integral equations :

The

Gà

^are the Green functions for the

unperturba-

ted hamiltonian

The X _ray response will be

given by letting

^{-c --+ t’}

and T’ - t in

({Jij( ’r, ^{i’ ; t, t’),}

^we shall call

Lij(t - ^t’)

the limit of

({Jij(t’, t ; t, ^t’),

^so

^{that, according}

^to

^(3),

Defining

^{the function}

C(t) by

we

get

^also

where À is ^a

coupling

^{constant to the}

potential Vo,

which can be defined

by

(5)

becomes in matrix notation

Before

solving (4)

^and

(6),

^{we must obtain}

Gà(t)

explicitly

^and

investigate

^{the influence of}

symmetry

properties.

c)

EVALUATION OF THE

GP.(t).

^-

Using

^{the trans-}

formation

the hamiltonian

Ho

^becomes

In

(7),

the summation is extended over the first Bril- louin zone. The

corresponding

Creen functions are

given by

of

(p

is the Fermi

energy),

^{and the}

G °(t) ^by

where

QIN

is the atomic volume and

G°(q, t)

^the

Fourier transform of

G°(q, E)

( Y(t)

^is the Heaviside

function).

By introducing

one can write

Integration

ⁱⁿ

(9)

^{is limited}

by

^a maximum energy 8M

given by

^{the band structure.}

This

expression

^is

formally

^{similar to the one which}

has been used in

[5].

If we consider

only

^{the behaviour}

of the X _ray

spectrum

^{in the}

vicinity

of the upper

edge,

^{we can take for}

(9)

^its

asymptotical

^{form cor-}

rected at the

origin by

^{a term}

proportional

^to

£5(t)

(see [5]),

^we

^{get finally :}

with

and

the

exponential

^factor

^{e - ip,t}

^which appears in all pro-

pagators

^{can be} eliminated

by

^a proper choice of the energy

origin

^{at the Fermi level.}

2.

Symmetry properties

and resolution. ^-

a)

^SYM-

METRY PROPERTIES. ^- As in _many other

impurity

studies in

tight binding [10], [11],

^{we assume that the}

potential Vo

^{has the}

point symmetry

of the lattice and that its effects are limited to the rth nearest

neighbours,

if N is the number of

perturbed sites,

the

system (4)

^{is a set of N x N}

coupled equations.

However, symmetry properties

^will

partially decouple

these

equations.

For this _purpose we substitute for functions

Pi.

a set of N

symmetrical

^functions

P cxç.

^{Index a means}

an irreducible

representation

^{of the}

point

group

and (

is used to

distinguish

the functions of a

given

repre- sentation. After this basis

change, only

^{the matrix}

elements between two functions of a

given

represen- tation must be considered. The

system

^{becomes :}

The index n means that we use a new basis.

Up

^{to this}

point,

^our

development

^{has been}

quite general.

^{To make}

things

^{clearer we} shall consider the solution of the

equations

^{in the}

particular

^{case of an}

FCC _array when the influence of a

deep

^{hole located}

on the

origin

^{is limited to the 12 first}

neighbours.

Atomic states are

supposed

^{to be non}

degenerate.

By numbering

^the

symmetrical

^{wave functions}

we notice that the

ri representation

appears

twice,

(! 0 >

^and

12-’/’(11 > + --- + 12 »)

^and

^r15,

r12, r25’

^and

r25 only

^{once with the}

following degree

of

degeneracy :

nr ⁼

3, 2, 3,

^{3. Therefore} the initial

system

^{of 13 x 13}

equations

^can be reduced to 4

independent equations with,

^for

instance,

^for

r15

and two

systems

^of

coupled equations

^for

ri,

^bet-

ween qJOO,n ^and qJl0,n ^{on the one}

hand,

and between qJ 11,n ^and qJ 01,n ^on the other hand. The first of these two

systems

^can be written :

Let us notice that the elements of the new matrix

Go

are linear combinations of the elements of

GO

and that

they

^can still be written :

Now we shall omit index n.

b)

^THE INDEPENDENT EQUATIONS. ^- Let us

analyze

first _eq.

(12) ;

^{in this case} the Nozières and De Domi- nicis treatment can be

exactly reproduced.

^{The solu-}

tion is

with

According

^{to the}

preceding definitions,

^{to obtain}

Lrl s(t - t’)

^{we have to let 1:} = t’ and 1:’ = t in

~r15

the

divergence

^{which occurs then can} be avoided

by introducing

^an

imaginary

^cut-off

(iço) -1

^where

Ço

is an energy of the order of the band width.

Thus,

we have

where

Y(t)

appears from the condition t’ - t > 01 and where we have restored the function

exp(iut).

Due to our

approximation

^{on the}

G°,

^{this result.}

is

only

^valid

for 03BEot 1

> 1.

It is

shown,

^{in the}

appendix,

that the function

C(t),,

defined

before,

^{is the sum of} two terms :

where A is the shift of the hole energy due ^to then transition.

Only

^the

Cl(t) part

^is

responsible

^{of the}

power

dependence

^of

F(t). By applying

^{the basis.}

change 03A6i -> 03A6a03BE

in formula

(6),

^we isolate each irre- ducible

representation

^{and the}

Tl 5

contribution to

C’(t) (that

^{we calculate as in}

[5]),

^{can be written :}

c)

^THE SYSTEM OF COUPLED EQUATIONS. ^- The pro-- blem is more

complex

^for

system (13).

^{We shall first.}

write it in matrix form.

By introducing

^{the vectors.}

1084

and the matrices

we see that

(13)

^becomes

The solution of these

systems

^is

given by

^Muskheli-

shvili

[12] ;

^{in the}

appendix

^{we show that it can be}

written as

where,

^{if we call}

Ài

^the

eigenvalues (that

^{we assume}

distinct) of (A - B) (A

⁺

B) -1

^{we have}

fi(03C4 - i’)

^{is a matrix} whose elements are

In formula

(17)

p is no

longer

^a

vector,

^{as in}

(16),

but the matrix :

the

justification

^{of this}

change

^is

given

^{in the}

appendix.

The

ôi

^defined

by (18)

^are the usual

phase-shifts, they satisfy

^{the relation}

[13] :

We have also

(Im

^means

imaginary part.)

Let us notice

that,

^when

t’ --> + oo

and t ^{- -} _oo, solution

(17) approaches

^{the matrix whose elements}

are the

starting

^Green functions of the emission

problem,

^{i. e. the case where the}

deep

^{hole has a}

permanent

effect. To prove this

point,

^{let us write}

the

equation

^which

gives

^Ge.

According

^to

(16),

^we

have

This

system

^{can be solved}

directly by

^{Fourier trans-}

forms

(the -

^{and +}

correspond respectively

^to

and E

0).

If we use

equation

where A * and B* are matrices which are defined

by

(A *

^{is real and} B* pure

imaginary),

^{we have :}

which, by using

is transformed in :

a result which is also

given by (17)

^{when we take the}

limit.

The matrix

L(t),

^{limit of}

~(03C4, i’ ; t, t’)

^when

ï -+ t’ ⁼ 0 and i’ --+ t, ^{can be} obtained

by

^{the same}

method as for the

independent equation

^{case :}

by introducing again

the cut-off

(iço) -1

^{we obtain :}

with 1 Ço t 1

^{» 1 and where gi is a matrix with elements}

We know that

only

^the

Cl(t) part

^of

C(t)

^contri-

butes to the power

dependence

^of

F(t).

In the appen- dix it is shown that the

rl

contribution to

C’(t)

^is

and

collecting

^{all the} contributions

with

(the

^{factor 2 comes from the}

spin degeneracy), finally :

as before the factor

Y(- t)

^{comes from the}

asborption

condition t’ - t > 0.

3. The transition

probability

for the transition of an

électron from a bound level to the conduction band.

- The W elements of formula

(1)

^{take non zero}

values

only

for sites nearer than one

given

^limit

(say

^{the sth nearest}

neighbours).

^{As before}

symmetri-

cal functions must be built and this new basis will contain the

03A6a03BE ^system

^{if s} > r. In this new basis we

write the elements of W :

WT.

To be able to write

S(t)

we must

complete

^{the set of}

Lij,

^{which concerns}

^only

the rth nearest

neighbours, by

the Green functions of the

problem

^with

Vo

⁼ 0 for sites

lying

^between

the sth and the rth sites.

Moreover, only

transitions which

satisfy

the conservation of momentum rule will be allowed. To make

things

^{clearer we} shall examine the

particular

^{case where s = r and where the}

deep

hole is such that 1 ⁼

1,

^then

S(t)

^is

with coo = Jl -

Eo -

d . We notice that

by taking

into account

only

^{the electric and}

magnetic dipole

transitions the

r2 5

^term

disappears

^{in the sum}

(27) Setting a = co + Eo + A - y,

^{the real}

part

^{of the} Fourier transform

gives

^{the behaviour of the} spec- trum near the

edge

with

r is the usual Gamma function.

4.

Shape

^{of the}

spectral

^{line in a} transition between two bound levels. ^- The

preceding

^{results can be}

applied

^to other transitions

problems.

^{We shall exa-}

mine now the

absorption

^{of an X} ray line in a elec- tronic transition between two bound

levels,

^a

pheno-

menon in which the conduction electrons intervene

by asymmetrizing

^{the line.}

The hamiltonian is

and,

^for

calculating

^the

asymmetry

^{index the know-}

ledge

^{of the Green functions}

(with Fij(1:, 1:’; t, ^t’)

⁼

({Jij(7:, ^{T’ ; t, t) G(t - t’),}

index 1 and 2 refer

respectively

^{to the}

empty

^level in the initial and final

state),

^is

^required.

Let us suppose that the

potentials

^{due to the holes}

have the lattice

symetry

^{and the same range. Then} matrix

Tl

^and

T2

^{are both}

partially diagonalized by

the

symmetric

^wave functions in the same way. Thus in the conditions discussed before of ^an FCC lattice with hole

potential

effects limited to first

neighbours

there will be one

integral equation

^{for each} repre- sentation

F15, rl 2, 1’2-5" F25

^{and one}

system

^{of two}

coupled equations

^for

ri.

1086

In this last

representation

the matrix

(p(,r, i’ ; t, t’)

is

given by

in which

G1

^{is the matrix of the Green} function in the initial state

given by (22)

Ai

^and

BÎ

^are

given by (21)

^{in which 0 is}

replaced by 0i

^defined

by Fi

⁼

Àel.

Eq. (28)

^{can also be written}

By introducing

and

we see that

(29)

^{takes the same form as}

(16).

^{Thus the}

solution can be undertaken in the same way : ^as in

(19),

^{we have}

If we take into account relations :

A 12 - B12

^can be written :

therefore

(30)

^becomes

or

Moreover, using (20), (24)

^and

(31)

^{we have also :}

These results have been established for

coupled equations,

^{of course}

they apply

^{also for}

single

equa-

tions,

^thus the power

dependence

^of

G(t) (see ⁽²⁵⁾

and

(26))

^is

with

Formula

(33)

^{can be}

directly

^{used to} calculate the

asymmetry

^index.

According

^to

[7],

^{if a lifetime}

ci

yx ’

^is introduced the

shape Yx(,r)

^{of the}

spectral

line will be :

with

and where 8 is measured relative to the maximum energy in the absence of a lifetime

broadening.

From

(34)

^{one deduces}

easily

^the

asymmetry

^index.

Conclusion. ^- In this paper the

shape

^of the spec- trum of two different X ray transitions have been studied. In

spite

of the matrix form of the

integral equations

the calculation has been undertaken ana-

lytically

^{for the two}

problems

and it appears that the method could also

apply

^to other transitions or to

problems

^{in which one would consider}

systems

^{with of}

larger degree (potential

^with

larger range).

Our results show a

simple dependence

^{on the}

phase

shifts

br.

^These

parameters

which intervene in nume- rous other

impurity problems

^{can be obtained}

easily

when one knows the

density

^of

unperturbed

^states

in each

representation [11].

Electronic transitions in transition metals have been

experimentally

^studied

extensively,

^e. g. X ray

absorption

^{or emission}

([14], [15]), photoemission

([16], [17]),

^ESCA

[18], Auger

^effect

([19], [24]).

Some of these

experiments

^have

already

^{revealed the}

singular

behaviour in the upper

edge spectrum [5].

This calculation should allow a more

precise

compa- rison between

theory

^and

experiment.

Acknowledgments.

^{- It is a}

pleasure

^{to thank}

Pr. J.

Friedel,

Dr M. Natta and Dr G. Toulouse for

reading

^the

manuscript

^{and usefull comments.}

APPENDICE

The calculation of

e(T, r’ ; t, t’).

^- The solution of

system (16), given by

Muskhelishvili

[12],

^is

with

where the

Ài

^{are the}

eigenvalues

^{that we assume dis-}

tinct of

(A - B) (A

⁺

B) -1.

The other

coupled equations system

^between ({Jl0 and _(pl, take also the form

(16),

^{therefore we can}

write

(35) again

^{in which we}

change

the former defi- nitions of _({J and

f by

The calculation of the

integrals

ⁱⁿ

(35)

^{can be done}

by

the method used in

[5]. Then,

^{if we take into account}

the

following equations

we get

where

f’

^{is a matrix} defined in the text.

By using

ç

finally

^becomes

Eq. (36)

^and

(37)

^are consequences of

Sylvesters

theorem

[25].

The calculation of

C(t).

^{- Let us} take the limit of

(17)

^{when z’ - 1 -}

^0,

^{we obtain}

where Ge and gi ^are

given

^{in the text.}

If we take into account the

following equations

we have

Two terms appear in

(38) ;

^we shall call

Clr,

^the

contribution to C

arising

^{from the second term}

of

(38), Fi

indicates that we

study

^{here the}

Tl

repre- sentation.

By using (6)

^{we obtain :}

Moreover,

^{we have the}

equations

therefore

or

As, according

^to

(20),

^{we can write}

1088

we have here

A term similar to the first term of

(38)

appears for each

representation.

^{All these terms, treated}

together,

give

^a contribution to

C(t) proportional

^{to t which}

represents only

^a shift of the

deep

^{hole energy. We}

shall

put

^{in the}

general expression

^of

G(t)

^{a factor}

where A is the energy shift.

References

[1]

^FRIEDEL

(J.),

^Phil. Mag.,

1952, 43,

^153.

[2]

^FRIEDEL

(J.),

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[3]

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(B.),

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(J.)

and NOZIÈRES

(P.), Phys.

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1969, 178,

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and ROULET

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MUSKHELISHVILI

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