Elements of tight-binding method in terms of graph theory

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Elements of tight-binding method in terms of graph theory

N. Cotfas

To cite this version:

N. Cotfas. Elements of tight-binding method in terms of graph theory. Journal de Physique I, EDP

Sciences, 1993, 3 (11), pp.2269-2284. �10.1051/jp1:1993244�. �jpa-00246867�

J.

Phys.

I £Fonce 3

(1993)

2269-2284 _NOVEMBER 1993, PAGE 2269

Classification

Physics

Abstracts

0210 6170R

Elements of tight-binding method in terms of graph theory

N. Cotfas

Department

of Mathematics,

Faculty

of

Physics, University

of

Bucharest,

P-O- Box 76-54,

Bucharest,

Romania

(Received

29 April 1993, revised 19

July

1993, accepted 28 July

1993)

Abstract. A description of the diamond structure in terms of grapb theory obtained by

using

_an additional axis

(similar

to the additional axis _{one uses} in the _case of the

hexagonal lattice)

is presented. The group of all

graph-automorphisms

of the

graph

_{we use as} model is

isomorphic

to the _{space group}

O( (the

symmetry group of the diamond

structure)

and it admits natural representations in _some mathematical _spaces that _can be associated to the

graph.

These spaces allow _us to obtain

advantageous

mathematical formalisms for certain models and offer

new

possibilities

to extend them to

crystals

witb

impurities

_or defects,

quasi-crystals,

etc. We

exemplify by considering

_some elements of the

tight-binding

method.

1 Introduction.

The

advantages

of _an additional axis in the

description

of the

hexagonal

lattice _are well-known.

Similar

advantages

_can be obtained in the _case of the diamond structure if _{we use} both the usual Cartesian reference

system (or

the

corresponding

orthonormal basis

I, j, k)

and _a

system

of reference

having

four _axes

(corresponding

to the vectors eo = -I

j k,

_ei

= -I +

j

₊

k,

e2 " I

j

₊

k,

_{e3 "} I ₊

j k).

Thus,

beside the usual Euclidean _space

1t3,

_{we use as} model the factor vector space P

lt~/W,

where W ₌

((I, I, I, I)

I _E

lt)

_11,

2].

=

Denoting by _[To,

_{xi, x2,}

x3] "((To

₊

I,

_xi

+1,

_x2

+1,

_x3 +

1)

1 _E

lt)

the coset

correspond- ing

to

(To,

_{xi, x2,}

x3),

the

correspondence

between the two models is the linear

isomorphism

q:P-it~

n

[To,

xi, x2, x31 =

(-To

_xi + x2 + x3, -To + xi x2 + x3, -To + xi + _x2

x3) (1.1)

n-~(x,v,z)

=

1°,(v

₊

z)/2,(x

+

z)/2>(x

+

v)/21.

For each

point

^of the space we have two

descriptions ((x,

_y,

z)

E 1t3 and

[To,

xi, x2,

x3]

E

P),

and the

isomorphism

_q allows _us to choose in each situation the most convenient of them.

Particularly,

the usual scalar

product

ⁱⁿ the two

descriptions

is

((xo>xi>x21x3j>ix(,x[,x[,x())

"

((x,y,z),(x',y', z'))

"

=x.x'+y.y'+z.z'=3~jxj

3

x(-~jxj xl. (1.2)

j=0 j#1

The usual

representation

of the

symmetry

group

O( (Fdlm)

of the diamond structure in the _space P coincides

[I]

^{with the}

subgroup generated by (A)

U

(A,

_{a E}

G4)

in the _group of all isometries of

P,

where G4 ^is the _group of all

permutations

of the set

(0,1, 2, 3)

and

A,A,.P-P

A (To> Xl X2>X3j "

(~X0

+

1,

-Xl1~X2>

~X3j

l~«

_{iX0>Xl >X2>X3j " [X«(0)>}

Tail)> X«(2)>X«(3)j (1.3)

The _set

13

ti = n =

(no,

_{ni, n2,}

n3)

E

Z~ ~j

nj ^E

(0;1)

j=0

where Z is the set of all

integers,

_can be identified with the

O(-invariant

subset

([no, ni,n2,n3] (no,

_ni,

n2,n3)

E

ti)

of P and the set R of the

corresponding points

of the space form the structure of diamond

ill.

The

representation

of

O(

in P subduces _{a represen-} tation

by permutations

^of

O)

in ti. The group

O)

is

isomorphic

to the

subgroup (denoted

also

by O() generated by (A)

U

(A,

_a E

G4)

in the group of all

permutations

of

ti,

where

A,A,.ti-ti

A

(no,

_{ni, n2, n3}

=

(-no

+

I,

-ni, -n2, -n3

,

A, _(no>

nl> n2>

n3)

"

(n,(0)> "«(1)1"«(2)> n,(3)) (1.4)

3

Let

So

= m =

(mo, ml,m2,m3)

E ti

~jmj

= 0

and,

for any m E

tic,

let tm P _- P

j=o

(resp.

tm ti _-

ti)

be the

mapping

N°11 TIGHT,BINDING METHOD IN TERMS OF GRAPH THEORY ₂₂₇₁

~m (To Xl X2>X3j " (To + m01Xl + ml1X2 + m21X3 +

m3j (1.5) (resp.

tm

(no,

_{ni, n2, n3}

"

(no,

_{+mo, ni + ml, n2 + m2, n3 + m3}

))

In the

representation

of

O)

in P

(resp. ti),

the

subgroup

of all translations

belonging

to

O(

coincides

[I]

^with

(tm

P _- P _{m E}

Sol (resp. (tm

ti

- ti _{m E}

Sol)

and it will be denoted

by So.

The

mapping

3

b ti _x ti

-

N,

b

(n, n')

=

~j _{jnj n( (1.6)}

j=o

is _an

O(-invariant

distance _on ti. We will _prove

(Sect. 3)

that the _group

ig

_. ti _- ti b

(g(n),

_g

(n'))

= b

(n, n')

for all _n,

n'

_E

tip

of all isometries of the metric space

(ti, b)

coincides with

O).

The

point

^n' E ti is _a

neighbour

of order

j

of the

point

_n E ti if and

only

if

bin, n')

=

j.

For each

point

_{n =}

(no,

_{ni, n2,}

n3)

E

ti,

the

points

"~

"

("0

+

6(")1

"1> n2>

n3)1 n~

=

(no>

nl +

6(n),

_n21

n3)>

"~

"

("0>

"1> "2 +

6(")> "3)> ^n~

=

(no>

nl>n2> n3 +

6(n)), Il.?)

where

61n)

₌

(-1)"°+"~+"~+"3,

are the first

neighbours

of _n, the

points

~ji

~

(~J)~,

where

j #1>

are the second

neighbours

of _n, the

points

n~~P

=

((nJ)~)~,

^where

^{j #}

ⁱ

^#

p, are the third

neighbours

of _n, etc.

The

O)-metric

_space

(ti, b) represents

an abstract model for the diamond structure. It will

help

_us to describe the diamond structure in terms of

graph theory.

The usual

applications

of the

graph theory

in

crystal physics [3,

4] can be extended _to the _case of the diamond structure

by using

this

description,

but _we will consider

only

_some elements of the

tight-binding

method.

2.

Theory

of the

quasibound

electron for diamond

type

semiconductors.

The diamond

(C),

silicon

(Si), germanium (Ge)

and tin

(a-Sn)

have the _same structure

(namely,

the structure of

diamond)

and

they

will be called diamond

type

semiconductors. A solution

of the

Schr6dinger equation

for the electron in the

periodic

field of such _a

crystal

has _[5] the form of the Bloch _wave

lbk(x)

₌

exP(I(k,x))~Jk(x), (2.I)

where _~Jk P _- C satisfies the condition

~Jk(x

+

n)

=

~Jk(x)

for any n E

So, (2.2)

and k E P

(the corresponding

_wave

vector).

Certain values of the _energy E

corresponds

_to each _wave vector k _E

P,

and _an

important problem

is _to establish the

dependence

E

=

E(k).

In the _case of the diamond

type semiconductors,

there _{are more} accurate methods for

solving

this

problem [6-13],

^but

important

informations _can be obtained in _a

simple

and

general

_way

by using

^the

theory

of the

quasibound

electron

[5, 14].

This

theory, developed

here in classical terms, ^{will be} reconsidered

(Sect. ₅₎

in _{a new} mathematical

formalism,

suitable for extensions to distorted

crystals, crystals

with

impurities

or

defects,

etc.

Let k ₌

[ko, ki, k2,

_k3] E

P,

and let

Kj

₌

(k,ej)

₌ 3

kj ~j ki, (2.3)

1#j

for _any

j

E

(0,1,2,3). Following

the

general theory

of the

quasibound

electron

[5, 14],

we

consider the Hamiltonian

/~2

H = -6 +

~j _{((x n)}

₊

_{W(x), (2.4)}

where

H~

/~2

= --6 ₊

((x) (2.5)

is the Hamiltonian in the _case of _an isolated atom and

W(x)

is _a "correction" necessary for the

self-consistency

of lattice

potential

field. We choose _{as zero}

approximation

for the solution of the

Schr6dinger equation

H~fi

= E~fi

(2.6)

a sum of atomic _wave functions ~fi~(x

n), namely

1fi° P _-

c, q~°(z)

=

~ exp(I(k, n)) q~~(z n), (2.7)

where

Halba(x)

=

Ea lba(x). (2.8)

N°11 TIGHT-BINDING METHOD IN TERMS OF GRAPH THEORY 2273

One _can remark that

~b°(x

+

n)

₌

exp(I(k, ni)~b°(x), (29)

foranynEtio.

It is known

that,

in this _case, the first

approximation

for _energy

E(~)

_{satisfies the}

_equation

lfi2 ^GA

^~

^jj

^v&(X

^"')

⁺

^{w(X) 16°(X)}

^"

^{Ei~~ '16°(X). (2.1°)}

~,

As usual

[5],

we assume that

~fia(x n)

is _non-zero

only

in the

vicinity

of _n

(the overlap

of the

wave functions

~fia(x n)

_can be

neglected

_even for _nearest

atoms)

and the electron

exchange

between _{any two atoms} of the

crystal

takes

place by

_way of _a chain of _nearest

neighbour exchanges.

More

exactly,

_{we assume} the existence of _two real _constants A

(the exchange

energy between nearest

atoms)

and B

(B

m

0)

^{such that}

/~fi] ^{(x nJ) ~fia(x n)dx}

⁼

^{B, (2.ll)}

P

/

~fi]

(x

nJ

~j

( (x n')

₊

W(x) _~fia(x n)

dx

= A

(2.12)

~ n'#n

foranynEti, jE(0,1,2,3).

Since

~o*j~)~oj~)

~~

2

p

n ~

+

£ exp(-I(k, n))~fi](x n) exp(I(k, n))~fia(x n)

₊

~j

_exp

_{(I (k,}

nJ

))

_~fia

(x

nJ dx 2 =

~

n j=

3 3

=

~j

_{I + B}

£

cos

((k,

n nJ

))

₌

£

_{I + B}

~j

cos

Kj

m

N,

" J"° " J"°

where N is the number of atoms of the

crystal,

from

(2.10)

_we

get

in _a similar way

El~)

=

/ ~fi°*(x)H~fi°(x)dx

₌

£ Ea

+

/

~fi]

(x n) £ _{( (x} n')

₊

(x)j

~ ~ ~ x

"

~ "'#"

x~fia(x n)dx

+

(

exp

(I (k,

nJ

n) Ea /

~fi]

(x n)~fia (x

^{nJ dx} +

/

~fi](x n)

_x

~~~ P P

x

~j _{( (x} n')

₊

(x)l~fia _{(x nJ)}

_dx

+

f

exp

(I (k,

_n nJ

))

x

'#nY

~ j=0

x

Ea /

~fi]

(x nJ) _~fia(x n)dx

₊

/

~fi]

(x nJ) ~j ₍ lx n')

₊

W(x) ~fia(x n)dx

~ ~

"'#"

~~jlEa+C+(B.Ea+A)fcosKj

mEa+C+AfcosKj,

~

n j=0 j=0

where

C

=

/ ~fi](x n) ~j _{( (x} n')

₊

(x)l~fia(x _{n)dx. (2.13)}

~ ~l#~

Particularly,

El~~

=

Ea

+ C + 4A for k

=

[0, 0, 0,

0]

and

E(~)

=

Ea

₊ C 4A for k ₌

[0,1r /2,1r/2, _0].

One _can remark

that,

in the

crystal,

the _energy level

Ea

is lowered

by

the amount C and

split

into _a band of width

8(A(

3

El~~

=

Ea

+ C ₊ A

~j

cos

Kj. (2.14)

j=o

3. A

description

of the diamond

type

semiconductors in terms of

graph theory.

It is usual to describe _a

crystal

in _terms of _an Euclidean vector space and to think of _a

crystalline symmetry

in terms of _a

rigid displacement,

but _a less

rigid description

in _terms of

graph theory

_may be

enough

accurate to be useful in

modelling

and

enough general

to be

suitable

for extensions to distorted

crystals _[15]

_or to

crystals

with

impurities

_or defects.

Let G

=

(ti,L)

be the

graph (we

_use the

terminology

^from

[3])

whose set of all

points

is ti and whose set of all lines is

L

=

((n,

nJ

j

n e

s, j

e

jo,1, 2, 3j)

,

(3.1)

N°11 TIGHT-BINDING METHOD IN TERMS OF GRAPH THEORY 2275

and let G be the _group of all

graph-automorphisms

of

G,

that

is,

^{of all}

bijections

_g ti _- ti such that

(gin), gin'))

_E

L,

for _any

(n, n')

_E L.

Evidently,

_a

k-step

walk

having

_{n E} ti _as

starting point

_can be written

In,

n~i

(n~i,

n~i~2

(n~i~2...~k-1,

n~i~2...~k-i~k

,

(3.2)

where

iii12, ...,ik

E

(0,1, 2, 3).

It will be denoted

by (n;ii12.. ,ik)

A closed walk

(n; ijkijk)

with I

# j #

k

#

I will be called _a

hexagonal

walk.

One _can prove

[1,

2] ^that

b(n, n')

is the

length

of the shortest walk

having

_n as

starting point

and n' _as end

point.

Particularly, (n, n')

_E L if and

only

if b

(n, n')

= I.

Obviously, A, A,

E

G,

for _{any a E}

G4,

^{and hence}

O)

_C G. We _prouve that G

=

O).

Let _{g E}

G,

_m E ti be fixed elements and let m'

=

g(m).

Since _g

((mJ j

E

(0,1,2,3)))

=

(m'J j

E

(0,1, 2, 3)),

It follows that there exists _{a E} G4 such that

g

(mJ

₌

m"lJ) (3.3)

for _any

j

E

(0,1,2,3).

We _prove that there exists h _E

O)

such that

h(m)

=

m',

h

(mJ)

=

m"lJ),

for _any

j

E

(0,1, 2, 3),

and then _we prove that _g

= h.

The

mappings

Azj

^ti -

ti, _Azj

=

Ajoj~

o A _o

Ajofl

o

A(oz)

^{o A} o Aoz

(3.4)

where

I, j

E

(0,1, 2, 3), (0k)

E G4

represents

^the

identity

for k

= 0 and the

corresponding transposition

^for k E

(1, 2, 3),

_can be written

nk

+ I if k ₌

j Azj (no nl1n21n3)

_"

(n~> "~> "~> "~)

i

"~

" "k l if k

~ ~

(~.~)

nk if k

f (I, j)

Since the group

O)

is

generated by (A)

U

(A,

_a E

G4),

the

mapping

h _can be

easily

obtained

by using

the

mappings Azj.

Evidently,

the groups G and

O)

leave invariant the class of

hexagonal

walks

(see Fig. I).

Let m" be _a fixed

point

with I

# j.

The

images by

_g or h of the two

hexagonal

walks

(m; ijkijk),

k E

(0,1, 2, 3) ( (I, j),

which _pass

through

_m,

m~,

m" and do _{not pass}

through

mJ must be the two

hexagonal

walks

(m'; a(I)aba(I)ab)

which _pass

through m', ^m"l~)

and do not pass

through m"(J)

This is

possible only

if _a

=

a(j),

that is

g

(m"

=

m"l~l'(J)

= h

(m") (3.6)

By using

this

result,

_{one can prove} that

g

(m~i~2...~~)

=

m"l~il'l~2).."(~kl

= h

(m~i~2...~k)

,

(3.7)

for any

ii,

12,

,

ik

E

(0,1, 2, 3),

and hence _{g =} h.

A diamond

type

semiconductor

crystal,

considered infinite and without

impurities

_or

defects,

can be

regarded

_as a

graph (

=

(R,£)

whose set of all

points

is the set R of the

equilibrium positions

^{of all} atoms of the

crystal

and whose set of all lines £ is the set of all covalent bonds in the

crystal (identified

with the set of all

pairs

of nearest atoms

(A, A')).

One _can prove

[1,2]

^{that there exists} a

family

A of natural

graph-isomorphisms

between

(

and

G,

that

is,

_a

family

^of

bijections

Aj

A

Aj

Fig.

1. A

regular

_or distorted diamond type semiconductor is

regarded

_{as a}

graph (R,£)

and is described by

using

the "numerical"

graph (§, L).

~j.R-ti

such that the

image

of each covalent bond

(A, A')

E £ is _an element

(~Jj(A),

_~Jj

(A'))

E L.

For _any two

isomorphisms

_{~Ji, ~Q2} E A the

mapping

_{~J2 °}

_~oi~

ti _- ti is _a

graph-automorphism

of

G,

and hence it

belongs

to

O).

More than that 11,

2],

we have

O)

= (~a2 o

~a/~

^ti - ti _~ai,

~Q2 E

A). (3.8)

The considered

crystals

_can be

regarded

_{as a}

crystallographic variety

modelled _over the

"numerical"

graph

G with the structure group

O) by using

the atlas A. The

isomorphisms

~aj ^E A

represent

^the

corresponding

charts.

A small deformation

(due

to _a small _pressure,

etc.)

of the

crystal,

which do not

destroy

the covalent

bonds,

leads to a distorted

crystal _[15]. Evidently,

the structure of

crystallographic variety

of the

regular crystal

can be extended _to the

corresponding

distorted

crystals,

that

is,

the "numerical"

graph

G is _a model for

regular

and distorted diamond

type

semiconductors.

Since b ti x ti _- N is _an

O)-invariant distance,

that is

b

(g(n),g (n'))

₌ b

(n,n') (3.9)

for any g ^E

O),

_n, ^n' E

ti,

it follows that

b

(~gi(A),

_~gi

(A'))

= b _{(~g2 O}

~gi~)

(~gi

(A)),

_{(~g2 O}

_~gi~)

_(~gi

(A')))

= b

(~g2(A),

_~g2

(A')) (3.io)

and

hence,

_{we can} ^define a distance _on R

b R _x R _-

N,

b

IA, Al)

= b

(~gi(A),

_~gi

(A')) (3.ii)

by using

a chart _~Ji and

independently

^{of the chart} we choose.

N°11 TIGHT-BINDING METHOD IN TERMS OF GRAPH THEORY 2277

Generally,

each

O)-invariant

mathematical

object

_{we can} consider _on the

graph G,

_can be

brought

_on

(

in _an invariant way and it may be useful in

modelling.

We shall

give

_some

examples

in the _next section.

For _{n E}

ti,

k _E

N,

k

# 0,

the _group of all

graph-automorphisms

of the

subgraph (S,L)

of

(ti, L),

^where

S =

(n'

E ti b

(n', n)

<

k),

L

=

((n', n")

_E L

n', ^n"

E

S) (3.12)

is

generated by (A,

a E

G4),

and hence it is

isomorphic

to the

complete

tetrahedral group

Td il].

This _means

that,

in the _case of the

graphs (ti,L)

and

(R,£),

the local

symmetry

is the

tetrahedral

symmetry.

4. A class of

O)-invariant selfadjoint operators.

Certain mathematical spaces can be associated to the

graph (

in _a natural way, and

they

_are useful in the

modelling

of the

physical phenomena occurring

inside the

crystal.

oJ

Let _a N _- ti be _a

bijection,

and let _u ti

- C be _a

mapping.

If the series

~j _(u(o(j))(

_is

j=o

convergent,

then the

family (u(n))~~~

^{is called}

absolutely

summable

[16],

^{and its} sum is

~j U(n)

=

~j _{U(a(J)). (4.i)}

nes

~0

One _can prove that this definition does not

depend

_on the

bijection

_a N _- ti _we choose.

The vector space

l~(ti)

=

(u

^ti - C

((u(n)(~)

_~~ is

absolutely summable)

n

(vi

+

u2) (n)

₌

ui(n)

+

u2(n)

(1 U)(n)

= I

(U(n)), (4.2)

Considered

together

with the scalar

product

(ui,u2)

"

~j _{ui(n) u2(n) (4.3)}

neti

is _a Hilbert _space. A

unitary representation

^of

O)

in

l~(ti)

_can be obtained

by associating

the

mapping

Tg 12(s)

_-

12(s), Tgu

_{= u o}

g-~ (4.4)

to each _{g E}

O).

Let

A,

C _E lt be two real constants, and let V

l~(ti)

_-

l~(ti)

be the

operator

3

(V u)(n)

₌ C

u(n)

₊ A

~j

u

(nJ) (4.5)

j=0

By using

the

Cauchy-Schwarz inequality _[17],

_{we prove} that the operator ^V is defined _on the whole _space

l~(ti)

~

2

~j _((Vu)(n)(~

=

~j

_C

_u(n)

_{+ A}

~j

u

(nJ)

^<

neti neti

J=o

3

<

(C~

+ 4

A~) ~j _(u(n)(~

₊

~j ~j _{ju (nJ)}

^~

= 5

(C~

+ 4

A~) ~j _(u(n)(~

nE§

J"°

nE§

nE§

and it is _a

selfadjoint operator (V

_ul

u2) 13

"

~

C _ul

(n)

₊ A

~

ul

(nJ u2(n)

=

~

C _ul

(n) u2(n)

₊

neti J=° neti

3 3

+A

~j £

vi

(nJ)

_u2

In)

=

£

_C

vi

(n)

_u2

In)

₊ A

£ £

vi

In)

_u2

(nJ

=

J=° neti neti J=o ~~§

=

(vi,

V

u2)

One _{can see} that for _any k

=

[ko, ki, k2, k3]

E P such that

~j

3 _sin

_Kj

= 0

(4.6)

j=0

where

Kj

=

(k, ej)

= 3

kj ~j ki, (4.7)

1#j

the

mapping

uk . 5 _-

C, uk(n)

=

exp(I(k, n)), (4.8)

which

belongs

to an extension of the space

l~(ti),is

_an

eigenfunction

of V

corresponding

to the

eigenvalue

3

uk = C ₊ A

~j

cos

Kj. (4.9)

j=0

For each _{w :} N _-

lt,

such that

(j

E N

w(j) # 0)

is _a finite

set,

we consider the

operator

V~V

l~(ti)

_-

l~(ti), _(V~vu) (n)

=

~j

w

(b (n, n'))

_u

(n'). (4.10)

n'eti

The

operator

V considered above is of this

type,

^{and it}

corresponds

to _{w :} N _-

lt, w(o)

=

C,

w(I)

=

A, w(j)

₌ 0 for

j

> 2.

Computations

similar to those

presented

for V prove that the

operators

_V~V are defined _on the whole _space

l~(ti)

and _are

selfadjoint.

N°11 TIGHT-BINDING METHOD IN TERMS OF GRAPH THEORY 2279

The

operators

_V~V are

O(-invariant operators. Indeed,

((Vw

O

Tg U) (n)

=

~j

W

(b (n, n')) _(TgU) (n')

=

n, es

=

~j

w

(b (n,n'))

_u

_(g~~ (n'))

=

~j

w [b

(g~~(n), g~~ (n'))]

_u

_(g~~ (n'))

=

~,~§ ~,~§

=

~

w

(b (g-~(n), n'))

_u

(n')

=

(vwu) (g~~(n)I

₌

((Tg

o

vw) u) (n),

n> es

for _{any n} E

ti,

_u E

l~(ti)

and _{g E}

O(.

The Hilbert _space

l~(R)

is defined similar to

l~(ti),

and for _any chart _{~J E}

A,

the

mapping

p 12js)

_-

12jn)

_{u - u o ~}

j4.ii)

is _an

isometry.

The

operator

i~v l~(R)

-

l~(R), _i~vfi

= (V~V

(fi

o

~J~~))

o ~J

(4.12)

can be associated to the

operator

_V~V ^l~

(ti)

_-

l~(ti), by using

_a chart _{~J E} A and

independently

of the chart _we choose.

Indeed,

let ~Ji, ~Q2 E

A,

and let g = ~J2 o ~Jp~ ^E

O).

Since

(V~VOT~)u=(T~OV~V)u,

it follows that

(~wU)

_{° ~9l ° ~92}^~ "

~w Ill

° ~9l ° ~92~)

(~.l~) whence,

for _u

= fi _o

~J/~,

we

get

(V~V

(fi

o

~J/~

o ~Ji = (V~V

(fi

o

~Jp~)

o ~J~.

(4.14)

5.

Quasibound

electron

theory

for

regular

and distorted diamond

type

^semicon- ductors.

In this

section,

we reconsider the

theory

of the

quasibound

electron in _a mathematical formalism based _on the

description

in terms of

graph theory presented

in section 3. As

usual,

_{we assume}

that the electron may be

located,

with _some

probability,

near any ^atom and it _moves

"freely"

through

the

crystal jumping

from _{one atom} to another

by exchange

_{process [5].} In

addition,

_we

assume that the

description

_{we use} in the _case of _an isolated atom _can be

kept

in the

vicinity

of each nucleus in the

crystal.

Let

A,

C E lt be two constants and let d be the Euclidean distance between nearest atoms

in the

crystal.

We _{use as} mathematical model the Hilbert space

7i =

1b

P _x 5 _- C

~ /~ ^{lib(x, n)l~dx}

^{< OO;}

^{1b(x, n)}

^{= 0 if ((x((}

² (1(5.i)

nes

with

(till>1fi2)

"

~ / _till(Xin)

1fi2(X>n)dX (5.2)

nEli

^~

and the Hamiltonian

[18]

H=Ha+V, (5.3)

where

Ha

is the Hamiltonian in the _case of _an isolated

atom,

and

3

(V~fi)(x, n)

= C

~fi(x,n)

₊ A

~j

~fi

(x, nJ) (5.4)

j=0

3

Let k

=

[ko, ki, k2,

_k3] E

P,

such that

~j

sin

Kj

₌ 0

(see (4.7)),

and let

Ea

be _an energy j=o

level in the isolated _atom

corresponding

to the

eigenfunction

_~fia, that

is,

H~q~~ =

E~

_q~~.

j5.5)

The function

1b : P _x ti _-

C, lbix, n) =lba(x) exp(ijk, n)) 15.6)

is _an

eigenfunction

of H

(H1b)(x, n)

=

(Halba) (x) exP(I(k, n)) +lba(x) V(exp(I(k, n)))

=

3

=

Ea +C+A~cos ~j

.q~(x,n),

j=o

corresponding

to the

eigenvalue

E=Ea+C+A~jcosKj.

3

_(5.7)

j=o

One _can remark that the result is similar to that obtained in section

2,

and it is _an

argument

for the existence of the _energy bands in

regular

and distorted diamond

type

semiconductors.

This _new formalism is _more suitable for extensions to

crystals

with

impurities

_or

defects,

zinc blende

type crystals,

_{or, more}

general,

to

crystals

with

regular

_or distorted tetrahedral structure

[15].

At the _same

time,

it may be used _{as a} basis for other _more accurate models.

N°11 TIGHT-BINDING METHOD IN TERMS OF GRAPH THEORY 2281

6.

Regular

and distorted diamond

type

semiconductors with

impurities

_or defects.

In the

previous sections,

_we showed that _a diamond

type

semiconductor

crystal (regular

_or

distorted)

_can be

regarded

_{as a}

graph (

₌

(R,£),

and described

by using

the "numerical"

graph

G

=

(ti,L)

and the atlas A. Some elements of _a

possible

extension of these ideas to

crystals

with

impurities

or defects will be

presented

in this section.

Let

(E,fi4)

be _a

graph,

_{ao E} E be _a fixed

point,

and let B C E be _a fixed _set. A

graph (E, M)

will be called _a

subgraph

of

(E, fi4)

if E C E and M

=

((a, a')

E fi4 _a, a' _E

E).

The

subgraph (E, M)

will be called _a

vicinity

of _ao if

(a (ao, a)

_E

fi4)

C E. The

graph (E, fi4)

will be called _a

B-graph

if the

points

of B _are considered _as

privileged points

of

(E, fi4).

Let

(El, Mi ), (E2, M2

be _two

subgraphs

^of

(E, fi4).

A

graph-isomorphism

_g

El

-

E2

will be called _a local

automorphism

of

(E, fi4).

A _pseu-

dogroup

of

automorphisms _{[19, 20]}

of

(E,fi4)

is _a set r of local

automorphisms

of

(E,fi4) satisfying

the

following

conditions:

(a)

r contains the

identity automorphism

Id _: E _-

E, Id(a)

= a, for any

subgraph (E, M)

of

(E, fi4)>

(b)

if _an

automorphism

_g:

El

_-

E2

is in

r,

then

g~~ E2

_-

El

is in

r,

(c)

if the

automorphisms

_g

El

_-

E2, g' E[

_-

E[ belong

to

r,

and

E2

n

E[ # 4l,

then the

automorphism

belongs

to r.

Let

(I, _fir)

,

(E, fi4)

be two

graphs

and let r be _a

pseudogroup

of

automorphisms

of

(E, fi4).

A

family

A of

graph-isomorphisms (called

local

charts)

_~J

^I

_- E from

subgraphs (I,$i)

of

(E, fi4)

^to

subgraphs (E, M)

of

(E, fi4)

will be called _a r-atlas for

(E, fi4)

^{if it satisfies the}

conditions:

a)

for each _{a E}

I,

_{there exists}

~J

I-

_{E in fir such that}

I

_is

a

vicinity

of _a,

b)

if_~Ji

ii

-

El,

_~Q2

12- _{E2 belong}

_{to fir and}

ii n12 _#

_{4l then}

_~J2o~Jj~

~Ji

(ii _n12)

-

~J2

(ii

_n

_{E2) belongs}

_to r.

Evidently,

the

family

r of all local

automorphisms

of G is _a

pseudogroup

^of

automorphisms

of G and A is _a r-atlas for

(.

We consider _a diamond

type

semiconductor

crystal doped

with donor

impurities.

Let

no

C R be the subset of all

points occupied by impurities.

The

crystal

_can be identified with the

no-graph (

and modeled

by using

the

(o) -graph G,

where _o

=

(0, 0, 0, o).

The set of all local

automorphisms

_g

El

_-

E2 satisfying

the condition

g(o)

= o if _{o E}

Eli

is _a

pseudogroup

of

automorphisms ro

of the

(o)-graph

G. If _{we assume} that the minimum element of the set

(b (A, A') A, ^A'

E

no

is

greater

than _one, then there exists _a

ro-atlas Ao

for the

no-graph (,

whose local charts _~J

I

- E

satisfy

the condition _~J

(I

_n

_no

=

lo).

The

ro-invariant

mathematical

objects

_{we can} associate to the

(o)-graph G,

can be carried _on the

no-graph ( by using Ao

and

they

_may be useful in

modelling.

Now,

_we consider _a diamond type semiconductor

crystal doped

with

acceptor impurities.

Let

(Aj j

E

J)

C R be the set of all

points occupied by impurities

and let

((Aj, A') _j

_E

J)

be the set of all absent covalent bonds

(Fig. 2).

^~

JOURN~L DE PHYSIQUE I -T 3, h'll NOVEMBER 19Vl

Fig.

2. -A diamond type semiconductor

doped

with acceptor impurities. The covalent bond

(Aj,

A is absent.

The

crystal

can be identified with the

graph (R, £i),

where

£i

" £

( ((Aj, _{Aj j}

_E

J),

and described

by using

the "numerical"

graph (ti, Li ),

where

Li

₌

L( ((o, o')),

_{o =}

(0, 0, 0, 0),

and o'

=

(1, 0, 0, 0).

The set of all local

automorphisms

_{g :}

El

_-

E2

of

(ti,Li) satisfying

the condition

g(o)

= o

if _{o E}

El,

and _g

(o')

= o' if

o'

_E

Eli

is _a

pseudogroup

of

automorphisms ri

of

(ti, ^Li

If _{we assume} that the minimum element of the _set

(b (Az, Aj I, j

E

J)

is

greater

^than

three,

then there exists _a

ri-atlas AI

of

(R, £i),

whose charts _~J

fl

- E

satisfy

the conditions:

~J

(I

_n

(Aj(j

E

J)) ^=(o)

^and _~J

(I

_n

_{(Aj j}

_E

_J))

=

(o').

The

ri

-invariant mathematical

objects

_{we can} consider _on

(ti, Li

_can be carried _on

(R, £i by using Al

and

they

_may be useful in

modelling.

In _a similar way, for _a

crystal

with vacancies

(Fig. 3),

_{we can use as} model the "numerical"

graph (ti ( (o), L2),

where

1~2 =

ll~l llo,O°I,lo,o~l,lo,o~l,lo,o~ll)

_U

llo°,o~l,lo~,o~ll.

7. Conclusions.

The

descriptions

in terms of

graph theory

offer _new

possibilities

to extend _some elements of the usual models from

perfect crystals

to real

crystals.

The mathematical

apparatus developed

for diamond type semiconductors _can be

easily

extended to zincblende type

crystals (regular

_or

distorted).

Extensions to other

regular

or

distorted tetrahedral structures

[15]

seem also to be

possible.

All the notions and results

presented

in this article _can be extended _to

higher

dimensional spaces

by using (0,1, 2,..

,

k

I)

instead of

(0,1, 2, 3),

_Gk instead of G4.

N°11 TIGHT-BINDING METHOD IN TERMS OF GRAPH THEORY 2283

Fig. 3. - A

vacancy in

a

diamond

type

emiconductor. The atom

A and

the bonds

(A,

lk-I

n =

(no,

_ni,

,nk-i)

_E

Z~ ~jnj

_E

_{(o;1) (7.1)}

j=0

instead of

ti,

_etc. This

possibility

and the ideas

presented

in section 6 may be useful in the

study

of the

quasicrystals [21-23].

The

physical phenomena

in _a

crystal (for example,

in _a semiconductor

device) depend

_on the local

physical phenomena occurring

inside the

crystal.

The mathematical elements

presented

in section 6 may be useful in the

description

of the

physical phenomena

in _a

crystal by taking

into consideration of the local

phenomena.

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ill

Cotfas

N.,

J.

Phys.

Condens. Matter 3

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[2] Cotfas

N.,

J. Geom.

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lo

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107.

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Kasteleyn

P-W-,

Graph Theory

and

Crystal Physics,

F.

Harary Ed.; Graph Theory

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M.,

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York, London, 1964).

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Cycon

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Kobayashi S.,

Nomizu

K.,

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Kobayashi

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