A simple approach to covalent surfaces

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A simple approach to covalent surfaces

F. Flores, C. Tejedor

To cite this version:

F. Flores, C. Tejedor. A simple approach to covalent surfaces. Journal de Physique, 1977, 38 (8),

pp.949-960. �10.1051/jphys:01977003808094900�. �jpa-00208662�

A SIMPLE APPROACH TO COVALENT SURFACES

F. FLORES and C. TEJEDOR

Instituto de Fisica del Estado Sólido

(CSIC

^and

UAM)

Univ.

Autónoma, Cantoblanco,

Madrid

34, Spain (Reçu

^le

23 fevrier 1977, accepté

le 19 avril

1977)

Résumé. ²⁰¹⁴ La surface d’un système ^{covalent a} été étudiée avec un modèle unidimensionnel dans le cadre de

l’approximation

d’une bande interdite étroite. Notre analyse ^{montre :} i) ^{que la}

distribution et le

potentiel

^{dans la}

région superficielle

^se comportent ^de façon

comparable

^{à ceux}

d’un métal, ^et ii) que l’état

superficiel

correspondant ^{à la} liaison rompue ^est déterminé par ^une

règle

^{de somme basée sur} la neutralité de la charge. Nous discutons le cas d’une surface 111 et il

apparaît que les conclusions de l’analyse précédente ^sont valables pour ^cette surface. En utilisant

ces résultats, ^le

potentiel

de surface du Si-111 a été obtenu et

comparé

^de façon satisfaisante à d’autres calculs

théoriques.

Abstract. ²⁰¹⁴ A 1-dimensional covalent surface is first studied

selfconsistently

within the narrow

gap

approximation.

^Our analysis ^{shows :} i) ^{that the} potential ^and charge distribution in the surface

region behave rather like those for the case of a metal, ^and ii) ^{that the}

dangling-bond

^{surface state}

is determined by ^a charge neutrality sum-rule. A covalent 111-surface is then discussed. Our arguments strongly suggest ^{that the same} conclusions apply ^to this surface. By using ^these results, ^{the 111-Si}

surface

potential

^{has been obtained and}

compared satisfactorily

with other independent theoretical calculations.

Classification

Physics ^Abstracts

8.322

1. Introduction. ^- Current studies of the electronic

structure of covalent semiconductor surfaces

usually

involve a selfconsistent numerical solution of the

Schrodinger equation

^{for a}

given

assumed surface

geometry,

with the bare ion

potentials

of the semi- infinite

crystal plus

^a suitable local

potential

^{to take}

some account of

many-body

^effects

[1].

While a great ^deal of information has been obtained about surfaces in these calculations

[2, 3], perhaps

^it

could be of interest to look for some

general

^relations

between different surface

properties

^which

might

be contained but unnoticed in the selfconsistent numerical

computations

^{and which}

might

^also

apply

to other situations. This has been the

point

^{of view}

motivating

this work.

In this _paper we first show the two

following

^results

for a 1-dimensional covalent surface :

i)

^The

potential

^and

charge

distribution near the surface

region

behave rather like those for the case of a

metal

(the

necessary

qualification

^{will be}

given presently).

ii)

^The

dangling-bond

^{surface state} is determined

by

the condition of surface

charge neutrality,

^with

independence

of the surface

potential

^details.

Later, ^we suggest ^{that the same results can be}

generalized

^{to a}

( 111 )

covalent surface.

These surfaces will be discussed in the next two

sections,

^{while an} illustration of these results will be

given

in section

4,

where the Si-111 surface

potential

is obtained in

good

agreement with selfconsistent calculations.

2. A 1-dimensional model. ^- Let us start with a

1-dimensional

model,

^{where our results can be}

exactly proved.

^{First of}

all,

it is of interest to consider

a metallic surface and derive its

density

^{of surface}

states, ^as well as a type of Friedel sum rule. In

figure

^{1 a}

we have drawn this model with a

hypothetical

^surface

potential.

In the

region

z >

0,

^well inside the

crystal,

^the

wavefunction for a

given energy E

^can be written as

with E ⁼

2 ^À,2.

To

study

^the

density

of surface states we

proceed

ⁱⁿ

two steps. First ^we

imagine

^a

hypothetical

^infinite

barrier at ^{z =}

0,

and another one at z ⁼ L, ^with L

as

large

^as necessary. In this

hypothetical

^{case, the}

wavefunctions are

given by

and the

eigenvalues

^are obtained from :

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01977003808094900

FIG. 1. ^- 1-dimensional model of (a) ^{a metal and} (b) ^{a two-band}

semiconductor with a non-abrupt potential ^across the interface.

Notice that there is no bulk state for n ⁼ 0, ^since this is the conduction band

edge.

^{What we are inte-}

rested in is the

density

of states, which ^can be obtained

by comparing

^our system ^{with one of}

length

^{2 L}

with Born-von Karman

boundary conditions, i.e.,

with no surface effects. The allowed

eigenvalues

^would

be

given by

2 LÀ= 2 nn, ^with n=O, ± 1,

± 2, ..., + N.

If we denote the

corresponding density

^{of states}

by A’2L,

^{then the}

density

^{of states for a} 1-dimensional metal of

length

^{L and} infinite barrier on both sides is :

Now the defect in the

density

^of

states -1/ 2 6(E)

is to be shared between both surfaces. Thus the

density

of surface states

(S.S.)

^{on one} surface is :

Notice that this result

implies

^{a defect of}

2

^{of the}

charge

^{of an electron at} the surface of a 1-dimensional

metal with an infinite barrier at z ⁼ 0. This can be

easily

^checked

by looking

^at the total surface

charge

of this model

p(z),

^{since :}

where po and

kF

^{are the bulk}

charge

and the Fermi momentum

respectively,

^and

Now we can

imagine

^{that the}

given

system ^{with an}

arbitrary

^surface

potential

^can be obtained

by lowering

the

initially

infinite barrier.

Knowing

^the

density

of states for the case of infinite barrier at z ⁼

0,

we can

easily

evaluate the

density

^{of states for the}

system ^described

by

the wavefunctions

(1).

In this _case, the

eigenvalues

^must

satisfy

^{the condi-}

tion

no

being

^the

phase-shift produced by

^{the surface}

potential.

Comparing (6)

^and

(3)

^{we find the extra}

density

of states introduced when the infinite barrier is lowered :

Then

Now the

important point

^{to be} noticed is that, ^since

charge neutrality

^must

always

^be

obeyed,

^for any selfconsistent surface

potential

^{we must have}

a sort of Friedel sum rule.

Let us now consider the semiconductor case.

The model is shown in

figure

^{1 b. In the} bulk, ^{a weak}

pseudopotential ^Veigz

^{+ c.c.}

(g

⁼

2 7r/a)

^{creates an}

assumedly

^narrow gap. This

potential

^varies

smoothly

across the

interface,

_up ^{to the vacuum} level outside the

crystal.

^{We take a metallic}

plane

^{at z = a, where the}

effect of the surface barrier is

negligible.

^{In order}

to

study

the wavefunctions inside the

crystal (z

>

a),

we measure

energies

^{s from the}

midgap

energy

t(g/2)2

^and

distinguish

^the

following

ranges :

o s ) I I

^V

1.

^Inside the gap the evanescent wavefunction is

. 1 V I I s I 2-(g/2)2.

Just inside the lower band.

Measuring crystal

^{momentum A from the B.Z. boun-}

dary g/2,

^{we have}

gA

^{= 2}

E2 - ^V2.,

^{and ’}

1 V I , I s I i(g/2)2. Further

down inside the lower band we have

The wavefunction

(11)

becomes the free electron wavefunction :

Outside the

crystal (z a), going

^down

smoothly

from the vacuum

level,

^{as z}

approaches

^the

matching plane,

^{we have a}

region

^{in which the}

potential begins

to oscillate so as to become the weak bulk

pseudo- potential.

Under these circumstances, in the

region

close to ^{z =} a, within a

length

^{of order a, since}

the weak

pseudopotential

satisfies the condition

I v I a

^{g, the} effect of the small

ripples

^is

negligible compared

with the main smooth

potential profile.

We can then

write,

^{to a}

good approximation,

^that

for z :5 a the wavefunction is

where

and il’

^{is the}

phase

^shift

produced by

^the

potential

which starts from the vacuum level and then

joins smoothly

the dashed line in

figure

^lb.

Knowing il’

^as

a function of _energy we use this

i)

^to find the S.S.

eigenvalue,

^and

ii)

^{to find the}

phase shift il

^of

(11).

Thus, matching (13)

^and

(10) :

where

Ev

^and

Ec

^are the lower and _upper band

edges,

and V

g2

has been used. This is the secular

equation

for the S.S.

eigenvalue. ^Also, matching (13)

^and

(11) :

This

gives q

and hence fixes the wavefunction

(11).

We now

study

^the

density

^{of S.S.}

proceeding

^{in two}

steps, ^{much in the same} way ^{as we} used to

study

^the

metallic surface. In the first step ^we

imagine

^two

infinite barriers at z ⁼ 0 and z ⁼ L, ^{with L =}

N(2 n/g).,

and N very

large.

The wavefunctions of the lower band are :

and the

eigenvalues

^are’ obtained from

Now there are no bulk states for n ⁼ 0, N,

i.e.,

the two

band-edges. By using

^an argument ^similar to the metal-like _case, it is

straightforward

^{to obtain}

the surface

density

^{of states} for the 1-dimensional

crystal

^{with an} infinite barrier

Knowing

^the

density

^{of states for an} infinite barrier at z ⁼ 0, ^{we can}

easily

evaluate it for the system described

by

the wavefunctions

(11).

^{Here, the}

eigen-

values must

satisfy

the condition

and

finally

^we

^conclude,

ⁱⁿ

analogy

^with eq.

(8),

that the surface

density

^{of states} for the valence band is

given by :

In the semiconductor case it is

interesting

^{to notice}

how il changes

^{across the band.} For E --> 0, q ^-+

0,

while for E ⁼

Ev,

^{a = 1}

and q

^{= nn}

(see

^eq.

(15)).

. This

gives

^the whole ^surface

charge

in the valence band

by using

eq.

(20) :

where we have taken into account

spin degeneracy.

The value of n

depends

on - the surface

potential.

As the barrier is lowered

and/or displaced

^{to the}

left,

a S.S. _goes down _{in energy} into the lower band and the associated

charge changes by.

^two electronic

charges.

Thus n takes on

successively

^{the values}

0,

^{1. This}

implies

^{a constant}

occupation of

the S.S. for reasonaoie

changes

in the surface

potential, i.e.,

^{not so}

large

^that

n

changes by

^{one unit. For a surface}

potential

^{of the}

kind shown in

figure ^{1 b,}

^{and with a S.S. near the}

midgap

^{it is}

easily

^{seen that n = 0. This}

implies 1/2 occupation

of the S.S.

After

discussing

^the

density

^{of states,} it is also

interesting

^{to look at the}

charge density

disturbance in real _space near the surface

region.

^{We recall}

that,

as

regards density

^{of states in} energy, the

change

^due

to the appearance of S.S. in the _gaps is

exactly

^can-

celled

by accompanying

distortions in the

density

^of

states in the continuum of the bulk

bands,

^{so that the}

change

in the total number of states is zero

[1, 4].

We now concentrate

specifically

^{on the} consequences of

having

^{S.S. in} semiconductor surfaces and inves-

tigate

^a similar but different

question, namely

^the

changes

ⁱⁿ the local

density

^{of states associated}

with

(a)

^the gaps, and

(b)

^{the two}

neighbouring

^bulk

bands.

The

problem

^can be studied

starting

^{from the wave-}

function

(11), assuming that I’

^{is known}

(see

^eq.

⁽¹³⁾

and

(15)).

From this wavefunction it is a

straight-

forward matter to obtain the local

density

^{of states}

N(E, z)

in the bands. The details ^{and the}

integrated density

are

given

^of

states

ⁱⁿ

appendix ^{N(E, z)}

^I.dE

The result is that the above two local densities turn out to cancel each other at every

depth

^{z. Thus, in our}

2-band

analysis,

^{if a} gap is in a range of

occupied energies,

then all distortions in the local densities

(a)

and

(b)

cancel out, ^so that the

specific

features of the semiconductor case in the local

density disappear entirely.

An

interesting

^case appears when the S.S. is in the

midgap.

Here, the distortions in the local

density

^of

states in the valence and conduction bands ^are

equal,

due to the symmetry ^{of the narrow} gap.

Then,

^the

charge

cancellation takes

place

also between the

charge defect

ⁱⁿ the valence band and a

t occupied

^S.S.

With all these results we now

proceed

^to

study

^self-

consistently

^the surface of a 1-dimensional semi- conductor model.

Assuming

^{a narrow} gap, ^we

imagine

the

system

^built up in successive steps ^{as follows :}

we start with a

jellium

model and then switch on a

crystal pseudopotential.

^{This has two effects on the}

charge

^{near the}

^surface, namely, (i)

^the consequence of a modification in the bulk

charge density

^which

appears both for metals and

semiconductors,

^and

(ii)

the

specific

features of the semiconductor _gap. While for a small

pseudopotential

the first effect is

small,

the second one can be much more

important

^since

the

charge rearrangement

between the S.S. and the valence band includes a very strong ^surface

dipole.

This is illustrated in

figure 2,

^where this surface

dipole

has been drawn as a function of the S.S.

position

^with

parameters adapted

^{to the Si case.} The S.S. is

2 occupied

^{and the}

figure

shows how

strongly

^this

dipole depends

^on the surface

phase-shift n’.

However, we are now

going

^to argue that this effect

disappears

^{in a} selfconsistent surface. First we

postulate

^that the surface

potential

^{can be}

reasonably

calculated in a first

approximation by neglecting

^the

specific

features of the semiconductor case. The idea is to

obtain q’

from the surface

potential,

^{hence the}

S.S.

itself,

and then see whether the first

approxima-

tion needs further corrections. We are interested in the upper energy range of the valence

band,

^{close to}

Ev.

With a narrow gap this means

energies

^{close to the}

Fermi level

EF. But,

^{for the reasons}

just stated,

^we

start

treating

^{the surface}

potential

^as in the metal case.

Here we have shown that

ilo(EF) = c/4,

^{the value}

ofyy’

then

being - n/4 (see

^eq.

⁽¹³⁾

^and

(1)).

^{With this}

value the S.S. as

given by

eq.

(14)

^is

placed

^{at the}

midgap,

^{and we} conclude that no net

change

^{in the}

local

density

^{of states} appears ^{as a} consequence of the S.S. itself. Hence there is no need to correct the

starting point

^{of our first}

approximation.

FIG. 2. ^- Induced surface dipole ^{as a} function of the S.S. level for

a 1-dimensional model. The S.S.

is i

occupied ^and

This establishes the main result of this section : the selfconsistent surface

potential

^{of a} 1-dimensional two-band semiconductor is rather like the surface

potential

^{of a} 1-dimensional

jellium

case, the

only

modifications

coming

from the induced bulk

charge,

this effect

being

^{the same} in both metal and semi- conductor surfaces.

Moreover,

^with

independence

of the details on the surface

potential,

the S.S. is

placed

^{at the}

midgap with 2 occupation.

3. The

(111)

^{covalent face. - We now} discuss how the

previous analysis

^can be extended to a

(I 11)

^covalent

face.

We assume a local one-electron

potential

in the form

where G is a 2-dimensional

reciprocal

^{lattice vector. For}

given

^{k vector}

parallel

^to the surface ^- henceforth K

- the

Schrodinger equation

^{to solve is :}

and we want ^to

study

^all energy ranges, those of the bulk band included.

Very

detailed selfconsistent solutions have been obtained

[2, 3]

in numerical form. Since the

computational

^task involved is rather

heavy,

^the

problem

is sometimes

simplified by studying only high symmetry points

of the 2-dimensional

B.Z.,

^{such as}

T and X

in

figure

³

[2].

^Surface

charge

^and

potential

^are then constructed ^on the basis of the information obtained in the

study

^{of these two}

points.

The results of

(2)

^{show that one can} obtain in this _way ^a reasonable

description

^of

the surface

potential. ^However,

^{if the}

problem

^is

to be simplified

^{in this manner, it} may be more

appropriate

to

study

^other

high symmetry points.

^For

example, T would

^{seem to} be the least convenient _one, since it

actually

has the

highest

^{energy of the}

^dangling

bond band and is therefore empty. ^{A more} convenient choice would

seem the J

point

^{which in some} calculations turns out to have the lowest _energy of the

dangling

bond band and

which,

moreover, has further surface states in the lower _gaps

[5, 6]

^{which can} be relevant in the

interpretation

^of

experimental

^{data. We thus choose}

J

as more

representative

^{of the surface}

properties

and will later substantiate this in the context of a more

complete analysis.

Figure

3 shows the

(111) projection

of the Jones zone

(J.Z.)

of the diamond lattice and the B.Z. for this surface. As shown

by

Elices et al.

[7],

^{in order to}

study

S.S. it is sufficient to include all

plane

^waves

corresponding

to those

points

^{on or} inside the J.Z.

boundary,

such that their surface

projections

map ^onto the

point

of the B.Z.

under

study.

^{For the J}

point

^{this means}

just

^the wavevector which

project

^{onto the three}

equivalent points

ⁱⁿ

figure

^{3. The}

potential

^{is then}

consequently approximated by :

and we look for solutions of the form

FIG. 3. ^- 2-dimensional B.Z. (inner hexagon) ^and projected ^J.Z.

(outer hexagon) ^{for the} (111) face of the diamond lattice with

special points shown. Other authors use the following ^notations

for special points : r XJ - rMK (3) -> rJK (2).

where (o ⁼

exp - 1 .2 3 n ),

^and

^only

^{the three} wavevectors icj, K 2 and x 3 ^are involved. The functions

f(ll(z) satisfy

^the

equations

where

Thus the

matching problem

^{for the J}

point

factorizes into three

independent

1-dimensional

problems.

For

large positive

^{z -} inside the

crystal -

^the

potentials V(i)(z)

^become

where

V3, V8, V11

^{are the}

pseudopotentials

associated with the

components (111), (220), (311),

and h is the modulus of

(111).

^{In the J.Z.}

approximation

so that

FIG. 4. ^- Bulk _energy bands in Si for k varying perpendicularly ^{to a} (111) ^{face - J} point. ^Dotted lines : real _energy loops ^for complex ^k.

The

pseudopotential components originate

gaps ^at the wavevector 4

h/3, h, h/3

ⁱⁿ the extended zone scheme.

The one at 4

h/3

^{is the} fundamental _gap. As noted

by

^{Heine and Jones}

[8]

^{its width to} second order is determined not

only by V8

^{but also to a}

large

^extent

by V3,

^{A similar} effect appears with the gap ^at 2

h/3,

^{which to second}

order is created

by V 3 ex p i 3 hZ . 3

^{The band structure}

^resulting

^from

⁽²⁷⁾

^and

⁽²⁹⁾

is shown in

figure 4,

^reduced

to the first

B.Z.,

^{whose border is at}

h/3.

^The

complex

^{band structure} is also shown.

We remark that the

independent

1-dimensional _eq.

(26)

here obtained

correspond

^{to the three}

equations

of

[7]

^{derived in the}

study

of the fundamental _gap at the J

point.

^{It was} shown there that a two-band model

gives actually

^{rather accurate results}

compared

with those based on a more

complete description

of the bulk band structure. The same

applies

^{when each} gap ^- besides the fundamental ^{one -} is studied

separately, i.e.,

an

adapted

^two-band model which focuses on each _gap

provides

^{a rather accurate basis to}

study

^different gap ranges. We shall use here this

approximation,

^so that all 1-dimensional

problems

will involve a two-band model in the ^manner

explained,

^{and the}

complete

3-dimensional

study

^{of the J}

point

will be built _up from the three

independent

1-dimensional

problems.

Now, ^we concentrate on the

study

of the fundamental _gap, since the effects on the surface

potential

^{of the}

other three _gaps for each 1-dimensional

problem disappear completely.

^{This was} shown in the

previous

^section

for a 1-dimensional _gap in a range of

occupied energies.

^{In each one} of the three 1-dimensional

problems,

^we

concentrate in the fundamental _gap within a two-band

approximation

^{with an effective}

pseudopotential Y8 [7].

The three surface

potentials

^{become :}

which

correspond

^{to three}

independent

1-dimensional

problems,

^{with the}

pseudopotentials displaced by 0,

±

a/3.

Now, ^{we can}

analyse selfconsistently

the surface of the 3-dimensional

crystal

^{as in the}

previous section,

the

only

difference

being

the simultaneous consi- deration of three 1-dimensional cases. To

begin with,

we need to know the

phase-shift

’10’ which ^was deter- mined in the 1-dimensional case

by

^{a sort of Friedel}

sum rule. In our

present

^case, we can use a genera- lization of this sum rule as shown

by Langreth [9]

and

Appelbaum

and Blount

[10].

Their results state

that,

^{in a}

metal,

^the average

value ( n 0 >

^{for all wave-}

functions at the Fermi level is determined

by

^the

condition of surface

charge neutrality,

^{its value}

being n/4.

^{This is a rather evident}

generalization

^{of the}

1-dimensional result.

We use this value

for n0 >

ⁱⁿ

studying

^{the contri-}

bution of

(30a),

whereas for the other

1-dimensional

n 2 n

problems

^{we must} take no

= 4

^±

3

^{due to the}

displacement

^{of the}

pseudopotentials by

±

a/3.

^We

shall discuss later the

validity

^{of this}

approximation.

Proceeding

^{in this} way ^we find that

(30a) yields

^{a S.S.}

eigenvalue exactly

^{at the}

midgap,

^{whereas no S.S.}

arises from

(30b, c).

^But in the latter ^two cases we find results which are

symmetric

^with respect ^{to the}

midgap, i.e.,

the local surface

density

^{of states in the}

valence

band/conduction

^{band of one case is}

equal

to that in the

C.B./V.B.

of the other case

(Fig. 5).

This result is of paramount

importance

ⁱⁿ

establishing

the

occupation

^{of the}

^S.S.,

^{because now we have}

detailed local cancellation of the actual surface

charge density

in the V.B.

arising

^from

(30b)

^and

(30c) (Fig. 5).

^{Therefore we are left with}

just

the 1-dimen- sional

problem

^of

(30a)

^{and with no extra}

charge

^or

potential

^{due to}

(30b, c).

^{After the}

previous

^section

this means

occupation 2

^{for the}

^S.S.,

^{and no net}

change

in the local

density

^{of states as a} consequence of the S.S. itself. This

finally

^{extends to the}

(111)

^covalent

faces the results of the 1-dimensional semiconductor model as stated

in

^the

previous

^section.

FIG. 5. ^- The density of surface states associated with each one of the three different 1-dimensional _gaps at J. In a) the S.S. is at the

midgap, ^{while in} b) ^and c) ^the density ^{of states} in the valence (V)

and conduction (C) ^{bands are related} by :

bNb

⁼

6Nf

^and

bN6

⁼

6N?.

Notice that in the three cases the total density ^{of S.S.}

around the _gap is zero.

We finish this section

by discussing

^the

validity

^of

the

approximations leading

^{to our} results. These are :

i)

^{the use}

of n0 > ^{for J,}

^and

ii)

^{the use of}

J

^as a

representative point

of the whole B.Z.

To understand the

meaning

of the first

approxi-

mation let us consider the 3-dimensional surface of a

jellium

^{model. In} this surface the

only

^non-zero

potential

^is

Vo(z),

^{and the}

phase-shift

110 for the different 1-dimensional

problems changes

^conti-

nuously

^{from a maximum at K = 0}

(T point),

^{to a mini-}

mum at K ⁼

kF.

^{The average value}

n/4

^{can be}

expected

to occur at some intermediate

point

^{near J where}

k 1. = 3/4

^{h. This can} be illustrated with a

simple model,

an infinite

potential

^barrier

placed

^{at a} distance b from

the jellium edge,

^{so as to} neutralize the surface

charge.

In this case, ^{it is an} easy ^{matter to} find the

phase-

shift

at J

^--

This

expression gives

for the Si

density

very close to

n/4.

Let us now discuss the

validity

^of

using J

^{as a}

representative point

of the 2-dimensional B.Z. This

can, be clarified

by looking

^{at the}

complex

^band

structure in the

(111) direction,

^{as we move} off the J

point (7).

^{In the same J}

point

^we find three identical

loops,

^{one of them}

giving

^{a S.S. As we leave this}

point,

the three

loops split and, depending

^{on the}

selected

direction,

¹

(or 2)

^{of these}

loops

^move up in

energies,

while the other 2

(or 1)

^move down. At the

same

time,

^{the S.S.}

changes

^its

position

in relation to

its supporting loop,

^{in such a} way

that,

^for

instance,

at

T

the

S.S.

is lower than its

corresponding midgap,

while at X the

opposite

^{situation occurs}

[2, 3, ^7].

Under these

conditions,

^the

validity

^of

using J

^{as a}

representative point

^means

that,

^{on the} _average, ^the

weight

^{of the}

loops

with the S.S.

placed upwards

^its

midgap

is counterbalanced

by

^the

weight

^{of those}

with the S.S.

placed

downwards. An idea of this

compensation

^can be obtained

by estimating

^the

centre of the three

midgaps averaged

^{over the different}

points

of the whole B.Z. When this is

done,

^we find that this centre is at 0.5 eV above

’the midgap

^at

J.

We think that this result is a

good

indication of the

compensation existing

between different

2-dimensional points. Moreover,

^we remark that

using J

^{as a} repre- sentative

point

^we have found

a 2 occupation

^{of the}

S.S., demonstrating

^a

general

^theorem

proved by

Kleinman

[11].

^{It seems to us} that this result

exemplifies

the afore mentioned

compensation

between different 2-dimensional

points.

4. The

(111)-Si

^{surface. - In} this section we check the

previous results, applying

^{them to} obtain the surface

potential

and surface states at

( 111 )-Si

^sur-

face. The idea is to

study

this surface

using

^{a method}

employed

^elsewhere

[12]

^to obtain the interface

potential

^{at metal}

surfaces, including

^the discreteness of the lattice.

Following

^the

point

^{of view}

given

^{in this}

approach,

we start from an

equation

which relates the actual surface

potential V(G, z)

^to the bare ion

potential Yo(G, z) through

^the dielectric function

~(G, G’ ;

z,

z’)

of the surface system :

In

principle

the difference between metal and semi- conductor surfaces lies in different functions

E-1.

After our

previous

arguments ^{we shall use for}

E-1

the same

approximations

^as in the metal case. We summarize here the

procedure

^followed

[12] :

i)

^We

neglect

the effect of

crystallinity

^{on the}

exchange

and correlation electronic

potential,

^so

that we take the

Lang-Kohn [13]

solution in the surface

region.

An idea of the error introduced in this _way can be obtained

by comparing

the value of this

potential

^{for the} average

density

of Si with the average value

resulting

^{from a}

density varying

^accord-

ing

^{to the bulk}

periodicity.

^{The errors} introduced are

between 0.1 eV and 0.2 eV at most.

ii)

^{We use for}

^E-1

^an extension of the semiclassical method as discussed in

[12].

The limitation of this

approximation

^{- based on}

jellium

^{models - for}

semiconductors lies in the

spatial

variation of the

charge density,

in the bulk as well as near the surface.

We have corrected for this

by using

^the response function evaluated not for the _average bulk

density

po, but for the G ⁼ 0 component ^{of the}

density

^{at z = 0.}

This amounts to

replacing

po

by

0.67 po,

using

^data

of Bertoni et al.

[14]

^{which we} have done in

general

for all information needed about the bulk.

Using

^this

effective

charge,

^{we must} calculate the _response to a

surface

charge

^{and a surface}

dipole.

^{In the first} case, we use an

interpolation

of the results as

given by Lang-Kohn [13]. Furthermore,

^{we also need the}

response ^{to a} surface

dipole,

^{which in}

[ 12]

^{was obtained}

in a linear _response. Here instead of the linear _response

we have calculated the

screening by using

^{the same}

approach [12]

^without

linearizing

^the

potential.

We have thus calculated the surface

potential

^for

the ideal 111-Si surface.

Figure

⁶ shows the G ⁼ 0

component

^{of this}

potential,

^and

figure

^{7 shows this}

potential

^{as a} function of z

along

^{two lines}

going through

^{the atoms in the} first and second atomic

layers respectively.

These results can be

compared

with those of Schluter et al.

[3], although

in the second

case these authors

give

their results in the form of

equipotential

^{contours. On}

extracting

the information from their results and

comparing

^{with ours, we find}

large discrepancies

in the bulk

potential

^{near the ions.}

This is

simply

^{due to the use of} very different bare

FIG. 6. ^- Surface potential ^for Si(111), averaged ^over the surface G ⁼ 0 as a function of z.

FIG. 7. ^- Surface potential ^for Si(111), ^{as a} function of _z, along

two lines going through ^{the atoms} in the first and second atomic

layers.

pseudopotentials (1). However,

^the

potential shape

is _very similar in the

regions

away from the ions

and,

in

particular,

^the

profile

of the surface

potential

barrier turns out to be _very

similar,

^thus

providing

^a

rather

satisfactory

^{check on our}

approximations.

It is also

interesting

^{to look at} the results for the ionization _energy

(the

difference between the vacuum

level and

Ev). Taking

^{for the}

position

^of

Ev -

^cor-

responding

^{to T - relative to the zero of bulk}

energies

the value 10.8 eV

(E. Louis, private communication),

we obtain an ionization _energy of 4.5 eV which compares with the

experimental

^{value of 5.15 eV}

[15].

Moreover,

^it

might

be in order to discuss

briefly

^the

S.S. that our model

yields.

^We concentrate on the

J

point.

^The calculation was done

following

^{the same}

general

^{lines as in}

[7],

^and

using

^{the same detailed}

many band bulk structure, but with the

following important

differences :

i)

^{We did not use a} step

potential,

^{but the}

potential profile

obtained in the

previous calculation,

^and

ii)

we studied the fundamental _gap and the lower _gaps

as well. Our results have been

published

^elsewhere

[1]

and shall not be

given

^here.

However,

^{we remark} that the S.S. in the fundamental _gap is

placed

very close to the

midgap

ⁱⁿ agreement ^{with our}

simplified analysis

and other calculations

[2, 3].

^{At the same}

time,

^two other S.S. _appear at lower

energies,

^over-

lapping

with the valence band. The lowest one agrees with the state found

by

^Louis

[5]

^and

by

Falicov and Yndurain

[6].

^{This state}

belongs

^{to a} band of S.S.

which,

^as found

by

^these

authors,

^{exists over domains} of the 2-dimensional B.Z. close to the

J point.

In

figure

^{8 we} have drawn the

charge density

^asso-

ciated to the S.S. of the fundamental _gap at

J.

^This

can be

compared favourably

^{with the same}

density given by

Schliiter et al.

[3], although

here these authors have a relaxed

geometry.

The relaxation must enhance the localization of

charge

^{near the}

surface,

^{this will}

explain

^{the small} differences between the two

charges.

FIG. 8. ^- Charge density associated to the S.S. at J along ^{a line} going through ^{the atom} in the first atomic layer. ^{The dots} represent

plane layers.

In

conclusion,

^we find that the

previous

^results

show a

general

agreement ^{with more elaborate}

computations, giving

^a strong support ^to the conclu- sions of section 3.

At this

point,

^{it is}

tempting

^{to use the}

general

method outlined in

[12]

^to obtain the ionic relaxation at the Si surface. This has been done and we have found a very small

tendency

towards relaxation.

Of course this is

only

^a

plausible

indication and must not be taken too

literally

after the _many

approxi-

mations involved.

However,

^{it is}

interesting

^{to note}

that in a recent paper, Shih et al.

[16]

have estimated from LEED

experiments

the inward relaxation of the last atomic

layer

^as

only

^0.12

^A.

(1) ^We have used the data of Bertoni et al. (based ^{on a Heine-}

Abarenkov pseudopotential) kindly communicated to us by ^the

authors.

Finally,

ⁱⁿ

support

^{of the}

viewpoint expounded

in

paragraph 3,

^{we refer to a recent}

publication [17],

where it has been shown how the

dangling-bond

^S.S.

at the

(111)

Si and Ge surfaces can be obtained

quite

well with a

suitably displaced abrupt

barrier. The idea is to

displace

^the

abrupt

barrier in order to achieve metallic

charge neutrality.

This fixes the _average

phase

of the wavefunctions at the metallic Fermi level in its correct value

n/4,

^thus

giving

^{the best}

approxi-

mation to the S.S. The results of this work

give

^{a new}

support

^{to the}

point

^{of view}

expressed

^here.

5. Conclusions. ^- In the first

part

^{of this} paper ^a 1-dimensional covalent surface has been studied.

Our

analysis

shows that :

i)

^the

potential

^and

charge

distribution in the surface

region

behave rather like in the case of a

metal,

^and

ii)

^the

dangling-bond

^S.S.

is determined

by

the condition of

charge neutrality.

This

gives a i occupation

^{of the}

S.S.,

^{and a S.S.}

placed

^{at the}

midgap.

A covalent

(111)

surface has been discussed later.

Our arguments

strongly

suggest ^{that the same conclu-} sions can

apply

^to the surface of this 3-dimensional semiconductor. This allows us to

study

the surface

potential

^{in a}

simple

way ^as in the case of a

metal,

the results

comparing quite

well with the ones of elaborate

computations.

Finally,

^{we stress that the} type ^of

analysis

^{shown in}

this work can be used to

study

different surface condi- tions. For

example,

^{in a recent work}

[18]

these ideas have been

applied

^to

analyse selfconsistently

^{the Si-}

metal

junctions

^with

quite good

^results.

Acknowledgments.

^{- We wish to} thank Professor F. Garcia-Moliner for _many

helpful

discussions and the critical

reading

^{of the}

manuscript.

Appendix

^{I. - In this}

appendix

^we calculate the local

density

^{of states}

N(E, z)

^{and the}

integrated density

of

states fN(E, ^z)

^{dz, near a} 1-dimensional semiconductor surface.

The wavefunction in the valence band is

given by :

while in the conduction band we have :

The

phase-shift ’1

is determined

by matching

^the wavefunction to the one

given by

eq.

(13) :

where it has been assumed A’

g/2,

^{and we have defined}

Once

#(z)

^{has been} obtained and

properly normalized,

^one has the local

density

^{of states :}

Here

and

ps

giving

the surface contribution. In these

expressions

the +

/

^-

sign standing

^{for the}

valence/conduction

^{band. It is}

interesting

^{to note} that far from the _gap, a -> oo or zero, in such a way

that p’

^{-+ - cos}

2 2 9 + , (z - zo) , like in the

^{metal case. In order to obtain the}

typical

contribution for the semiconductor

surface,

this last metallic limit has to be subtracted. This leaves us

with the

expression :

The

integrated density

^{of states} in the valence band is

given by :

The contribution

coming from p’

^{can be}

simplified using

^the symmetry

in A,

^since

a( - A)

⁼

1 /a(A).

^{In this} way

we can write :

with

Now, to obtain

(1.10)

^we

change

^the

independent

variable from A to a :

and

Thus

The conduction band can be

analysed following

^{the same} steps, ^{in such a} way that we can write :

and

the

only

difference

being

^{that now}