Structural study of bismuth films and its consequences

Structural study ^{of bismuth films and its} consequences

on

their electrical properties

J.

Buxo, M.

Saleh and G.

Sarrabayrouse

Laboratoire d’Automatique ^et d’Analyse ^des Systèmes ^du C.N.R.S., 7, ^{av. du} Colonel-Roche, ³¹⁴⁰⁰ Toulouse, ^France

G. Dorville

Société Jules-Richard, 116, quai ^de Bezons, ⁹⁵ Argenteuil, ^France

J.

Berty

^{and M. Brieu}

Laboratoire de Physique Structurale, Equipe de Recherche Associée au C.N.R.S., Cinétique Cristallochimique des Couches Minces, Université Paul-Sabatier,

118, route de Narbonne, 31077 Toulouse Cedex, ^France

(Reçu ^{le 13 mars} 1979, révisé le 4 février 1980, accepté ^{le 8} février 1980)

Résumé. 2014 Les auteurs étudient les

propriétés

structurales des couches minces de bismuth à 1’aide des

techniques

suivantes : diffraction des électrons par réftexion, diffraction des rayons X,

microscopie électronique

par

balayage

et diffraction et

microscopie électronique

par transmission. Ils

rappellent

^les

propriétés électriques

des couches minces de bismuth (mesures de conductivité et du facteur

RH). Il

apparait que les états de surface

jouent

^{un rôle}

déterminant dans les définitions ^des

propriétés

^{du facteur} RH ^{et de la} conductivité. Cette conclusion est étayée

par les résultats de 1’etude structurale. Une

propriété importante

^{du mécanisme} de conduction telle que le libre parcours moyen des porteurs ^a été estimée et sa valeur est aussi en accord avec les résultats de l’étude structurale.

La densité des états de surface ainsi que la hauteur de barrière associée aux joints ^de

grain

^ont été évaluées en

fonction des différentes techniques ^de

preparation.

Abstract. ²⁰¹⁴ The structure of bismuth thin films prepared according ^{to three different}

deposition techniques

^is

inspected

by : reflection electron diffraction, X-ray diffraction, scanning ^electron

microscopy,

transmission

microscopy

^and electron diffraction. The electrical conduction

properties,

RH and conductance measurements, of these films are then discussed. It appears that surface ^states

play a major

^{role in}

dictating

^the

properties

^{of both} RH ^and

conductivity.

^{This is} clearly confirmed with the aid of the conclusions of the structural

study.

By

comparing

size effect theories with the

conductivity

data the carrier mean free path ^{can be estimated to} be of the order of 1 000 Å in good agreement with the observation of the film structure. The surface state

density

and the energy barrier that controls the carrier transport ^{across the} grain boundaries have also been evaluated as a function of the preparation

technique.

Classification

Physics ^Abstracts

73 . 60D

1. Introduction. ^- Some electrical and electro- mechanical

properties

of thin bismuth films were

presented previously [1].

^The

annealing process

^and

the substrate

temperature during evaporation

^have

appeared

^{to have an influence on these}

properties.

^In

order to get ^a

deeper insight

^{into the}

physical

pro-

perties goveming

^these

phenomena,

^a

systematic

structural

study

^{of the films}

corresponding

^{to three}

different

processing

^{methods has} been carried out.

The conclusion is that the

annealing

process

(pro-

cess n°

2)

^{does not}

bring

^about any

significant

^struc-

tural

modification ;

^{the films are}

highly

textured with

crystallite

size dimension

lying

^between

^{1000 Å}

^and

2 000

A. Nevertheless,

films _grown on hot substrates show a marked

tendency

^to coalescence

thereby giving

^{rise to}

polycrystalline grains

^of very

large

dimension. The same conclusion is reached when two different substrates are

used, namely,

^{a resine}

layer (for

^the

study

^{of thick}

films)

^and

amorphous

^carbon

(for

^thinner

layers).

The

preceding

conclusions have made

possible

^to

obtain some new and

quantitative

features of the

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/rphysap:01980001505096100

electrical

properties

^{of bismuth}

films, particular

attention is

paid

^{to the Hall Constant}

(RH)

^{and con-}

ductivity (0)

measurements.

The effect of thermal treatments either after

depo-

sition

(process

^n°

2)

^or

during deposition (process

^n°

3)

on the

dependence

^of

RH

^on

temperature

is discussed in detail. The structural

study

has shown that no

marked structural différence is obtained after annea-

ling

^which puts ^{forward the}

major

role of surface states to account

for,

^the influence of the

annealing

treatment on the

conductivity values,

^{on the one}

hand,

^{and the} marked différences between the

RH(T)

curves for the three différent processes, ^on the other hand.

The

conductivity

variation with the film thickness has been

compared

^{with both the surface}

scattering theory (Fuchs-Sondheimer) [21] and

^the

grain

^boun-

dary scattering theory (Mayadas-Shatzkes) [22].

^It

has been found that the

conductivity

^{of films} pro- cessed

according

^to process ^{n° 2 can} be accounted for

by

^the

predictions

^{of the F-S}

theory

^{with the}

mean free

path

^{of the} order of 1000

A

which is in

good

agreement ^{with the}

grain

size estimate from the structural

investigation

^{with the} aid of the

Scanning

Electron

Microscope. Nevertheless,

films from _pro-

cesses n° 1

and

^{n° 3 cannot be}

reasonably

^described

by

^the

preceding

theories and this _appears to be due to ^the presence ^a

high density

of surface states.

Indeed,

^the

comparison

between the

conductivity

values of films from

processes

^{n° 1 and n° 2 has allowed}

to evaluate this surface states

density

which for films of _process n° 1 appears ^to be of the order of 2 x

1012 ^cm-2.

^{With an} associated _energy barrier

height

of 1.3 eV. The

annealing

^treatment

brings

about a reduction in the surface states

density

^which

in turn causes the

barrier energy

^{values to decrease.}

The positive

value of the

temperature

coefficient of the

conductivity - 1 (semiconductor-like)

^for

films from process ^{n° 1 can then} be

interpreted

ⁱⁿ

terms of the

increasing probability

^of

overcoming these

barriers at the

grain

boundaries.

Conversely

the observation

according

^{to which the size}

^of

^the

disoriented

polycrystalline grains

^increases

^with

^thick-

ness for films from

process

^{n° 3} is consistent with

the experimentally

^observed

tendency

^{for the}

mobility

to decrease with thickness

[1].

2.

Experimental

^{devices. -} The structural

study

has been carried out on

squared

^{aluminium substrates}

of dimensions 40 ^x 40 mm. An

insulating

¹⁰ ym thick resin

layer

^was

deposited

^{on the}

^substrate using

the method of

auto-regulated electrophoresis

pre-

viously

^described

[1].

Bismuth films of thicknesses

comprised

between 1 000

A

and 3 000

A

were

depo-

sited onto the resin

layer by’

^thermal

evaporation

under vacuum of

10-6

_torr, at a rate of

evaporation

of 50

Â/s.

^The

purity

of the metal used is 99.999

%.

Different thermal treatments were

given [1]

^and

their effects

with regards

^{to their} structural

properties

have been

carefully analysed by using

^the

following

three

techniques :

1)

Reflection electron diffraction.

2) X-ray

diffraction.

3) Scanning

^electron

microscopy.

The

experimental

conditions of

evaporation

^were

those

already

described in

[1]

^{that we} recall hereafter :

A fourth

technique

is used for the structural

study

of bismuth films of

small

thicknesses

(100-400 A)

prepared by

^thermal

evaporation

^{carried out in the}

interior of an electronic

diffractograph

^as

depicted

ⁱⁿ

figure

^{1. The} substrate material

is,

^{in this} case, ^amor-

phous carbon,

^{and the rate of}

evaporation

⁵⁰

Â/s.

Fig. ^{1. -} Diffracto graph object chamber bolt-equiped.

By using

^{a heated}

substrate-holder,

^{it has}

been

pos- sible to obtain the two

following preparation

^tech-

niques :

1)

Condensation on a substrate

kept

^{at room tem-} perature.

2)

Condensation on a substrate heated at 100 °C.

These

experimental

conditions are similar to those of the aforementioned films of

greater

thicknesses.

3. Structural

study.

^- This section

describes _the

experimental

^results

obtained by using

^the

^différent

techniques

^of

analysis

^{as well as} the conclusions obtained with

regards

^{to the films structure.}

3.1 REFLECTION ELECTRON DIFFRACTION. ^- The films have been examined inside an electronic dif-

fractograph operating

^{at an} acceleration

voltage

^of

80 kV.

Figure

^{2a shows the} diffraction pattern ^cor-

responding

^to process ^n° 1. The

diagram

^{has been}

completely

^{indexed as shown on}

figure

^{2b. This}

indexing

^as discussed in

Appendix, clearly

^{shows that}

each

point

^{in the bismuth}

reciprocal

lattice has

actually

been found on

figure

^{2a. The}

quality

^{of the}

picture

^{allows to} conclude that the bismuth film is made _up of

microcrystallites

^that arrange themselves in a

high quality

^texture. Identical results were

obtained for

films prepared according

^to

process

^{n° 2.}

Fig. ^{2. -} a) Reflection electron diffraction pattem ^{for Bi} films,

3 000 A thick. Preparation process n° 1. b) ^Indexed diagram of (a).

It should be noticed that the

diffractioii-diagram

for films

prepared according

^to process ^n° 3 and shown

on

figure

³ presents ^{a different} aspect, ^the presence of

rings

^{should be}

interpreted

^as

being

^caused

by

^the

grains

^with

completely

random orientation.

3.2 X-RAY DIFFRACTION. ^- This

study

^{was rea-}

lized with the aid of a CGR

type

¹⁰ diffractometer

operated

^{at 40 kV and 10}

mA.

^The

recordings

^{of the}

Fig. ^{3. -} Reflection electron diffraction pattem ^{for Bi} films,

3 000 A thick. Preparation process ^no 3.

curves of azimuthal distribution were carried out

by using

^the

Kai

radiation of copper ^whose

wavelength

is  ⁼ 1.540

5 A.

Fig. ^{4. -} Intensity of diffracted X-ray ^{for 3 000} A thick Bi films.

a) Preparation process n° 1. b) Preparation process n° 2. c) ^Pre- paration process ^no 3.

The

sample

^{holder turns in a vertical}

plane

^{with a}

speed

^{of 1}

degree/min.

The diffracted intensities were

graphically

recorded and are shown on

figure

^{4 for}

the three different processes. The Y-axis shows the diffracted

intensity

and the X-axis the azimuthal

angle

ⁱⁿ

degrees.

^{Table I} summarizes the obtained results.

Table 1.

It should be noticed that the films

corresponding

to processes ^n° 1 and n° 2

only display

^the

0003, 0006, 0009, ...

reflections which are

typical

^{of a}

perfect

texture, whereas for _process ^no 3 a

supplementary

reflection

corresponding

^{to the}

1012

direction is observed in addition to the

preceding

^{ones. Table II}

summarizes :

a)

^The different hkl reflections of bismuth as des- cribed within ’the three

crystallographic

^systems,

namely, hexagonal,

rhombohedric and

pseudo

^f.c.c.

b)

^{The values of the}

interplanar spacing (dhkl

^calc. > 1.118

À)

and those of the different intensities as calculated within the kinematic

approxi-

mation for electron diffraction

(the

value 100 has been attributed to the

largest reflection).

c)

^The values 0 of the

Bragg angle

^{as calculated}

for

X-rays by using

^the

Ka

radiation of

Copper.

The

comparison

of the calculated values in table II with those found

experimentally

^and

presented

ⁱⁿ

table 1 is _very

satisfactory.

The

X-ray study

shows therefore that films from process ^{n° 1 and 2}

display

^a

perfect

texture, whereas those from process ^{n° 3 are}

probably

^made up of

crystals

of random orientations

superimposed

^on

crystallites showing

^the aforementioned texture.

Table II.

3 . 3 SCANNING ELECTRON MICROSCOPY. - The bis- muth films were also examined with the aid of JEOL JSM-25 S

Scanning Microscope.

^Even

though

the resolution related to this

technique

^is

fairly

^weak

it is nevertheless

possible

^to obtain information about the

shape

^{and the} dimensions of the

grains

^or

crystallites

^that make up the films.

Figure

^{5 shows}

the results for films 1 500

À

thick.

Fig. ^{5. -} Scanning ^electron micrograph ^{for 1500 A thick Bi films.}

a) Preparation process ^n° 1. b) Preparation process n° 2. c) ^Pre- paration process ^n° 3.

For films

corresponding

^{to process n° 1 and n° 2}

grains

^of

homogeneous shapes

^with

grain

^{size of the}

order of 2 000

A (Fig.

^{5a and}

b)

^{were observed. For}

films

corresponding

^{to process n° 3}

(Fig. 5c)

^two

types

^of

grains

^can be observed :

-

homogeneous grains

^of

shape

^{identical to those}

in

figure

^5a

^{and b,}

-

larger grains

with thicknesses ^as

large

^{as 6 000}

^Â.

3.4 INVESTIGATION OF BISMUTH FILMS OF SMALL THICKNESSES

(100-400 Å)

^BY TRANSMISSION MICROS- COPY AND ELECTRON DIFFRACTION. ^-

Figure

^{6 shows}

the

micrographs

^{and the}

corresponding

diffraction

diagram

^{for the two}

preparation techniques

^described

in section 2 in connection with their

particular

^inves-

tigation procedure.

Fig. ^{6. -} Investigation ^{of Bi} films of thickness 100-400 À by transmission microscopy and electron diffraction. Technique ^{1 :} a) microphotography, b) diffraction pattern. Technique ^{2 :} c) microphotography, d) diffraction pattern.

For films condensed on a substrate

kept

^{at room} temperature ^the

micrograph

ⁱⁿ

figure

^{6a shows that}

the film is continuous and is made _up of

grains

^of

polygonal

^{forms with} dimensions

comprised

^between

1 000 and 2 000

Á.

The

corresponding

diffraction

diagram

^{shown on}

figure

6b indicates that the films

almost

exclusively

consist of

grains presenting

^the

texture

(0001).

For films condensed on a hot

substrate,

^{the micro-}

graph

^{shown on}

figure

^{6c shows} the presence of

grains

identical to those in the

micrograph

^on

figure

^6a

together

^{with other}

large crystals

of diameter about 4 000

Á.

The appearance of these latter

crystals

^can

probably

^{be due to the}

high temperature

^{of the} substrate which causes

during

^the condensation process, the coalescence of certain zones of the films.

We should also notice on the same

micrograph

^the

presence of white zones

surrounding

every

large crystal

^{which can be}

interpreted

^{as due to some}

discontinuities in the films as a result of the coalescence process.

All

rings

^on

figure

^6d

correspond

^{to the}

completely

disoriented

polycrystal.

^{If the}

comparison

^{is made}

between

the calculated and the

experimental

^intensi-

ties,

it appears that the reflections

corresponding

^to

the texture i.e.

(1120, 0330, 2240,...) give

^{rise to}

larger experimental

intensities. It should then be concluded that when

depositing

^{bismuth on a hot}

^substrate,

two

types

^of

crystallites

^are obtained : a first lot of

grains

^with

planes (0001) parallel

^{to the}

^substrate

^and

another lot of

completely

disoriented

crystals of larger

size.

In conclusion the three

types

^{of film}

preparation

allow the

following description

of the film structural

properties :

1. Process n° 1 leads to films

which always present

a

high quality

^{texture with}

homogeneous grains

^of

polygonal

^{form. For} thicknesses of the order of 1 500

A

the

grain

^{size is}

approximately

^{2 000}

^Á.

However it appears that the

grain

size tends to

increase with the film thickness.

2. Process nD 2

gives

results identical to those obtained

for process

^{n° 1.}

3. Process n) 3 leads to two

types

^of

crystals : a) homogeneous grains

^with

perfect

^{texture iden-}

tical to those in _process n° 1 and n°

2,

b) grains

^of

larger

dimensions

reaching

^{10 000}

^Á

for the

thickest

films. These

grains

^are

randomly

oriented with

respect

^to each other.

These

conclusions are also confirmed

by

^{the iden-}

tical

results obtained from the

complementary

^struc-

tural

study

^realized

by

transmission

microscopy

^and

electron diffraction on films of lower thicknesses.

, These

^{results are in} agreement with those

reported by

^{numerous authors}

[2-11]. Indeed,

it has

appeared

that the

(0001) planes

^{tend to lie}

parallely

^{to the}

substrate whenever the films are

evaporated

^{on an}

amorphous

substrate of

carbon,

^{collodion or}

glass

and also when

they

^are

epitaxially

grown on

both,

mica and sodium chloride.

Contrarily,

^if

epitaxially

grown ^on

potassium

^{chloride or}

potassium

^bromide

substrates the

deposit

^of

planes (0112) parallel

^to

the substrate seems to be enhanced.

4.

Conséquences

of the structural

study

^{on some}

conduction

properties

^{of bismuth films. - In what}

follows,

^a

general

discussion of some

important

electrical characteristics of bismuth films obtained

by

different authors is

presented.

^The

interpretations

^will

be

compared

^{to the} results of the

preceding

^structural

investigation.

This discussion leads to a new conduc- tion model and to a better

understanding of

^the

annealing physical

^mechanism.

4. 1 CONDUCTIVITY. THE « SURFACE STATES » MODEL.

- The main

properties

^{of the}

conductivity

^{of thin}

bismuth

films

^{can be} summarized as follows :

i)

^{Its value is}

proportional

^{to the} film thickness

[l,14,16-19].

ii)

^The

conductivity

^value

increases

^for films inclucl-

ing

^thermal treatments

(process

^{n° 2 and no}

3) [1, 16].

iii)

^The

temperature

coefficient

of

the

conductivity

is

positive (semiconductor-like)

up until a

critical temperature

^value

Tc

^{where it}

changes

^into

negative

values

(metal-like) passing through

^zero

[1, 14, 16, 20].

The value of

Te

^decreases for films of process ^{n° 2} and no 3

[1, 16].

4. 1. 1 The classical « size

effect » approach.

^-

For better

understanding

^the transport ^mechanisms

as a function of the films

thickness,

^a

comparison

^of

the

experimental

results with the

predictions

^{of the}

geometrical scattering theories, namely,

^F-S

theory [21]

and M-S

theory [22]

has been made. It is

worth

to note that the F-S

theory

^can

only

^be

applied

^{to mate-}

rials with

spherical

energy surface.

Nevertheless,

^the

case of an

ellipsoidal

energy surface has been formu- lated

by

^{P. J. Price}

[23]

^{and its}

application

^{to bismuth}

films was done

by

^{A. N.} Friedman et al.

[24]

^{and M.}

Subotowicz et al.

[25]

who observed that the results obtained

by using Price’s

^model

only

^{differ a few}

percent from those obtained with the F-S

theory.

The relative

change

^{of the}

conductivity 03C3F/03C30

^with

thickness,

^where ao, is the bulk

conductivity that

^has

been assumed to be

at 300 K and 77 K

respectively [26]

^{and 6F} is the film

conductivity,

^was

analysed

within the classical size effect

approach by

^Fuchs

[27]

and Sondheimer

[21] ]

and for two extreme cases the

following expressions

were found

[21] ]

for K >

1,

^{where K =} and

for K 1.

Where d is the film

thicknesse, 1.

^{the mean free}

path

of the bulk bismuth and ^P the reflection coefficient of the carriers at the film surface.

Figures

^{7a and b show the}

experimental

^results

corresponding

^to process ^no

l, 2

^{and 3} for the thickness

dependence

^{of the}

conductivity

^ratio

UFlao at

^{300 K}

and 77 K. The

comparison

^between

theory

^and expe- riments has been carried out with values of

10 ranging

from 1 000

A

to 3 500

A

for

experiments

^{at 300 K}

[28]

and from 35 000

A

to 50 000

A

for

experiments

^at

77 K [18].

From

figures

^7a

and b,

^{it can} be concluded that the best fit is obtained for measurements made at room

temperature

and for films

prepared according

^to

process ^n° 2. The deviation from the

theory

^{is more}

marked at 77 K for the other two processes.

This

discrepancy

^{cannot be}

explained by giving

^P

a

higher

value which in _any case would

simply

^come

to

give 10

^still

higher

^{values. It}

^is, by

^{the way, very}

interesting

^{to note that the}

^value, 10

^{= 1 400}

A,

which

gives

the best fit at 300 K for process ^{nD 2 is} consistent with a conduction _process

through grains

of dimension

ranging

between 1 000

A

and 2 000

A

as was

directly

observed in the

preceding

^SEM

observations.

1

Neverthless,

^{the F-S}

theory

^{does not take into ,}

account the

scattering

of carriers

by grain boundary

surfaces and

imperfections,

and the size

dependence

of the

transport properties

^{can be}

significantly

^modi-

fied

by

^the presence of ^these

scattering

processes

[29].

The

problem

^of

scattering

^{of carriers at}

grain boundary

surfaces was

carefully investigated by Mayadas

^and

Shatzkes

[22] by including

^the

change

^of

potential

^at

grain boundaries ; they

have found the ratio of the

conductivity

6g of ^the

polycrystalliné sample

^{to the}

conductivity

Qo of the

single crystal

^{for a 0 to be}

independent

of film thickness and to be

given by :

where,

dg

is the linear dimension of the

grain

^{and R the}

probability

^{for a carrier to} be reflected at a

grain .boundary (0 R 1).

Using

^the results of

Mayadas

^and

^Shatzkes,

^Mola

et al.

[30]

^have found the

following approximate expression

^{for the ratio}

UF/UO in

^the presence ^{of the}

grain

boundaries and for the range

0.2 K

^{5 :}

which have the

advantage

^of

being

^thickness

depen-

dent.

It is clear that the

expression

of Mola et al.

only

differs from the one obtained

by

^F-S

by

^{the factor}

Fig. ^{7a. - Thickness} dependence ^{of the room} temperature ^conduc- tivity ^ratio for different processes. The continuous curves are

theoretical according ^{to F-S} theory for different mean free path

value : _03C30 ⁼ 7.35 x 105 MKS [14], ^{P = 0.}

Fig. ^{7b. - Thickness} dependence ^{of the} conductivity ^{ratio at 77 K}

for different processes. The continuous curves are theoretical

according ^{to F-S} theory, ^{for different mean free} path ^{value :}

03C30 ⁼ 3.03 x 106 MKS [14], ^{P = 0.}

1 - R ;

^{and it is}

^possible

^{to obtain the same}

^quality

of fit

by choosing

^{the values of}

10

^{and R as in table III.}

Table III.

Consequently

^and

taking

^{into account} the measured value of the

gain

^size

(1 000 ^A-2

⁰⁰⁰

Â),

^{it seems that}

for bismuth films

prepared according

^to process ^n°

2,

the presence of

grains

^is

conveniently represented by a

value ofthe mean free

path

of the order

of l0 ~ 1000 ^Á

and a very low reflection coefficient associated with the carrier

transport

^{between two}

grains (R 0.1).

Even

though

^M-S

theory

includes the

grain

^boun-

dary scattering,

^{it fails to take into account the}

scattering

^effect

opposed by

defects such as surface states which are

expected

^{to be} present

predominantly

at the

grain boundary

surfaces for films

processed according

^to process ^n° 1. With the aid of the model hereafter the

quantity

of surface states associated to this _process will be

computed.

4.1. 2 The surface states model. - In what follows

a

quantitative study

of the surface states influence

on the film

conductivity

^{is outlined.}

Property

^is

according

^{to which} U2 > a,, where the

subscripts

²

and 1

respectively

stand for _process n° 2 and n°

1,

will be accounted for within the framework of this model.

Indeed,

^the structural

study

^has

provided

identical results for these two processes, ^{as a conse-} quence, the increase in the

conductivity

^after

annealing

will be

entirely

attributed to the removal of a surface states

density.

In

considering’

the details of a

grain-boundary scattering mechanism,

the idea of quantum ^mecha- nical tunnel has been

given quantitative

and detailed examination

by

Drumheller

[31]

^who

showed,

ⁱⁿ

particular,

that the tunnel mechanism for

grain boundary penetration

^can

predict

the films ohmic behaviour.

The

theory [31] ]

^{leads to the}

following expression

for the

conductivity :

which

provides,

^{in terms} of the notation in

figure 8,

the

conductivity

^{value in}

^{03A9-1 .}

^m-1

if m*,

^{the effective}

mass of the

tunneling

carrier is in

kg ; ô,

^{the effective} thickness of the excess

charge layer

^{in the}

crystallite

in

m ; d,

^the film thickness in _{m ; a,} the

intergrain spacing in m ;

ç, the barrier

height

ⁱⁿ

joules

^{and K}

the relative

intergrain

dielectric constant.

The numerical

application

^has been carried out

by assuming

^a hole conduction mechanism and therefore m* ⁼ 0.16 ^x mo

[32].

^{As in}

[31],

^the

following

^values

have been chosen : k ⁼

3, ô

⁼ 3 ^x 10-’ and

EF

^{= 0.23 eV. The most sensitive} parameters ⁱⁿ

(5)

are a and _ç that have been chosen to be _~2 ⁼ 1 eV and a ⁼ 7

A

to

provide

^{the most} convenient fit to

experimental

^{values of} U2 in

[1].

Fig. 8.

^{- Energy band diagram at the} grain boundary ^{between two} crystallites.

For the value of _03C31 to be obtained with

[5]

^{it will be}

assumed,

ⁱⁿ agreement with the structural

analysis,

that all the

preceding

parameters ^remain

unchanged excepted

(p which takes on a

larger

^value

(çi > ~2),

as a result of the

larger density

^{of surface states in}

process ^n° 1

(Fig. 8),

^{which causes the}

systematic

reduction for Q observed with unannealed films.

The ratio

03C32/03C31

^{is then}

given by :

This

equation

^has been solved

graphically

^for

03C32/03C31 = 2, in

agreement ^with

(1)

^and ~1 ⁼ 1.3 eV has been found.

This result is consistent with the situation

depicted

in

figure

^{8 where} the presence of a

density

^{of donor-}

like surface states

NS

^{causes the build} up

of compensat- ing

^{electrons at} the surface of the

crystallite.

^{For these}

electrons ^to be

supplied

the energy bands bend at the surface

thereby causing

the barrier

height

^{to increase.}

For

NS

^{to be}

computed

^a

relationship

^between

~1 - CP2 and

Ns

will have to be found

by solving

Poisson

equation

^{in the}

crystallite

^surface

region.

The evaluation of

dECV/dX

in S can then be

computed

in a way similar to

that_used

in semiconductor surface calculations

[33]

^{with the result :}

where _Eo ⁼ 8.85 ^x

10-12 F . m -1

and

Ne

^{the effec-}

tive

density

of states in the bismuth conduction

band,

is

given by :

if the electron effective mass

density m:e

^{is taken to be}

0.05 ^x mo

[32]

^{the value}

of NC

^{is 2.62 x}

^{1023 m-3.}

For

computing

^the

integral

ⁱⁿ

(7)

^the

assumption

of

complete degeneracy

^{has been}

^made,

^{so that}

[34] :

the relation between

NS

^and

(EF - ECB)

^{then takes}

on a

quasi-linear particularly simple,

^{form :}

Therefore the surface state

density co-rresponding

^to

the films of _process n° 1 will be

given by :

and for _{~1 - ~2} ⁼ 0.3

eV, (11) gives

This

is a very reasonable number which is consistent with the

evaporation

^{condition of the films}

(50 Â/s

and 10-6

torr) [1].

^A

satisfactory

^linear

approximation

of

(11)

within the _{range :}

appears ^to be

given by :

which allows to derive ^a

particularly simple

^{and useful}

relationship

^between

NS

^{and 03C31. In the case of bismuth}

(~2 = 1 eV, a = 7 À

and m* = 0.16

x mo)

^this

expression

^{is :}

4.2 HALL CONSTANT

RH.

^-

Figure

^{9 summarizes}

some results

obtained

for the Hall Constant value as a

function of

temperature

^for unannealed films

(process

n°

1) deposited

^{on different types} of substrates and from different authors

[12, 13].

^A

general

^{feature of}

these results is the

change

^{in the}

sign

^of

RH

^for tempe-

ratures

ranging

^{between 100 K to 200 K.}

Fig. ^{9. - Variation of the Hall constant versus} temperature ^{for Bi} films prepared according to process ^n° 1. According ^to (2), glass

was used as substrate ; According ^to (3), glass ^{was used as} substrate ; According ^to (1) ^{resin was used as substrate.}

Figures

^{10 and 11} also summarize results

corres- ponding

either ^to films

including

^a thermal treatment after

deposition (process

^n°

2, Fig. 10)

^{or to those}

grown ^on hot substrates

(process

^n°

3, Fig. 11).

^Their

most salient

properties

^{are :}

i)

^{The value of}

RH

remains

positive

for the whole range of

temperatures examined.

ii)

The values of

RH particularly

^{at lower}

tempe-

ratures are

highly dependent

^{on the film thickness and}

can

attain

values 10 times

larger

than those in _process

no 1 for the thinner films.

Figure

^{10 shows}

that

^our earlier results

[1]

^{can be}

continuously

extended into the thicker film domain

by including ^those

^{of Ivanov}

et al.

[14].

This extension tends to show that ^the pro-

perties of RH

^{do not seem to be}

signifiçantly

^influenced

by

^{the substrate material on the} one hand and that

for very

^{thick films}

(15 000 Á)

the effect of the

annealing

process, if

measured

^{in terms of}

RH,

^is

practically destroyed

^on the other hand.

Fig. 10. ^{- Variation of the} Hall Constant versus temperature ^{for Bi} films prepared according ^to process no 2. Continuous lines corres-

pond ^to data obtained by (4), ^{mica was} used as substrate with

annealing temperature 240 OC. Other curves as in (1) ^{resin was} used

as substrate with annealing temperature ^{240 °C.}

The

change

^{in the}

sign

^of

RH

^{at low}

temperature

has been observed and first

interpreted by

^{Le Traon}

et al.

[4]

^{in terms} of structural

changes, namely,

^the

quality

^{of the texture would be}

affected by

^{the film}

Fig. ^{11. -} Variation of the Hall Constant ^versus temperature ^{for Bi} films prepared according ^{to process n°} 3. Continuous lines cor-

respond ^to data obtained by (2), glass ^{was used as} substrate, substrate temperature 150 °C. Other curves as in (1), ^{resin was} used as

substrate, ^the temperature of the substrate was 220 °C.

thickness which is in marked contrast with the main conclusion of the structural

study.

^More

recently, however,

Inoue et al.

[12, 15]

^and Kochowski et al.

[13]

have

interpreted

this result

by

^the presence of surface states formed

during

condensation of the films. The

respective

^{values of}

ny’

^and

p03BC2h

govem the

sign

^of

RH

^{in :}

where the

subscripts

^e and h refer to the electrons and holes

respectively ; n

^and p are

their

concentrations

and y

^their

mobility.

However,

^even

though [14]

^{does not}

apply

^to

anisotropic

^{materials such as bismuth it has the} great

advantage

^of

showing

how sensitive the value and the

sign

^of

RH

^{can be to} the influence of a surface state

density. Indeed,

^the evaluation of the variation of n and _{p at} the

grain boundary

^{can be}

directly

^derived

from the

preceding surface

^{states model.}

The

surface

^states model also accounts for

properties i)

^and

ii) ;

it has been shown that the _presence of donor-

type

^{surface state causes a decrease} of the hole current contribution to the value of the

conductivity

^which

is due to the

higher

energy barrier related to the valence

band

(~1). Therefore,

^{it is}

likely

^that the barrier

associated

to the free electrons in the conduction band is lowered and

thereby

^{the electron current contri-}

bution can be enhanced. This effect accounts for the

change

^{in the}

sign

^of

RH

^{at low}

temperature

^for unannealed films. Moreover property

ii)

^{can then be}

interpreted

^as

follows :

^{surface states introduced}

during

^the

growth

of thicker films are removed less

easily

than for thinner ones

by

^{the same}

annealing

treatment. This is in agreement ^{with the result of}

figure

^{10 where the} annealed thick films

(15

⁰⁰⁰

A)

appear ^to

display

^{a behaviour in}

RH(T)

^{which is}

typical

of unannealed films.

In conclusion because of the

important

^role

played by

surface states, ^the size

dependence

^{of the}

transport properties

^are

drastically

altered for bismuth films

prepared according

^to

process

^n° 1 where their

density

has been evaluated to be of the order

of 2 1012 cm-2

and hence the deviation of

experimental

^{curves from}

those

predicted by

^{F-S and M-S} theories is accounted for.

As stated above the low value

(0.05

^x

mo)

^{of the}

electron effective mass in bismuth has a great ^influence

on the

sensitivity

^{of its}

conductivity properties

^{to the}

presence of a surface state

density

^{at the}

grain

^surface.

Films

processed according

^to process ^n° 3 have shown to be made _up

by polycristalline grains superimposed

to an array of textured

crystallites,

the size of the disordered

grains increasing

with thickness. This result is

perfectly

consistent with the

experimental

observation

according

^{to which the}

mobility

^{of these}

films decreases _very

rapidly

with their thickness

[1].

The

large

disoriented

polycrystals

^{are the}

limiting

agent of the conduction _process

mainly

for thick films and for this reason no agreement ^{between the}

predic-

tions of either the F-S or the M-S theories and the

experimental

results should either be

expected

^for

these films.

It has been stated that surface states will cause the energy ^at the

grain

^{surface to rise. The} temperature

dependence ^of

^the

conductivity

^{and in}

particular

^the

value of

Tc

^{at which a maximum}

conductivity

occurs, where

dQ/dt

⁼

0,

may be discussed in terms of the

probability

^of

overcoming

^{these barriers.}

Indeed,

^this

probability

increases with

increasing temperature

whereas at T >

Tc

^a

larger

role should be attributed to

scattering

^from

phonons leading

^{to a metallic}

behaviour. It was

shown, indeed, [1 ] that

^{the maximum}

conductivity temperature

^{for films}

prepared according

to process ^n° 1 with the

highest

^barrier

height (~1

^{= 1.3}

eV),

^{was not} obtained within the _rage

Tc

^{300 K} while for films

processed according

^to

process ^{n° 2}

(~2

^{= 1}

eV)

^{the value of}

Tc

^{was attained}

within that range.

Appendix.

^- The bismuth films obtained

by

^ther-.

mal

evaporation

^on

amorphous

^substrate

kept

^at

room temperature

(293 K) display

^{a texture such that}

plane (111)rhombohedric

^or

(000 1) hexagonal

^is

^parallel

^to

the substrate. If we consider one of the

single crystals making

up the texture, ^its

reciprocal lattice

^{is a}

point

lattice.

Figure

^{la shows} every

point

^{with the same}

intensity.

^{A more strict}

figure

should modulate the

point intensity

^{as a} function of the structure factor.

The

reciprocal

lattice around the OZ* direction of the fibre axis. Each C*

reciprocal lattice plane

^consists

of a set of concentric circles

[35, 36].

It the electron beam is

perpendicular

^to the surface of the

films,

^the

corresponding

diffraction

diagram

is made _up of concentric circles and differs from the

Debye

^Scherrer

diagram

^{of the}

polycrystal by

^the

absence of certain

rings. Indeed, only

remain the

rings (hkl),

^such that their indices are related to those of the fibre axis

(HKL) by

^{the relation :}

i.e. for the fibre axis

(OOOL),

^the

rings (1120), (0330),

(2240), (1450),

^etc...

(the

^{indices are here}

represented

in

hexagonal notation).

If the

sample

is tilted with

respect

^{to the}

beam,

^the diffraction

diagram corresponds

^{to the} intersection of the

sphere

of reflection and the

reciprocal

^lattice

circles.

A

particularly interesting

^case is that for

which

^the

normal to the

sample

is tilted about 900 with

respect

to the electronic beam i.e. when the

sample

is examined

by

reflection. The OZ* axis is then in the

plane

tangent

to the

sphere

of reflection. On the theoretical

diagram, figure 2b,

the diffraction spots will be found on

the Debye

^Scherrer

rings

^at the intersection of the

planes C*,

²

C*,

³

C*,

^{... and the}

generant

^of

cylinders reclining

^on the circle lattice.

Fig. ^{lA. -} Reciprocal lattice of bismuth.