Discussion of the band structure effects on thermoemission as deduced from a simple model

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Discussion of the band structure effects on thermoemission as deduced from a simple model

A.K. Bhattacharjee, B. Caroli, D. Saint-James

To cite this version:

A.K. Bhattacharjee, B. Caroli, D. Saint-James. Discussion of the band structure effects on ther- moemission as deduced from a simple model. Journal de Physique, 1976, 37 (2), pp.149-158.

�10.1051/jphys:01976003702014900�. �jpa-00208397�

149

DISCUSSION OF THE BAND STRUCTURE EFFECTS

ON THERMOEMISSION AS DEDUCED FROM A SIMPLE MODEL

A. K.

BHATTACHARJEE,

^{B. CAROLI}

(*)

and D. SAINT-JAMES

Groupe

^de

Physique

^des Solides de

l’E.N.S.,

^{associé au}

C.N.R.S.,

Université Paris

VII, 2, place Jussieu,

75221 Paris Cedex

05,

^France

(*) Also, Department ^of Physics, UER Sciences Exactes et

Naturelles, Université de Picardie, ⁸⁰ Amiens, ^France.

(Reçu

^{le 15 mai} 1975, ^{révisé le 19}

septembre

^1975,

accepte

^{le 10 octobre}

1975)

Résumé. ²⁰¹⁴ La formulation exacte de l’effet tunnel de Caroli et al. ^est utilisée pour étudier l’émis- sion thermoionique. ^{Le courant est calculé en} présence ^de potentiel

image

^en incluant des correc-

tions quantiques. ^{On montre} que l’effet de ^ces corrections est négligeable. ^{On rend} compte ^{des effets} de structure de bande à l’aide d’un modèle de Krônig-Penney ^{à une} dimension. On montre alors

qu’il n’est pas suffisant de

remplacer m

par m* dans l’expression ^{du courant} déduite d’un modèle d’électrons libres. De

plus,

^{le courant}

dépend

fortement de la

position

de la limite entre le métal et le vide. Enfin, ^{on montre} que suivant la

position

du niveau du vide par rapport ^aux bandes inter- dites la loi de Richardson-Laue-Dushman est ou n’est pas vérifiée.

Abstract. ²⁰¹⁴ The exact treatment of tunnelling ^of Caroli et al. is used to study thermoemission.

The current is

computed

in the presence of the image-force

potential

^and quantum mechanical correction. The effect of this correction is shown to be

negligible.

^{Band structure effects are taken}

into account

through

^a one-dimensional Krônig-Penney ^{model. The current, in this} situation, ^is

not obtained

simply by replacing

^m

by

^{m* in the} expression ^{for the current} obtained when the metal is described by ^a free electron gas, ^even close to band

edges.

Moreover, ^{the current}

depends strongly

on the location of the

boundary

between the metal and the vacuum. It is also shown that,

depending

on the location of the vacuum level with respect ^to the band gaps, the

well-accepted

Richardson- Laue-Dushman law may ^or may ^not be valid.

LE JOURNAL DE PHYSIQUE ^TOME 37, ^FEVRIER 1976,

Classification

Physics ^Abstracts

8.340 ^- 8.940

1. Introduction. ^- Since the

original

paper

by

Richardson

[1],

the thermoionic emission current has been

expressed

^{in terms} of the Richardson-Laue- Dushman

(R.L.D.) equation [2, 3, 4] :

where T is the

temperature,

^T the work function of the

metal,

^{r the} average reflection coefficient of the surface and

Ao

^{a universal constant} which has the value

[3] :

In

using (1),

^{it is}

commonly accepted

^{that the}

reflection coefficient r is

temperature independent

and much less than

unity

if the barrier is

fairly

^well

described

by

^the

image-force potential.

^This assump- tion based on quantum mechanical calculations

by

Nordheim

[5]

and Mac Call

[6],

^leads

experimentalists

to use the rather

simple

^{formula :}

where

AR

^and QR ^{are constants} that are determined

experimentally.

^{It is}

implicitly expected

^that

AR

should not be _very different from

Ao. ^However,

the observed

[3, 4]

values of

AR

^are

systematically

smaller than the theoretical _one,

and,

^{in some} cases,

large

deviations are observed. To try ^{to reconcile}

theory

^and

experiments,

considerable work has been

done, mainly

^{in two} directions :

i)

^The

periodic

^nature of the surface

potential

and the value of the

potential

inside and outside the metal

[3]

has been studied in connection with the fact that _eq.

(1)

^assumes that the electron mass is the same in the metal and in the _vacuum,

assumption

which is

only

valid if the

potential gradient parallel

to the surface is zero.

ii)

The effects of the

non-uniformity

of the surface have been

studied,

^and

promising

^{results were obtained}

using

the so-called

patch-theory [3]

^: the actual

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01976003702014900

surface of the metal is viewed as a collection of small

areas

Si,

each of them

contributing

^to the thermo- current an

amount ji given by (3);

^{the total current}

is then looked at as the _average of the contributions of the different areas. Refinements of this method lead to the more

sophisticated micro-patch theory [7, 8].

In contradistinction with what is

usually

^done

when

using

the R.L.D.

law,

^we

focus,

ⁱⁿ this paper,

our attention on the reflection coefficient

r,

^{or more}

precisely

^{on the} transmission coefficient r defined

as :

The common

assumption, r

^{= 0 or r =}

^1,

^amounts

to

neglecting

all information about the band structure of the metal.

In section 2 we

give

^the

expression

of the thermo-

current j

^{as a} function of the transmission coefficient

T(CO),

^{where ro is the} energy of the

particle,

^{as mea-}

sured from the vacuum level.

Assuming

^{the metal}

to be

represented by

^a free electron _gas, we show

that j

follows the R.L.D. law

only

^if

r(0+) #

^0.

In order to

study

the band-structure effects on the

thermo-current,

^we consider in section 3 an

exactly

soluble model for the metal : the

Kronig-Penney

model. We show that

taking

^{into account the} energy

dependence

^{of r}

leads,

^not

only

^{to a} lower value of the effective Richardson constant

A,

^but

also, depend- ing

^{on the band structure of the}

metal,

^{to a} tempe-

rature

dependence

^of the thermoionic

current j

which

might

differ from the one

given by

eq.

(1)

or

(3).

We also show

that,

^even when the bulk _pro-

perties

of the metal are

properly

^described

by

^the

effective-mass

approximation,

the thermo-current is

not obtained

by simply replacing m by

m* in the free electron metal formula

(1) :

^more

complicated

^band

effects occur even in that case.

2. The thermoionic current and its

temperature dependence.

^{- From the}

microscopic theory

^of

tunnelling developed by

Caroli et al.

[9, 10],

^{one can}

easily

^{find the}

expression

^{for the}

tunnelling

^current

(along

^the

x-direction)

^{for a} three-dimensional _sys- tem,

provided

^that one assumes translational inva- riance in the _y,

z-directions,

i.e. that the one-electron

potential

^is

given by : V(x,

y,

z)

⁼

V(x).

^The

resulting expression

^{for the current is :}

where :

T(CO)

^{turns out to} be the conventional transmission

coefficient, given by :

and

when one assumes that interactions are

negligible

ⁱⁿ

the metal. _xB defines the

separation plane

^{between the}

left (x xB)

^and

right (x

>

xB)

^submedia.

is the Fermi

function,

YL

and pR

the chemical

poten-

tials of the isolated

left

^and

right submedia, separately

at

equilibrium, WL

^and

WR

^{are the solutions of the}

Schrodinger equation :

which behave as

outgoing

^{waves at x = - oo and}

x ⁼ + oo

respectively.

Since the Wronskians which appear in _eq.

(6)

^are constant, xB ^can be located _any- where on the x-axis. It is however most convenient to choose for _xB the

boundary

between the two

physical

parts ^{of the}

system (e.g.,

^for

thermoemission,

^the

vacuum-metal

interface).

Eq. (4)

^{is an exact}

expression

^{for the current in an}

out-of-equilibrium

^situation

corresponding

^{to two}

electrodes of chemical

potentials

ilL and _,uR. The formulation is

readily

^{extended to} thermoionic emis- sion

by simply remarking

^{that it} may be vizualized as a current

flowing

^{from a metal}

(left submedium)

^into

vacuum

(right submedium). Clearly

PL = ,u,

where p

is the Fermi energy of the metal measured from the

vacuum level

(i.e. ,u

^{= -} cp, T

being

^{the metal work}

function), while,uR = -

^{oo. It is seen}

that,

^as

expected, only

the electrons

thermally

excited above the vacuum

level contribute to the current.

Indeed,

the transmis- sion coefficient

-r(ro)

^{is zero for ro}

0,

^{since no}

running

^{wave can} propagate, ^{in the vacuum}

region,

below the vacuum level.

Note that _eq.

(4)

^is

completely general

^{insofar as}

it is valid for

arbitrary V(x),

^and

that,

when interac- tions are

neglected (i.e. when T(co)

^is

given by eq. (6)),

it

gives

the well-known kinetic model of the one

electron

theory [11].

Taking

^{into account that} (p >

kB T,

the thermo- ionic current is

readily

obtained from _eq.

(4)

^and

(5)

as :

For x --+ oo, the

x-component

^{of the}

outgoing

^wave

151

behaves as

e:tikx u t k(x),

where k is related to the

argument

^{co of}

T(co) by :

Thus it is a function of k ~

w1/2.

This shows that

T(CO) (which

may be

computed

^at any

point

^{x, and}

particularly

^{at x -+}

oo)

^{is a}

single

function of

w1/2.

Then :

Eq. (10) clearly

demonstrates that one can

expect

the R. L. D. law

(1)

^{to be valid}

only

^if

r(0+ ) # 0,

^{i. e.}

if

T(a))

is discontinuous at the vacuum level.

However,

this _necessary condition is

by

^{no means}

sufficient,

since the existence of the R. L. D. law is linked to the

validity

^{of the} power series

expansion

^for

t(ro). If,

^for

ro >

0, -r(ro)

^varies

sharply

^{in a} range of order

kB T,

the

expansion

^{will be no}

longer valid,

^{and the}

tempe-

rature

dependence

^{of the current will be more involved.}

This is for

example

^{the case when the vacuum level}

m ⁼ 0 lies near the top ^{of an} allowed band of the

periodic potential ^V(x)

^{of the}

^metal,

^{or in a forbidden}

band.

Let us leave for the moment this last case out of our

consideration and concentrate on the behaviour of

T(oi)

^{close to co = 0.}

The

computation

^of

T(co) requires

^the

knowledge

^of

the wave functions

Y’L(X)

^and

YR(x) which,

in turn,

are determined as soon as the

potential V(x)

^{is known.}

Since the aim of our calculation is to compare the

commonly

used R. L. D. law

(eq. (3))

^{with a more}

accurate one which includes band

effects,

^{we will}

assume for the

potential V(x)

^the

following

^{form :}

where W is the

height

of the conventional step poten- tial

representing

^{the metal}

surface, VL(x)

^and

VR(x)

the

potentials

^seen

by

^electrons

respectively

^inside

and outside the metal. _xo is the mathematical surface defined

by writing

^the

continuity equation

^for

V(x).

Note

that,

^{if we}

neglect

^the metal and vacuum

potentials ( VL

⁼

VR

⁼

0) :

so that

’t(0+)

^{= 0}

[11]

and the thermocurrent reads :

Consequently

^{there is} no reason to

expect

any kind of R.L.D. law in this situation. Moreover it

can be shown from multichannel

scattering theory [ 12J

that

z(0+)

is indeed zero

(i.e. ^’t(w)

continuous at

w ⁼

0)

^{when the}

potential V(x)

^{decreases at}

infinity (for x -+ + oo)

faster than

x-lor

is

periodic.

^There-

fore the R.L.D. law for

evaporation

may ^occur

only

^{in the} presence of a Coulomb

potential,

^i.e.

. cannot hold for

uncharged particles.

Let us

emphasize

^{that in most}

physical

^situations

involving uncharged particles,

^a

one-body approxi-

mation cannot be

used,

^{so that no}

simple quantitative prediction

^{can be made.}

In the conventional treatment of

thermoemission,

one retains for

VR(x)

^the

following expression [5, 6,13] :

which represents ^{the usual}

image-force potential

sup-

plemented by

^the quantum mechanical corrections due to

exchange

and correlations

[14].

^The

phenome- nological parameter il

^is

supposed

to account for the surface

properties

of the metal. The location _xo of the mathematical surface is now related

to ’1 by (ao

is the Bohr radius and R the

Rydberg constant) :

Although

it has been shown

[15, 16, 17]

^{that such a}

form for

YR(x)

is incorrect for

light charged particles,

we believe that the detailed

shape

^of

V(x)

^{in the}

vacuum

region

^{will not} affect the

qualitative

^features

of our

results,

^{i.e. the}

dependence

of the transmission coefficient _upon the band structure of the metal.

Moreover,

^{such a} choice for

YR(x)

will allow direct

comparison

^with eq.

(3).

Using

the results of references

[5]

^and

[6],

^one

finds

(1) :

.where :

(1) Wz,,,(z) is the Whittaker function and Hy(z) the Hankel function of order v. The prime denotes derivation with respect

to the argument.

Therefore,

^{if we}

neglect

^the

crystal potentiel (i.e.

VL

⁼

0), T(O +)

^{’# 0.} This is indeed a consequence of the

x -1

behaviour of

VR(x)

^for x -> + oo, i.e.

of the existence of the

image potential.

^If

VL(x)

^is

such that

t’(w)

^is

expandable

in power series of a)’/2 for m >

0,

^{we then recover} the usual R.L.D.

law,

i.e. the

T2 exp(- QlkB T)

temperature

dependence

for the thermocurrent.

Figure

^{1 shows} the value of

r(0+) plotted

^{as a}

function of

R/ W

^{for the two}

limiting

^values

of 11

^:

11 ⁼ 0

and 11

⁼

x 1 /4,

^{and for}

YL(x)

^{= 0. We observe}

that the quantum mechanical correction

(n = 0)

to the

image

^force

potential

introduces no

qualita- tively

^new effects. And

since,

^for

VL

^{= 0 and for}

actual values of

W(-

¹⁰

eV), r(0+)

^is

already

^close

to

1,

^the

commonly accepted assumption, r(0 + )

⁼

1,

is

justified

when band effects _may be

neglected.

FIG. 1. ^- Plot of 1(0 +) ^{as a function of} R/W (eq. (16)) ^{for the two} limiting cases : q ⁼ 0,ju

= 2 1

^{and q = x,/4, p}

1 (

^{1 +}

^{4 R W} ji12

^.

limiting ^cases: 11 = 0, J.l.

2

^{and 11 = x J.l.}

= 2 1 +

4 W ^°

However,

^{we shall see in the}

following

section that the band structure of the

crystal

may have

important

consequences ^on

z(0+),

^so

that,

^{even for}

charged particles,

the R.L.D. law _may not be valid.

3. Band structure effects : the

Krdnig-Penney

^model.

- As is well

known,

band effects in metals are linked to the

periodic

^{nature of the}

.potential V(r).

^{In order}

to

investigate

the influence of this

periodicity

^{on the}

thermocurrent,

^we shall solve out a

simple

^case,

namely

^the

^familiar

^one dimensional 6-function

Kronig-Penney

^model

[18] (K.P. model).

For the sake of

simplicity,

^{we assume that :}

In this

expression

^{P is a} dimensionless

parameter representing

^the

strength

^{of the 6}

potential, a

^{is the}

lattice

spacing along

^{the x}

direction,

^{and b is a} para- meter which we allow to vary from - a to zero.

This means that we describe the surface

by

^a step

potential

the location of

which,

^with

_respect

^{to the}

last 6

potential,

may vary within a unit cell.

To be _{sure, eq.}

(18)

^{does not} represent ^satisfac-

torily

^an actual metal in the _presence of its

surface,

but it has the

advantage

^of

leading

^{to a}

completely

soluble situation and to show that the thermocurrent is not

only

^modified

by

the effect of

periodicity,

but also

depends

^rather

strongly

^{on the}

position

^of

the surface

(i.e.

^on

b).

In the metal

region (x xo)

the relevant electron

wave function

VFL(X) compatible

^{with the}

boundary

condition at x ^{= -} oo is

readily

calculated and one

finds from _eq.

(6) :

where K is defined

by

eq.

( 12b),

and use has been made of the well-known

dispersion -

relation :

Note that the allowed _energy bands

correspond

^to

those values of K for which the left-hand side of eq.

(21)

lies between - 1 and + 1. It is also seen

that surface states

[19] (Tamm’s states)

^which appear in the _gaps

only

^{for co 0}

(corresponding

^{to ima-}

ginary

^{values of}

K)

^{do not} contribute to the thermo- current.

The

complete computation

^of

T(w) and j requires

also the

knowledge

^of yR. In order to

disentangle

somewhat the effects of surface

location, periodicity

and Coulomb

potential,

^{we shall}

investigate

^the

two situations of no

image

^force

(i.e. VR

^{= 0 for}

x >

xo)

^{and of a vacuum}

potential

^as

given by eq. (14).

3.1 TRANSMISSION COEFFICIENT IN THE CASE OF NO IMAGE POTENTIAL. ^- We set

VR(x)

^{= 0 for x} > 0

(i.e.

xo ⁼

0).

^The transmission coefficient is

readily

obtained,

^{and reads :}

153

while the

density

^{of states}

[20],

^{for one}

spin direction,

^is

given by :

Note that there is no

simple

relation between the

density

^{of states} and the transmission

coefficient, and, consequently,

^{between the}

density

^{of states and}

the

thermocurrent,

in contradistinction to what is sometimes assumed.

3.1.1

Effect of surface

^{location. - It is}

interesting

to remark that

T(w) depends strongly

^on the surface

position

parameter ^{b. As an}

example

^{we have}

computed T((o)

^{as a} function of co for P ⁼ 3

n/2, a

^{= 4}

^A

and W ⁼

13,5

^eV. The results are

displayed

^on

figure

^{2. The two curves}

correspond

^{to the two}

different values of b : -

a/2,

^0. The effect of the

surface

parameter ^{b is seen to be}

quite

^drastic.

(Our

choice of the parameters a and W is such that the bottom of the third band lies 0.4 eV below the vacuum

level.)

FIG. 2. ^- Plot of T(co) ^{as a} function of (o (eq. (22)) ^{for different}

values of b : W = 13.5 eV (P ^{= 3} Tc/2, ^{a = 4}

A).

3.1.2 The

effect of

^band structure : the

effective

mass

approximation.

^- The effect of the

periodicity

appears in _eq.

(22) through periodic

functions of Ka and _pa. As well

known,

^{close to a band}

edge,

^the

dispersion

relation is

quadratic,

^{so that one can}

introduce an effective mass m*. In the K.P. model the band

edges

^are

given by :

the

corresponding energies

^cv°

being given by

eq.

(21).

In the

vicinity

^{of such a band}

edge

^{we have :}

where

is the effective mass.

At first

sight

^{one can be}

tempted

^{to take}

advantage

of the

quadratic

^relation

(25)

ⁱⁿ

computing T((D), by describing,

^{close to a band}

edge,

^{an electron}

inside the metal as a free

particle of

^mass

m*, moving

in a constant

potential wo,

^{the outside}

potential being

^{zero. The}

resulting Le.m.(ro)

^{reads :}

This

commonly

^used

approach

is however incorrect.

Indeed,

^{close to the band}

edge (dp - 0) 1(cv),

^as

given by

eq.

(22),

^{reads :}

It is seen that

T(W)

^{vanishes at the band}

edge.

^This

is to be

expected ^since,

^{for this energy, the group}

velocity

^is zero, ^so that no

particle

^can

propagate

in the metal. However,

i(cv)

differs from its

effective

mass value

Te.m.(W) by

^a

multiplicative

factor which

depends explicitly

^on the characteristics of the

periodic potential. Figure

³ shows the variation of

T((,O)/Te..,.(co)

^{as a} function of b. For

instance,

^with

FIG. 3. ^- Plot of T(60)/T’...((O) (eq. (27) ^and (28)) ^{as a} function of the

parameter b ^{for W = 13 eV and coo = 0.09 eV} (P ^{= 3} 7r/2, a ^{= 4}

A).

W = 13

eV, P=37r/2, a=4A

^and

b = 0,

^this

factor is 0.14 for

roO

⁼ 0.09 eV. Note that

depends strongly

^on

b,

^the

surface

parameter, ⁱⁿ

agreement

with what was

already

remarked when

discussing

eq.

(22).

However,

^{it must be}

kept

in mind that the

descrip-

tion of the surface

by

^a

step potential

^becomes ques- tionable close to the first atomic

layer.

^{In other}

words,

^it may be somewhat

misleading

^{to draw}

quantitative predictions

from the above

calculation,

since the variation of b takes

place

^{in a}

region

^where

the

potential description

^is

questionable.

^{The reader}

should conclude from our results that the transmis- sion coefficient _may be

quite

^{sensitive to} the surface

description

and that it is rather

involved

^{to obtain}

information on the surface

by simply studying T((O).

3.2 TRANSMISSION COEFFICIENT IN THE PRESENCE OF AN IMAGE POTENTIAL AND VALIDITY OF THE R.L.D. LAW.

- We now assume that the

potential

outside the

metal is

given by

eq.

(14),

^{so that} yR is

given by

eq.

(16a)

and

(16b). r(co)

^is

easily computed

^from eq.

(16), (19)

^and

(6a).

It is clear

that,

^as in section

3.1, T(ro)

^{vanishes at}

a band

edge

and remains zero inside ^a band _gap.

This

immediately

shows that one cannot expect

i(cv)

^{to be} discontinuous at co ⁼ 0 if the vacuum

level lies in a forbidden band. We shall therefore

investigate

^two

possibilities :

a)

^{the vacuum} level falls into an allowed band.

b)

^{the vacuum} level falls into a forbidden band.

a)

^{The vacuum level}

falls

^{into a} allowed band. ^- The situation is

pictured

^on

figure

^4.

i(cv)

^{is now}

discontinuous at co ⁼ 0.

However,

^as discussed in section

2,

the R.L.D. law is obtained

only

^if

i(cv)

varies

slowly

^{on a} range of order

kB

^{T. Since}

Aco)

will vanish at the

top

of the band _ro1, we can distin-

guish

^{between two cases}

(2) :

i) co, >> kB

^T

(Fig. 4a).

The variation of ⁱ takes

place

^{on an} energy range much

larger

^than

kB T,

we can then

replace

ⁱⁿ eq.

(8) 1(cv) by T(O ’),

^and

obtain :

FIG. 4. ^- The metal band structure is such that the vacuum level lies within an allowed _energy band : a) kB ^T w, ; b) kB ^T > col.

Therefore,

^as

expected,

the thermocurrent follows the R.L.D. law with ^a Richardson’s constant which

depends

^{on the band structure and on the}

position

of the surface of the metal

through

the transmission

(’) ^{We assume from now on} that the width Jf of ^the forbidden band above _wl is much larger ^than kB ^T.

155

FIG. 5. - Plot of i(0+) (K.P. model) ^{as a} function of b for

W = 13.5 eV, a = 4 A, P = 3 7r/2,?l = 0.

coefficient

1(0+).

^{We have}

plotted

^on

figure

^{5 the}

variation of

1(0 +)

^{as a} function of b for W = 13.5 eV

(P

^{= 3}

n/2, a

^{= 4}

A).

^Note

^that,

^here

again,

^the

effect of the surface

parameter

^{b on} the thermocurrent is _very

significant.

ii) kB

T >> w1

(Fig.

^4.

b).

The variation of z takes

place

^{on an} energy range much smaller than

kB ^T,

so that a power

expansion

^{is no}

longer

valid. Therefore

one does not

expect

^an R.L.D. law for the current.

Moreover,

since the thermal _energy

kB

^{T is small}

compared

with the work

function,

^{the current will}

be small.

Anyhow

^{it is}

interesting

^to

study

^this

situation,

insofar as it shows how band effects ^can lead to drastic

changes

in the thermocurrent temperature

dependence.

Since w1 is close to the vacuum

level,

^{we can assume}

that it is a

good approximation

^{to use for yR its}

ro ⁼ 0+ value

(eq. {16b)).

^{Near the} top ^{of a band} eq.

(25)

^reads

[20] :

Using

eq.

(19)

^and

developing

eq.

(6)

^to first order in

bp,

^{we obtain :}

where :

(n

⁼

1, 2,

^{... is the band}

index).

^Since

we can, to first

order, replace exp( - wlkb T) by

1 in the

integrand

^of eq.

(8).

The thermocurrent takes the form :

We have written _eq.

(33)

under the conventional form. It is seen that the R.L.D. law is not

obeyed,

since the Richardson constant

C(.)(T) ^depends

^on

temperature

(in

^this

case j

^{is even}

proportional

^{to T}

instead of

T2.!)

Note

that, similarly

^{to the case}

VR

⁼

0,

^{the effect} of the

periodic potential

^{does not} appear

only through

the conventional effective mass correction

(mlm*)’I’

but,

^{in a} rather involved _way, via T(.). ^{Note also} that the value of ’t’(n)’ ^hence

C(n.)(T), ^depends

^{on the}

index n of the allowed band considered.

Figure

⁶

shows the variation of ’t’(n) ^{with b for}

FIG. 6. ^- Plot of t(n) (eq. (32)) ^{for n = 2 as a} function of the surface

parameter b whence 1 kB T. ^{Here : W = 9.39} eV, pa = Kl ^{a = 27r}

FIG. 7. ^- The metal band structure is such that the vacuum level lies within an energy gap. The thermal energy is such that coo kB ^T wo ^{+ A.} Hatched areas correspond ^{to allowed}

energy bands.

b)

^{The vacuum} level lies within an energy gap. ^- The situation is

displayed

^on

figure

^{7. Let us call}

again

roo the _energy of the bottom of the allowed band

lying just

^{above the vacuum}

level,

^{and A its}

width. Then from _eq.

(8) :

Eq. (34) pictures

the fact that

r((o)

^{is zero for a)} coo and co > coo ⁺ A.

For kB

^T coo the current prac-

tically

^{vanishes as the}

exponential

^{in the}

integral

will be _very small in the

region

^of

integration.

^For

kB

^T > coo ⁺

A,

^one would have to take into account the

explicit dependence

^of

-r( OJ)

^{on co. Let us how-}

ever note that this situation is not realistic

since, usually,

the bandwidth is much

larger

^than

kB

^T.

Therefore,

^the

physically interesting

case corres-

ponds

^to coo

kB T

^{d . In this case} the thermo- current will be

governed by

the behaviour of

i(cv)

near the bottom of the band. In this energy range yR is

given by eq. (16.a)

^{and the}

co-dependence

^of

T(W)

is

governed by

the behaviour

of yL (eq. (19))

^{near the}

bottom of the band. To first order in

bp (eq. (25))

the transmission coefficient reads :

where _coo is the _energy of the band

bottom, n

⁼

1, 2, ...

the band index and :

Thus the thermocurrent is

given by :

This

expression

^for the thermocurrent is valid

only

when the band structure can be described

by

the effective mass

approximation (3),

^{i.e. for cv z} wo.

The value D of the _upper limit of

integration

^is

roughly

^{the energy} for which the effective mass

approximation

^{fails. In} thermoionic

emission,

^this

energy ^can be assumed to be

larger

than the thermal energy

kB ^T,

hence the value of the thermocurrent

j

^is

roughly independent

^{of the exact value of D and}

one can write :

(3)

^{Let us once} again emphasize the fact that s(m) ^{is not} computed

in an effective mass approximation.

Thus,

^{one obtains}

(cvo « T) :

This result is not

surprising for,

^as

long

^{as we are}

interested

only

^{in the}

temperature dependence of j,

we are in a situation _very similar to the one described

at the end of section

2, namely

free electron model and a constant

potential

^outside the metal.

Indeed,

as we

pointed

^{out, the}

temperature dependence

^of

the thermocurrent is

strongly governed by

^{the beha-}

viour of

i(cv)

^{in the}

low-energy region :

^this

explains why

^{the T}

dependence of j

^{is the same in eq.}

(39)

as in _eq.

(13);

in the first case

I(cvo)

⁼ 0 while in the second we have

-r(0)

^{= 0.}

Moreover,

^the

exponential

factor in _eq.

(39)

is of the same nature as in _eq.

(13)

since

only

^{electrons with} energy ^w > a)o ⁺ (p ^~ Q

can escape from the metal.

The effective mass and

periodic potential

^effects

are described

through (mlm*)1/2. ten)’

^where

ten)

^is

given by

eq.

(37).

^{We have}

plotted _ten)

^{as a function}

of b in

figure ^8,

^{for W = 13 eV}

(P

^{= 3}

a/2, a

^{= 4}

A).

4. Conclusion. ^- We find that the Richardson law is not

universally

valid for thermoemission. The

157

FIG. 8. ^- Plot of t(., (eq. (36)) ^{for n = 3 as a} function of the surface parameter ^{b. Here W = 13} eV, po ^a = 2, Ko a ^{= 2.36 7r so that}

mo ⁼ 0.09 eV and (m*lm) 1/2 ⁼ 0.31 ; tj ^{= 0} (P ^{= 3} n/2, a ^{= 4}

A).

thermocurrent

depends quite strongly

^on the surface barrier as well as the

periodic potential

inside the emitter.

While the _presence of the Coulombian

image

^force

outside the emitter is _necessary

(but

^not

sufficient)

for the Richardson law to be

valid,

the first-order quantum mechanical correction

[14]

^{to the}

image

force does not introduce _any

qualitatively

^new

effect on the thermocurrent. When the vacuum

level at

infinity (cv

⁼

0)

falls inside an allowed _energy

band,

^the top of which lies much

higher

^than

kB T,

we find the

following temperature dependence

^of

the current :

The latter remains valid if the vacuum

potential

vanishes faster than

x-1

at

infinity.

This result for the thermoemission of neutral

particles should,

^how-

ever, be

interpreted

^{with care. For}

^instance,

^{in the}

case of

evaporation

of excitons from electron-hole

drops

ⁱⁿ

semiconductors,

^the

many-body

^effects

should

presumably

^{be non}

negligible.

In the _presence of the

image force,

the effects of location of the

surface,

^{as well as of the}

periodic potential

inside the

crystal

have been

investigated

in the frame of the

Kronig-Penney

model. The main results of this

study

^are enumerated below :

1)

The thermocurrent is

extremely

^{sensitive to the}

location of the surface with respect ^{to the last}

spike

of the K.P. _array, and

depends explicitly

^on the lattice

spacing

^{for a}

given potential strength.

2)

^{Even near a band}

edge,

^the current cannot be obtained

by simply replacing m by

^{m* in the free}

electron

expression,

^{and the} temperature

dependence

of the current is sensitive to the

position

^{of the vacuum}

level

(w

⁼

0)

^with respect ^{to the band structure of} the emitter. For instance :

a)

^{When ro = 0} lies inside an allowed

band,

^the top ^{of which is} at w 1

b)

^{When co = 0} lies in the _gap at a distance -

kB

^T

below the bottom of ^an allowed band

Even

though

^the

limiting

^cases

(a-ii)

^and

(b)

^corres-

pond

^{to small}

thermocurrent, they

illustrate

signi-

ficant

departures

from the Richardson law.

It should be remarked that our formulation of the emission

problem

^is

effectively one-dimensional,

^since

we have made the. conventional

assumption

^of

translational invariance in

planes parallel

^{to the}

surface. The fact that the thermocurrent

depends explicitly

^on the lattice

spacing

^{and the band structure}

along

^the

perpendicular

^to the surface

implies

^a

very

interesting

result : the

anisotropy

^{of thermo-}

emission from

crystals.

The

problem

^of

description

of the surface is illus- trated

by

^{our K.P.} model result that the thermocurrent is _very sensitive to the location of the surface

step.

The

representation

of the surface

by

^a step

potential (plus

^the

image

^force

outside)

^is

certainly questio-

nable when one is interested in effects on distances of the order of the lattice

spacing. Moreover,

^we

cannot rule out the

possibility

^{that the K.P. model}

be

pathological

in relation to the location of the

step.

In _{any case,} whatever be the

model,

^{it should be}

noted that the transmission coefficient

depends

^on

the

matching

^{condition at the}

boundary,

^{and there-}

fore the effects of the location

of

^the

surface

^are

expected

^{to be}

significant.

^The

physical

^{state of the}

surface is

expected

^{to be of} consequence, ^even

though

the localized surface states do not contribute to the thermocurrent.

References

[1] RICHARDSON, O. W., ^Phil. Mag. ²³ (1912) ^594.

[2] DUSHMAN, ^{S., Rev. Mod.} Phys. ² (1930) ^381.

[3] ^{HERRING C. and} NICHOLS, ^M. H., ^{Rev. Mod.} Phys. ²¹ (1949)

185.

[4] NOTTINGHAM, W. B., Hand Phys. ^{Band 21} (1956) ^1.

[5] NORDHEIM, ^L. W., Proc. R. Soc. 121 (1928) ^626.

[6] ^MAC CALL, ^L. A., Phys. ^{Rev. 56} (1939) ^699.

[7] GARDNER, ^F. M., GIROUARD, F. E., BOECK, W. L. and COOMES,

E. A., Surf. ^{Sci. 26} (1971) ^605.

[8] NEEHAM, P. L. and COOMES, E. A., Surf. ^{Sci. 44} (1974) ^401.

[9] CAROLI, C., COMBESCOT, R., NOZIÈRES, ^{P. and} SAINT-JAMES, D., J. Phys. ^{C 4} (1971) ^916.

[10] CAROLI, C., COMBESCOT, R., LEDERER, D., NOZIÈRES, ^{P. and} SAINT-JAMES, D., ^J. Phys. ^{C 4} (1971) 2598.

[11] DUKE, ^C. B., Tunnelling ⁱⁿ Solids, Solid State Phys. Suppl. 10,

H. Ehrenreich, ^F. Seitz & D. Turnbull eds. (Acad. Press, N. Y.) ^1969.

[12] NEWTON, R. G., Scattering Theory of ^{Waves and} Particles, Chap. 16, 17, (Mc ^{Graw Hill} ed., N.Y.) ^1966.

[13] CUTLER, P. H. and GIBBONS, ^J. J., Phys. ^{Rev. 111} (1958) ^894.

[14] SACHS, ^{R. G. and} DEXTER, ^D. L., ^J. Appl. Phys. ²¹ (1950) ^1304.

[15] APPELBAUM, ^{J. A. and} HAMAN, ^D. R., Phys. ^{Rev. B 6} (1972)

1122.

[16] FEIBELMANN, P. J., DUKE, C. ^B. and BAGCHI, A., Phys. ^{Rev. B 5} (1972) 2436.

[17] HEINRICHS, J., Phys. ^{Rev. B 8} (1973) ^1346.

[18] KRÖNIG, ^{R. de L. and} PENNEY, ^W. G., Proc. R. Soc. 130 (1931)

499.

[19] TAMM, I., Phys. ^Z. Samj. 1 (1932) ^733.

[20] DUKE, ^{C. B. and} FAUCHIER, J., Surf. ^{Sci. 32} (1972) ^{175. The}

same K. P. model is used there to study ^{the band structure}

effects on the field emitted current at T ⁼ 0 K. In the _range of _energy concerned in field emission, ^the shape ^{of the} potential ^{in the vacuum is} mainly governed by its field-

dependent part 2014 eEx, ^{so that one can} neglect the contri- bution of the image ^force potential (See ^also B. Caroli : Surf. Sci., 51 (1975) 237). ^Thus 03C4(03C9) ^as given by eq. (22)

is the E ~ 0 limit of Duke and Fauchier’s result.

[21] ^{We recall} that, ^{at the} top ^{of a band} (03C90 ⁼ 03C91) ^{one has :} K1 a = pa ⁼ n03C0, n ⁼ 0, 1, 2, ... is the band index and K is given by eq. (12b).