Classical electrodynamics of non-specular conducting surfaces

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Classical electrodynamics of non-specular conducting surfaces

F. Flores, F. García-Moliner

To cite this version:

F. Flores, F. García-Moliner. Classical electrodynamics of non-specular conducting surfaces. Journal

de Physique, 1977, 38 (7), pp.863-870. �10.1051/jphys:01977003807086300�. �jpa-00208649�

CLASSICAL ELECTRODYNAMICS OF NON-SPECULAR CONDUCTING SURFACES

F. FLORES and F.

GARCÍA-MOLINER

Instituto de Fisica del Estado Sólido

(CSIC

^and

UAM),

Universidad

Autónoma, Cantoblanco,

^Madrid

34, Spain

(Reçu

le 15 décembre

1976,

^{révisé le 14 mars}

1977, accepté

^{le 18 mars}

1977)

Résumé. ²⁰¹⁴ Pour les milieux conducteurs dispersifs le rôle de la diffusion

superficielle

d’électrons est étudié dans une approximation semi-classique ^{en utilisant un} paramètre spéculaire phénomé-

nologique p

(0 p 1). ^Le problème ^de l’accumulation de charges ^non physiques ^{sur la} surface, qui apparait ^{souvent dans les}

développements

^en p, ^est

explicitement

^{discute et} évité. La relation de

dispersion

^du

plasmon

^de surface de mode-P dans la limite

quasi-statique

(c ^~ ~) ^est obtenue pour p et 03BA (vecteur ^de propagation) arbitraires. Ceci permet ^{une étude de} l’effet de diffusion sur l’atténuation du

plasmon

de surface et aussi des différents rôles des fonctions

diélectriques longitudinales

^et

transversales. Ceci est illustré par

l’application

d’une formule

d’interpolation simple

^{pour les fonc-}

tions

diélectriques.

Abstract. ²⁰¹⁴ For

dispersive conducting

media the role of surface scattering of electrons is studied in a semiclassical

approximation

using ^a

phenomenological

specularity parameter p (0 ~ p ~ 1).

The problem ^of

unphysical charge

accumulation at the surface, ^which often appears in ^treatments in terms

of p,

^is

explicitly

discussed and avoided. The P-mode surface

plasmon dispersion

relation in the

quasistatic

^limit (c ^~ oo) ^is obtained for

arbitrary

p and 03BA

(propagation

vector). ^This

permits

^a study of the effect of surface

scattering

^{on surface} plasmon

damping

^{and also of the} different roles of the

longitudinal

^and transverse dielectric functions. This is illustrated with an

application

^{in terms}

of

simple interpolation

formulae for the dielectric functions.

Classification

Physics ^Abstracts

8.170

1. Introduction. ^- This _paper follows a

previous publication

^- henceforth denoted as

[1]

^{- which}

discussed the

general problems

^of

electromagnetic matching

^at the surface of a

dispersive

^non

conducting

medium. The case

of dispersive

conductors will be discussed

here, concentrating

^on the surface

plasmon dispersion

^relation

(S.P.D.R.)

^{in the}

quasistatic

limit c - oo. The idea is to

investigate

^the effect of ’ surface

scattering

^{for non}

specular

surfaces. It is well known that the

theory

of surface

plasmons

^{in metals}

needs to be extended

beyond

^the semiclassical

approxi-

mation.

Quantum

mechanical interference effects between incident and reflected electronic wavefunc- tions and details of the surface

potential

^{barrier are}

very

important.

While all this is

commonplace

ⁱⁿ

current literature on the

subject,

the effect of surface

scattering

has been studied much less. A serious

investigation

of surface

scattering

within the semi- classical

approximations

^was undertaken

by

Zaremba

[1] using

^a Boltzmann-Vlasov

approach

^and

all the structure of the dielectric function of the electron _gas which is contained in the random

phase approximation. Explicit

^{results were} obtained for the

long

^{wave limit on} the basis of detailed numerical

computations

^{carried out for different}

assumptions

about surface

scattering,

ⁱⁿ

particular

^{for the}

specular

and diffuse cases. Zaremba

[1]

very

appropriately

stressed that diffuse

scattering

^must be treated with

care. There must be some way in which the

incoming

current ^- here denoted

by J - reaching

the surface must return into the bulk.

It is customary ^{to use a}

phenomenological

^{model to}

describe surface

scattering

^{in terms of a} parameter p which can take values between 0 and 1 and is

something

like the total fraction of

specularly

reflected electrons.

Of course this

description glosses

^over the actual

problem

of the detailed

probability

^of

scattering

^{of an}

incoming

^{state k into and}

out-going

^state

k’,

but for the time

being

^{it seems to be} very difficult to go

beyond

this model and still have a

fairly

flexible method to

perform

^a

physical analysis ending

ⁱⁿ

practical

calculations. For the

S.P.D.R.,

which is the concern

of this _paper, Zaremba’s

[1]

work is the most articulate attempt ^at

studying

^{the effect} of surface

scattering, taking

^{care to} avoid the accumulation of

charge

^{at the}

surface.

Precisely

^{for this reason} the formulation in

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01977003807086300

864

terms of the parameter p ^was

rejected

and different

assumptions

about surface

scattering probability

were made

separately.

^These consist in

assuming

different forms of

angular dependence.

Between the extreme cases of

specular

^and

completely

^diffuse

scattering

^two

plausible specific

^{forms were}

^assumed,

in some way

representing

intermediate situations.

Thus, although

these calculations bear out the effect of surface

scattering, they

^{are not} calculations from first

principles

^and

ultimately

^{amount to different}

phenomenological

models. It would seem desirable to have a solution in terms of the

phenomenological

parameter p which can then be varied

continuously

or even treated as an

adjustable

parameter. ^{It is also}

clearly

^{desirable to have a solution to the}

problem

^for

arbitrary

wavevector K

parallel

^to the surface.

This will be the _purpose of this _paper and will also be

a natural continuation of the

investigation

^undertaken

in

[I].

It is of value to

recapitulate briefly

^the

approach

to the

problem. (i)

Given the medium in z > 0 and the

vacuum in z

0,

^a

hypothetical

extended medium

(M)

is defined and likewise a

hypothetical

^extended

vacuum

(V). (ii)

The definition of

(M)

^includes

fictitious ^- surface and volume ^- stimuli and has a

field

Tm everywhere.

Likewise for

Tv. (iii)

^{The real}

system is reconstituted

by matching

^{the real}

(p(zO) = (pv(zO)

^{to the real}

(p(z>O) = qJM(Z>O).

This

provides

^the

matching

^{or secular}

equation.

(iv)

The model is

prescribed by imposing

^{a definite}

relationship

^between

q>M(Z 0)

^and

q>M(Z 0).

^This

provides

^the

subsidiary equations

necessary ^to elimi- nate the

parameters

introduced in the definition of the fictitious stimuli.

It is

important

to stress that

simply taking

^the

limit c -+ oo in the results of

[I]

^{would be} incorrect for conductors. The reason is the

problem

^of

charge

accumulation’

just

discussed. This is most

clearly

seen

by considering

the electron distribution function

f (r,

v,

t),

^{which can} be written as :

f i being

the correction to the

equilibrium

distribution

fo(v).

^{In standard} treatments

fl(r,

v,

t)

is determined

as a function of the electric field E

by solving

^the

linearized Boltzmann

equation,

^{with some}

boundary

conditions. Here we are concerned with the surface condition on

11 (r,

v,

t).

^In the usual _way

[3]

^this

condition states that a

fraction p

of the incident electrons ^are reflected

specularly,

while the rest

undergo

^diffuse

scattering.

^This

implies

where

6fi

represents the distribution for electrons

scattering diffusely.

Notice that we cannot write

bfi

⁼

0,

since this would

imply

^{that a fraction}

(l-p)

of electrons is

disappearing

in the surface instead of

being diffusely

scattered. Similar considerations are

made in

[1, 2]. Moreover, bfl

^{must have such a value}

that the total

perpendicular

component ^{of the current}

density

^{is zero at} the surface ^{z =} + 0.

However,

^this is a condition

determining

^a scale factor but ^not the form of the distribution function

bfl

itself. Different authors have

postulated

different forms of

bfl.

^For

instance,

Keller et al.

[2]

^have

proposed

A

being

^oc

exp[i(K.p - rot)],

^{where k =}

(K, A)

^and

r -

[p, z] ;

^while Zaremba has used other alternatives such as

or

0

being

^the

angle

^{between v} and the normal to the surface. These last distribution functions ^are

reflecting

a certain

degree

of surface

anisotropy

^{as Greene’s}

work

suggests [4].

Here we have taken

bf as :

This choice allows us to

simplify

^the

problem

^asso-

ciated with the surface

boundary condition,

^since in this _way the condition on the distribution function

can be simulated

by

^a surface electric field. The idea is to use an extended

medium,

^and substitute the

boundary

^{condition at} the surface ^{z =} + 0

by

^a

properly adjusted

^{electric field} zo

E+ b(z) ei(K.p-rot).

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

perturbed

distribution

function, by using

^a formulation

[5]

equivalent

^{to the} Chambers method

[6],

^{in which}

the

perturbed

distribution function is

given by

where a past local time t’ for the electrons is introduced

by using

^the

path

variable method

[6] (e

is the electron

charge).

^{It can be}

readily

^{seen that}

introducing

^a

6-function field ^{- as} indicated above ^- into this formula

by

^itself

produces

^a distribution

This

expression

^{bears out the cos} 0 factor and shows how this form for

bfl

results from a fictitious electric field normal to the surface. This trick will be used later to obtain a zero current

density

^{at the}

surface, by conveniently adjusting

^a surface electric field in the fictitious extended medium. In order to understand this

point clearly

^{it is}

interesting

^to

analyse

^{in detail}

the structure of different types of surface stimuli.

This will be done

in §

^2.

2.

Symmetric

^and

antisymmetric

surface stimuli. ^- The wavevector will

always

^{be of the form k =}

(K, 0, A),

where A will be

ultimately integrated

away and x will be the

propagation

^vector

parallel

^{to the surface. The}

symmetric

^case is trivial. One

puts

^a surface stimulus of the form

uM(K, (0) b(Z), i.e.,

in Fourier

transform, e(K, W)

⁼

^e. Then,

^{for an}

isotropic

^medium

[I]

from

and the

divergence equation

the scalar

potential

ⁱⁿ the extended

hypothetical

system

(M)

^is

given

in Fourier transform

by

This

generates

^a field with mirror symmetry. ^Hence- forth the

explicit dependence

^on

arguments

^{such as r,}

k,

etc., ^will

only

be indicated whenever _necessary.

Likewise for the extended vacuum

(V)

In the real

system Dz

^{must be}

continuous, i.e.,

Ð;( - 0) = Dt(+ 0),

or, with

t? = ! I z I 0,

whence

a’ = - a.

All the

integrations

^{are under-}

stood from - oo to +00. The

dispersion

^relation

(S.P.D.R.)

^{follows from the}

continuity

^of cp in the real system,

i.e., T’(- 0) = Tl(+ 0),

^whence

a well known solution for the

specular

^surface

model

[7].

^{This is} also the limit c - oo of the result obtained in

[I],

^and

corresponds

^{to the case} p ⁼ 1.

It is also

interesting

^to discuss the

antisymmetric

case p ^{= -}

1, although

^{we stress} that this is an

unphysical example

for conductors since _p,

being

^a

real

fraction,

^takes

only

real values in the interval

(0,1 ).

However,

^{this case is} discussed here because the formalism thus

developed

will be used later as

part

of the method

employed

^{to obtain a}

physically

^valid

solution for

arbitrary

^values

of p.

An

antisymmetric

^{field can be} obtained much in the

same way ^as

explained

ⁱⁿ

[I].

Since in this case

E( - 0) = Et( + 0),

^and.

EM( - 0) = - Et( + 0),

one can write :

This

signifies

^a fictitious surface stimulus of the nature of a

magnetic

^current

creating

^an

antisymmetric

^field.

We stress that _eqs.

(4)

^{reduce to}

^{O. DM}

⁼

^VAEM

⁼

0,

for z >

0,

the electrostatic

equations

for the real

system,

^{for all z} > 0. In other

words,

^{when we} extend the medium in z > 0 to the whole _space, we can

extend

similarly

the electrostatic

equations,

^{with the}

sole constraint that the extended

equations

agree with the real ones for z > 0.

Now, taking tt

ⁱⁿ

the y

direction and

integrating

the curl

equation

DM

is now

purely

transverse :

whence

Hence there exists a vector

"’M

⁼

(0, tf¡M, 0)

^{such that}

Then

whence,

^from

(4),

and,

in Fourier

transform,

For later reference :

Equations (5)

^and

(6) with ET

^{= 1 describe}

(V) and,

as

usual,

since in the real system, ^{at z =}

0,

866

it follows that

whence jV = - jM,

which used in the condition

yields

the S.P.D.R.

This is different from

(3),

^{since it}

corresponds

^{to a}

different ^-

unphysical

^{- model. It} is also the limit

c - oo of the result obtained in

[I]

^for p ^{= -} 1.

However the

point

^to notice is that this

provides

^an

antisymmetric

^field

but,

^{as can be seen from the fields}

given by

eqs.

(6),

the real electric current thus cons-

tructed fails to

satisfy

the condition

j_,(+ 0)

^{= 0. In}

order to achieve this it is _necessary to invoke another

antisymmetric

field. This other

antisymmetric

^field

can be built _up

by using

^{a surface}

dipole, i.e.,

^{a surface}

charge

^{which is}

proportional

^{not to}

b(z)

^{but to its}

derivative. In this _way, instead of _eqs.

(4)

^{we have the}

following

^extended

equations :

The

corresponding antisymmetric

^scalar

potential

^is

where

ÐM

⁼ 4

n’J)M(K, w),

and likewise for

(p’

^with

’6L ⁼ L Notice that an

antisymmetric

^{model is}

being

constructed but now the _response is

governed by EL

instead of _GT.

Now,

the scalar

potential (8) gives

^the

fields EM and DM. In

particular

and also

From the

continuity

^of

D,,

in the real

system, 5)v = 9)m

and from the

continuity

^{of T in the real}

system,

This

dispersion

relation is different from

(3)

^{and also}

from

(7). Furthermore,

this field also

yieldsj,(+ 0):o 0.

We can ^{now see how one} could build _up a formal

solutions

for the _{case p} ^{= -}

1, satisfying

^{the condition}

jz( + 0)

⁼

0, by

^a proper combination of the two

previous antisymmetric

^{cases. In order to achieve}

this,

let us discuss a

peculiarity

^{of the fields}

given by

eqs.

(9).

^{The field}

D’

^{can be rewritten as}

The constant term

Om

in

DI(A, z) gives

^{a ð-function}

in real _space,

i.e.,

^a

singularity

^{in the z}

component

^of

Dm

for the second

antisymmetric

field. This

singularity

appears

equally

^{in EM}

giving

^a contribution to the

perturbed

distribution function of the form discussed

in §

^{1. This source}

gives

^{rise to a} contribution to the normal current

jz(+ 0),

^{which can be used to account}

for all the backflow of

charge

which would otherwise be lost in the

diffusely

^{scattered fraction.}

It follows from this that the formal construction of the

antisymmetric

extended medium

(M)

^would

require

^a combination of the two

antisymmetric

^cases,

where the surface

dipole

^and the surface electric field

are

adjusted

^{in order to achieve}

j,, .(+ 0)

^{= 0. This}

would

yield

^a

formally

valid solution for _{p = 2013 1}

which, however,

^{is of no}

physical

interest and will ^not be discussed _any further.

Instead,

^we shall

proceed directly

to construct the

complete

solution for the

case

0 p 1.

It is now clear that the construction of the extended medium

(M)

^{for a}

general

^model of surface

scattering requires

^a combination of three types of fictitious

stimuli, corresponding

^{to the}

symmetric

^{and to the}

two

antisymmetric possibilities,

^{the last two}

having

different roles and

involving

^different parameters.

It is rather cumbersome to have to treat the two

antisymmetric

^{terms on a different}

footing,

^{as describ-}

ed in

(4)

^and

(8),

^{but this can be}

easily

^avoided

by

means of a formal device as follows.

Consider the electric field

EM obeying (4).

Its curl is

zero

everywhere

except ^{at z = 0.}

Thus,

there exists

some scalar

potential rp’

^{such that E’ = -}

V(p’

^{is the}

same as

EM

of

(4)

for all z =A ^0. The two terms will

only

^{differ in a}

possible

6-function

singularity

^{at z = 0.}

In Fourier

transform,

This will have the same behaviour as EM of

(8)

^if

T’

^is

chosen to have the form

Then

This field will therefore be

equal

^{to the first}

antisymme-

tric field

EM obeying (4), making

^the identification

aK

= jM

^and

remembering

^{that the}

singularity

^{at the}

origin, corresponding

^{to the}

Â.-independent

^term

in

(11),

^must be removed when

writing

^{down the}

field

in real _space. With this device all terms

contributing

to the field in

(M)

^can be described on the same

footing,

which makes the

analysis

^a great ^deal

simpler.

3. Surface

plasmon

^{relation for}

arbitrary

p. - After the discussion

in § 2,

the extended system

(M)

^is

defined to have the

following

^field

We stress that the first and third terms,

although formally similar,

^are

going

^{in fact to}

play

^different

roles.

The first term is

antisymmetric.

Notice that the dielectric function of the medium does not appear in this term. Its role is to

provide

^the

appropriate

functional

dependence

^{on A to}

give

^an

antisymmetric

field. The 6-function

singularity

^must be removed

before writing

^{down the}

corresponding field

^{in real} space.

In

practice

this is achieved

by making

the substitution

before integrating

^over ,1,. The second term is the

symmetric

^{one. Since its}

À.-dependence

^is

different,

^so

is its

z-dependence

^{in real} space. Thus with the surface stimulus

uM(K)

^{alone a fixed}

relationship

^between

(P’(z 0)

^and

cpM(Z

>

0)

^{cannot be} achieved for all z. For this reason it is necessary ^to add a bulk

charge

distribution measured

by f

⁼

f(", A). However,

since

T’(z

>

0)

^{must be}

equal

^to

T(z

>

0)

^{for the}

real system,

f

^{must not}

give

^{rise to} any bulk

charge

for z >

0, i.e.,

^{must not have}

poles

^{in the’} upper

half plane-complex

^,1,

plane.

^{The last term is the} response of

(W

^{to a surface}

dipole stimulus,

^{as discussed}

in §

^2.

It is also

antisymmetric

^{and of course} this surface stimulus must be screened with _EL. The role of this term is to

produce

^{the surface source to ensure}

conservation of

J,.

As for

(V),

it suffices to use

The _programme can now be carried out as follows.

The S.P.D.R. is obtained in

general

form from the conditions that in the real system cp ^must be continuous

at z = 0, i.e.

and likewise for

Dz.

^The first condition can be written down ^at once. Thus

The second condition

requires

^{some care, in}

particular,

^the substitution indicated in

(13)

^must be made in the

D,,

component ^derived from the first term of

lpM

ⁱⁿ

(12).

^This

yields

Introducing

^the parameters

equations (14)

^and

(15) yield

^the

general

^form of the S.P.D.R.

The term in _X2 is the new-source-term which would be absent from the limit c - oo of the results obtained in

[I].

^{The task now is to find} xl, X2

and f

^{This comes from the}

following

conditions :

(i)

^The definition of the model in

terms of p, i.e.,

868

(ii)

^The

requirement

^{that the}

net Jz

^{must be zero at z =}

+ 0, i.e.,

These conditions can be written down

knowing qJM

^from

(12)

^{and hence E’ and}

^DM.

The result

depends

on

integrals involving EL

^{and ET. At this}

stage

^the

following

model will be

chosen,

with

and

with

Then,

ⁱⁿ

evaluating

^{terms like}

there are two contributions. One comes from the

pole

^{at A = ix and}

gives

^{a term in}

exp( - xz). The

^{other one}

comes from A = iL and

gives

^{a term in}

exp( - Lz). By itself, f gives

^{no extra} contribution since it has no

poles

in the _upper half

plane. Thus,

^on

closing

^the

integration

^contours in the lower half circle at

infinity,

^{the A}

depen-

dence of

f

^{must not leave a} residue of its own, ^so that the same two

exponentials

appear for z 0. This is satisfied for a functional

dependence

of the form

Then, imposing

^{the condition}

(18)

^{for the two}

exponential

^functions

of I z separately yields

^{the two}

subsidiary equations

and

where E =

E(k

⁼

0, w)

^{= 1 -}

cv§/cv.

For condition

(19), Ez

^and

Dz

^must be derived

again

^from

cpM

ⁱⁿ

(12) remembering

the substitution

(13)

^for

the contribution of the first term. This

yields

The three fictitious parameters ^are eliminated

by using

^{the three}

subsidiary

conditions

(24), (25)

^and

(26).

It is

trivially

easy ^to rewrite these as a set of three

inhomogeneous,

^linear

algebraic equations

for the unknowns xl,

X29 B

^{and to see that the}

inhomogeneous

^{term is}

proportional

^to

(1 - p).

The determinant of the coefficients is

always 0 0,

^for

arbitrary

^x

and p,

^{and in}

particular for p

^{= 1.}

Thus,

xl = x2 ⁼ B ⁼ 0

for p

^{= 1.}

Using

^{this in}

(17) yields again

the trivial result

(7)

^{for the}

purely specular

^model.

For

arbitrary

p, the values of _{x 19 X2} and B obtained from

(24), (25)

^and

(26)

^{are to} be used in

(17) which,

^for

these model dielectric

functions, yields

^the

following explicit

^result for the S.P.D.R. :

valid for

arbitrary K and p.

This result is

significantly

different from the limit c - oo of the result of

[I],

^due

to the source term in _X2-

The S.P.D.R.

for p :0

^{1 has been studied}

by Quinn [8]

for small values of

(1 - p), ^i.e.,

^for

nearly specular surfaces,

^{as a} power series in

(1 - p).

Unfortunately

^no

comparison

^{with the}

present

^results

can be made because the

analysis

^{made no}

provision

for conservation of

J.

As indicated

in § 1,

Zaremba’s

[11

^{work does conserve}

J,.

^Since

explicit

results were

given only

^{for the}

long

^wave

limit,

^this

limit will be studied

in §

^4.

4. The

long

^{wave limit. -}

By studying

^the

disper-

sion relation

given

ⁱⁿ

(27)

^{it is seen that} ii this is written

so that the factor

(2 ^{QJ2 -} (02)

^{is isolated on the}

I.h.s.,

then the

following practical

^{rule must be}

observed for all the terms

appearing

^on the r.h.s. : Put 8 ⁼

1,

^{co =}

(t)p/,/2-,

^evaluate

^{B, l,}

^{L and t to}

lowest order -

i.e.,

^{for x = 0 - and} xi ^to

first

^order

in K. Care

^must be exercised for

10

^and to, ^since

J-=t

^{can be ± i. The correct results are}

and,

^{to lowest}

order,

while to first order

This

yields

^the

long

^{wave S.P.D.R.}

The next

question

^is the choice of

PL

^and

PT.

^The

forms used

for 8L and &r

^can be obtained from the

complete

formulae of Lindhard

[6] taking VF k

^fixed

and

considering

the lhnits cv « _Up k and CO >> _VF k.

This suggests

for 8L

^and &r

interpolation

formulae of the form of

(20)

^and

(22)

^with

PL

⁼ VF

,,/3-/,/5-

^and

PT

⁼

VF/J 5. ^Then,

^{with úJo =}

úJp/j2, ⁽²⁸⁾

^{can be}

written in the form

where

The real

part

^{does not}

depend

^on p. This is an

artifact of the

simple

model used for dielectric func- tions.

However,

^{it is}

interesting

^to compare with the far more elaborate

computations

of Zaremba

[1]

who used the full R.P.A. We recall

from §

^{1 that our}

model of surface

scattering

^is

specular

^when p ⁼ 1 and

corresponds

^{to Zaremba’s cos 0}

scattering

^model

when _p ⁼ 0. Thus our result

(29)

^{must be}

compared

with these two cases. Zaremba’s actual results are

C ⁼ 0.394 - i 0.002 1 for the

specular

^{model and}

C ⁼ 0.362 - i 0.131 for the cos 0 model. It is _gra-

tifying

^{to see} that rather similar

figures

^{are obtained}

in the present

simplified analysis

^{which is based on}

very

simple

dielectric functions and can be carried

through

ⁱⁿ

analytic form,

^not

only

^for

long

waves, but also for

arbitrary K

^and p.

5. Conclusions. ^- In the

spirit

^of

[I],

^this paper shows that it is not necessary ^to invoke additional

boundary

conditions in order to solve the

problem

of surface

matching

in the classical

electrodynamics

of

dispersive

media. The

proof

consists in

giving

^the

explicit

^{solution - in this case} for the S.P.D.R. ^-

by proceeding directly

^{from a}

physical ^model,

^here

described in terms of the surface

scattering

parame- ter p.

Although

^the form of the dielectric functions for reasonable models _may be the _same, the ^case of conductors is

physically

different from the ’ ^case of dielectrics or excitonic

systems

ⁱⁿ

that p

^{takes real}

values in the interval

(0, 1)

and in that for all values except ^{p = 1 one must face the}

problem

^{of current}

conservation in the direction

perpendicular

^{to the}

surface. For this reason the results of

[fl

^{cannot be}

simply

^extended

by letting

^{c - oo. It has been}

necessary ^to redo the

analysis including

^{a surface}

source term to cancel out the artificial sink of

charge

which would otherwise be hidden in the formalism.

This is not a trivial matter. It is instructive to see

what one would obtain without the source term.

In this case the

dispersion

relation has still the form

870

of

(27),

^but X2 is now zero and _x, and B are different.

The result for IOW K is then

For all

but p

^{= 1} this differs

significantly

^{from the}

correct result

(28).

^In

particular,

this would lead to the

erroneous conclusion that one has found a

dependence

of the real

part

^{of cv on} the model for surface

scattering.

Of _course, there is a small

dependence,

^as evidenced in

a more accurate

computation using

^{the full structure}

of the dielectric function

[1].

This difference is a small order effect which is missed out in a crude estimate such as the

present

^{one based on the}

simple interpo-

lation formulae

(20), (22).

^The

origin

^{of the difference}

between

(28)

^and

(30)

^is

physically interesting

ⁱⁿ

itself. In the random

phase approximation

^{one can}

show

[9]

that the coefficient of K is

proportional

^{to the}

dipole

^{moment of the}

charge density

fluctuation associated with the surface wave. It has been further shown

[10]

that this holds on

general grounds, irrespec-

tive of the

model,

^{if one does not take account of} surface

scattering.

^More

generally

^{one finds}

[10]

^the

dipole

^term and also another term

associated

with the current

density

fluctuations

parallel

^to the surface.

More will be said

presently

about this extra term, which is

mainly responsible

^{for the}

damping

^contained

in the

imaginary

part.

Now, looking only

^{at the real}

part ⁱⁿ

(28)

^and

(30)

it is clear

why they

differ. An incorrect

charge

^{sink is}

responsible

^{for a} false des-

cription

^{of the}

dipole

^{moment in the}

region

^{of the}

surface wave. For p ⁼ 0 this reduces Re C in

(30)

^to

one half of its correct value. The smaller difference between

(28)

and Zaremba’s

[1]

^{more accurate result}

is due to the same reason. The

charge

^{densities - and}

hence their

dipole

^{moment -}

differ, although

^{not too}

much,

^because different dielectric functions are

being

used. One could

always

carry

through

^the present

analysis

^{with a more} elaborate dielectric

function,

but this is outside the _scope of this _paper, which aims rather at

having

^an overview of the

problem.

^{It is}

more instructive to

study

the effects of

spatial

^dis-

persion by looking

^{at the}

imaginary part

^for

which,

as stressed elsewhere

[10],

the effect of surface scatter-

ing

^{on the current}

density

fluctuations is

mainly responsible.

^{It is seen in}

(29)

^{that this is zero for}

p ⁼

1,

and is of the same order as Zaremba’s result

[1]

for the case

corresponding

top ^{= 0. Indeed Zaremba’s} result for

specular

^surface

scattering

^-

corresponding

to p ⁼ 1 in

(29)

^{- is an order of}

magnitude

^smaller.

Bearing

in mind that the model used in all cases is collisionless in the

ordinary

sense, the

meaning

^{of these}

results is

quite

clear. Of course the smaller order contribution is the effect of Landau

damping,

^which

again

is missed out when the

interpolation

^for-

mulae

(20), (22)

^{are used. The main effect comes from}

the lack of surface

specularity,

which introduces a loss of momentum

parallel

^to the surface.

Again

^{it is seen}

that an

approximate interpolation

^formula

for h

can

produce

the main contribution to this term.

The essential

point

is that it is

dispersive.

^{If one uses}

instead a local dielectric function 8 ⁼ 1 ^-

w;1 W2

one obtains - i

0.023(1 - p)

^{tor Im}

C, i.e.,

^an

order

°

of

magnitude

smaller than the value obtained from a

simplified dispersive

^function

ET(k, w).

The lesson

, from this

simple

exercise is that the current

density

fluctuation term is _very

important

^{for the}

decay

^of

long

^{wave surface}

plasmons

and that it

images

^rather

conspicuously

the effects of

spatial dispersion

^{of the}

medium.

This _paper for conductors

together

^{with the one}

previous [fl

for dielectrics

give

^a

complete

^{solution to}

the classical

electrodynamics

of the surfaces of

dispersive

media valid in the classical

approximation.

We have

actually

discussed here

only

^the

quasistatic

limit because this is in

practice

^{most often the case of}

interest when

studying

^surface

plasmons

in metals.

Work is now in _progress on the extension of this method to include retardation effects in order to

study reflectivity

and related

electrodynamical

pro- blems in metals.

Our

analysis

^{focuses on} the effect of the surface as

described in terms of the parameter p, which allows for a

plausible phenomenological

model of surface

scattering.

Further necessary

improvements

^{of the}

theory require going beyond

the semiclassical

approxi-

mation and

beyond

^the

simplest

barrier models for the surface electronic

potential.

^{For the electron} gas these

problems

^{are the}

subject

^{of a}

great

^{deal of current}

activity,

without however

incorporating

^an

explicit study

of the effects of surface

scattering.

^The present work is in this sense

complementary.

^{It is}

hoped

^to

undertake a further

investigation

in which the ideas

expounded

^{here can be}

incorporated

^{in a more}

advanced scheme.

References [I] GARCÍA-MOLINER, ^{F. and} FLORES, F., J. Physique ³⁸ (1977).

[1] ZAREMBA, E., Phys. ^{Rev. B 9} (1974) ^1277.

[2] KELLER, ^J. M., FUCHS, ^{R. and} KLIEVER, K. L., Phys. ^{Rev. B}

12 (1975) ^2012.

[3] ^{MAXWELL, J. C., in The} scientific papers of ^{James Clark} Maxwell, ^{W. D. Niven} (Ed.) Dover, N.Y., 1965, _p. 703.

[4] GREENE, R. F., Crit. Rev. Solid State Sci. 4 (1974) ^477.

[5] FLORES, F., GARCÍA-MOLINER, ^{F. and} NAVASCUES, G., Surf.

Sci. 24 (1971) 61 ;

FLORES, ^{F. and} NAVASCUES, G., Surf. ^{Sci. 34} (1973) ^773.

[6] CHAMBERS, ^R. G., ^{in The} physics of ^{metals. I :} Electrons, J. M. Ziman (Ed.) (Cambridge University Press) ^1969.

[7] RITCHIE, ^R. H. and MARUSAK, ^A. L., Surf. ^{Sci. 4} (1966) ^995.

[8] QUINN, ^J. J., Solid State Commun. 11 (1972) ^995.

[9] HARRIS, ^{J. and} GRIFFIN, A., ^{Can. J.} Phys. ⁴⁸ (1970) ^2592.

[10] FLORES, F. and GARCÍA-MOLINER, F., Solid State Commun.

11 (1972) ^1295.