V,

IC/70/23

INTERNATIONAL CENTRE FORTHEORETICAL PHYSICS

ELECTRON IN DIELECTRIC FILMS.

QUANTUM THEORY

OF ELECTRON ENERGY LOSS AND GAIN SPECTRA

INTERNATIONALATOMIC ENERGY

AGENCY

UNITED NATIONSEDUCATIONAL.

SCIENTIFICAND CULTURALORGANIZATION

A 4 A. LUCAS

E. KARTHEUSER

and

R.G. BADRO

1970 MIRAMARE-TRIESTE

IC/70/23

INTERNATIONAL ATOMIC ENERGY AGENCY

and

UNITED NATIONS EDUCATIONAL SCIENTIFIC AND CULTURAL ORGANIZATION

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

•_ ELECTRON IN DIELECTRIC FILMS.

QUANTUM THEORY

OF ELECTRON ENERGY LOSS AND GAIN SPECTRA

A. A. LUCAS

E. KARTHEUSER **

and

R . G . BADRO * * *

MIRAMARE - TRIESTE

April 1970

t To be submitted for publication.

*•" Charge de Recherches au Fonds National Beige de la Recherche Scientifique.On leave of absence from Department of Physics, University of Liege, Belgium.

** On leave of absence from Department of Physics, University of Liege, Belgium.

**•* On leave of absence from Department of Physics, Lebanese University. Beirut,Lebanon.

ABSTRACT

The classical analysis, due to Kliewer and Fuchs, of the optical

polarization modes of an ionic crystal film in the long wavelength limit

is used to develop a quantum mechanical treatment of the interaction

"between an electron and the optical phonon field of the film. Retardation

effects are neglected. The analogue of Frohlich's polaron Hamiltonian

is obtained hut with explicit inclusion of surface effects.

As a first application of this theory the case of a fast electron

is treated and the electron-crystal energy exchange spectra are derived.

The classical energy loss spectrum is recovered and involves processes

resulting in one-phonon excitation. Quantum mechanical two-phonon

processes are evaluated. The gain spectrum is obtained for the first

time and shows a strong temperature-dependence. The results are in good

agreement with experimental spectra in LiF crystal films.

-1-

I. IMTRODUCTIOU

The study of the interaction "between electrons and various

elementary excitations of a solid in the neighbourhood of a "boundary

surface is of primary interest for the understanding of a number of

phenomena involving surfaces,such as transport properties of thin

films, photoemission, electron diffraction, etc.

A particularly clear case of strong surface effects on the

electron behaviour has been demonstrated "by Boersch et al. ^ in their

measurements of electron energy loss and gain spectra in LiF crystal

films. Their results show that for thicknesses up to a few thousand

A , energy exchange between the electron and the crystal film is mainly

due to the strong coupling between the electron and the surface optical

phonons, while for a thick crystal slab the more efficient coupling is

with the usual bulk longitudinal optical phonons. A similar coupling

has been exhibited recently in LEED and photoemission, in the plasmon

energy range , The same pattern is indicated in recent tunnelling

experiments in semiconductor-metal boundary layers ,where,again, the

electron is strongly coupled to the surface plasmon .

Several authors have developed a classical theory to

describe the energy loss spectrum of fast electrons in thin films in

both the plasmon and phonon energy ranges. For lower energies where

the quantum nature of the electron and the elementary excitation should

be taken into accountt it would be desirable to set up a Hamiltonian

formalism which would properly include surface effects.

In the present paper a quantum mechanical theory for the

interaction electron-long-wavelength optical phonon of a dielectric slab

will be developed. It will incorporate explicitly the features of the

phonon modes associated with surfaces.

In Sec.II, the classical Hamiltonian of the problem is written

using the polarization eigenmodes of the slab already obtained by Fuchs121and Kliewer ;j then (Sec.II and III) tfie phonon field is quantized in the

standard manner hence obtaining a Hamiltonian similar to Frohlich's

in polaron theory . As far as the coupling to the longitudinal

optical phonons(LO) is concerned, the only difference from Frohlich's

Hamiltonian is the "space-quantization" of the LO modes. More important

is the fact that the electron is also coupled to the so-called

-2-

surface phonons although the polarization associated with these' modes is

divergence free. These and other features .to be discussed may

"be of considerable importance for the polaron theory in finite-size1A)

crystals .

Sec.IV deals with the case of an electron at rest imbedded in

a dielectric. This case corresponds to the calculation of the screening

of a fixed charge "by the phonon field,

Sec.V treats the situation where the electron is sufficiently

fast for its velocity to remain essentially unaltered by the interaction

with the phonon field (yet not fast enough to include relativistic

corrections). This case is exactly soluble and provides a suitable

quantum mechanical model for calculating the energy exchange spectrum

in the phonon energy range.

The spectrum which is obtained in the framework of classical9)electrodynamics 'is recovered here as resulting from one-phonon

prooesses, Multi-phonon processes are evaluated and they are found

to give negligible contributions beyond the two-phonon excitation

threshold. The gain spectrum is also calculated and shows the observed

strong temperature dependence '* , .

The quasi-static approximation for the polarization field will

be used throughout this paper; no account of retardation effects is

taken care of. These could be included in a more general theory

based on the classical treatment of the retarded polarization eigen-

modes

II, FEEE POLARIZATION IN THE SLAB

A, Integral equation

The problem of finding the long wavelength polarization eigen-

modes of a dielectric slab of a given uniform dielectric function £(")

can be solved either as a problem of classical electrodynamics with

matching boundary conditions, etc.; or by incorporating the boundary

conditions of the problem into an integral equation for the polarization.

Both methods have been used '* ̂ ' for discussing the optical properties

of ionio crystals. The integral equation method will be outlined here,

mainly to introduce our notations in a form suitable for the subsequent

analyses.

-3-

As stated in the introduction, only the non-retarded equations

of motion will be treated. The equation of motion for the displacement

fields U(r,t) and^ r , t ) of the (continuous distribution of) positive

and negative charges ±e , respectively,is

-jjjx.t)] --/i«g ( U + - U J+ q

2

where /u is the reduced mass of the ion pair, JU UQ is a short-range

force constant (excluding Coulomb fields) and E(r,t) is the local electric

field x t ) .In the dipole approximation, the local field is given "by

slab

where n is the ionic pair density. The integral gives the dipole field

propagated "by the dipolar tensor T

T(r) - & " 3 £-£-" (2.3)

where the superscript 0 indicates a unit vector. _E^ (a?) is the Lorentz

local field contribution

• • - \

which depends on the polarization P at r. The origin of this term is

related to the pathological behaviour of dipole lattice sums, as has

been discussed at length by de Wette and Schacher \ The polarization

itself is defined by

and, from (2,1) and (2.2), P(r) satisfies the integral equation

•/3 . . \

. -2/3

2n e

f £(£"£')* p^rl^ d«' - ^2'6^

- 4 -

*Assuming tha t £ ( r , t ) = £ ( j ) e * t equation ( 2 . . 6 ) oan "be wr i t t en as

(AEE - A ) P ( r ) = f T(r - r ' ) . P ( r ' ) dr' (2.7)

swhere B is the unit matrix and where

A = 4* 4 ' x n = 4 * ^ t (2'8)•P P

and

T

A = [ . XT

Xm and A? are the bulk TO and LO frequencies, respectively, in units of the

ion plasma frequency

2

The symmetries of the slab will be used now. First the

translational invariance for continuous displacements parallel to the

slab is exploited by introducing two-dimensional Fourier transforms

(2.11)

where A is a unit area of the slab surface and (p,z) are the surface and

normal components of r. , Since P(r) is real, i ts Fourier trans-

form must satisfy the following condition:

P(k,z) - P*(-k,z) . (2.12)

The two-dimensional Fourier transform of T(r-r') is derived from

ik.-e-klz\

f 3 >1 r

7 - J

-5-

By successive differentiations one obtains

£ ' f JI i3c-P i \< -klzl /„ . . \1 = J d k e ~ ~ _ K e I > (2.14)

*g J * rir,, D - / dk e ~ £ - ^ - ^ * e K ' a | (2 15}

where

f +1 if z > 0K - tk , i e (z )k ] , 0(z) = J . (2.16)~ | - 1 if z < 0

Substituting (2.11) and (2.15) into (2.7), one finds

(XE-A) P(k,z) - - ^

-a

Next, the rotational invariance round the a-axie is exploited. Using

the k-reference frame ' the polarization is written:

P(k,z) »'p(k,z) k° + P (k,z) z° + P (k,z) n° (2.18)^ ** . #•* z n̂ r n ^

where0 O, . .O

n = z Y k

is splits eg.(2.17) into

and

a

Pn (k'z) = ° (2#19)

X 7r(k,z) => f dz1 MCz-z1)- ^ ( k ^ 1 ) (2.20)J a* ~

-a

y(k,z) is a two-dimensional polarization vector defined "by

T(k,z) = [P(k,z) , Pz(k,z)J (2.21)

which, as - a consequence of (2.12) and (2.18), must satisfy

T(k,z) - / • J r*(-k,z) < (2

- 6 -

In eq..(2.20) the kernel M i s given "by

M(z-z') * f 6(z-z') + 2rrk e |Z"Z '

X /(2. 23)

B. Polarization eigenmodes

The solutions of (2.19) are trivial Ji any function of z

defined in the interval (-a, +a) satisfies this equation with the

degenerate eigenvalue X = XT« An arbitrary pattern of surface

polarization may be expanded in terms of a complete set of ortho*

normalized eigenfunotions in the interval (-a, +a)# One can choose,

for example,

icos ^ z (2.24)

P . (2) = i s in J i z (2.25)

where j = 0 , ±1, ±2ttt, * These transverse modes of vibration (S

polarization) are completely decoupled from the* P polarization modes

and,moreover, they do not interact with the electron, as will be shown

later .

Differentiating eq,.(2.20) twice, and provided that det (X~A) / 0,

one obtains the differential equation

H 2 ' O

^-r £(k,z) = k -irCk.a) (2.26)dz "*

whose solutions are the eigenvectors of the surface polarization waves.

If, on the other hand,

det(X- A) = 0 (2.27)

then eq.(2.20) is satisfied by the ordinary sine or cosine bulk

polarization waves with the important difference that only a discrete

set of wave vectors k are allowed as a result of the finite thicknessz

of the slab.

-7-

Due to the existence of the z - 0 plane of mirror symmetry,

all the modes can "be further classified as even or odd with respect

to that symmetry. The detailed analysis is given in Ref.12. We

reproduce the results in Appendix A, The eigenvectors of eq«(2,20)

have "been orthonormalized according to

, 6 ,m m ' pp'

f dz T* . z , i = 5 , 6J ~mp "m'p1 mm1

-a

They also satisfy the closure relation

Z T* (z) * (z

In terms of the Fourier components of P(r) defined in (2*ll)

these relations become

hence

J.z). Pfk',*

yielding

J dk iAr 1

i -

+a

-a

2

IzI

j

^ £< A i l J

P

( 2' 3 4 >

2-a

Prooeeding in the standard manner, the polarization operators £(k,z)

are expanded in terms of a complete set of orthonormal eigenvectors

P.(k,z) = [TTmp(k,Z) , Pnj(k,O] .'here the 7rmp and 7^ have been

defined above :

,1/2

I72' (2.36)

and

The coefficients of these expansions are the creation and annihilation

operators of the corresponding eigenmodes and, from (2.34) and (2,29)»

they satisfy

[a.(k) , aj(k')] = \ 6(k-k') 6±. . (2.38)

Substituting (2.36) and (2.37) into (2.35) yields

where H is the completely independent Hamiltonian of the S polarization

waves

-9-

and where H .̂ i s the P polarization Hamiltonian

mp

Here the summation extends over a l l the TT-eigenmodes l i s ted in

Appendix A. All the LO and TO modes are degenerate and only the

surface modes have a spat ia l dispersion.

I I I . ELECTRON-PHOUOU INTERACTION

If r i s the position of the electron (charge - a ) , the

polarization at r sees an additional f ield

r - rBQ(r) = -e

Therefore the dipolar coupling^will result in the following electron

phonon interaction energyt

HT =e fdr Ll is . . P(r)

Using the Fourier expansions (2.1l) and (2.13),this transforms into

** 3.

Prom this expression and due to the particular form (2.16) of the

vector K - it is immediately apparent that the electron couples only

to the P polarization and not to the completely transverse S polarization

P « Finally, use of second quantization yields

-10-

H, = A / d k e - " 6 £ r.(k.ze) (a++ a.)(3.4)

i

where the coupling functions I*, are defined "by

where

. X = [i, -0(z-ze)] . (3.6)

Inserting in (3.5) "the various eigenvectors of Appendix A, one is left

with elementary quadratures to obtain the explicit form of the coupling

functions I.. These functions are listed in Appendix B, It turns

out that the electron does not couple to the TO modes of P

polarization any more than to the S polarization waves. As for the

case of an infinite dielectric medium, this can "be seen to result

essentially from the faot that the polarization field associated with

any TO mode is divergence free. Indeed,

div P(r) = f dk ê 'ftttk, I-) -P(k,z) (3.7)

(3.8)

and the integrand is identically zero for the TO modes. The z -

dependence of the various coupling functions T.(k,z ) is sketched in

Fig.l.

Comparing the maximum strengths of the surface and LO coupling

functions, one finds (see Appendix B)

V 2 A —.< i F t (3.9)

4

1/21/2sinh 2ka \ ' -ka; e

Noting that g~ a g , we see that the eurfaoe modes, In suite of their

divergence-free character, are coupled to the electron just as strongly

as any LO mode. One could argue that due to the larger number of

-LO modes for . each. k̂ in a relatively thick

slab ', the surface effects studied here on the electron properties

will "be negligible. However, this is certainly not the case for

properties which depend on a selective and particular phonon frequency

(such as the electron energy loss or gain spectra considered "below).

In fact, two features, peculiar to the surface vibrations, may "be of great

importance in the behaviour of the electron close to the surface. First,

there is a continuum of surface state energies available for transitions

between W_ and &>, whereas in the bulk only a sharp state round OJ^

is coupled to the electron. Second, the surface coupling function dies

off as k s for large k (eq.£}.3O)) whereas the LO coupling function goes

as k"1 (eq.(3.-9)'j this is well known in Frohlich'sHamiltonian ' ) . -

Therefore,for the case of conduction electrons in polar dielectric slab

thinner than,say,5000 A, it does not seem justifiable to neglect the

surfaoe effects associated -with the existence of these surface phonons.

The application of various techniques of the theory of large polarons14)to the present Hamiltonian is being carried out *

IV. SCREENING OF A FIXED CHARGE

As a first simple application of Hamiltonian (2.41)^(3.4);the

screening of a perfectly localized point charge provided by the phonon

field will be considered. For this case the coupling functions

r. (k,z ) are constant. The phonon field operators are then statically

displaoed (assuming £Q =* 0) •.

1 1 THii.

This yields a Hamiltonian diagonal in

T™1

H '- A f dk £ [^.{a+.*i + \) ~ *t \

-12-

The seoond term of (4»2)

Es = " A / d k Z i > 3 )i 1

gives the infinite classical self-energy of the electron in the polarizationfield it induces. As an illustration, the ground-state average

20)polarization ' at the centre of the sla"b due to a point charge at th

surface ZQ =» +a will he computed. Using (2.1l), (2.36) and (4.1),one

l/2 ^ ,

f Wfinds

=

.1/2 ^ T (k,a)

j / kdk ) ~r^Z— f-(k '°) *i

The z-component of £ will receive contributionsonly from the (0+) surface

mode; that is

/ P ( 0 ) \ = i - r*k d k e'ka (4.6)

\ z t UV 2 T J k d k o + e-2ka

where 2

0 = 2 1- + 1 , (4.7)

This result can also "be derived from classical electrostatics or,

equivalently, "by solving the inhomogeneous integral equation obtained

when setting K O in eq.(2.2O) and by adding a source term TT due to

the point charge

(4.8) 'dz' M(z-z') > jr(k,z' )

-awith

-kjz-z |s e . - t l , i0{z-z )] . (4.9)

Bq.(4.8) is solved "by expressing TC in terms of the complete set of

eigenvectors TT. (z);

-13-

dz (4.10)

-aa

The factor in square brackets essentially gives T7, (see eq,(3«5)).

Therefore (4.10) is equivalent to (4.5)*

V. FAST ELECTRON CASE

A, .Evolution of the, johpnon states

Here i t will "be assumed that the electron is so fast that any

momentum transfer -3fk to the phonon field is much smaller than the electron

momentum p = yimK? where EQ is the (essentially constant) electron energy*

Then the electron can be treated as a classical particle of constant

velocity v and as the source of a time—dependent perturbation of the

slab. In the continuum approximation, an upper bound for one-phonon

momentum transfer may be -ftk ;v 5 % 10 gr,om/sec (corresponding to

a wavelength of *- 100 A ,̂ Therefore, for one-phonon

•processes, the constant velocity approximation may hold for electron-19 /

momenta higher than p rj 20 -fik /v 10 ' gr.cm/sec , i^e . , for energieshigher than p2/2m vt (5*1)

the Hamiltonian (2.4l) and (3.4) takes the form

H = A J dk ̂ [ *%, (% % + i) * rmp(t) (.;p + amp)m P (5.2)

Consider a particular (k,mp) phonon mode with the Hamiltonian

h = ^ w (a+a + £ ) + T(t) (a+ 4- a) ; (5.3)

then the phonon state vector in the interaction representation

-14-

2i\satisfies the evolution equation '

£ > = r(t) (a+ eiWt + a a"1 *) | ^(t) > . .(5.5)

On integration,

|^(t)> = exp [ - I j X dttrXf) (e iu ) t 'a++e"iUt Ia)].|0 I(to) > . ( 5#6)

The V functions of the present problem have a definite parity p, so

that by lett ing t̂ —•» -oo and defining

^ T(t) e" i W t dt (5.7)

- 0 6

one has

- j - I(pa + a)* ' '/V«) > (5.8)

o r

l p a + ! a

Let |n / represent the initial state of excitation of the phonon mode;

then one finds,on expanding the exponential operators,

(+00) rn {-T* {=rA ml nl (n" - n)I

X p m | n ° -n+m£> . (5*10)

To carry out the sum, the summation order is changed such that

m = n + r > 0

m + n > 0 (5.11)

hence

-15-

oO

r =

where

* 2 T

(n+r)! n! (n°-n)JX

i tr -,r+n

P J. (5.13)

is the probability amplitude of finding the phonon mode in i ts 1 n + r)>

excited state when the electron has travelled through the slab.

The total state vector of the slab at t a +00 is given by

1 r.:

where i stands for (k,m,p), Eq.(5.14) can be rewritten;

(5.15)

where

C = TT C o (5.16)1 * / 1 1 1

and

l + r i > ' (5.17)

i

is an eigenstate of the Hamiltonian, with a total energy*

E =in ,v) T

In (5«l5), 21 means a multiple integration (index fc) and summationin

(indices m,p) over all the eigenmodes of the slab and over all their

possible excitations r.

-16-

B. Loss and gain spectra. One-phonon processes

The coefficients jC, 0 . j of eq.(5.15) give the probability of

finding the phonon field in the eigenstate I (n°+rj )> of total energy

^(hofr\ a t ^ =* + oo > knowing that before the electron passage, i . e . ,

at t = -OQ , the phonon field was in jn I >̂ eigenstate and characterized

by an energy E, „, . Therefore, these coefficients would also give the

probability for the electron to exchange energy Erno+1-{ - EfMo? - ±.lfa>

with the slab. Depending on the sign of this energy difference, one

obtains by definition the loss or gain spectrum, respectively.

The ini t ia l state of the slab is determined by the Boltzmann

statist ical factor Z~ (T) exp(-Ef^l/kT) which gives the probability of

finding the phonon field in the Er̂ oi energy state, in a statist ical

ensemble at temperature T (Z is the partition function of the phonon

field). Thus, the observed energy exchange spectrum is proportional

to the product of the quantum mechanical probability /.Ci^o^i j by

the stat ist ical factor

p < ± f l u » •

where the delta function takes care of the energy conservation and where

the upper and lower signs refer ;respectively,to the loss and gain

phenomena.

Except for the temperature factor, eq..(5»19) is Fermi's golden

rule for the transition probability between two states of energy Er o|

and Ej oj ± •fi c*) . ^n the context of this work, this relation provides

the exact result since tlae interaction causing the transition is linear

in the phonon field operators.

The loss spectrum (upper sign in (5*19)) will be examined f i r s t .

I t is clear that the contributions to P (iTw) can be classified as due to

one, two-, or multi-phonon processes. This can be seen by comparing

the value of "fi&> to the minimum energy loss, i . e . , "Bwm« For

"£«_ < 1ito< 2tft>m energy conservation requirement allows excitation

involving ont phonon only. For ĉJ™ < •;5to< 3Et*y, ,t wo -phonon processes

may ocour as well, and so on. Ifhen the temferature-dependent factor is

taken into account, the leading term in the summation over the ini t ia l

states clearly comes from the ground state of the polarization field in

- 1 7 -

which no phonons are excited: E , , = •§• V_, otJ. . At sufficiently low

temperature this leading term is essentially temperature independent as

a result of

lim Z(T) =i e

Hence, in the whole temperature range where one has kT < "TTuL,, the loss

spectrum will not "be very sensitive to changes in the temperature. This

is the case for LiF and other alkali-h&lide crystals for which

•̂nw —0,05 eV ( — 600°K). The next term in the summation (5»19) over

in i t ia l states would have the extra temperature factor e;cj>(-"fcu?/kT) ^ 0>l

• at room temperature and therefore may be neglected in first approximation.

Therefore, in the case of one-phonon processes and when the slight

temperature effects just discussed are neglected, the normalized loss

spectrum reduces to one term;

PfHl«) = P0A J dk I f"P2

k J [ - - " m p W l (5) « 0 (below the threshold: of one-phonon excitation).

/ e +1 \V^i i ) i j ^ (A> < W 1 where WT = ( 0 ) is the limiting surface mode

frequency for large k. In this range the o-function of (5»2l) selects

the «„ mode as the only possible excited mode. Introducing in (5.21)

the expression for I-, (k) given in Appendix B, one finds (dropping the

constant exponential factor T )

2 2 w a

" mC O S —vP(«u>) = p „ / dk — 7-2—r tanh ka

(k2 +-f) u2 mp

-18-

bk

(5.23)

where k_ is the zero of

To evaluate (5-29) the expressions for I given^'in Appendix Bean "be

introduced and summed term by term. However, a simpler way to get the

result in closed form is to use the closure relation (2*29) for the eigen-

vectors* setting i s (mp)}then for any k

I l*«r 2_A dt dt1 e-iuL(t-t')

/ /dz dz' e

where

X(z-z ) .Vis, e £_,

i

z1e

(5.30)

(5.31)

right-hand side of the ahove equation, one mayTo the sum 2_, °^i y' T* T

add the tensor / , T£- (2) TT- (*') : this merely introduces the coupling

functions V' of the transverse modes which are known to "be identically

zero. Using now the closure relation (2,29)

5T. (Z) T (Z1)" 1 rml J

=

which in fact leads to t r iv ia l integrations already met in the evaluationof I . The final result is

QP

2 rJ= 2

C ^ (5.34)

where

2u V2 ' • (5.35)C =

IT W . V

Integrating (5,34) with respect to k yields

Pffio,T ' 0 9 v -2 9,, "y" WT I (c T

Comparing this result with expression (5.2l), one finds that, to lowestt

order in temperature effects, the gain spectrum Vill "be the mirror image

of the loss spectrum multiplied "by the weight factor expt-JIfo/kT). In-2addition to giving an overall reduction of the order of e « 0.1 at

room temperature, this temperature factor will have the effect of smoothing

out the high—energy features of the loss spectrum such as the resonant

peak at cÔ. . However, the gain spectrum is sensitive to small variations

in temperature since it depends directly on the initial thermal

population of the excited states of the phonon field. The spectra

described by eqs.(5.27), (5.28), (5,36) and (5.38) are sketched in Fig.2.

To conclude this section, it must be pointed out that in the present

theory no account is taken of the damping of the phonon modes due to

either anharmonicity of radiation damping (retardation effect, see

Ref.15).

A small anharmonicity amounts to the inclusion of a small

imaginary part in the dielectric constant. In the classical theory,

this has been shown ' to lead to an important term in the loss spectrum

around "fin,1 the effect of this is to reduce considerably the resonant9) 10)

peak at (0_, Concerning radiation damping, recent calculations >y '

have shown that this and other retardation effects may "be neglected.

C, Multi-phonon processes

Finally, the relative importance of multi-phonon processes on

the loss spectrum will be evaluated^ the region studied is restricted

to 2&u>rj1 < lieo < 3EUm, i . e . , between the thresholds of two- and three-

phonon losses. In LiF, this energy range includes the LO phonon

excitation energy "fito, and i t should be interesting to compare this purely

quantum-mechanical two-phonon loss to the classical contributions studied

so far.

According to eq.(5.19) (upper sign) and if the temperature factor

is neglected, the two-phonon loss spectrum is

1 Wkl> *•ft

6[U - (fa) (k) + u , ,(k'))]

- 2 2 -

Only the energy region close to "£«, (see Fig. 3 ) will he investigated

where the only possible two-phonon process is through the excitation of

two low-energy surface modes WQ_(k). This is sufficient ' since>'beyond

u, , the experimental spectrum dies off quickly.

Using the o-function condition one can define a function K(k,tJ'such that

(5.40)

Bq*(5«39) may then he rewritten (dropping the normalization factor PQ)

ffiff dk dk' F(k) F(k') 6(k'-K)(5.41)

vhere

F(k)

2 2 wo-(k) a2 k tanh ka cos ( )

[ k - + • " J w , (5.42)

The range of integration over k in (5*41) i s from k = 0 to a maximumvalue k determined ~by the relation

w = 2uQ_(k) (5.43)

(see Fig.3)*, therefore,

f1 dk F(k) F[K(k,w)]

0a.K

(5.44)

In LiP, for C o # i o , , k defined by (5.43) is very small (ka Z 0.02) and

hence the integrand can "be expanded in powers of ka. Start ing from

the dispersion relat ion (A*5) one has

V -0 (5.45)

- 2 3 -

fience from (5.40), (5.43) and (5.44) one can write

K = 2k - k (5,46)

and

/v . i

(5.48)

This result can "be compared to the maximum loss due to one-surface phonon

excitation as given by eq.(5.23) (see Ref.9)J

where

U I (5.50)

A typical numerical example appropriate to the experiments on LiP '

will "be considered :

*0

10 om/sec, e « e = 4.8 x 10 cge/ ; n = 0,05 x 1O24 cm"3 .

These lead to

~ 2 x 1 0 X 4 s e c* (5-51)

- 2 4 -

For a Blab of thiokness, say,500 A' one has

2k % \ ~ 104 cm"1 . (5.52)

Henoe, the second factor of the integrand in (5.48) is a decreasing

function of k, Eeplacing this function "by its maximum value, one obtains

the upper "bound

where

2 2

leading to

4(5 .

(5

54)

.55)

This estimate shows that the irwo-phonon excitation probability, and

certainly high^brder processes, cannot modify significantly the loss

spectrum due to one—phonon processes.

;VI. COITCLTJSIOFS

The interaction between an electron and the optical polarization

of a homogeneous crystal slab has been studied in the long wavelength

limit (continuum approximation), neglecting retardation. The

quantum-mechanical Hamiltonian of the system is obtained.

The electron interacts with the longitudinal optical phonon

modes in the same way as in the infinite crystal (Prohlich's Hamiltonian ' ) .

It also couples, with comparable strength, to the surface modes of

vibration although the polarization associated with these modes is

divergence free. This lat ter coupling is likely to play an important

role in determining the transport properties of thin films of polar

semiconductors or the surface conductivity of thick samples. It might

also lead to a new kind of surface state (polaron bound state) .

- 2 5 -

The new Hamlltonian haa been studied for the oaae of a fast

electron. Here, the quantity of interest is the probability of ex-

changing a given energy -tfus, in the phonon energy range,when the electron

travels through the slab. The results of the classical theory of the

energy loss spectrum are recovered as involving the loss of one'single

phonon energy. Excitation of the surface phonons proves to be the most

efficient process for thicknesses up to a few thousand A units. Many-

phonon excitations ocour beyond the threshold 3 5 ^ where (O^ is the TO

phonon frequency, but they have a completely negligible effect on the

energy loss spectrum.

The gain spectrum is derived for the first time. As it requires

an initial thermal excitation of the phonon field, it turns out to be

strongly temperature dependent and vanishes at aero temperature. This

is in agreement with the observations in LiP .

The overall exchange spectrum obtained in the present theory

exhibits the main features of the experimental spectrum in LiF. To

obtain a quantitative, agreement, however, requires the inclusion of

enharmonic damping of the phonon field as has been discussed in the

classical theory '' «

ACKNOWLEDGMENTS

This work was performed while the authors participated in the

1970 Winter College and Research Workshop in Solid State Physics at

the International Centre for Theoretical Physics, Trieste. They wish

to thank the International Atomic Energy Agency and UNESCO for

hospitality at the Centre. Two of us (AAL and EK) thank "Administration

de l'Enseignement Sup§rieur" (Belgium) , "Patrimoine de l'Universite

de Liege and FFRS for financial support.

-26-

* • • * • ! « • •

APPENDIX A

In Tahle I are listed the eigenvalues-eigenvectors of the

integral equation (20) of Sec.II. The relation "between A. and the

frequency of the mode i s ( see eq . (2< ,8 )'•)

The normalization constants GQ and C are given "by

c V/2

)0 V sinh2ka ) == (A.2)

2 2 v-1/2k2a2 + -S2-2- ) 1

4 ^ (A. 3)

and they are chosen so that eq.(2.28) is satisfied. The 6losure

relation ( 2.29). can he verified hy expanding the hyperbolic functions

of the surface mode components in Fourier series of z in the interval

[-a, +a] .

If the slah is made of point ions the dispersion relation of the

surface phonon modes is given "by (see Ref.12)

22 2 T "p ,. , -2kaxu l + . ^ ( l ± e ) . (A.4)O i T L ; J

If the electronic polarizahilities of the ions are taken into account,

then (seeRefj.12 and 7)

2 2 v1

whioh reduces to (A.4) when e ^ = 1 and

2 *

The eigenvectors are the same for "both cases.

- 2 7 -

APPENDIX B

In Table I I are given the coupling funct ions F (k,z ) defined

in eti#(3.5) sn

REFERENCES.. AUD FOOTNOTES

1) H. Boersch, J, Geiger and ff. Stickel, Phys. Hev. Letters

379 (1966).

2) A.V. MacRae, K. Muller, J.J, Lander, J. Morrison and J.C. Phillips,

Phys* Rev. Letters 2£, 1048 (1969).

3) D.C. Tsui, Phys. Rev. Letters 22, 293 (1969).

4) K.L, Bgai, E.1T. Eoonomou and M.H, Cohen, Phys. Rev, Letters

,22, 1375 (1969); Phys. Rev. Letters 24_, 61 (1970).

5) R.H. Ritchie, Phys. Rev. 106, 874 (1957).

6) R.H. Ritchie and H^B. Eldridge, Phys. Rev. 126 1935 (l96l).

7) T, Fujiwara and K. Ohtaka, J, Phys. Soc. (Japan) 2£, 1326 (1968).

8) H. Boersch, J, Geiger and ¥. Stickel, Z. Phys. ZL2, 130 (1968).

9) A,A, Lucas and B, Kartheuser, Phys. Rev, in press.

10) J.B, Chase and K.L. Kliewer, guTDmitted for publication.

11) A preliminary report of the present work is-to appear in Solid

State Communications. f

12) R. Fuchs and K.L. Kliewer, Phys. Rev. 140, 2076* (1965),

13) J.M, Ziman, Electrons and Bhonons (Clarendon Press, Oxford

1967) p.208.

14) Work in progress.

15) K.L. Kliewer and R. Fuchs, Phys. Rev. 150, 573 (1966).

16) E.N. Economou, Phys. Rev. 182, 539 (1969).

17) For simplicity, only the field due to the point ions is considered

explicitly here. Inclusion of the electronic field arising from

the atomic polarizabilities is straightforward and can be. done

in the final result for the eigenmodes themselves (see Eef.12)..

18) F.W. de Wette and G.E. Schacher-' Phys. Rev. Al37i 78 (1965).

19) In the continuum approximation and for a %1000 A, m can be as

high as 50.

20) One can also use (4.1) and (4.2) to compute the temperature

dependence of screening.

21) D. ter Haar, Seleoted Problems in Quantum Mechanics (Infosearch

Ltd., London 1964) p.152.

-29-

Table I

Eigenvalues

\

X

\

\

XT

Eigenvectors

»»*

ff = C (i cosh kz, sinh kz)

ir .= C (i sinh kz, cosh kz)

L „ f, i m i r m i r inir 17T = C ( i ka sin r r - z , -— cos -r— z)~m- m\ 2a 2 2a '

L „ A , i"f nnr a mir ^Tr = C (i ka cos -r— z , —— sin —- z /-ra+ m V 2a 2 2a

T „ /Imir mir . , rnir -\•n - C 1 —T— cos —— z , ka sin —— z)- m - m^ 2 2a 2a .

T „ i imff . mn , mir sv = C 1 •-— sin -— z , ka cos —- z J~m+ m > 2 2a 2a '

m

0

0

2 , 4 , 6 , . . .

1 , Ot Ot * * »

2,4,6, . . .

1 $ t J | O | • * *

Table II

Eigen modes

X

V .

V

xLm

(m = 1,3,5...)

xL

m

(ra = 2 ,4 ,6 . . . )

m

(all m > 0)

ze

Ve

V-e

< -a

0

0

0

Coupling Functions F.

-a < z < +ae

-kaecoshka e

-ka1 : • ' : sinhkzesmhka c

mircos—- z

2a e

sin • z2a e

0

z > +ae

-kze+ e

0

0

0

Integrals J.

k

2 J

cos—

Vcosh ka

. . UJa.

k 1 > U 1 ~2 y sinh ka

2 aik + —

v i ; 2 2 2w m ir

v2 4a2

m , mir

t l\2 ' a' *' 2 2 2

w m ir32~ . 2v 43

0

V

. uasin —

V

- 3 0 -

FIGURE CAPTIONS

Fig.l Spatial dependence of the coupling functions

T\ (k,z ), (a) and (b) give the coupling strength

for the two surface modes. (t>) and (c)

correspond to the LO modes. The TO modes are

not coupled to the electron (see Appendix B),

Fig. 2 Electron energy exchange spectrum in LiP as

calculated from eqs.(5.27), (5.28), (5*36)

and (5*38), The gain spectrum corresponds

to room temperature.

Pig. 3 Range of k-integration for two-phonon excitation

processes. When k varies from 0 to k defined

in (5*43)j the function K(k) given by (5.40)

decreases from K 2 2k to k. Here the two-,

phonon excitation probability is evaluated for

u - w . ,

TABLE CAPTIONS

Table I Eigenmodes of the P-polarisation. Because of

relation (2.22), the first eigenvector components

are pure imaginary and the second are real.

Table•II Coupling functions P.(z = vt) of the electron

.to the phonon modes of the slab as functions of

z and their Fourier transforms J,(CJ).

-31-

0-

-a

0 ..

e

1s - a

\

+ a

*

Z \e

i

0+

. •

/

I

Cb.V

. • . ! •

- u .

Fig. 2

0

k k K(k) K(0)

Fie. 3

.-3ir