-
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