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-
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