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Surface aspects of excitons in semiconductors
Progress in Surface Science, Vol. 36, pp. 179-287 Printed in the U.S,A. All rights reserved.
0079-6816/91 $0.00 + .50 Copyright @ 1991 Pergamon Press plc
SURFACE ASPECTS OF EXCITONS IN SEMICONDUCTORS A.E. CHEREDNICHENKO and V.A. KISELEV Leningrad State University, Solid State Department, Institute of Physics, 198904, Staryi Petergoff, Leningrad, U.S.S.R., and, A.F. Ioffe Physico- Technical Institute, Academy of Sciences of the U.S.S.R., 19402 I, Leningrad, U.S.S.R. Abstract This review considers the factors which influence the exciton behavior at the semiconductor surface and, consequently, are revealed through excitonic optical spectra. The basic knowledge of excitons in semiconductors is first presented. The experimental exciton reflectance spectra of OdS and CdSe crystals are discussed followed by an analysis of the results which make use of the near-surface "dead-layer" model. In particular, the effect of the electron irradiation on the exciton reflectance lineshape and exciton photoluminescence is considered with reference to the physical factors which control optical spectra. Various mechanisms of localization of excitons in a semiconductor surface region are presented. Relevant experiments on low-temperature photoluminescence and reflectance are discussed. Finally, the effect on the exciton reflectance of the electric field of the space-charge layer considered in detail.
Contents
Page
1.
Introduction
181
2.
Wannier-Mott excitons. Energy considerations
182
3.
Excitons: Mechanical, Coulomb, real (polaritons)
186
4.
Exciton reflectance spectra
191
A.
191
Thomas-Hopfield near-surface "dead" layer and the exciton reflecfivity
179
180
A.E. Cherednichenko and V.A. Kiselev
5.
B.
The effect of keV-electron irradiation on exciton reflectance spectra
198
C.
Interpretational problems
205
D.
Interference effects in thin crystals. Size quantization of excitons
211
E.
Exciton transition layers and a formal model
215
Crystal defects and localization of excitons
218
A.
Bound excitons
218
B.
The effect of keV-electron irradiation on the exciton photoluminescence spectra
220
C.
Surface and near-surface excitons
225
D.
Localization of excitons in rectangular potential wells
229
E.
Localization of excitons in defect-rich surface layers
237
E
Localization of excitons by fluctuations of a near-surface
6. A.
random potential
241
Band bending and its relevance to excitons and exeiton spectra
244
Band bending in semiconductors
245
B.
The effect of a uniform electric field on excitons
248
C.
The effect of a non-uniformelectric field on excitons
254
D.
Exciton relectivity in the ease of thin SCLs
257
E.
Quasilocalization of excitons in the SCL field and the effect of damping
267
E
The effect of SCL on the phase of the reflected light
273
G.
Exciton reflectance in the case of wide SCLs
276
References
283
Abbreviations
ABCs AWs DL EB ERL ETL SCL
Additional boundary conditions Additional waves Dead layer Electron bombardment Exciton reflectance lineshape ~xciton transition layer Space charge layer
Excitons at Semiconductor Surfaces
181
1.Introduction In surface physics, the most sensitive probe is the electron, be it in Auger spectroscopy, in low energy electron diffraction, or in photoelectron spectroscopy. However, the photon also provides information on surface phenomena, though its penetration into the bulk of the material usually exceeds that of the electrons. Optical techniques are, for instance, ellipsometry, spectroscopy of the vibrational structure of physisorbed chemisorbed molecules, spectroscopy of surface polaritons.
IR and
In this review we intend to discuss one more optical technique which can be used to investigate surface (near-surface) layers of semiconductors. The purpose of the paper is to present an exoiton as a probe in the studies of surface (near-surface) properties of real semiconductor crystals, either as grown or subjected to various surface-sensitive treatments (e.g., electron irradiation). We will conoentr~ate mainly on A2B 6 - compounds (CdS, OdSe) and summarize the present state of understanding of the problem based on
our
experimental
and
theoretical
studies
and
also
on
the
results obtained by other researchers specializing in the field of the exoiton behavior at a semiconductor surface. We shall show that some of the data on the exciton reflectance spectra
may
serve
a
useful
guide
to
understanding
surface
properties of real semiconductors. In particular, the overall shape of the exciton (polariton) reflectance signal appears to be a sensitive function of the presence below a semiconductor surface of the exciton-free layer (the so-called "dead" layer). The width of this layer can be varied from several tens to thousands of ~gstroms by a number of techniques (e.g., by keV-eleotron bombardment). A theoretical lineshape analysis of the reflectance will be given which evidence the role of the exciton-free layer and space charge layer (SOL) in the formation of the exciton
182
A.E. Cherednichenko and V.A. Kiselev
reflectance lineshape (ERL). Studies
of
the
so-called
bound
exciton
luminescence
and
its
modification under electron bombardment will be presented in Sect. 5 and various physical mechanisms of the exciton localization at a semiconductor surface will be discussed. Finally, in Sect. 6 the effect on the exciton reflectance of the SOL will be considered in detail. We also note that surface aspects of exciton photoconductivity phenomenon are not considered in this review. A detailed account of
a
surface
sensitivity
of
photoconductivity
spectra
is
in
preparation. 2. Wannier-Mott Excitons. Energy Considerations An
exciton
is
the energy
quantum of
an electronic
propagating in insulating or semiconductor crystals.
excitation In case the
energy band model is well applicable, it can also be regarded as a composite particle hole
in
the
of an electron
valence
band
bound
in the conduction band and a to
each
other
by
the
Coulomb
attraction between them. If the electron and hole are so far apart that
the atomic
structure
of
the
crystal
can be
ignored
their
mutual potential can be written as v = -e2/~
,
(2.1)
where p is the electron-hole separation and ~ is the macroscopic (long
wavelength)
dielectric
constant
generally
taken
as
the
square of the refractive index. Such a composite particle, which we
call
Wannier-Mott
exciton
(see
[1,2])
may
analogy with the hydrogen atom. The potential Coulomb potential
reduced by
the dielectric
be
visualized
(2.1)
by
is just the
constant,
and there
will be bound states of the exciton system having total energies lower than the bottom of the conduction band. By analogy with the hydrogen (n=1)
atom
state
the binding (the
energy
energy level
of
the
referred
exciton to
the
in
the ground
bottom
conduction band) is given by the m o d i f i e d R y d b e r g formula:
of
the
Excltorl lit 6ernk~nductor Surfaces
183
4
~e ~1 = - ~
= (~/=o)'s-2"~1~
,
(2.2)
where ~ is the reduced mass of the exoiton ~-1 = m;1 + ~ 1 , formed from
the
effective
masses
m e , mh
of
the
electron
and
hole,
respectively, m 0 is the free electron mass and IEIH i = 13.6 eV is the ionization potential of the free hydrogen atom. Pot a simple case of isotropio non-degenerate baud extrema at the zone center the kinetic energy of the carrier is E k = ~2k2/2m = where m is the effective mass (me for the electron and m h for the hole) and Phk is the momentum at the extremum. The resulting spectrum for the exciton energy consists of the sum of the internal energy and the exciton center-of-mass kinetic energy ~e 4
Q¢K)
=
2~2~2n2 +
~2K2
2.
'
¢2.3)
where I = m e + m h, h~ is the momentum of the center of mass and n = 1,2,3 .... o0. £n is measured from the band edge EG. Thus, the
Fig. 1. Energy of the exoiton bands for Wazluier Mot t excitons vs waveveotor K. The d~mhed line represents the energ~ vs waveveotor dispersion curve 0~ ~ o o u p l e d photon.
184
A.E. Cherednichenko and V.A. Kiselev
exciton spectrum consists of a series of parabolic bands below E G which merge into a continuum at high energies, The
binding
energy
of
the n=1
exciton
electron-hole at the band edge given by
see Fig. I.
relative
to
the
free
(2.2), often called the
"exciton Rydberg", can also be expressed in terms of the "exciton Bohr radius" given by
Rex c = ~ 2 s / ( ~ e 2 )
= (~/mo)-l.
8-R,I:I
,
(2.4)
where RH = 5.29"10 -9 om is the radius of the first Bohr orbit for the free hydrogen atom. (2.2)
and
(2.4)
Note that the factors ~/m 0 and 8 entering
reduce
the
binding
energy
and
correspondingly
increase the radius by a factor of 102 or even 103 as compared with the hydrogen atom. As a result, typical values of the exciton Rydberg
in A2B6-compounds
are
of
the
order
of
10 meV
and
the
exciton radius is about a few tens of ~ugstroms. By virtue of its large
size
the
Wannier-Mott
exciton
is
extremely
polarizable.
Consequently the applied electric field has a pronounced effect on exoitons. In what follows we shall show that the exciton spectra are
very
sensitive
to
the
surface
electric
field
which
is
an
scheme
is
an
essential point to be emphasized here. As
shown
infinity
of
transitions
in
Fig.
I
the
overlapping create
exciton continua;
excitons
of
energy
level
nevertheless,
well-defined
its
energy
optical and
can
therefore be sharp. This a consequence of momentum conservation. In
the
case
of
direct
transitions,
i.e.,
vertical
on
the E(K)
diagram, a photon of energy ~L0 can only create an exoiton with K = nt0/c, where n is the refractive cm -I so that ~2K2/2M ~ IO-SeV.
index.
For ~
~ eV, o/c
~ 105
Thus the only exoitons which can be directly created by a photon have
a
well-defined
negligible.
energy,
Transitions
electron-hole pairs,
and
involving
their
kinetic
the
creation
on the other hand,
give
energy of
is free
continuous spectra
for E > E G, since momentum conservation only puts one condition on k e and kh, leaving the relative motion of the electron and hole free to be specified by energy conservation. For the exciton, the
Exdtons at Semiconductor Surfaces
185
requirement that the electron and hole sts~V together eliminates this degree of freedom. The existence of sharp optical transitions in a crystal is evidence for the existence of exoitons. It is difficult to produce excitons in sufficient concentrations to observe directly the transitions amor~ the exoiton levels, but it is possible to observe the transitions between the valence band edge and an exoiton level. A series of sharp lines occurring on the long wavelength side of the absoz-ption edge and merging into the continuum has been first observed in cuprous oxide Ou20 (see, e.g., [3]). The spacing of the exoiton levels has been found to be in surprisingly good agreement with the Rydberg formula for n > 2. Later, large-radius Wanniez--Mott exoitons were revealed in a great number of semiconductors (for the A2B6-compounds see, e.g. [4]). It is evident that it the energy surfaces are not isotropio and non-degenerate the exoiton problem is considerably more complicated. A sketch of the energy band structure of A2B6-oompounds (wurtzite) is shown in Fig. 2. In wurtzite-type crystals the conduction band is simple, whereas the valence band consists of the three close-lying bands F 9, r 7, and r 7 which arise due to crystalline field and spin-orbit splitting effects. The valence bands are termed A, B, O, starting from the uppermost valence band, respectively. An appropriate optical excitation yields A-, B-, and O- exoiton series.
Fig. 2. A sketch of the energy bands in wurtzite-type crystals.
B
C
r, x(ooo)
186
A.E. Cherednichenl(o and V.A. Kiselev
Note, that the A-exciton is only allowed for light polarized perpendicular to the hexagonal axis ElO (~ is the electric field vector of the light and o is the unit vector along the hexagonal axis (c-axis)). A free exoiton at liquid helium temperatures has a velocity N of about 106 cm/s, so for an impurity concentration N ~ IO150m -3 its lifetime before capture iS expected to be (Nov)-I ~ I0-9s. Thus the exciton is a short lived excited state of the crystal observed at low temperatures near the band edge (below ~5000 ~ for OdS and ~7000 ~ for CdSe) so that virtually the only way of studing it is by optical spectra, v~Z., transmission, refleetlvity and luminescence. (We note that the exeiton photoconductivity phenomenon is b~¥ond the scope of the present review), To conclude, we note two more points important from the experimental point Of view. (a) Optical experiments are essentially performed at loW temperatures (moStly at 4.2K) insofaras the exciton binding energies are very small which results in an easy thermal dissociation of exoitons. (b) The exciton absorption coefficient in the cr~stal with allowed direct transitions may be of the Order of ~IO 5 om -I . This has two consequenoies. (i) Ex0itons thus excited are essentially generated in the near-stu~f~ce region of a cr~stal. (ii) Small penet~mtion depth of the exoiton does not allow transmission studies, hence, exciton reflectance ~easurements need to be carried out instead.
3, Exoitons! Mechanical, Ooulomb, Real (Polaritons) Figure I corresponds from an experimental point of V i m to a low-resolution spectroscopy, whe~ the intersection of the photon dispersion curve and the eXciton band is given by a point. In the case of a high resolution this approach would be inappropriate, especially for the ground (n = I) states of dipole-active exoitons [5] which are characterized by a large oscillator strength. More accurately the exciton states in the region of their intersection with the photon dlspe2sion should be considered as in Fig. 3.
Excitons at SemiconductorSurfaces
~
b
I I ! !
/
187
o
I wL~--" 7
/ k
~
k
Fig. 3. Exciton dispersion curves. (a)- mechanical approximation, (b)- Coulomb approximation which allows for the longitudinal field, (c)- with allowance for longitudinal and transverse fields.
Figure 3a corresponds to the approximation mentioned above, when the longitudinal electric field, produced due to the exciton wave propagation is not taken into account and the interaction with the transverse electromagnetic field is negleQted. Within that approximation the exoitons are referred to as mechanical [6], therefore, the approximation is termed mechanical. The inclusion of the longitudinal electric field arising already in a more simple system of the Lorentz oscillators [7], when longitudinal vibrations (exoitons) are excited, leads to increase in the energy of these excitations by the value h(~ L - ~0 ) ~ ~ L T (Fig. 3b). The longitudinal exciton, that is the exciton with P~K, P being the vector of polarization, is refem~ed to as the Coloumb exoiton [6], so that the approximation is a Coulomb one. The exoitons with the polarization PIK that do not interact with the longitudinal field
188
are
A.E. Cherednichenko and V.A. Kiselev
still
mechanical
in
this
case.
Lastly,
an
account
of
the
interaction with the transverse electromagnetic field brings about modification
of
yielding real The
above
the
transverse
excitons
[6]
(photoexcitons
considerations
vibrations by Born
exciton
were
applied
dispersion
(Fig.
3c),
[8], polaritons to
crystal
[10], and to excitons by Pekar
[9]).
lattice
[8], Hopfield
[9] and Agranovich [ 1 1 ] . Figure
3 can
also
dielectric
theory.
associated
with
be
The
the
treated
from
the point
of view
of
the
excitation of a dipole-active exciton polarization
of
the
medium
P(R,~)
is
which
satisfies the following equation [6,8,12]: B(~° where B = ~O/M, is
the resonant
+ q2(~))
= _~ 2 ( R , ~ )
~ = (2%)-1~O~LTSO , q2= _ B - I ( ~ frequency
the mechanical
~(~,~)
of
the mechanical
exoiton damping;
,
(3.1)
_ ~2 _ ¢ ~ p )
exoiton
at
a n d ~0
K = O; F i s
s 0 is the background dielectric
constant which contains contributions from all interactions except the exciton in question. A right-hand part of the electric field may be mechanical
field vector, treated as
exciton,
connected with
the driving
since
for
the
(3.1) incorporates
the exciton wave.
force with respect
latter
we
have
E
=
derivation of (3.1) was suggested by Pekar [8]. Equation a material
equation
for
the medium which
Maxwell's set of equations for fields.
is
additional
The
to
the
O.
The
(3.1) is to
the
For a homogeneous medium
one can make use of the Fourier transform and write down, in the case of an isotropic medium, the relation of the following type: 1
~(k,~)
= ~--~[s(E,~)
where the dielectric function ~(k,~) ~(k,~)
(3.2)
- s012(1~,~) is of the form:
= ~£O((fl) - 4'Fu30/((o2 - (,o2
- B k 2 + ~,~r)
,
(3.3)
For the mechanical excitons ~. = O, P # O, and, as it follows from (3.2),
~(~,~)
= ~,
or
Excitons at Semiconductor Surfaces
~(k)
~ (~0 + t ak2/(2M) -
(F/2
189
,
(3.4)
which corresponds to Fig. 3a. For the Coulomb exciton there is only a longitudinal component of the electric field E~k. Since the displacement field B = 80 ~ + 4~7 is transverse we have case D = O. From (3.2) follows 8(k,~) = O, or ~(k)
~ ~0 + %_,T + '%k2/(aM) -
61/2
,
in this
(3.5)
which corresponds to the longitudinal exciton in Fig. 3b. Lastly, for the real excitons (polaritons) there exists only a transverse component of the field EIK, for which the Maxwell equations are related to D by = (02k2/~2)E Solving
(3.6)
together with
(3.6)
(3.2) we arrive at the dispersion
relation: 8(k,~) = c2k2/602
.
(3.7)
It is convenient to solve the equation for k or n (n = ck/~ is the index of refraction). Resulting is the Pekar's which defines the polariton dispersion:
nl,2(~)
I = {60 + 2-
[Q ± (a2 + b ) 1 / 2 ] } 1 / 2
Mc 2 where G = ~°O~ 2 (~2 _ ~02 + ¢~F) - 60;
equation
,
[8]
(3.8)
Mc280~LT b = 8
t~02
These relations define the two polariton branches in Fig. 3c. The polariton approach will be utilized below. In some cases, however, the mechanical exciton approach appears to be more convenient for discussion. Such is the case of the exciton localization in potential wells, which we intend to treat at some length below. As to the calculation of the crystal optical properties, it will be
190
A.E. Cherednichenko and V.A. Kiselev
performed making allowance for the photon-exciton coupling,
i.e.,
within the framework of the polariton theory. In case the exciton mass M is not infinite the mechanical exciton band has dispersion and ~ depends not only on ~ but on k as well. The dependence of on k is referred to as spatial dispersion. Actually this implies a possibility for the excitation to propagate through the crystal in a purely mechanical way when no macroscopic [6,12].
This
also
implies
that
reference point of a medium
the
fields are involved
electric
is not
polarization
in
completely specified by
a
the
field at the same point. The situation is represented by the first member in (3.1) which contains the second derivatives with respect to coordinates.
Therefore a solution to the problem requires an
introduction of the boundary conditions. They are additional with respect
to the Maxwell's
theory and,
hence,
are referred to as
additional boundary conditions (ABCs). Additional boundary conditions have been intensively studied and were a subject of much discussion.
The efforts of the theorists
were essentially concentrated on three main problems: ABOs
appropriate
from a phemenological
point
(i) Type of
of view
(we note,
that even the necessity for introduction of the ABCs was sometimes under
question),
(ii)
The
comparison with experiment, making
use
of
various
attempts
to
specify
the
AB0s
from
(iii) The attempts to derive the ABOs
microscopic
models
of
the
surface.
The
results of the theoretical efforts in these fields were discussed in several reviews and monographs (see, e.g., [6,8,13,14]). The ABO can be given in the form of a linear relation [6]
c
(P + 7 ~ 0
c
dz
)lz:o
: (~
+ ~ ~o ~z
)lz=o
,
(3.9)
where 7, ~, and ~ are functions of frequency (usually ~ = ~ = O is supposed).
Specification of the ABC
is dictated by the physical
properties
of a medium
and,
and
surface
hence,
the ABC
is not
universal. To solve the problem one should specify the properties of the medium and its surface.
Excitons at Semiconductor Surfaces
191
Most common is the ABO introduced by Pekar [8]
#(z
= o) = o
(3.1o)
.
I n a number of papers the calculations of the exoiton reflectance lineshapes were compared for several different ABCs with the experimental spectra. Patella et al. [15] have concluded that the theory based only on the ABC approximation gives a poor fit to the experiment.
4. Exoiton Reflectance Spectra A. Thomas-Hopfield near-surface refleotivity
"dead" layer and
the
exoiton
A theory of exoitons in confined media based on the boundary conditions, the ABOs included, suggests that the exciton state (the exciton binding energy, oscillator strength, damping, etc.) is unperturbed up to the very boundary or perturbed in a very thin layer. The effect exerted on exoitons by the latter is, to a fair approximation, usually neglected. Such an approach, though, may prove inappropriate insofaras, in reality, there exists a near-surface layer where excitons are drastically perturbed. The part of the layer where exoitons are totally damped is referred to as the exciton-free layer or "dead" layer (DL). The concept of the DL was first introduced b y H o p f i e l d a n d Thomas [12]. 0onsider a simple hydrogenio exciton in its ground state. The interaction between the exoiton and its image results in a potential energy of the exoiton ( in a z-plane)
v(z)
1
=~.--
8-1
(--g~)
~
('
Rexo3
.--~--.
,
(4.1)
where E B is the exciton binding energy. Since 8 is greater than !, the force is repulsive. If all the effects of the surface could be
192
A.E. Cherednichenko and V.A. Kiselev
represented by a potential U(z) for the exciton, U(z) would have to be sufficiently repulsive to cause a free exciton to be totally reflected from an effective barrier a finite distance inside the surface. Thomas and Hopfield were first to replace the potential U(z) by an infinite potential barrier, as indicated in Fig. 4. The layer
1
is
referred
to
as
a
Thomas-Hopfield
dead
layer
for
excitons. We note that
the binding energy of the ground exciton vs the
distance z of its center of mass from the surface has been first calculated in
[16] by taking into account the influence of the
image
on
approaching
the
surface. The main conclusion arrived at by the authors was
forces
both
the
electron
and
hole
that
the exoiton binding energy was found to equal the bulk effective exoiton Rydberg only far away from the surface. For z ~ 1.5 Bohr radius, exciton ionization occurs. Oorrespondingly one has in this model
a
layer
decreasing
where
from
its
the bulk
exciton value
approximation depicted in Fig.
to
polarizability zero.
The
is
gradually
infinite
barrier
4 by a dashed line suggests
that
the Pekar's boundary condition P = 0 apply at and to the left of such a barrier.
ZL,
Fig. 4. A formal adiabatic potential (solid line) and the infinite barrier approximation (dashed line).
Excitons at Semiconductor Surfaces
Actually,
193
three spatial regions should be considered,
as applied
to excitons (see Fig. 5): z < 0 (I), 0 < z < 1 (II), and z > 1 (IiI). The first two are characterized by classical indices of refraction n = I and n O = ~ 0 ' respectively. The usual Maxwell boundary conditions plus the P=O boundary condition determine the connection between the second and third regions.
n-1 10
k~
R2 0
1,
z
Fig. 5. A schematic diagram of the normal-incidence exciton reflectacne problem with a dead layer of thickness i. I O, R I , R2 are incident and, respectively, reflected beams.
The exciton reflectance lineshape
(ERL) is primarily determined
by the phase delay between the two interfering beams R I and R 2. The first analysis of the experimental reflectivity measurements using a spatial dispersion approach was carried out by Hopfield and Thomas [121 who pointed out several anomalies in the reflectance spectra to be discussed below. The DL of Thomas and Hopfield, thus depicted is clearly of intrinsic origin and bears no connection with the particular conditions of the surface, taken as an ideal geometry boundary. In general, there will be other contributions to the DL to be discussed further. Whatever the origin of the DL may be, it allows, within a hard wall approximation, treatment of its effect on the reflectance in terms of interference between the light
194
A.E. Cherednichenko and V.A. Kiselev
beams reflected at the vacuum - semiconductor
(z = O) and at the
DL - bulk (z = l) boundaries, respectively. The hard wall DL model has been generalized to arbitrary values of 1 not related to Rex c by Evangelisti et al
[17,18] and it can
be depicted as in Fig. 6.
0.61-
m
-
o ,
,
,
,
I
i
I
I
l
I I
l
I I
g I
I
I i
Fig. 6. Calculated exciton lineshape of the normal-incidence reflectance for the A (n=l) exoiton in CdS. The DL depth 1 was used as a parameter. 1,nm: a- O; b- 4.7; c- 7.1; d- 9.5; e- 11.8; f- 14.2; g- 19.O; h- 26.1; i- 37.9; j- 52.1; k- 71.1; l- 80.6. The positionsof the transverse and longitudinal exciton frequencies are shown by arrows. [18].
Excitons at Semiconductor Surfaces
195
The DL is supposed to be uniform and is characterized by the parameter 1 and a real non-resonant index of refraction, equal, for instance,
to the background index of refraction of a crystal
n O . The crystal bulk is characterized by the dielectric function (3.3). The incident light I 0 generates
in the crystal
with the wave numbers k I and k 2 defined by (3.8). incidence reflectance is given in the form [18] P12 + P23
e~e
R
The
normal
,2 I
1 + /"12r'23e ~'e
two waves
I
,
(4.2)
where ~12 is a constant, ~23 is a function of a resonant character which defines the ERL, and the round-trip phase delay (or the phase angle) e is e = 4'11~_01/~,
,
where k is the wavelength of light in vacuo.
(4.3)
The value of R is
seen to be a periodical function of 1 with a period k/(2n O) which is of the order of 102nm for semiconductors. A reversal of the ERL for the An= I exoiton in OdS is shown in Fig. 6 [18]. With increasing 1 the reflectance minima give place to maxima (and u~ce uersa) and a reversal of the lineshape occurs. With further increase of 1 the maximum is shifted towards long wavelengths. A prominent feature of the ERL is the refleotivity peak at the frequency ~L - the so-called "spike". It is nominated, for the sake of brevity, the "right" spike as referred to the spectral position of the reflectivity minimum. The emergence of the right-spike structure in the case of a homogeneous DL model is accompanied by the decrease of the main reflectivity maximum. The spike problem has been treated in detail by Benemanskaya et al. [ 1 9 ] who attacked the problem of the exoiton reflectance spectra modification under keV-electron bombardment
(see Sect. 4B)
The authors made use of the approach, developed in [12,18,20] and calculated the ERLs in OdS using the two-layer model depicted in
196
Fig.
A.E. Cherednichenko and V.A. Kiselev
5 with
spatial
dispersion
taken
into account.
In addition,
the exciton damping parameter F was introduced and its effect on the ERL was investigated in a wide range of variations o~ 1. The calculation
was
carried
out
for
the
An=l-exciton
in p-geometry
(~,Io, klc) or normal incidence. The following formula was used to calculate
the
reflectivity
coefficient
(with
allowance
for
the
Pekar's ABO [8]) •
2
1 -n o
¢2knol no+n
/ e
R =
(4.4)
,
1 +
e ~2kno 1 I +no
no+n*
where n o = W ~oI' n = (nln2 + ~ol)/(nl + n2)' ~o1 is the background dielectric constant for light with Eic, k is the wave vector of the exciton,
i is the depth of the dead layer,
and n 1
and n 2 are the refractive indices of the transverse waves. For the computer
calculations
the
values of the parameters:
authors
of
[19]
used
the
following
the longitudinal - transverse splitting
~LT = 2 meV, Eoi = 8.3, M = 0.9 m O, and 0~0 = 2.5524 eV. The DL depth 1 was varied from 0 to 1000~ and the damping F from 0 to 1 meV.
Fig.
7
shows
the
calculated
reflectance
spectra
for
a
negligible increase of l, with ~F = 10 -5 eV corresponding to that value
of
experiment at ~L
the
damping
which
(see below).
yields
Note,
the
(the spike) appears at 1 ~ 60~,
2Rex c. At
layer values
best
agreement
with
that a subsidiary reflectance
1 > 150~ and
peak
i.e. at the value close to small
rtF ( < 10 -4 eV)
results were found to be close to the data of
the
[18] shown in Fig.
6. The inversion of the ERL occurs at 1 ~ 200~, and the return to the initial lineshape at 1 ~ 850~. The results make it possible to estimate
the
values
of
1
and
I~.
A
comparison
between
these
calculations and experiment will be done in Sect. 4B. Alterations of the DL depth observed to occur under application of the external
field,
surface-sensitive
keV-electron techniques
will
irradiation be
shown
to
and result
other from
the
Excitons at Semiconductor Sudaces
197
changes in the near-surface electric field value and also from the emergence of the defect-rich near-surface layer, which is evidenced by the studies of low-temperature exciton photoluminescence spectra and, in particular, of their modification under electron irradiation (Sect. 5B).
I
0.6
0.¢
•
~'\
0.~
Fig. 7. Calculated ERLs for the A (n=1) exciton in CdS as a function of the DL depth 1
Note,
on
the other hand,
that
the outlined model
suggests
a
completely transparent homogeneous DL, though it is evident that field-induced changes in the value of the exciton damping P should be taken into account. Large values of P yield an exciton-free layer which, in fact, is responsible for the observed alterations of the lineshapes under electric field. It is also evident that the layer can not be derived straightforward from the DL in the spirit of Hopfield and Thomas [12] by merely increasing i since the latter is of intrinsic origin and is due to increase in ~0 but not in r as in the former case. On the other hand, it is clear that the exoiton-free layer caused by the abrupt increase in the damping value should likewise result in the rotation of the ERL
198
A.E. Cherednichenko and V.A. Kiselev
when 1
is varied.
Sect.
6 where
field
is
This
allowance
made.
The
issue for
latter
shall the
be
considered at
inhomogeneous
brings
about
the
length
character of diffusion
of
in the the
boundary of the exciton-free layer which may be so strong that the ERL rotation will not proceed at all. In what follows it will be shown that the electric field may lead to localization of excitons near a semiconductor surface.
B. The
effect
of
keV-electron
irradiation
on
exciton
reflectance spectra The
exciton
reflectance
spectra
of
semiconductors
with
the
direct allowed transitions appear to be drastically influenced by the surface. The ERLs are markedly affected by the surface states, near-surface defects, and, especially, electric fields. It is well kwown that strong variations in the lineshapes can be found among different materials and, in a given material, among various levels of the hydrogen-like excitonic series. Quite often, a given line may
look
rather
different
semiconductor material.
from
This
sample
is
true,
to
sample
of
in particular,
the
same
of A2B 6 -
compounds in which the ERLs may vary depending on the conditions for crystal growth and generally uncontrollable surface conditions in a given specimen. The for
electron bombardment chapping
the
(EB) has proved a suitable
reflectance
spectra.
In
discussed in this section keV-electrons were
the
technique
experiments
taken advantage of
(the penetration depth into the material being of a few hundreds of ~ngstroms).
The
(CdS [19,21], CdSe oil-free
vacuum
experiments [22], ZnSe
using
the
were
performed
on A2B 6
crystals
[23]). The EB was performed in an cryostat
permitting
optical
and
electrical measurements at 4.2K. Irradiation-induced changes in the exciton spectra vs electron fluence were registered ~ s~tU without warm up. Near-to-normal-incidence reflectance spectra were measured from a plane containing the C-axis for light polarized both parallel and perpendicular to it. By the effect on the ERL
Excitons at Semiconductor Sudace$
199
the electron irradiation can be classified into a low-dose (1013-1015) e/cm 2 and a high-dose (9 1016) e/om 2 one. The low-dose EB produced reversible changes in the ERL which are discussed below. Fig. 8 shows the changes of the exciton reflectance spectra of a 0dS crystal for the An= I- exciton that occurred under low-fluence electron irradiation with 3 keV electrons. The as-grown crystal is seen to reveal the right-spike structure (curve I), whereas after EB the left-spike structure appears instead (curve 2). (Note that the left spike is the reflectance peak at ~L which is situated to the left from the reflectance minimum (also see Fig. 7). Further exposure to electrons results in a reversal ("rotation") of the ERL.
R
a~ a~ ?i 8. 8. Experimental exciton reflectance spectra of CdS at T=4.2K vs 3-keV electron irradiation dose (a low-fluence case).
0,I
200
A.E. Cherednichenko and V.A. Kiselev
R
A
O_
Ill,
0.2! 0
8"=1
,
I
,
,
,
2 I
~1
I
I
I
I
I
~o wl.
2.550
2.570
2.650
2.570
E,eV
Fig. 9. Experimental exciton reflectance spectra of CdS at T=4.2K vs electron irraditaion dose [19].
Figure 9 shows the modification of the exciton reflectance
for
the An= 1-, Bn= 1-, and An= 2- excitons in CdS [19]. (Note that according to the sketch of the energy bands presented in Fig. 2 along with the An= I- excitons, a more short-wavelength structure associated with the B-band and excited states of the excitons may arise). The initial ERL of the An=1-exoiton also features the
ExcilOnsat Semiconductor Surfaces
~,n'rn 680.0 6"/8.0 676.0 67#.0
A..I
670.0 668.0
201
666.0
8n=2
O#
An=2 O~ 0.2!
0.31-
Ill
A.
0.~
0.~ b, 0.2
4 1+?00 1#'/~ f#?80 ¢ t#820
1#900 ~ I#9#0" 1#g80 0251
Pig. 10. Experimental ERL of 0dSe vs dose of EB [22].
202
AE. Cherednichenko and V.A. Kiselev
right-spike structure whereas following low-fluenoe electron irradiation the left spike appears. This is accompanied by the enhancement oZ the main reflectance maximum. Similar effects are readily revealed in the spectra of 0dSe crystals
(Fig. 10, curve
2). It is pertinent to note that these findings are peculiar of the A-exciton. The low-fluence EB has only a slight effect on the B-exciton (compare curves I and 2). The reason for this difference is discussed in Sect. 6). The analysis of the exciton reflectance in CdS was performed in [19] using the DL model (see Sect. 4A). As seen from Fig. 9 the initial spectrum features the coefficient at the main maximum
right spike, the reflectivity is 0.42. A comparison with the
calculated spectra yields for the Afl=S-exciton a DL value liA SO~. After a low-dose EB (curve 2) the right spike is converted to the left one and the main reflectivity maximum is markedly increased spectra
(by approximately
O.1).
According
to
the
theoretical
(Fig. 7), the ERL in this case corresponds to a DL liA
60~, which is close to the intrinsic DL thickness (~2Rex c) in this compound. This was the smallest DL thickness that the authors could obtain experimentally in CdS (for CdSe, see Sect. 4C). With further increase in the EB dose, the ERL tends to return to the initial state (curve 3), i.e., the depth of the DL begins to increase. Curves 4 and 5 reveal a gradual "rotation" for all of the exciton resonances indicating that
of the ERL the DL is
increased. A comparison with the theoretical spectra yields liA 160~, ~ = 10 -4 eV and lIA ~ 200~, and ~ = 5-10 -4 eV for curves 4 and 5, respectively. With increasing electron
irradiation
dose
the
spike
evolves
towards a maximum and the ERL has an inverted appearance (see ctu~ve 5 in Fig. 9, curve 4 in Fig. 10). Note that curve 6 in Fig. 9 agrees in shape with the initial spectrtum (which corresponds to a phase shift of about 2~, see Fig. 6) though the reflectivity is markedly damped. This implies that the high-fluence EB increases both the DL depth and the value of the damping (~U ~ 5.10-4eV). It appears that the boundary between the DL and the crystal bulk can no longer be represented by a plane which results in an additional
Excitons at Semiconductor Surfaces
smearing
of
the
spectrum.
(Note,
that
the
203
shortcomings
of
the
DL-model have been touched upon at the end of Sect. 4A and will be considered
in
what
follows).
The
changes
in
the
reflectance
induced by the EB persisted for hours provided the sample was not heated.
Annealing
of
a
moderately
irradiated
sample
at
room
temperature with an exposure to air usually recovers the initial lineshapes.
The authors of
[19,221 concluded that the effect of
the EB was reversible. Following high-dose EB, when the total dose reached 1018 or more an overall decrease of the exciton structure was revealed. This effect was partially reversible. An incomplete of
the
high-fluence
recovery
EB
spectrum which
was was
observed
following
assumed
to
be
the
repeated
due
to
the
subthreshold-irradiation-induced defect creation. Similar reversal of
the
ERL
under
EB
was
observed
for
the
excited
states
of
excitons each featuring its characteristic phase of rotation. The analysis of the ERL and its modification under EB in CdS single
crystals
has
been
performed
by
Benemanskaya
et
al.[19]
within the framework of the DL-model (Sect. 4A) with low values of the exciton damping F, spatial dispersion having been taken into account.
The experimental
lineshapes were
compared with
the the
calculated ones. The changes of the spectra under low-fluenoe EB were interpreted as a result of a decrease of the DL depth to the value of about
60 ~
(Fig. 9, curve 2) which
is
typical of
the
intrinsic DL with the ERL featuring the left spike.
It is evident
that
the
reversible
reflectance
and
spectra
irreversible
are
of
changes
different
of
origin.
The
exciton
former
are
associated with oxygen desorption and recharging of surface and near-surface
centers under low-fluence EB resulting in the band
bending decrease. In the initial state there occurs at the surface of the CdS crystal a field corresponding to
the depletion.
The
transition I ~ 2 is corresponds to the decrease of the electric field,
which
takes
place,
apparently,
as
a
result
of
the
electon-irradiation-induced oxygen desorption (direct evidence of the band bending
decrease under low-dose EB has been obtained in
[22] for the case of 0dSe crystals). This means that curve 2, on the DL model, corresponds to an "ideal" surface, when the DL is
204
A.E. Cherednichenko and V.A. Kiselev
equal to the intrinsic DL of Thomas and Hopfield. No such state, however,
was observed on natural
crystal surfaces.
with further increasing of the EB dose an
appearance
of
the
initial
ERL.
We note
that
(see curve 3) the ERL has
Only
then
corresponding to an increase in 1 relative
do
the
changes
to the initial value
occur in the ERL (see curves 4,5). Thus, successive effect of the EB
is first
depth.
The
to reduce
and
then,
latter case p~oceeds
gradually,
to
from either
increses
the
the DL
electric
field
increase or, as it will be shown in Sect. 5, from the formation of the defect-rich near-surface layers. The increase in the amount of irradiation results
leads
in
the
to
the
storage
modification
of
of
the
the
electric ERL.
field
Note,
which
that
the
experimental data on the effect produced by EB on the ERL of CdS crystals result
of
at
SOK
the
reported by
Ratsch
[24] were
electron-beam-induced filling
interpreted as of
surface
a
states
with electrons from the bulk, producing a negative surface charge and a depletion layer below the surface. Thus,
that part of the
effect is reversible which is due to the inhomogeneous field and Franz-Keldysh effect at the surface
(Sect.
6). Annealing at room
temperature in atmosphere surroundings recovers the spectra. The irreversible decrease of the exciton reflectance following a repeated
high-dose
long-lived defects,
EB,
is
e.g.,
more complex defects
predominantly
sulfur
(see Sect.
associated
with
the
(selenium) vacancies Vs(Vse)
or
5.E). To account for the overall
decrease of the reflectance resulting from the repeated high-dose electron
irradiation
the
authors
oZ
[19]
had
to
use
in
calculations of the reflectance a somewhat enlarged value for the bulk
damping
near-surface
parameter layer
is
r.
The
formation
evidenced
by
of
the
the
defect-rich
analysis
of
the
the
main
photoluminescence spectra (see Sect. 5E). Additional
features
on
the
long-wavelength
reflectivity maximum may be observed
side
of
(see, Figs 9,10). The lines
labelled I s and I se appear to be very sensitive to the EB. These features are also revealed in the photoluminesoenoe
spectra and
will be discussed below. In conclusion we wish to point out that the DL-model with hard
Excitons at Semiconductor Surfaces
205
edges enables a qualitative description of the features in the ERL under EB. A more adequate description which makes allowance for the effect of the inhomogeneous electric field on the exciton resulting in its field dissociation and quasilooalization at the surface will be given in Sect. 6. Still that seems to be only an approximate model,
since
the electron irradiation has a complex
effect on semiconductors and a non-trivial electron energy loss distribution should result in complicated field and defect distributions in irradiated specimens.
O. Interpretational problems As noted, Hopfield and Thomas [12] were the first to introduce a near-surface exciton dead-layer (DL) model. The original variant of the DL, namely, the one with hard edges could explain several features in the exoiton reflectance lineshapes (ERLs). Starting from mid-1970s much effort was given to investigation of surface aspects of the exciton behavior in semiconductors with a particular emphasis on the study of the ERL in various semiconductor crystals. The spectra were explored under application of the electric field, following etching, electron and ion irradiation, additional illumination. Studies of the ERLs and their alterations under various external stimuli revealed several important features. Some of them, such as the occurrence of a maximum at the frequency corresponding to the longitudinal exciton ~L (the so-called spike) and its spectral stability have been interpreted within the framework of the intrinsic DL model whereas a reversal of the ERL has been explained within the extrinsic DL with a varying depth [18] (see, Sect. 4A). The extrinsic DL observed to occur in A3B5-compounds has been treated in [17,25] as arising due to the exoiton ionization in nea~-surfaoe electric fields. This issue as applied to A2B6-compounds shall be discussed in detail in Sect. 6. It appeared, however, that a number of experimental data could not be understood within the DL-model: the occurrence of the
206
A.E. Cherednichenko and V.A. Kiselev
"left"
spike
pulsation,
(see Sect.
stretching
multy-spike
4B), of
structure
exciton reflectance
the
ERL
(see Sect.
accounted for by the DL-model. the
attention
to
those
along
6).
enhancement
the
These
frequency
features
and
scale,
can not
be
The aim of this section is to draw
features
in
the
ERLs
of
CdS
and
CdSe
crystals that are anomalous from a traditional point of view. A marked sensitivity of the excitonic spectra to various surface treatments
indicate
that
physical
conditions
at
a
semiconductor
surface dominate the ERLs. Possible mechanisms of the influence of a
surface
on
excitons
are
discussed
in Sect.
5C.
A
comparison
between the calculated ERLs and experiment allows in some cases to gain
information
about
near-surface
exciton
transition
layers
(ETLs),- see Sect. 4E. Several
authors
Patella et al. with
the
definite
attention
to
the
anomalies
in
the
ERLs.
[15] compared the experimental ERLs in CdS crystals
theoretical choice
experiment Close
paid
ones
between
for the
different ABCs
DL-depths
was
made.
The
and
ABCs.
fit
with
No the
was rather poor for the large values of the DL-depth.
correlation
between
the
magnitude
of
the
spike
and
the
magnitude of the main reflectivity peak Rma x has not been revealed which is at variance with the DL-model. calculated
ERLs
and
experimental
ones
Discrepancies were
also
between the
pointed
out
by
Brozer et a1.[26]. Of
particular
interest
is
indicated by several
authors.
of
only
a homogeneous
damping ~ ,
DL
for
a
complex
The left spike arises very
(see, e.g.,
small
values
with minimal
the intrinsic DL
spike
in the model
of
the
exciton
values
(1 ~ 2Rexc).
of
in CdS crystals could only be the DL depths
or more
only
the
structure
right could
equal to zero [22]. Hence,
spike be
corresponding
For the CdSe crystal
DL should be about 10 nm [28], but, as Fig. left-spike
a
[12,27]). The left-spike ERL (Figs. 7,8)
observed by Benemanskaya et al.[19]
nm
of
about lO-5eV, which is too low a value relative to the
commonly used revealed
structure
is
11 shows,
available,
obtained
for
1
for DL of 5
though ~
to
the intrinsic
4 nm
a
slight
setting
F
the left-spike ERL in CdSe crystals can
not be reproduced within the frames of the homogeneous DL model.
Excitons at SemiconductorSudaces
207
0.2
Wo
wL
Fig. 11. Calculated ERLs for the A(n=1 ) exciton in OdSe as a function of the DL depth i. 1,nm: (I)- O; (2)- 4; (3)- 11; (4)- 15; (5)- 30. P = O. Fig. 12 shows the fine structure of the spike observed in the exciton reflectance of OdSe at T=4.2K by Batyrev et al. [29]. The spike consists of the two maxima near ~L; a more complicated structure was reported by Davydova et al.[30] on OdSe crystals with "shifted" exoiton spectra (Fig. 13). In Sect. 5D we shall discuss the nature of the shifted exoiton spectra in CdSe crystals and give comparison between the experimental and calculated lineshapes. In addition to the anomalies in the region of ~L' 0dS and CdSe crystals show various features on the long-wavelength side of the main reflectance maximum (see, e.g., Figs. 9,10). Prominent features seen in the figures result from exoiton localization at the surface (near-surface) centers which is also evidenced by the studies of the photoluminesoenoe spectra (see Sect. 5).
208
A.E. Cherednichenko and V.A. Kiselev
0.5
/
O,4 {z
O.3
/ 0.2
R 0.4
0.1
0.3
-0
/, b
Fig. 12. Experimental exciton reflectance spectra in the vicinity of the A (n=1) exciton in OdSe at T=4.2K [29]. (a)- initial spectrum; (b) - after a 3-keV low - fluence electron irradiation A complex structure o f the spike is seen.
0.1~
o.1
0
!
i
!
~t700 14720 74740
4
It is pertinent to note that exciton spectra are measured at low temperatures where the exciton damping due to temperature is insignificant. With temperature rise, one should expect a spectral shift, line broadening, and other temperature effects. As for the EP~L, a number of authors (see, e.g.,[23]) reported a reversal of the lineshape upon heating from T=4.2K to 77K which has been ascribed to a thermally induced increase in the DL depth value.
Excitonsat SemiconductorSurfaces
209
R ¢u~. un,.
Fig. 13. Experimental exoiton reflectance spectra in the vicinity of the A (n=1) exoiton in CdSe at T=4.2K [30].
I 682.0
R,, n m I 678.0
I
F, rt~V
I
~
t
0,8
t tA 0.*
"
/~°'2"' / I" /.I/.,.,'-~ the ERL of OdS vs temperature [27]. A solid curve represents a theoretical dependence on I~.
/RmLn'°l°
t
O.a
8or, K
210
A.E. Cherednichenko and V.A. Kiselev
Pevtsov et ai.[27] studied the exciton reflectance in CdS single crystals in the temperature range 2K
the
reflectivity
signal
were
observed,
especially
spectral region of a minimal reflectivity value 2 m n i The magnitude of the minimal reflectivity R m n i zero at T ~ 3OK increased. model
of
To
(plane " R mni
account
for
the
dropped to nearly
- T") and afterwards monotonically
this
Hopfield-Thomas
in
(Fig. 14).
[12]
observation has
been
the
homogeneous
utilized.
A
DL
distinct
step-wise change in the reflected wave phase by 2~ at about 3OK [31] allowed to ascribe this finding to the Brewster effect 0). Note,
(R =
that the values of the exciton damping r derived from
the experiments were
two orders
of magnitude
as large as those
obtained with making allowance for the exciton-phonon scattering. In Sect. 6 an alternative explanation for the observed anomalous effect is suggested. Permogorov et al.
[32] examined the exciton reflectance spectra
of CdS
crystals which
random
illumination
induced
the
spike
temperature for
a
initially revealed no
of in
a
sample
the
long
at
liquid
spectrum,
which
period
of
time
spike
structure.
helium
temperature
persisted
The
A
at
occurrence
this
of
the
illumination-induced spike has been ascribed as due to initiation or alteration of the SCL. Profound changes in the ERLs of several CdS samples have been reported
in
[33]
following
illumination
with
k
~
51OO~.
Preliminary illumination of a sample was performed at T = 4 . 2 K u s i n g a 250W halogen lamp. The light beam passed through a monochromator and
a
heat
filter.
ERLs
were
taken
with
increasing
amount
of
illumination (time of illumination). The intensity of the probing light was low enough not to affect the ERL. A pattern of
illumination-induced changes of
the lineshape of
the An=1-exoiton is given in Fig. 15 for a number of expositions (in minutes).
Initial spectrum is seen to have no peculiarities,
whereas after exposure to a 5-min illumination an enhancement of the
reflectance
exposition
time
was
revealed.
the
marked
With
changes
further in
the
increase
of
the
magnitude
of
the
reflectance as well as in the lineshape were observed and at the
Excitons at Semiconductor Surfaces
same
time
the
left
spike
appeared.
As
a
211
result
of
a
30-min
exposure a reversal of the ERL to the initial shape was observed. Alterations
of
the ERLs
with
increasing amount
of
illumination
will be discussed at some length in Sect. 6.
R
0.4
20
30m£~
O.Z
Fig. 15. Evolution of ERL of OdS with illumination time. T=4.2K.
D. Interference effects in thin
crystals.
Size
quantization
of excitons The
discussions
in
the
previous
sections
have
indicated
the
usefulness of the refleotivity measurements in bulk crystals for estimation of the magnitudes of the dead layer the
interference
features
(DL) which govern
in the exciton reflectance
lineshapes
212
A.E. Cherednichenko and V.A. Kiselev
(ERLs). This section shortly discusses a more complicated case of interference
features observed
to occur
in the vicinity of
the
exciton resonance transition in thin crystals. An important contribution to the exciton optical response comes from the so-called additional or anomalous waves
(AWs)
[6,12]. We
note, that inherent in the concept of AW's is the problem of the additional boundary conditions (ABO's) which was discussed at some length in Sect. been
3. The
demonstrated
by
existence
a
number
of additional of
light waves has
techniques
(see.
e.g.,
the
reviews [34,35]). The direct verification of the theory of the AW has been performed by Kiselev et al. experimental
and
theoretical
[36-38] as a result of the
studies
of
the
reflectivity
and
transmission in thin CdS and 0dSe crystals. The
computer and experimental results
for the reflectivity
in
the neighborhood of exciton A in a CdS platelet 0.34~ thick are compared in Fig. for
the
16. For
two-wave
the calculations,
interference
in
a
Pekar's
slab
were
formulas
used.
[8]
The
ABC
employed was that of Pekar eliminating the exeiton polarization at the boundaries of the medium. The computation involved a set of parameters common for different thicknesses but different damping terms.
The
calculations
following
the
ABC
scheme
enabled
authors to reproduce the essential features of the spectra, the side
one-period Fabri-Perot and
the
two-period
high-energy side of ~L" Fig. 16 shows that on interference
feature
(FP) (FP
interference plus
AW)
the low-energy
can
be
observed.
on
the
the ~z.
low-energy
interference
on
the
side of e L a distinct This
is
due
to
the
FP
interference of excitons from the lower polariton branch (see Fig. 3c).
On
the high-energy
side
of ~L
two
series
of
interference
peaks occcur. The first one corresponds to the FP interference of exciton polaritons from the upper branch; the peaks in this series are well separated. Another series superimposed on the first one consists
of much
closer
spaced peaks
and
is due
to
the mutual
interference of exciton polaritons from both of the branches. Thus, optical
the
AWs
spectra
produce of
thin
several crystal
interesting platelets.
features The
in
the
interference
Excitons at SemiconductorSudaces
213
structureadditional to the one that should be observed with M = is a direct proof of the fact that the AWs do exist. Studies of interference due to AWs provide evidence for proximity of the ABO to
that
of Pekar
for
low-energy
waves arise as a result of
exeitons
[37,38].
Additional
excitation of polaritons. Nonetheless,
it seems also useful to consider the problem within the mechanical approximation [39]. The mechanical exciton is depicted by (3.1) at E=O.
In
the
case
of
a
plane-parallel
conditions should be imposed (e.g., boundaries
platelet
the
boundary
the Pekar's ABO P=O) on both
(let them be z = 0 and z = a). In other words, a thin
plane-parallel semiconductor platelet constitutes a quantum well for excitons which is approximated by a rectangular potential well with
infinitely high
walls.
It
is
evident
that
the mechanical
exciton band (3.4) is size-quantized in the direction z normal to the crystal plane and the solutions of (3.1) along z would depend on z as
sin(N%z/a), where N = I, 2, 3 ....
part
Pig.
of
17
rectangular well
which
corresponds
is shown with
to
In the right - hand
real
values
of
k=k z
a
the levels spaced by the values
(P~V~/a)2/(2M) as referred to the bottom of the well. These levels are
represented
on
the
mechauical
exciton
dispersion
(dashed
curve) by points, or, more strictly, the levels and corresponding points indicate the bottoms of the 2D subbands in the x,y - plane. A much more complicated problem arises when the exciton - photon coupling is
to be
taken into account
[39] as
in the case when
transmission and reflection coefficients of a crystal platelet are to be derived (see Fig. 16). The exciton levels in a rectangular well can be shown [39] to govern the interference structure which is
the
result
of
both
the
FP
interference
within
the
entire
frequency range explored and the interference between the AWs for ~ ~L" A theoretical account of the size quantization of excitons fits fairly well the features of the experimental spectrum 16).
In
what
follows,
we
shall
discuss
the
(Pig.
long-wavelength
structure at ~ < ~L which is due to the exciton localization in a quantum well. We note, that the levels of the mechanical excitons in the well do not coincide with corresponding features optical specti~m [40].
in the
214
A.E. Cherednichenko and V.A. Kiselev
Rtheo,
Rez~r
- 0.6
0.# 0.6 0.2 0.4 0 0.2 0
486.4
486.0
#85.2
485.6
#84.8
~8~. ~ A, , n m
Fig. 16. Reflectivity from a CdS crystal (0.34 ~un thick) in the region of the A(n=1) exciton. The upper curve is experimental (in arbitrary units) taken at T=I.6K, the lower curve is theoretical.
I
O" z~'Ct ~z
z =O
k--link
k=Rek
Fig. 17. Schematic representation of the optical response of excitons localized in: rectangulsr well ( a ) ; surface 6-potential (b).
Excitons at Semiconductor surfaces
215
This observation is accounted for by Fig. 17. The points in Pig. 17a [40] o o ~ e s p o n d to mechanical exoitons and the arrows give the positions of the expected features in the optical spectrum. Such a spectral shift arising due to the polariton effect enables the observation of the exoiton quantization in much thicker platelets than in those where quantization of carriers, i.e., of electrons and/or holes is possible.
E. Exoiton transition layers and a formal model In sections
4A and 4B we have
demonstrated
that
the simplest
model to calculate the ERL is that of the exoiton-free layer, supplemented by the ABO (usually of the Pekar's type). Yet, there exists evidence that more sophisticated descriptions of the excitonic optical response need to be incorporated. In factj we should also consider more complicated models of the near-surface transition layers and, consequently, of the exoiton transition layers (ETLs). The ETL may be of intrinsic or extrinsic origin. Intrinsic mechanisms are those stemming from the internal structure of excitons and are due to (i) the interaction of electrons and holes with their mirror images at the semiconductor surface
(see Sect. 4A) and (ii) the effect of the surface-induced
confinement
of
the
electron
and
hole
motions.
A
number
of
theorists (see, e.g., [12,16,41-47]) considered the details of intrinsic mechanisms of formation of the intrinsic ETL. They found that the mechanisms yield the exoiton-free layer of about ~Rex c. Extrinsic ETLs for excitons arise due to external, with respect to excitons, mechanisms and they are motivated by various surface (near-surface) imperfections (see Sect. 5) including the electric field (see Sect. 6). (Note, that, contr-ary to the intrinsic ETL, the extrinsic one may have macroscopic dimensions, well exceeding the exciton wavelength ~/nex o ~ 102nm). Theoretical aspects of the perturbation of exciton states near the surface were discussed in a number of papers (see, e.g., [48-54]). Most drastically excitons are perturbed by the electric field of the SOL. This issue will be
216
A.E, Cherednichenko and V.A. Kiselev
discussed in detail in Sect. 6. Whatever the origin of the ETL may be, the ERL can be calculated by formally introducing a near-surface potential for excitons. A method of calculating the exoiton reflectance in the case of an arbitrary potential has been put forward by Kiselev [55]. Assuming that the adiabatic approximation (see, e.g., [42]) holds, the elastic scattering of an exciton by a considered. A smooth adiabatic potential
surface potential was was approximated by a
step curve. The limitations on the size of the steps were as follows: the height of the step is small compared with the energy of the exciton being scattered or the width small compared with the exciton wavelength. On the other hand, the number of steps must
be
finite.
The
problem
is
thus reduced
to the problem
of
calculating the optical response of a layered medium where the potential is not dependent, within each layer, on the coordinate which enables the usage of the crystal optics of a thin slab with spatial dispersion [6,8]. Note, that it is also possible to formulate
the problem within the mechanical
exciton quantization
approximation [39,40]. Principally, one should arrive at the same results insofaras the phenomenology of the layered medium takes into account
both the above-barrier
[39] and below-barrier
[40]
solutions for the mechanical exciton. A number of different model potentials V(z) are presented in the left-hand part of Fig. 18 (see curves 1-17). The results of calculations of the reflectivity R are shown against corresponding numbers on the right. The spectra were calculated for the A(n=1) exoiton [12,18] in OdS (curves 1-16) and OdSe [37] (cuzwe 17). The analysis of the calculated spectra shows that the reversal of the ERL results from the effect of the repulsive potential experienced by longwavelength exoitons with kinetic energies from within the interval of 0 - h~LT. The reversal is exemplified by oases 1,3, and 5 for the potentials in the form of the steps of different widths. Note, that the height of the step (see 2-4) does not affect the reversal. The reversal is seen to occur also for the potentials featuring tails (6-8) and for the linear potential
(9,10).
Exc#ons at Semiconductor Surfaces
217
¢~,,6z|
Tig. 18. A number of potential s V and re 1 evant calculated refleotivity coefficients R in the vicinity of the A(n=l) exciton in OdS (I-16) and OdBe
{~7).
i~
~Y
Y]:., "-'-""
I Z
0,2
218
A.E. Cherednichenko and V.A. Kiselev
Inspection of curves 2-4 shows that the spike (in the spirit of Hopfield and Thomas [12]) occurs only for a sufficiently high step. In other words, the potential must have a hard core which is inpermeable
even for the high-energy excitons
is enough transparent to the light wave). The spike, however, appears for curves
(the surface layer
11-16
which
represent
near-surface potential wells for excitons. By insignificantly varying the depth and shape of the well (11-13) prominent spikes can be observed. Curves 14-16 show the optical response of the surface mechanical exoiton, i.e., the exoiton having the bound state in a fairly deep well. The spectra are seen to reveal the additional maximum which shifts to longer wavelengths with increasing binding energy (for the details, see [40]). The outlined method for calculating the exciton optical response has been applied by Kiselev
[56] to the Shottky barrier case at
low temperatures which is considered in detail in Sect. 6. To conclude, we note, that when finding the exoiton optical response one solves the direct problem of the exciton scattering only. The inverse problem, as is well-known, can not be solved uniquely, that is, only a family of potentials can be found. An appropriate potential can be chosen proceeding from physical mechanisms and from comparison between the calculated exciton reflectance spectra and experimental ones.
5. Crystal Defects and Localization of Exoitons A. Bound excitons Exeitons have a tendency
to forfn bound exciton complexes with
defects and these are also 0teated by the optical excitation of the crystals. This section describes shortly the properties of the bound W a n n i e ~ M o t t excitons. Lampert [57] pointed out the analogy between the hydrogen molecule and an exciton interacting with a shallow neutral donor (aoceptor) which can, like the exciton, be described by the hydrogenic model, Just as the hydrogen atoms bind
Excitons at Semiconductor Surfaces
219
to each other in the H 2 molecule, so should the exciton be bound to the neutral impurity. Since a bound exciton complex is made up of an impurity atom (or lattice defect) to which an intrinsic (free) exciton is bound and since the intrinsic exciton is a property of the crystalline host lattice, a question naturally arises concerning the relative intensities of absorption lines that derive from these two types of excitons. This is especially so when the bound exciton lines are observed to be nearly as intense as the intrinsic exciton lines, a frequent observation in both the absorption - compounds. The
theory
of
and
"impurity"
reflectivity or
defect
spectra
absorption
of
the
A2B 6
intensities
in
semiconductors has been developed by Rashba [58]. He found that if the absorption transition is direct and if the discrete level associated with the impurity approaches the conduction band, the intensity of the absorption line increases. The explanation offered for this intensity behavior is that the optical exoitatign is not localized in the impurity but encompasses a number of neighboring lattice points of the host crystal. Hence, in the absorption process, light is absorbed by the entire region of the crystal consisting of the impurity and its surroundings. As a result of an attack on the particular problem of excitons which are weakly bound to localized impurities , Rashba and Gurgenishvily [ 5 9 ] derived the following relation between the oscillator strength of the bound exciton ~ and the oscillator strength of the intrinsic exciton ~exc' using the effective-mass approximation Ig = (Eo/IEgll3/2~exo where
E0 =
(2~2/m)
(%/00)2/3,
Eg
is
the binding
(5.1) energy of
the
exoiton to the impurity, m is the effective mass of the intrinsic exoiton, and ~0 is the volume of the unit cell. Note, that the value of ~ , when substituting the parameters for the bound exoiton lines derived experimentally, may exceed ~exc by more than four orders of magnitude (the effect of the "giant" oscillator
220
strength).
A.E. Cherednichenko and V.A. Kiselev
Assuming
that
the
line
intensities
are
equal
to
a
simple product of density and oscillator strength (In ~ Nd'f~) and assuming an impurity concentration of N d ~1017cm-3~and intrisic exciton concentration of Nexc ~ 1022 one arrives at the conclusion that bound exciton intensities can be within an order of magnitude of those of the intrinsic exoiton lines, hence, they are observable in transmission and reflectance. We note, that the bound exoiton dominate the emission spectra. The emission results from the
disintegration
(radiative decay)
complexes
and
the
occurs
on
of the bound exoiton
long-wavelength
side
of
the
free
exciton transition as narrow (having half-widths of ~10-4eV) lines yielding information on the associated defects. Another important conclusion which can be drawn from equation (5.1) is that, as the intrinsic exciton becomes more tightly bound to the associated center, the oscillator strength (and, hence, the intensity of the exoiton-complex line should decrease and u~ce versa). The extensive studies of Thomas and Hopfield [60] on CdS have resulted in the development of a theory of bound exoitons. The theory is based on the band symmetry properties associated with the wurtzite structure of this compound. Several bound exciton complexes are postulated and observed experimentally. These are the exoitons bound to neutral and ionized donors. The transitions were designated as 12 and 13, respectively. The exoitons bound to neutral aooeptors, designated as 11 are also observed.
B. The effect of
keV- electron
irradiation
on
the
exciton
photoluminescenoe spectra Exciton photoluminesoence spectra are also dependent upon surface conditions of a sample. This section briefly reviews the effect on the exoiton photoluminesoenoe spectra of the electron irradiation of OdS and OdSe crystals [61,62]. Irradiation induced changes in the spectra were registered ~n s~tu at 4.2K as a function of fluenoe (or dose) of incident keV-electrons. Figure 19 shows the dependences on the irradiation dose [62]
Excitonsat SemiconductorSurfaces
I
221
ziO 20 10
1
,6~.
F~.
19. Intensities
o~
the emission lines in OdS at T=4.2K vs eleotron dose. 1- I2B; 2- I2A; 3- 11; 4- A(n=l); 5- 13 .
~'~
~+.,.--o-----I~. _+,+Z~"
-~2
~ ~~i~e.~.e..~e~ 0.6F\5
e
~i
0 0.5
4
~e~3
40 50 6O 70 8O
T
.............Iz
Pig. 20. Transmission speotra in the region of the 12bound exoiton in 0dSe exposed to inoreasing doses of 3.5 keV electron irradiation. 1initial speotrum; 2 - after 2"1015e/om2; 3- 8"1017e/om2.
681.0 ' 680.5
6B0.0
679.5 ~,,rmt
222
A.E. Cherednichenko and V.A. Kiselev
of various exciton emission lines typical ¢f CdS. With low-fluence irradiation (D ~ 1015 e/cm 2) an overall increase in the luminescence intensity was observed for the I2A-, I2B-,
Ii-, 13-
bound exciton emission lines and for the An= I- exciton emission line
as
well.
crystals
Similar
changes
occurring
seemingly results,
dose
dependences
were
found
for
0dSe
[61]. These findings indicate that the exciton emission due in
to
under a
low-fluence
surface
particular,
exoiton non-radiative
band
in
the
escape
electron
bending
irradiation
decrease.
decrease
of
probability.
a
The
The
are
latter
field-dependent in
the
amount of band bending is also evidenced by our studies of
the
reflectance spectra and surface photovoltage
reduction
[22]. Note, that the
transmission in the region of the I2-bound exciton line in OdSe crystals
shown
insignificantly,
in
Fig.
hence,
20
(curve
[61]
2)
the low-fluence EB
is
does not
changed
practically
produce defects. This conclusion is also supported by a reversible character of the exciton spectra changes. After electron irradiation to a fluence of above 1016 e/cm 2 the emission
lines
intensities
show
different
behavior
(Fig.
19),
which is indicative of their different origin. The most pronounced increase, about 1.5 order of magnitude, was observed to occur for the I2-1ines corresponding to excitons bound to neutral donors. A corresponding
increase
absorption
the
in
is
region
observed of
for
these
486.1-487.1
nm.
lines The
in
CdS
electron
in
beam
induced enhancement of the I2-feature in CdSe is seen in Fig. 20. These observations
indicate an increase
in the
intrinsic defect
(neutral donor) content in the near-surface region after EB. Figure 19 shows a sharp
drop
in the intensity of
the I3-1ine
(the Is-line, see below) which is due to the exciton bound to the charged
(ionized) donor. The anticorrelation between the 12- and
13- line behavior (also see [60,63,64]) and a reversible character of the changes show that a neutralization of ionized donors occurs under EB. At the same time the intensity of the It-line, which is due to the exciton, bound to a neutral acceptor is less dependent on the dose,
probably
because
a
major
part
of
acceptors
in
n-type
Excitons at Semiconductor Surfaces
223
semiconductors is charged and, hence, not influenced by EB. With increasing irradiation dose the effects due to long-lived defects become dominant which is evidenced by an overall bound exciton emission intensity drop (Fig. 19), an increase in the absorption in the region of the I2-I2"-lines (Fig. 20), the emergence of the electron irradiation induced Ie-lines (see below). Electron irradiation as well as other treatments were found to strongly affect the broad-line structure observed to occur in CdS crystals in the spectral range of 486.1-486.5nm. Initially this feature has been ascribed to the emission of the exciton bound to the ionized donor (the so-called 13- line). The reason for this assignment was the observed anticorrelation between the intensities of the 12 - and 13-1ines. Typical of this feature is its dependence on surface-sensitive electron irradiation, excitation light intensity. This surface-sensitive structure was labelled I s [65]. Travnikov [66] studied the exciton spectra from as-cleaved and air-exposed surfaces of OdS and found two lines in the spectral o -line (A = 486.4nm) may vary in region under consideration. The Isl intensity whereas the I°2-1ine may as well change the spectral position within the spectral range 486.4-486.2nm. One of the lines was ascribed to surface defects involving oxygen adsorbed at the surface, while the other one was claimed to originate from the space charge layer. The observed disappearance of the Is-feature under low-dose electron irradiation, resulting in the oxygen desorption [67] points to the fact that oxygen is involved in the centers yielding the Is-feature. A surface-sensitive Is-feature was likewise revealed in the reflectance and photolumin.escence spectI~a of 0dSe crystals,- see Figs. I0,2!, The Is-line appears in t~e initial spectrum as broad line with the maximum at k = 679.9nm, slightly varyir~ its spectral position from sample to sample within 0.1-0.2nm, With minimal amount of e l @ c t r ~ i~r~iation a short-wavelength shift (Fig. 21, OU~-'ve 2) ~ d subSeQuent quenohin~g of the Is-line (curve 3) were revealed. These observations indicate that the I s - feature is most likely associated with the excitons bound to ionized
224
A.E. Cherednichenko and V.A. Kiselev
15
Fig. 21. Exciton photoluminescence spectra of CdSe at 4.2K. (1)- initial spectrum, (2 4)- after exposure to increasing doses
(1.1016; 5.1016; 3.1017e/cm2)
of 2 keV electron irradiation.
680
679 j,nm
donor-type centers at the surface. The Is-line in CdS is seemingly a surface counterpart of the I3-type line in the bulk. It features a
large
half-width
electric
which
may
be
field value affecting
assigned
this
to
variation
exciton-impurity
in
the
complex at
the surface. The authors of ref.
[61 ] claimed that the Is-line in
CdSe
field-dependent
may
be
associated
with
a
localization
of
excitons in near-surface quantum wells. As
the
electron
irradiation
dose
is
raised
a
new
feature
designated as I e s arises in the spectrum. (i) This new feature is observed both in luminescence and reflectance spectra of CdSe (Figs.
10,21 )
and
CdS
(Fig.
9)
single
crystals
following
Excitons at Semiconductor Surfaces
high-fluence
EB.
In fairly
thin
samples
it
225
can be
observed
in
transmission as well (Pig. 20). (ii) Contrast to the Is-feature, the spectral position of this line is practically fixed and makes 485.9 and 679.6nm in CdS and CdSe, respectively. (iii) Following a repeated high-fluence EB, some samples reveal a reversal (rotation) of the ERL corresponding to this transition (see Sect. 5E). These findings suggest that long-lived defects produced by electron irradiation yield the I~-lines. The defects are located in the near-surface region whose width is governed by the electron irradiation dose and incident electron energy. A reversal of the ERL corresponding to the I~-feature proceeds much in the same manner as in the case of the intrinsic exciton although in the former case the centers responsible for the I~-feature are located in the near-surface region and are absent in the bulk (Sect. 5E).
C
Surface and near-surface excitons
Surface states and resonances for excitons are predicted by both dielectric and microscopic theories [6]. For ideal surfaces and interfaces, specific long-wavelength solutions, with the electric field localized within thin layers adjacent to the surface, can be obtained using a macroscopic dielectric treatment provided the dielectric function and the ABCs are specified (both, however, should be derived using microscopic models). The results of experimental studies of this type reviewed, for instance, in [68].
of
excitations
have
been
Conclusions of microscopic theories depend on the assumptions about the electronic structure of the surface. For example, in the case of molecular crystals the surface-site-shift model has been proposed [69] which leads to the exciton localization within one or two subsurface monolayers characterized by exciton resonance frequencies shifted with respect to the bulk value. The theory explains the fine structure observed in the reflectivity spectra of antracene. For interfaces, the excitons may be attracted to the surface by the image potential [70] provided the dielectric
226
A.E. Cherednichenko and V.A. Kiselev
constant of the overlayer is higher than that of the medium. Of
particular
carriers are
interest
is
the
case
in surface electronic
when
one
or
both
states or bands,
of
the
formed,
for
instance, from dangling-bond states [71]. Experimental aspects of the problem have been discussed in [72,73]. Theoretical aspects of 2D
surface
have
been
excitons
excitons
and,
discussed
in
with
surface
in particular, [71,74,75].
impurity
their optical
The
centers
response
interaction and
of
initiation
these of
2D
exciton - impurity complexes have been discussed in [76]. This
short
effort
of
review
shows
(for more,
see
[77])
that
those engaged in the problem of surface
the
main
excitons has
been concentrated on the mechanisms of surface excitons formation for
the
case
of
ideal
crystals.
New mechanisms
of
the
exciton
localization at the surface and within near-surface regions arise if we take into account the interaction of excitons with defects and impurities in semiconductors are
listed
below.
Note,
that
[78]. These
the
"defect" mechanisms
best-developed
model
for
the
interaction of excitons and near-surface charged imperfections is based on the Stark effect in an inhomogeneous electric field. The latter,
under normal
circumtanoes,
surface of semiconductors. excitons
are
localized
is always
present
below
the
It is important that quite frequently within
extended
layers
(102-
103nm)
exceeding in value those typical of the ideal surfaces by orders of
magnitude.
These
layer
thio~2qesses
are
close
to
polariton
wavelengths near resonance and also to the penetration depth of light in a crystal. Hence, exoiton spectra appear to be sensitive functions of near-surface exciton transition layers are
strongly
dependent
on
the
physical
(ETLs) which
conditions
at
the
defects
and
semiconductor near-surface layers. Actually,
the
interaction
of
excitons
with
impurities is always present both in the bulk and at the surface of semiconductors. However,
the defects and impurities themselves
may acquire new properties near the surface, and that may be the reason
for
the modified
or even new mechanisms
of
localization
(see [78)]. Some variants, discussed below, are displayed in Fig. 22.
Excitons at Semiconductor Surfaces
227
W w0
3,4r ~@@
Fig. 22. A s c h e m e for "defe c t" mechanisms of the excit o n localization near semiconductor surfaces (see text).
• 1
~
I@ @@ @@@ @@ @
@@@ •
@@ @@ •
• • @@@
..-- ".. @ .-.-~... .
;@41@ D @
I.
It
is well
known
that
defects
and
impurities
yield under
optical excitation various excitonic complexes whose radii are of the order of IO nm or more (see Sect. 5A). It is natural to assume that
within
magnitude
surface
new
layers
excitonic
with
thicknesses
complexes
may
of
arise
this with
frequencies other than in the crystal bulk. Moreover, may
be
even
thicker
if
the
effect
on
the
order
of
resonance the layers
complexes
of
the
near-surface electric field (see point 4 of our list) is strong. This influence is twofold. Firstly, it may change charge states of imperfections, and, secondly, to
the
Stark
responsible
effect.
for
the
In
the
binding
it may shift the energy levels due other of
limiting
excitons
are
case
the
centers
localized
on
the
surface of a semiconductor,- see, e.g., [66]. Provided
the
complexes
are
independent
(see variant
I)
their
contribution to the dielectric constant of the medium may be taken into
account
by
adding
oscillator
terms.
These
terms
should
include the Rashba enhancement of the oscillator strength [58,59] and inhomogeneous dependences of the parameters of the complexes along
the
z-axis
normal
optical properties of
to
the
surface.
Calculations
of
the
the media characterized by the dielectric
constant varying in one dimension may be performed exploiting the approach outlined in Sect. 4E [55].
228
A.E. Cherednichenko and V.A. Kiselev
2. Another mechanism of localization may result from an increase in
the
imperfection
qualitatively
concentration
new feature appears
that an exciton-defect thickness
of defects.
a nonstoichiometric surface
band
resonance resonance point
(or exciton-impurity)
of the localization
the distribution may
term
layer be
typical
oscillator
may be not of great
A
is so high The
this layer may coincide with
a
shown
into
free in Fig.
However,
the
importance
the
The exciton-defect
theory
exciton, 22,
term should be z-dependent
strength.
surface.
band is created.
(see the next point). of
as
the
layer in this case should depend on Actually
incorporated
frequency,
I this
towards
if the concentration
but
effects
2.
include of
for this band,
adding
for
variant
and
by
a
a
lower
Just
as
in
the enhanced
spatial
dispersion
and formally case 2
may be reduced to case I. 3. A semiconductor the surface,
crystal may have a nonstoichiometric
for example,
layer at
a layer enriched with Cd in the case of
CdS. Also real crystals may be characterized by chemically reacted layers
or/and
layers
one-dimensional gap
of
normal
the material The some
the surface
[55,79,29].
continuum deformation well
for
vanishing
exciton may
be
excitons into
described,
case
3
states,
discrete
importance
of
interaction layer
to
the
refer
may
crystal.
the
direction
be
treated
frequency,
~O'
to modifications the
variant
that
in
the energy
3
in
it will This
Fig.
cause
and
situation
near the
of
22. a
wave
by
of
deformation
of
a The
quantum
functions has
been
[29]. for near-surface
it also provides due
gap
eigenstates,
the
predominantly
to the localization
resonance
see
significant
is the charge of imperfections. 3 and 5, but
-
so
the bulk
lead
concerns
with
in, e.g., ref.
4. Of great
the
I and 2 which
states
of
in a graded
of
(a
which may change
semiconductor
variation
to cases
exciton
solutions
This may also
graded-gap
introducing Contrary
solid
is expected)
resulting
to the surface.
excitons.
bound
of
disorder
localization
of excitons
It is true for variants a specific
electric
field
(SCL). The latter acts on excitons
additional within
the
from I to
mechanism space
of
charge
through the Stark effect
Excitons at Semiconductor Surfaces
229
(see Sect. 6). The coordinate dependence of the field may be subdivided into regular
(averaged over fluctuations)
and irregular contributions
In Sect. 6 we shall show that the macroscopic electric field may cause exciton localization in the SOL. Note, that the width of the SCL, w, depends on the surface charge, Qs' and on the net space charge density, N + - N-. magnitudes.
The
highest
It may vary in a very broad range of sensitivity
should be observed for w ~ ~exc
of
the
exciton
reflectance
~ I0-I02nm where kexc
is some
characteristic wavelength of the exciton. Another mechanism may also be operating, when only one of the carriers, comprising the exciton is l o c a l i z e d b y the field with a subsequent
attraction
of
the
other
carrier.
This
claimed to be revealed experimentally (see, e.g.,
mechanism
is
[80]), but its
consideration is beyond the scope of the present review. 5. At last we wish to mention highly disordered surface layers (amorphous, sputtered layers of solid solutions, etc.) that may be represented
by
random
spatial
oscillators,
- see variant
5
and
energy
in Fig.
22.
distributions
This
variant will
of be
illustrated in what follows.
D. Localization of excitons in rectangular potential wells To account for some features in the ERLs of semiconductors appeared
appropriate
to
make
use
of
the
model
it
involving
near-surface potential wells for excitons with fairly sharp edges. It is evident
that the diffusion of the well's
exceed the exciton wavelength (~102nm). the exciton reflectance lineshape described
in
circumventing
[55] the
(see,
difficult
edge should not
A method for calculating
(ERL) for a given potential was
Sect. problem
4E). of
This
method
finding
the
permits
levels
of
mechanical excitons in the well and then taking into account the photon-exciton interaction, and it makes it possible to directly calculate the optical spectra of exoitons.
230
A.E. Cherednichenko and V.A. Kise ev
0
2
#
6
8
N 10 tZ
-/I /"
1 I
0
-5
T
.
111 e
1 0
250
2
$
6
2,rim
8
I0 I2 "~$ 02 0
iz '
R
Fig. 23. A scheme showing the fine structure o f t h e EP~ o f a CdS crystal (on the right) arising due to the potential well with hard edges (on the left).
Consider, well,
for
the
shown in Fig.
sake 23.
of
simplicity,
We note,
the
that the
right-angle
shaped
particular case of a
well with infinite walls has been presented in Sect. 4D. Here, inner wall of the well
is of finite height,
levels
finite.
in
the
well
is
These
levels
hence, are
the
the number of plotted
in
the
left-hand part of Fig. 23. From the figure we note that there are resonant
states
quantum-mechanical Bound spectral
and
above
the
well
which
are
dominated
by
interference of excitons above the well.
resonant
states
both
lead
density of states of mechanical
to
oscillations
exoitons.
in
The maxima
the in
the density of states are indicated on the dispersion curve by the dots plotted
in the middle part of the figure.
They indicate
positions of the bottoms of N = 1,2... 2D-subbands plane). Such a plot is
made
by
analogy
with
the
(in the surface
the
case
of
an
Excitorl~ at Semiconductor Surfaces
231
O.5
0.~ 0.5 0.2 0.1
f
I
0
0
0
w o
wj.
w
Fig. 24. The calculated ERL of a CdS crystal for different parameters of the rectangular near-surface potential well of Fig. 23 (see text).
infinite-wall well (see Fig. 17a) and, is, It has been pointed out in Sect. 4D that polariton effect the singularities in density of states do not coincide with corresponding to them in the optical
therefore, qualitative. in the presence of the the mechanical exciton the spectral features spectra, e.g., exciton
232
A.E. Cherednichenko and V.A. Kiselev
J /
!!
P~Jf
~03
k
Fig. 25. Schematic illustration of the origin of the spike features due to the near-surface potential well with hard edges (see text).
reflectance spectra
[40]. The singularities are shown in Fig.
to be related by means of upward and downward arrows
17
(see Fig.
17a), i.e., by means of projecting the dots onto the two polariton branches. Similar plot was made in Fig. 23. The right-hand part of the figure shows the fine structure of the calculated reflectance spectrum which is seen to be dominated by the exciton levels in the well, as well as by the above-well resonant states. The width of the well w was taken to be 250 nm, the depth of the well being given
as
the resonance
frequency
shift
A~O=
-5cm-~
Henceforth,
where necessary (see, e.g., Sect. 6), these two parameters will be given as put in parentheses without (250,-5), denoting the width and respectively. The spectra calculated for several shown
in
Fig.
24.
By
varying
w
one
units of dimensions, the depth of the different arrives
values at
the
e.g., well,
of w are following
Excitons at Semiconductor Surfaces
conclusions interference
(i)
With
features
the
increase
marked
in
1,2,3...
the begin
233
value to
of
appear.
w
the They
originate from the above-well resonant states which are subsequently captured by the well with increasing w. ( i i ) The interference structure shifts to long wavelengths away from the main reflectivity peak. (iii) The structure is spread out and, eventually, disappears with a new short-wavelength feature arising in the region of the spike. Such a modification of the spectra are easy to follow in Fig. 17 and Fig. 23. In fact, similar interference features can likewise be obtaind with a right-angled step (see, e.g., Fig. 18, variant 17) and this circumstance need to be taken into account when performing the analysis of experimental data. Still, the models involving potential wells quite often give a more close fit to the features revealed in the experimental spectra. Quite often, one may observe a long-wavelength shift of the main refleotivity maximum and minimum, stretching of the spectrum along the frequency scale, and appearance of a more pronounced spike as compared with that normally observed in the region of ~L (see Fig. 24 and text below). These observations are readily explained within the framework of the polariton approach, see Fig. 25. Lowering of the exciton resonance frequency in the near-surface layer (~OV ~ ~OS ) brings about the spectral shift of the polariton branches below the surface. As a result, the polaritons of the upper branch in this layer (shown by horizontal smrows) fall into the region of the longitudinal-transverse splitting for bulk polaritons, hence, they are reflected from the bulk of the crystal back to the surface. Actually,
the crystal bulk serves as a metal
mirror for near-surface exciton polaritons. As regards the origin of the wells in question, we note, that, at least, two mechanisms should be considered. The first one is due to a random field of charged centers in the SCL, where a sharp drop in the frequency ~0 below the surface is associated with the fluctuation part of the electric field of charged defects (donors) of high concentration. This case is exemplified by the spectra of Fig. 12 [29]. Relevant theoretical treatments are now in progress,
234
A.E. Cherednichenko and V.A. Kiselev
and a detailed
account
of the problem
is beyond
the scope
of the
present review. Another mechanism crystals [81].
The
crystal
experimental
is
spectrum
shown
in
features
compared
with
experimental about
a a
stretched along 13.4om -I
exciton
part
normal also
spectral
note,
that
the frequency
6.4cm -1
(which
shift
of
the
properties,
perturbation
is
of
spectra
as
the
that
of the surface.
a whole
Another
spectra
of the ERLs are value
used
of
the
[37]).
(A)
is is
associated a
X-ray analysis
with
result
of
Both of the shifts are assumed
due to one and the same reason.
are
to a commonly the
(A~L)
stretching
The
IOcm -I , as
[37].
"widths"
splitting of 7.7 cm -1
while
a.
a
curve a, is shifted by
compared
less
in
such
curve
experimental
The
of
about
position
the
scale.
26(I),
shift
of CdSe
observed
spectrum
of Fig.
short-wavelength
in some
solution
reflectance
and 25.5cm -1 , respectively,
of
The
to the occurrence
layer of a solid
the upper
longitudinal-transverse bulk
due
spectrum shown in Fig. 26(II),
49cm -I . We
value
arises
of a near-surface
indicates
a
to be
that the
crystals in question were heavily doped with S to a concentration of 1019-I020cm -3 . Actually, the samples represented solid solutions
of
CdSe1_xS x
Corresponding 1.5%
for
of
second
one
the
extrapolating
involving extended and
can
a fairly tail,
IIb,
deep well with
only
be
by
the
the of
order
first S
emploing
sharp
27. The upper
to calculate (Note,
6).
of
I%.
spectrum
was
accounted
regards for
edges shown in Fig. 27.
by
the
a
and
estimated
by
spectra and the model
that
the
potential
supplemented
(I) and lower of Fig.
see curve
feature
long-wavelength a
ERLs
are
is seemingly due to
by the inhomogeneous
incorporating
by an
(2) curves of 26,
theoretical
The tail of the potential
exerted on excitons As
edges
the ERLs
field which yields a multy-spike
Sect.
of
for
percentage
achieved
respectively.
the Stark effect electric
be
see Fig.
given as unshifted).
also,
(the
being
0.3%
agreement between the experimental
ones
the figure were used Ib
x
x were
the data of [82]).
A qualitative calculated
with
values
deep
surface
in the ERL feature, well
with
(see,
it
can
sharp
Ex¢itons at Semiconductor Surfaces
235
T R OL
0.#0.3- / 0.20.10 ! , I~720 , R
I
I
I
I
14700
I
14720
1"[ b
0.5-
0,#0.~- /
m
0.2o,1-
u~,cm 1 0
i
I
1#?~0
I
I
I
!
1#760 1~780
ua~c,m-t i
~
i
I~700
I
I
I
1#720
Pig. 26. (a) Experimental exciton reflectance spectra of OdSe(1-x)S(x) in the vicinity of the A(n=1) -exciton at 4.2K (see text); (b) Theoretical spectra calculated using near-surface potentials shown in Pig. 27. I corresponds to x =0.3% (see potential I in Pig. 27); II corresponds to x =1.5% (potential 2 in Pig. 27).
236
A.E. Cherednichenko and V.A. Kiselev
t" •,~ r ~ o , 4 --- .
.
.
crl, t"l
F .
"1
L,_.
0 -4 -'1 %.
0
Fig. 27. A resonant exoiton frequency shift A~ and exciton damping P vs coordinate z. (I) corresponds to curve Ib and (2) corresponds to curve IIb (see Fig. 26).
-8
-/2
z,~m
-16 I
I
I
I
0.2 0.4
0
In
i
view
of
I
0.6
I
I
these
(one-dimensional)
I
0.8 1.0
studies
near-surface
it
seems
natural
to
assume
a
fluctuation of the composition due
to a layer of a solid solution (only a one-dimensional fluctuation is
capable
estimates,
of
producing
sharp
a characteristic
be of about ~102nm.
features).
According
to
our
linear size of the fluctuation should
Sulfur content at the surface should be less
than in the interior, which yields a potential well rather that a step.
There
is
also
an
additional
that,
from the point
excitons
in
CdS(Se)
solid
does not seem to proceed. that,
depending
surface excitons.
on
fluctuation
the may
which
of view of solid solution physics,
these findings appear to be rather surprising, of
evidence
[81].
supports our suggestions Note,
experimental
solutions
Probably, growth arise
with
low
sulfur
content
the effect of surface is such
conditions,
which
since localization
allows
a for
one-dimensional localization
of
Excitons at Semiconductor Surfaces
237
E. Localization of excitons in defect-rich surface layers In
the
previous
section
the
exciton
localization
in
the
near-surface potential wells has been treated. Another important mechanism
of
localization,
U~Z.,
localization
field
the
space
layer
(SOL)
of
detail
in Beet.
charge
6. In the present
by
shall
the
be
electric
considered
in
section we consider one more
type of localization which is due to the binding of exeitons by near-surface defects. Several
types
of
bound
excitons
associated
with
different
centers can be excited in a semiconductor bulk (see Sect. 5A) but we
will
concentrate
herein
on
those
which
near-surface layer (Fig. 22, variants 1,2). impurities) occur
may exist
predominantly
in as-grown
as
a
result
are
peculiar
of
a
Such centers (defects,
(untreated)
samples,
of
treatments,
various
but
they e.g.,
etching, electron or ion irradiation. In Sect. 5B we have outlined that the electron irradiation of CdS and OdSe or~jstals to a dose of
IO17e/cm 2
spectra
yields
(taken ~
irreversible
modificaton
of
the
exciton
s~t~ at 4.2K) resulting from the electron-beam
stimulated formation of long-lived defects.
This
issue has been
treated in detail by Batyrev et al. [83] and this section reviews briefly the results of these studies. In
Sect.
5B
we
have
shown
that
after
a
prolonged
electron
irradiation of CdS and OdSe crystals some new features, designated e and 12* arise in the reflectance and emission spectra (see as I s Figs. 9.10,21].
0dSe samples, may be classified, with respect to
their photolumineseence spectra, into two groups; group I concerns "perfect" or~Jstals while group II include "defect" crystals. Fig. 28 shows
the evolution of
the exciton reflectance
spectra of a
defect CdSe crystal with increasing the irradiation dose. Photolumineseence spectra of a "perfect" CdSe crystal before and after
prolonged
electron-beam
irradiation
induced
line
are Ie s
is
presented seen
to
in shift
Fig.
29.
towards
The long
wavelengths by about O.Inm (curves 2,3). The donor-bound exciton, induced by the EB (see Fig. 20) is modified into a broad line (Ak ~0.6 nm) labelled I2s which also shifts towards long wavelengths.
A.E. Cherednichenko and V.A. Kiselev
238
IS
R~2
G
I2s l 1~
0.4
I2s
jv
I I 0/3 _I I I -I I I 0.2
b
J/
2
j~
I
I .I II
L-- I
6
\
6
0.~
G81
6~
681
680
~nm Fig. 28. Exciton reflectance curves of CdSe (2-61~exposed to increasing doses of electron irradiation ( x10 "e/cm). a: (1)-initial; (2)-I; (3)-2; (4)-8; (5)-20; (6)-60. b: the magnified spectra.
Excitons at Semiconductor Surfaces
239
~l Izs
Iz
I i}
Fig. 29. Exciton pho toluminesoence spec tra of a "perfect" CdSe sample (1)-initial spectrum, (2,3) - after exposure to increasing doses of electron irradiation, (4)-initial spectrum of a "defect" crystal (see text ).
x 10
~
J
j ......
,
" ";'"
, x3 '........-2",.;
°
\
i
i
681
i
i
i
i
679
i"--.,/.i
~,nm
Eventually, the I2s-line (~ ~ 680.65nm) becomes a predominant feature of the photoluminescence spetrum. Also seen is the Ie-line (~ u 679.7nm). The observed changes were found to be irreversible. s Note that the I s - and I2S- features can be revealed in the optical spectra of as-grown samples. Curve to curve 3, indicating that a prolonged yields modification of "perfect" (group I) (group II) ones. The ERL of a "defect" CdSe sample versus
4 is almost identical electron irradiation crystals to "defect" the electron dose is
shown in Fig. 28. Initially, a distinct structure associated with An=l-, I2S-, and I s- excitons is observed. Electron irradiation brings about the "rotation" of the ERL of the An= I- exoiton, which is indicative of an initiation of the exciton-free layer.
240
A.E. Cherednichenko and V.A. Kiselev
It
oi.;
° Fig. 30. Calculated ERLs of CdSe vs the width of the near-surface layer with an additional resonance, a 20nm; b - 50nm; c - 100nm; d lO00nm.
14710' "/4720
The reflectivity are
of
a
14730
e transitions features associated with 12S- and I s-
typical
dispersion
"rotation".
Furthermore,
the feature
associated
with
the
accompanied by an enhancement The
reflectivity
irradiation centers
dose
(defects)
that a defect-rich
features
point
character
and
also
reveal
the
curves 4-6 indicate a strong quenching of
to
capable
intrinsic
An= I- exciton
which
is
of the I2S- feature. and
the
their fact
of binding
reversal
that
the
excitons.
with
EB
increasing
produces
From
layer arises below the surface
local
this we infer
(the penetration
depth of keV electrons
is about ~102nm) whose width is governed by
the
energy
incident
Contrary excitons
to
electron the
and
conventional
the
electron
dead-layer
irradiation
(see
Sect.
excited at the centers of the near-surface
close to the surface and are essentially absent Relevant
calculations
in this case. The model
indicate
is considered
the
that the ERL can also be reversed
in question suggests to be
4A),
layer do exist
in the bulk.
the existence
layer of a finite width within and beyond of which constant
dose.
equal
to
of the
the dielectric
the background
dielectric
Excitons at Semiconductor Surfaces
241
constant ~0(~)" Within the layer, the dielectric constant is contributed by an oscillator term of the type: ~/(~I~0 + 47), where ~I is the excitation frequency of the e exciton (e.g., I2S and Is). The calculated 30, reproduce the effect of rotation of the in the width of the layer. The rotation is
relevant near-surface spectra shown in Fig. ERL with the increase observed for both the
intrinsic exciton and additional resonance outlined above. As regards the origin of the electron-beam-induced defects, latter seem to involve
the anion vacancies
Vse(V S)
the
and the I2s
line is a counterpart of the bulk 12 line. The I e s - and I2sfeatures show similar dependence on the irradiation dose and, in heavily treated samples they rather than local centers.
F. Localization of exoitons
are
by
associated
fluctuations
with
of
a
defect
bands
near-surface
random potential Low-temperature photoluminescence spectra also demonstrate the feasibility of surface localization of excitons (carriers) by fluctuations of a random potential. There is a variety of reasons for the occurrence of a random potential and corresponding in the density of states (see, Fig. 22, variant 5).
tails
In Sect. 5B we have indicated that the exciton photoluminescence spectra of CdS and CdSe crystals were modified by electron bombardment (EB). This section shortly reports a study of a broad band observed in the emission spectra of some CdSe crystals which was attributed to the luminescence of excitons localized at surface fluctuations of the potential (or charge) [84]. Figure 31 shows the photoluminescence spectrum of one of OdSe samples at T=4.2K typical of those reported in [84]. In the short-wavelength part of the spectrum (on the right of the figure) there
are
observed
luminescence
features
due
to
free
excitons
An=l, surface exciton I s (see Sect. 5B), and a strong line 12 due to exoitons bound to neutral donors. In the long-wavelength part of the spectrum ( on the left of the figure ) a very broad band
242
A.E. Cherednichenko and V.A, Kiselev
Zs
1
A .=Z
/jlAI:/ "i ')
~"=' (,=~)
/~
,l
/ilz~ ~. /:i~!):
/
21 ~,
[, I"
!.~x30
r 679 ~n.t
Fig. 31. Photoluminescence spectrum of a CdSe sample at T=4.2K (see ref.[84]). 1) initial spectrtun; 2),3) after low-dose electron irradiation (see text).
labelled X is revealed and against its background one can see oneand two-phonon replicas of the An= I- exciton. Several new ~eatures were observed for the new band. I ) The band
is wide with a halfwidth of at
least 20 meV.
The
spectral position of the band maximum varied from sample to samle within 6830-6850 ~. The intensity of the band was anticorrelated with the intensity of the exciton photoluminescence. 2)
The
band
X was
extremely
sensitive
to
surface
treatments.
Excitons at Semiconductor Surfaces
243
Figure 31 shows the photoluminescence spectra before (curve i) and after 1 sec ( ~2 1015e/cm 2) and 3 sec (~ 6 1015e/cm 2) of 3.5 keV electron
irradiation.
It
is
clear
from
the
figure
that
the
electron irradiation resulted in a marked quenching of the band and
in
a
characteristic
wavelengths.
The
shift
observed
of
its
quenching
maximum was
toward
shorter
accompanied
considerable increase in the exciton luminescence
by
intensity
a
(see
the right-hand part of the figure). 3) The band X depended on temperature in a complex mam~cr.
It
markedly shifted toward long wavelengths and reduced in intensity disappearing at about 20K [84]. These findings
(for more, see [84,85]) suggest that the X band
results from the radiative recombination of excitons localized at fluctuations of the random potential relief at the semiconductor surface.
The
fluctuations centers
fluctuations of
the
(defects,
result,
charge
due
impurities)
most
to
a
likely, random
from
spatial
distribution
of
in the surface layer. The presence
of these centers may give rise to wells
in the potential relief
and these wells may have localized (exciton) states similar to the exciton states in solid solutions of semiconductors. Small Sect.
electron 4B)
initially
and
doses also
reduce
reduce
localized
the
surface
charge
states
electric
fluctuations,
disappear.
This
efficiency of other luminescent channels, ~ Z . , and bound excitons, 31).
The
field
so
temperature
the
increases
the
those due to free
which is evidenced experimentally
influence of
(see
that
can be attributed
(see ~'ig. to thermal
delocalization of excitons. This results in a gradual quenching of the
band,
starting
from
its
short-wavelength
edge,
and
by
a
relative shift of its maximum toward longer wavelengths. To conclude, we note that surface fluctuations of the potential do not necessarily result in localization of excitons as a whole but
only
one
of
the
carriers
in
an
exciton
can
be
localized
instead. As an alternative to the model
in question we can also
consider
electrons
recombination
with
holes
of
localized
at
surface centers. This model is also popular and ft~ther studies are needed to identify the origin of the broad emission bands.
244
A.E. Cherednichenko and V.A. Kiselev
6. Band Bending and Its Relevance to Excitons and Exciton Spectra In
this
section
we
shall
discuss
extrinsic mechanisms by which
one
of
the
most
obvious
the exciton transition layer (ETL)
is formed, namely, the interaction of excitons with a non-uniform electric field of a near-surface space charge layer the near-surface and
modifies
drastically.
electric
the
field that perturbs
exciton
This
is
reflectance
particularly
compounds
With
the
direct
absorption
coefficient
at
the
of
(ERL)
the
most
semiconductor
transitions,
exoiton
It is
the exciton state
lineshape
true
allowed
(SCL).
resonance
where
the
very
high
is
(I04-I05cm -I ) so that excitons are excited close to the surface. Hence,
the
aim
of
this
section
is
to
consider
the
issue
in
detail and also to demonstrate the sensitivity of the ETL to the changes
in the SOL.
Many anomalies
of
the
exciton reflectance
spectra can be explained by this mechanism. The influence of the SOL
may
be
extremely
varied
depending
on
the
electric
field
strength at the surface F s and the field penetration depth into a crystal w which is related to the screening length. For calculations of the reflectance
the approach developed in
[55] has been employed (see Sect. 4E). The SCL was devided into a set of thin Pekar's slabs [8]. By means of this approximation the problem is reduced to the calculation of the optical properties of a layered medium. The effect of the potential is to increment the exciton resonant frequency in each layer by an a m o u n t e q u a l to the height of the corresponding step. The fact that within each layer the potential
is
independent
of
the
coordinate
simplifies
the
problem considerably, since for each layer it becomes possible to apply the crystal optics of a slab with spatial dispersion. At the inner
boundaries
conditions
between
(ABCs)
require
the the
slabs
the
continuity
additional of
the
boundary excitonic
polarization and its normal derivatives. The set of ABCs and Maxwell boundary conditions at all boundaries, inner and outer, form
the
set
of
equations
that
are
to be
solved.
Results
of
calculations shall be compared with data by other authors as well
Excitons at Semiconductor Surfaces
as with our own results on the
245
effect of preliminary illumination
with light, of electron bombardment and externally applied field on the exciton reflectance.
Characteristic
features of the ERLs
and their response to external stimuli may provide an estimate of SCL parameters: F s, w and the charge density e(ND-NA).
A. Band bending in semiconductors Bending of energy bands in the near-surface region is considered to
be
one
of
the
most
important
features
of
semiconductor
surfaces. By depleting the near-surface layer of majority carriers the band bending builds a potential barrier for them (the Shottky barrier) devices
which based
plays on
a key
role
in operation
metal-semiconductor
or
of
metal
semiconductor -
insulator
-
semiconductor structures. The band bending near a free surface is the result of a screening of the field by the surface charge. This charge
is held by surface
electronic
states.
A great
effort
is
devoted at present to achieving an understanding of the mechanism of Shottky barrier formation on an atomic level,
i.e. to clarify
the physical nature and energy structure of the surface states. The
surface
charge
density
determines
the
electric
field
strength at the surface F s. Given the space charge density value near a semiconductor surface,
p = e(N + - N-),
concentrations
of
positively
and negatively
respectively,
one
obtains,
coordinate
dependence
of
the
using field F
the and
N + and N- being charged
Poisson of
the
centers,
equation,
a
electrostatic
potential ~ . For a depletion layer of a n-type semiconductor we have 4%e + F(z) =--~-(N -N-)
(z-
~(z) =
w),
(F(O) = F s, F(z > w) = O ) , (6.1)
2~e ~ (N+ - N-) (z - w) 2 ,
where g is the static dielectric constant.
(6.2)
246
A.E. Cherednichenko and V.A. Kiselev
Q
bIFI C
W
Fig. 32. Distribution of the charge density p (a), of the electric field F (b), and electrostatic potential @ (c) assuming complete depletion.
W These
equations
Z
will
be
reflectivity coefficient. -
N-)
is
constant
(in
used
(6.1)
equilibrium
(N+
bending of free carriers The
approximation.
last
calculate
the
exciton
,and (6.2) apply if (N+ -
N-)
is
a
difference
aoceptor concentrations, N D and only one type of donors is present
the temperature is low enough,
depletion
to
Equations
between the donor and respectively), if, then,
excitons.
below
NA, and
so that a contribution to the band
( ~ kT) will be too small to affect
condition
defines
Variations
with
the z
so-called of
the
the
complete
space
charge
density p as well as of the quantities F and 9, calculated in this approximation,
are
plotted
in
Fig.
32.
This
model
is
able
to
account for many anomalies in the exciton reflectance spectra that could not be explained in the framework of traditional approaches (see Sect. level,
4) and, as well,
modifications
(illumination, variations,
of
electron
applied
field,
to explain, at least on a qualitative the
ERL
under
external
bombardment,
etching,
oxidation
others).
and
influence temperature
On
the
model
depicted by (6.1) any two parameters out of available three: F s, N + - N- and w may be chosen to describe these modifications, w being the depth to which the field penetrates the crystal width of the SOL).
It is clear
that external
factors
(or the
(including
Excltonsat SemiconductorSurfaces
247
probing light in optical investigations) may affect not only the three parameters mentioned but also the function F(z) (as well as p(z) and ~(z)). For instance, high concentrations of the electron-hole pairs generated in SOL by intrinsic light may disturb the constancy of p(z) across SOL in such a way that the holes are accumulated near the surface and electrons near the inner SOL boundary (see, e.g., [86]). A non-uniform charging of centers across the SOL will proceed and subsequent charge redistribution will be acting so as to flatten the energy bands near the surface. Note, that the authors of refs. [87,88] used other expressions for P(z) We have also tried to modify their model by adopting F ~ (z - w)~ where n = 2, 3, or more, but no improvement on the results was found. Direct observation of flattening of the energy bands by light is possible utilizing exoitonio spectroscopy (see [89] and further). Therefore, the light intensity has to be kept very low for the photogeneration of pairs to be negligible. This is a proper way in probing with light. In other kinds of experiment the light intensity should by sufficiently high to permit the amount of band bending to be contr~lled and the model depicted by (6.1) may be used only in a qualitative consideration of the effect (Sect. 6D). Note, that illumination with light from within an appropriately chosen spectral range enables one to charge the surface states in a controllable way [89]. The model in question still applies in this case but an appropriate change in the parameter F s is required. Excitonic spectroscopy is most suocessfull at low temperatures. Besides allowing to ignore free carriers while seeking to establish the form of the barrier, low temperatures lead to long relaxation times, such that certain non-equilibrium processes (externally provoked band bending, e.g., under illumination with light [89], depletion [90], charging of a near-surface layer by irradiation with electrons [19], charging of surface states [89,33] and others) get "frozen-in" and can persist for long enough to permit measurements of optical characteristics of the exoiton at any given band bending value. The effects of this kind
248
A.E. Cherednichenko and VoA.Kiselev
include the dependence of ERL on whether a crystal is being cooled when exposed to light or in the dark. To get reproducible results, it is extremely important to take account of such effects. One
more
bending
property
of
the
of
energy
the
B0L
bands
that
is
the
may
be
observed
quantization
of
at
strong
the
carrier
movement normal to the surface resulting in the formation of a 2D exciton
(see, e.g.,E80,91]).
A detailed account of the problem of
2D excitons is beyond the scope of the present review.
B. The effect of a uniform electric field on excitons Prior field
to a
of
useful
treatment
BCL
to
on
the
consider
assumptions
of
the
effect
exciton the
to be made
and
case
of
in Beet.
of
its a
a non-uniform
optical
uniform
field
of
an
proximation)
exciton
as
it
because
is the
6C will allow us to substitute
uniform field for the non-uniform one. One assumption movement
electric
properties
a
whole
is
slow
a
is that the
(adiabatic
ap-
and the other is that field variations over distances
comparable to the exciton radius are small
(the approximation of a
point
of small
increments
in
field does not mean small absolute values of the field, which,
in
fact,
exciton). may
Note,
be
Investigations
that the assumption
very of
strong
the
effect
near on
a
semiconductor
excitons
of
have been initially prompted by observations on
excitons
e.g.,J33)
by
Gross
and later,
et
al.
the electron-hole
interaction
electric
(the
field
"solid-state"
as
by results
far
electric
field
of the Stark effect
as
the
mid-50s
(see,
of the research on the effect of
on
the absorption
Franz-Keldysh
investigations
back
the
surface.
were
effect
carried out,
of light
in the
[92,931).
These
at least
for some
time, in isolation from traditional calculations of the electronic states of the h~vd~ogen atom in the electric field
(see references
in [94,95]) that can equally be applied to a theory of an exciton. Out only
of a large number of works those
energies
that
close
deal
with
in this
theories
to the ground
state
field we
applicable
in
shall the
energy of an exciton
review
range
of
and,
in
Excitons at Semiconductor Sudaces
249
addition, cover the case of strong electric fields, up to such that ionize a hydrogen-like system in the ground state. We note, that the ranges of the Coulomb interaction and field magnitudes outlined above rule out the usage of the perturbation theory. The problem was found to permit analytical treatment if either some special form of potential distribution is used instead of the actual one (see Fig. 33) [96-98] or a limit is imposed on the magnitude of the field [99,100]. The most reliable results for a wide
range
of
field
values
have
been
obtained
by
numerical
integration of the Schrodinger equation [101-103, 94]. Characteristic scales of energy, length and field strength are defined in this model as follows [103]. The exciton binding energy in the ground state has the form (2.2) and the exciton radius is given by (2.4). The field required to ionize a system is found by equating an increment in electrostatic energy binding energy:
over
the
exciton
radius
F I = IEll/(eRexc) = (~/mo)2. g-3Fi H
to
,
hydrogen-like the electron the
exciton
(6.3)
where FIH = 2.57"109 V/cm is the ionization field for the ground state of the hydrogen atom. Factors in (2.2), (2.4), and (6.3) containing the reduced mass ~ and the static dielectric permeability diminish
IEll and F I by several orders of magnitude
and increase Rex c compared to the CorTesponding quantities for the free hydrogen atom so that fields as high as F I (and even stronger) seem quite feasible in semiconductors. Some examples are given in Table I (for more see [103,102]). Major results obtained for the hydrogen-like system in an electric field may be summed up as follows. In weak fields, F/F I ~ f << I, the exoiton level is shifted to long wavelengths (the quadratic Stark effect ) by (9/8)f 2 in units of the exciton Rydberg: IEl(f = 0)I. This shift is due to some broadening of the Coulomb well under applied field, see Fig. 33 [103]. The field creates a potential barrier, such that the energy needed to pass it over would have dissociated the exciton. As the field is increased, the top of the barrier lowers
250
A.E. Cherednichenko and V.A. Kiselev
Fig. 33. A potential of the electron hole interaction in the presence of an extez*nally applied uniform electric field (solid curve) and without field (dashed curve).
TABLE
1
Rexc, nm
IEII, meV
F I, V/cm
GaAs
13.6
4.2
) -103
InP
12.5
4.9
4"103
OdSe
5 •I
16
3.104
OdS
2.8
28
1 • 105
Excitons at Semiconductor Surfaces
faster
than
the
ground
state
level
and
at
251
f
=
fc
~
0.13
a
classical transition of the barrier becomes possible (if the Stark shift is neglected,
then fc = 1/8 [1021). The finite probability
of the exciton dissociation (both tunnel and classical) the
exciton
fulfilled,
level. i.e.,
When
if
f
a
condition
<<
fc
the
for
density
deep
broadens
tunnelling
of
states
in
is the
resulting band is described by the Lorentzian distribution with a halfwidth (in units of IEll)
r/2
=
(6.4)
(81f) exp [-4/(3f)]
Despite the fact that the resonance happens to occur above the barrier for f > fc the
shift of the
level to long wavelengths is
nearly quadratic up to field strengths of about f ~ 0.25
[1031,
and the density of states distribution still retains the maximum, which can be followed up to f ~ I [101-103] (Pig. 34), i.e., up to the field value at which the bandwidth exceeds the exoiton binding energy (P =fEll for f ~ 0.75 [94]). The occurrence of the maximum in the spectrum of unbound states is due to quantum interference that an electron wave experience when passing over the well [102] (this effect for the case of a rectangular well has been discussed in Sect.
5).
It seems that few of those engaged in research on
excitons are aware of the fact that they often deal, in fact, with the unbound state at the above-barrier resonance rather than with the bound exciton state. The situation when in the semiconductor near surface region f ~ fc ~ 0.13 is met rather frequently. this
field value
the
shift
to
long wavelengths
and
broadening
amounts to a merely ~ 0.O2.1Eli and O.003.1EII, respectively, example, for the An= I exciton in 0dS
At (for
the respective values are
0.6 and 0.09 meV). As the field f ~ 0.4 is reached, the energy of the resonant exciton state assumes a value of ~ 1.1.E1(f = O) and with further field increase the resonance shifts in reverse, i.e., towards short wavelengths, - see Fig. 35 [103]. Hence, the maximum possible value
of
macroscopic field
the exciton resonance can produce
exoiton binding energy.
frequency
shift
that a
amounts to about one tenth of the
252
A.E. Cherednichenko and V.A. Kiselev
Eg.-ILll
Lg
Fig. 34. Profiles of the'density of states ~2(0) (relative units) of a hydrogen-like system for different values of the uniform electric field [101].
E,(,f)/IE,(O)I -0.2.
-0.6 -'1.0 O.q
0.5
2.0
Fig. 35. Variation with eleotrio field f of the ground state energ~y of a hydrogen-like system relative to the unperturbed oontlnuum [103]; dashed curve - perturbation theory.
Excitons at SemiconductorSurfaces
253
For the excited exciton states with the quantum number n above unity the exciton dissociation is possible at substantially lower fields, e.g., at fc(n) = fc(1)/n4 [102]. Therefore, at f ~ 0.2 essentially no peaks remain in the density of states spectrum except the one with n = I. This peak becomes broader and merges into the continuum, loosing its symmetry, longer be described by the Lorentz formula. At
stronger
fields
the 0oulomb
that
interaction gets
is,
it
can
no
progressively
less effective and for f ~ 10 the density of states calculations are but little influenced by an account of this interaction [103]. Most convenient for treating the effect of a non-uniform field on excitons happen to be the expressions, derived for the hydrogen atom using the results of ref° [94]. The calculations include strong fields and resonant energies, with the energy and broadening as major parameters of resonances. The field and energy values can be given in units of F I and IEII, respectively, of atomic units. To
solve
the
Schr8dinger
equation
for
the
instead
hydrogen
atom
subjected to the electric field the variables were separated by employing special "quadratic" pax~abolic coordinates and the problem thus reduced to solving a system of two differential equations. Numerical integration of the equations was performed using the method of series reexpansion. If a condition is imposed on one of the two equations that the wave function be finite at zero and decay exponentially to infinity, then a discrete spectrum of
eigenvalues
is
obtained
(with
the
energy
E and
field F
as
parameters). In our calculations, though, the demand of exponential decay was replaced by that of the wavefunction be zero at some point sufficiently remote from the origin of coordinate system. The other equation which depicts the transition through a barrier leads, in fact, to a problem on quasi-stationary states. Its solution was made to matah with an asymptote of the wavefunction also in a enough remote point, which yielded the asymptote parameters, the amplitude and phase, as functions of the energy E and field F. By analogy with the scattering problems, from the analysis of dependences on energy of the phase and
254
A.E. Cherednichenko and V.A. Kiselev
amplitude value.
the
The
EI
and
r
values
calculations
were
can
be
deduced
carried
out
for
along
every
the
field
lines
of
Breit-Wigner theory. The values of E I were deduced under condition that the squared asymptotic divergent
wave
is present)
parametrization phase
shift
Generally
amplitude
of
in
the
the
the P values
dependence
proximity
speaking,
parametrization
and
the
be minimum
on
of
obtained using
of
the
of
the
asymptotic state.
Breit-Wigner
is open to question.
of the ground state band [101-103]
only the
quasi-stationary
applicability
at high fields
were
energy
the
(i.e.,
An asymmetry
is indicative of a necessity to
go beyond the bounds of this theory.
However,
the probability
of
exciton dissociation for the noticable asymmetry of the band is so great,
that
the
bandshape
effects considered.
is
not
of
much
importance
Still, we intend to discuss,
for
the
on a qualitative
level, deviations from the Lorentzian lineshape evident in the ERL rotation above
(Sect.
deal
6G).
To conclude,
exclusively
calculations
to
Generalization
follow
we note
with
simple
will
be
of the results
that
the works
electronic
restricted
cited
bands.
to
to cover non-uniform
Our
this
case.
fields
can by
performed using the method discussed in Sect. 4.
C. The effect of a non-uniform electric field on excitons As
indicated
semiconductor considered
in
above,
the
may
strong
be
the previous
electric
field
enough
section.
to
within cause
all
Inhomogeneity
makes the density of states distribution
the of
SCL
the
of
a
effects
this
field
(Fig. 35) vary from point
to point along z. The effect on excitons of a non-uniform electric field near
the
crystal
surface
Gribnikov and Rashba
[104].
that
pulled
an
exciton
interaction result,
of
is
the
induced
was
first
In particular, into
the
dipole
treated
by
they have demonstrated
high-field
moment
in a paper
with
the quadratic Stark effect takes place,
region
this
due
field.
to
As
a
i.e., the resonant
frequency of the exciton near the surface gets lower. In ref.[IO4] the
influence
of
this
effect
on
the
exciton
diffusion
was
Excitons at Semiconductor Surfaces
analysed.
The analisys was later continued in
255
[105], emphasizing
the role the near-surface field plays in exciton dissociation. The possibility that the near-surface field might pull in and ionize the exciton was also suggested by Allen [106]. In the most definite manner the influence of the space-charge field on the exciton states in semiconductors was established in experiments o n reflectance and electroreflectance. For some crystals interference of light has been observed in the region of the exciton resonance. An additional reflection of light from the inner SOL boundary where an abrupt change of the dielectric permeability occurs was observed on CdS crystals at 273K [107] and 4.2K [121] as well as on GaAs [25,108,109] and InP [17] at liquid helium temperatures. At low temperature this interference could be observed without modulating the applied voltage, as a rotation of the ERL following alteration of the contact bias or under electron bombardment of the surface (see Sect. 4B). At room temperatures when the exciton peak in reflectance does not appear, voltage modulation is indispensible. (Analogous rotation can also be caused by an "excitonless" Franz-Keldysh effect in the SOL of a semiconductor [110], present review).
but
The ERL rotation was
this issue obtained
is beyond
the scope of
in terms of a theory
the
[17, 18]
which simulated the high-field region (F a F I ) by a uniform dead layer [12] of varying thickness (Sect. 4A). A m o r e adequate description based on adoption of the actual variation of the electric field with coordinate z (Sect. 6A) and assuming a gradual variation of the resonant frequency and of exciton damping with Z was proposed in [56,111], the calculation method has been given in Sect. 4E). In these studies an account was taken of the fact that the layer free of excitons is formed near the surface due to high probability of the field dissociation of the exciton (and not to an abrupt change of the exciton resonant frequency ~0 [17,18] ). Besides providing a reprodution of the effect of the ERLrotation, these calculations have also predicted or confirmed a variety of effects in the exciton reflectance spectra (see below). Other models were proposed to describe the exciton behavior
256
A.E. Cherednichenko and V.A. Kiselev
the near-surface field. In ref. [53] the field dissociation of the exciton
was
assumed
to depend
exponentially
on
coordinate.
The
probability of the field dissociation was found to decrease with flattening
of
the bands
near
the
surface.
Lagois
[54] using
a
three-layer approximation has derived the exciton potential, again relating Balslev
this
to a
strong near-surface
[87] used the exponential
screening
length,
to
describe
field.
Schultheis
term exp(-z/d),
the
coordinate
and
d - being the
dependence
of
the
field strength. Intrinsic mechanisms of the transition layer formation were
taken into consideration and variational method was
used to specify the adiabatic potential. Eleotrodynamic equations including
those
for
the
exciton
polarization
have
been
solved
numerically. The calculation has confirmed the possibility of the ERL
enhancement
and,
as
well,
Besides,
the
with the
variation
of
alternative
calculations
the near-surface
mechanism
performed
in
of
field
spike
[87]
formation.
yielded
structure typical of very wide and shallow wells
[111]
the
ERL
(see Sect. 6G),
the result to be expected in view of the assumptions made: d ~ 400 nm
and the field at
Sect.
6B).
near-surface
The
increase
excitons
essentially affect was
the surface not by
in
the
such
justifiably neglected
value
fields
the ERL, hence, [87].
in excess of 0 . 3 F I of
damping
is not
yet
(see
of
the
sufficient
to
to a first approximation,
it
Schultheis and Lagois
[88]
took
account of both the resonant frequency shift in the field, varying with coordinate as
(z - w) 2 and the damping, which,
was assumed exponential. The way in which the
as in [53],
ERL is affected by
the parameters of the SOL in GaAs crystals has been considered. Lowering of the exciton resonant frequency near the surface due to the Stark effect may lead to localization of the exciton as a whole in the potential well thus formed [78,29] (manifestations of the
exciton
localization
in
the
photoluminescence
spectra
have
been discussed i n s e c t . 5). Apart from field effects an impact ionization of excitons near the
surface
structures
can (see,
be e.g.,
scope of the review.
observed [53])
due but
to these
current effects
flow are
in
contact
beyond
the
Excitons at Semiconductor Surfaces
D.
257
Exoiton refleotivity in the case of thin SOLs In
this
section
calculations
the
results
will
for the An= I exciton
in a
[56,33]. The calculation procedure, proceeds the
from defining
exciton
resonant
presented
of
ERL
CdS crystal with a SOL
that has been outlined above
the functions
frequency
be
shift
A~o(Z) and
and r(z)
damping,
Outside the SOL, i.e., at z ~ w , one has A~ 0 = 0
which are
respectively.
and P = F O, the
latter being the exciton damping in the bulk of the crystal. If a field variation near the surface F(z) is known (see Sect. 6A) then the functions A~o(Z) and r(z) may be readily determined (see Sect. 6B). For a linear field variation two
values
of
w
the
resonant
(6.1), N D - N A = I016cm -3 and
frequency
shift
and
damping
as
functions of z are shown in Fig. 36 [56]. It is seen from this figure that A~o(Z)
forms a potential well
and r(z) increases towards the surface producing a physically dead layer
for
excitons.
For
larger
surface
fields
Ps
(which
is
proportional to w) the curves for A~ 0 and P are shifted as a whole along
the
surface
z-axis.
(Fig.
36b)
Eventually, where
A~ 0
a
layer may
and
r
are
adiabatic conditions are not warranted.
develop
not
adjacent
definable
However,
and
the the
this region is
inaccessible to the exciton and various attempts to determine A~ 0 and
r
there
have
had
little
effect
on
the
result
of
ERL
calculations. In calculating the ERL the following parameter values were used [38]: ~0 = 20590.2 cm -1 ~LT 15 cm -I ~0 9.5, M 0.78 mo, 9.3 [112], where ~0 is the resonant frequency of the An=1-exciton , ~LT' the longitudinal-transverse splitting, dielectric permeability in a region of translational optical
axis
mass O
of
, m O,
the
electron
the free
moving
~0' the background resonance, M, the at
electron mass,
a
normal
and ~,
to
the
the static
dielectric permeability. For the given parameter values the result of ERL calculation is defined by any two of the three SOL parameters: F s,
the SOL
thickness w,
and
the centers
the surface field
concentration N + - N-
258
A.E. Cherednichenko and V.A, Kiselev
r c~ t i
Z00 a
b
Z,~m
° V,.7°., -10 ~ I~
I
I"%1
I
-20 -,.Jx.L/ AUJO ~rlz "1
Fig. 36. Damping and resonant frequency shift A(0 of the exciton in the Shottky barrier on a CdS crystal; a -IV, b - 2.8V. which
defines
the
space
charge
density.
ERLs for a variety of combinations
of
We
have
(N +- N-,
calculated w) values,
the some
examples are given in Fig. 37 (for N D - N A equal to 2-1017cm-3(a), 5.1016cm-3(b), and 1 . 1 0 16cm-3(c)), more will be presented in Sect. 6G. With respect to their peculiar features all ERLs may be divided into types plane, Fig.
I, II, III, and IV. If mapped on the
38 by
bold
curves.
origin of coordinates (N+ - N-). certain of
(F s- w)
each ERL type is confined to a well-defined area shown in
Of
the
Thin
("rays")
sets
straight
lines
coming
out
of
the
correspond to different values of
of ERLs
in Fig.
37
each
corresponds
(Fs, w) points on a particular ray which means
to
constancy
(N+ - N-). The numbers at each curve indicate the values of w
in nanometers. fine structure,
Area I in Fig.
38 contains
spectra
exhibiting no
the examples are the spectra for w = 0.12
in the
first row, w = 10, 27 in the second and w = 20, 50 and 70 third row in Fig. spike
37. Area
II
contains
the ERLs with
in the the left
(see Sect. 4 ) i.e., the one to the left of the reflectance
minimum
( for the chosen direction of wavelength increase).
These
Excitons at Semiconductor Surfaces
259
a
ioI ? 0.6
0.4
Iz
0,2 0
0.~
Fig. 37. 0alculated ERLs for the A(n=l) - exciton in a CdS crystal calculated using a model of the Shottky barrier for different w values and at three values of N D - N A (see text).
are exemplified
in Fig.
the first row and
37 by the lineshapes
for w = 17,
18
in
w = 30, 31 in the second one.
It is important to note that the left spike is accounted for in our calculations assuming reasonable values for the bulk damping, say, ~
= 0.Scm -I ~ 0.1 meV
(the value used in
computing
all
of
260
A.E. Cherednichenko and V.A. Kiselev
F0 / 0.5_
05
0
,*5
5O
75 -~
w, nrn
Fig. 38. Contour plots of constant Rma x value (thin curves) and areas
(I, II,
III) of characteristic ERL behavior on
the (Fs-W) plane. Straight lines correspond to constant values of N +- N- in units of 1017 cm-3; ~F O = 10 -4 eV.
the
spectra
in
measurements
Fig.
of
37).
the
Close
forbidden
value
has
exciton
been
obtained
linewidth
from
[12],
and
longitudinal exciton linewidth [27] at liquid helium temperatures. On
the other hand,
the
spike
of
such a
shape
explain within the uniform dead-layer model substantially [19,22].
lower
values
In addition,
(~ 10-5eV)
in our model
of
damping r increases towards the surface very
high
values,
for
instance,
190
for the
respectively,
in Fig.
37.
The
difficult
the
exciton
to
damping
transition layer
the
(see Fig. 36) and reaches cm -1
linesh~pes with w = 18 nm in the first row and second,
is
[12] as it requires
and
30
cm -1
for
w = 31 nm in the
observation
that
for
the
Excitons at Semiconductor Surfaces
left spike to appear the damping should
261
not at all be too
low
attests
the mechanism under consideration as more realistic
than
the one
discussed
in
[12].
In particular,
a serious
discrepancy
between the values obtained from experiments on reflectance and on luminescence Area
III
[19] gets an explanation. in the F s- w plane plane
right spike
contains
the ERLs with
the
(see w = 20, 22 in the first, w = 33, 40 in the second
and w = 75, 80 and 90 in the third row of Fig. 37)
as well as the
"reversed" ERL. The effect of the ERL reversal (or rotation) demonstrated in Fig. 39 for N + - N- = 1016 cm -3. The effect
is is
seen to start with an appearance of the right spike and follows in its major features the predictions of the uniform dead layer model [18], though detailed agreement
is lacking. Area III, as well as
area IV, featuring small F s and large w values shall be discussed at length in Sect. 6G. Here we will concentrate on areas I and II, i.e.,
restrict
ourselves
to
fields F s around 104 V/cm
the and
cases
of
w
w ~ 25 nm
~
10 nm
with
with
the
F s around 105
V/cm, in conformity with this section's title. Besides
producing
the left
and right
spikes,
the near-surface
field changes reflectivity coefficients at the ERL extrema and Rmin)
stretches
the spectrum over greater
Fig. 37. These changes may be mapped on the F s-
w
drawing
values
curves,
parameters. constant
corresponding
As an illustration,
reflectivity values
to
(Rma x
energy range, s e e
constant
we show in Pig.
plane too, by of
ERL
38 contours of
at Rma x with a spacing between
the
curves ARma x = 0.05 . It is seen that Rm~ x as a function of F s and w has a peak value of 0.82 at F s = 0.25.105 V/cm and w = 27 nm, that is, Note,
the ERL is enhanced at these values
that at the values of F s and w close
of the parameters. to the above
quoted
there also occurs peak in R m n i (see Pig. 37). In the next section we demonstrate that the most likely cause of the ERL enhancement is the quasilocalization
of
the exciton at a near-surface
Stark
well. The presence
in the spectrum of
the anomalies mentioned above
(enhancement, left and right spikes) sets the following limits on SCL parameter values (for CdS):
262
A.E. Cherednichenko and V.A. Kiselev
Ri
~_
0.3 0.2 0.1 0 L~550
!i -
20600 20550
f
2060O oJ ~ C1~ t
Fig. 39. Reflectance of a CdS crystal in the vicinity of the A(n=1) -exciton for different values of the surface potential ~s V: a)- O, b)- 0.065, c)- 0.1, d)- 0.2, e)- 0.5, f)- 0.75.
104 <~ F s ~< 105 V/cm , 10 <~ w ~< 102 nm , 1016 ~< N + - N- ~< 5-I017cm -3
(6.5)
Many of the results of experimental investigations on exciton reflectance discussed in Sect. 4 can be understood in terms of the effect of SCL. For instance, two very similar F~Ls for w = 70 and 80 mm in the third row of Fig. 37, the latter featuring the spike and the former having none, may be compared to the spectra from the paper by Permogorov et al. [32]. In that stud~ the spectrum with the spike has been obtained umder intense illumination of the sample, which could, for example, produce charging of surface states [113] or cause a slow (frozen-in at low temperatures) recharging of the centers in the SOL. Below, we consider at some length the effect of preliminary illumination described in [33J. Short exposure to light also makes w somewhat larger but changes
Excitons at SemiconductorSurfaces
263
of the ERL will be different in cha1~aoter and similar to those in the first row of Fig. 37. It is natural to suppose that these different kinds of behavior arise from differences between the (ND - N A) values in the crpstals under study. Evangelisti et al. [18] and, especially, Patella et al. [15] stressed that the intensity of the spike is not rigidly related to that of the main reflectivity peak Rma x as the uniform DL model would have implied [12,18]. This subSect has been already discussed in Seat. 4, and a new mechanism of spike formation should be suggested, namely, the one that assumes the existence of a potential well for the exciton near the surface. In the model of the transition layer for the exciton considered here the above well apRears as the S~ark well produced by the nea~-surface field (see Pig. 36). Indeed, as seen f ~ m Pig. 37, with variation of the SOL pa1~ameters the relationship between Rma x and spike intensity might be most diverse. AS indicated in Sect. 4, by subjecting the surface to electron bombardment a varied and complex influence is exerted on the near-surface region and, consequently, on the exciton reflectance, It seems, though, that in some cases chauges in the ERL are q u i t e definite in character and ma~v be related to o h ~ e s in SOL parameters. So, after small (undestructive) doses of electron irradiation there is observed a conversion of the right spike into the left one and back, the ERL enhancement and at greater doses the ERL rotation is revealed (see Pigs. 9,10). The values of Rma x and Rmi n may also be changed by varying temperature (see Pig. 14). it is of special interest that as the temperature is raised from 2 to 30 K the value of R m i n may drop to practically zero, as reported by Pevtsov et al. [27]. This is contrary to the result of the calculations assuming an increase in PO provided no transition layer at the surface is taken into account, Bearing in mind that the SOL may undergo changes with temperature variation the outlined behavior is no longer surprising. For example, for w = 70 and 50 nm, in the third row in Pig. 37, the value of R m i n drops frc~n 4.6% down to 0.24% while the lineshape remains essentially the same. Another, more definite
264
A.E. Cherednichenko and V.A. Kiselev
interpretation of such behavior will be put forward in
Sect. 6E
where the dependence of the reflectance spectrum on the damping in the bulk PO is considered. There was investigated the influence on reflectance spectra [33] of preliminary exposure Fig.
to light with k ~ 51Ohm
(see Sect.
40,
15). Reversibility and reproducibility of the results prove
that
the major
various
effect
electronic
parameters.
These
of
states
illumination was with
experiments
that
resultant have
given
of
recharging
changes
in
evidence
SOL
of
a
quasilocalization of the 3D exciton revealed through reflectance spectra. Modifications of the ERLs with increasing amount of preliminary illumination Fig.
40.
The
(time of illumination in minutes) illumination-induced
changes
are displayed in
persisted
at
liquid
helium temperature for long enough, so that in the course of the reflectivity measurements, with the use of a probing light of very low
intensity,
the
SOL
configuration
remained
essentially
unchanged. As seen in the figure, the experimental spectra display all the major features
that appear in the calculated spectra of
Fig. 37, namely, the ERL enhancement followed by the emergence of the left spike, with an accompanying decrease (Rmin)
and by stretching of
the ERL along
(increase) of Rma x
the frequency scale.
Then the ERL is again enhanced and the reflectance damped. Such a complicated behavior of the ERL under illumination indicates that the corresponding trajectory in the
F s- w plane (Fig. 38) must be
very intricate as it should pass twice the area of high values of Rma x at exposures of 5 and 15 minutes. Modifications of the ERL at fairly low exposures to light
(0.5
and 10 rain) correspond best to a set of lineshapes seen in the upper row in Fig. 37 for which N +- N- = 2.1017 cm -3 and w = O, 12, 17 n m , though, of course, the trajectory might be, as well, both more steep (like the one for N + - N- = 5"1017 om -3 and more sloping
(as for F s ~ 105 V/cm).
trajectory starts
from Area
I,
Still,
it is certain that this
then passes
through a region of
fairly high Rma x values and finally enters region II where F s 105 V/cm. From here variations of SOL assume different character.
Excitons at Semiconductor Surfaces
R
0.3
F fO~n
265
f 2O
o.2[
/
p
Fig. 40. Modifications of the ERL of a CdS crystal at T=4.2K caused by preliminary illumination (of. Fig. 15).
One of the most likely reasons for the growth of F s and w at low doses of illumination is charging of surface states [113]. This seems
to be
minutes,
also
the reason
times of relaxation
for fairly
long,
to the initial
of
state.
the order
of
A 10 minute
exposure completely fills the surface states. Longer exposures lead to flattening of the bands, with the band bending region receding towards the surface. Flattening is effected by photogenerated pairs of carriers when these get separated in the near-surface field and driven toward the low field region (electrons) or the high field region (holes). The electrons will be captured by ionized donors thus decreasing the space charge density in the low-field region while the holes will be captured
266
A.E. Cherednichenko and V.A. Kiselev
R 0.5
0"3
n
0.2-
V 0.1
--
0 Fig. 41. Modifications of the ERL in the vicinity of the A(n=1)- exciton induced by various doses of 3.5 keV electron irradiation.
by charged aeeeptors, surface.
The holes,
increasing the space charge density near the in addition,
might be captured on the levels
of the 2D quantum well on the surface, acceptors
are
completely
filled.
As
both during and after the
calculations
show,
10 5 V/cm and w of the order of tens of nanometers series of two-dimensional To
establish
circumstances self-consistent
the is
a
subbands separated by
spatial
distribution
difficult
task
that
of
manner and requires knowledge
Therefore,
a
calculated
on the model
comparision
of
the
assuming
may turn out to be inadequate.
the
must
Nevertheless,
Fs ~
there exists a
~ 10 -2 eV. field be
in
solved
these in
a
of many parameters.
experimental a linear
at
ERLs
decrease
with of
those
the field
it is clear that the
Excitons at Semiconductor Surfaces
exposures average,
of
interesting control,
15,
hence
20
the
that
and
30 min
repeated
in
some
make
the
enhancement
267
field weaker of
experiments,
the
where
on
ERL.
situation
is
evade
this enhancement is accompanied by transformation,
that the left spike gives place to the right one,
the
It
such
see Fig.
40,
which implies that the descending part of the trajectory on the F s- w
plane transverse Area II in Fig. 38.
Very much doses
of
irradiation
crystals. this
similar modifications with
These are shown
case
the
of
the ERL result
electrons
of
the
surface
(for the same sample)
experimental
ERLs
compare
from of
in Fig.
best
to
a
small CdS
41. set
In of
lineshapes calculated for N + - N-= 5-1016 cm -3. It is evident that these lineshapes belong to Areas II and III of the F s - w plane as they
display
demonstrates
all that
the
typical
the surface
anomalies. is being
The
ERL
modification
charged negatively when
subjected to electron irradiation.
E.
Quasilocalization of excitons in the SCL field and the effect
of damping In the preceeding Section we have shown that for thin
SCLs ( w
~< 102 nm for CdS) the reflectivity in the vicinity of the An= 1exciton
might
be
expected
to
enhance.
An
anomalously
high
reflectivity may occur both in as-grown crystals and after some external
factor
has
been
applied.
The
reflectivity has been actually observed,
enhancement
of
the
some examples have been
given in Sect. 4, more will be discussed below. The effect under consideration was
found to be related to a feasibility for the
exciton to be localized by the SOL [78]. Fig. 42a shows grafs of A(~o(Z) and r(z) for N + - N- = 5.1016 cm -3 and w = 31 nm. A corresponding ERL in Fig. 37 is strongly enhanced and features the left spike. We note Fig.
that the lower part of the curve A~00(z) in
42b may be approximated with a triangular and show in this
figure two lowermost levels of the calculated using a known formula:
triangular well
gO
and £1'
268
A.E. Cherednichenko and V.A. Kiselev
r , cm -t
20~
a b
gl
f
/
0
/
-10 -2D
!
0
10 ,~U.)o~Cm-t
20
30
Z,~
Fig. 42. (a) The resonant energy shift A~ and damping F(z) for the first ERL of Fig. 46. (b) A~(z) for the potential approximated with a triangular. C O and C I are the bound states.
[
3~e
e~ =
Fs
3 ] 2/3
(~ + ~)
(6.6)
,
2v' 2M
where F s is not the electric field at the surface but a quantity defined as
~ o (z) i Fs : B ~-~ Iz=O
(6.7)
As seen in the figure, with the parameters as indicated,
the gO
level occurs somewhat below the edge of the real potential well. In
other
words,
an
enhanced
ERL
points
to
the
presence
of
potential well, sufficient to localize the exoiton. Simulation of a real potential with a triangular one gives, course, Yet,
only a crude estimate of the level position
a more
the magnitude
accurate of
calculation
P(z)
near
seems unnecessary.
the surface,
this
a of
in the well. To
level,
judge by due
to a
Excitons at Semiconductor Surfaces
269
high probability of the field dissociation of the exciton, spreads over the energy
interval
that exceeds
the binding energy of
the
exciton in the well. We have, in fact, dealt with the ERL enhancement due a single level. At concentrations N +- N- greater than 5.1016 cm -3 the enhancement looses in strength (see Fig. 38) as the potential well gets narrower while its depth is limited to a value ~ O.IIEII
(or
about
the
3
meV
contrary, of
for
CdS).
For
lower
values
of
N +-
N-,
on
the wells become wider and more levels for localization
excitons
may
arise.
At N +- N- = 5"1016
cm -3
the
levels
are
separated by more than I meV (see Fig. 42) and at N +- N - = I016cm -3 the separation between the levels is estimated at a few tens of a millielectronvolt. ERL
enhancement,
as
seen
With several levels in the well
in
Fig.
37,
is
less
pronounced
arises at larger w values. At still lower N + - N- values cm -3)
the levels
in the well
form a continuous
the and
( ~ 1015
spectrum and the
effect of enhancement is no longer observed. A
conclusion
to
be
drawn,
that
is
very
important
for
interpreting the ERL anomalies observed,
is that because of large
r
does
values
the
near-surface
detectable structure
field
not
produce
of the main reflectivity peak whatever
any the
value of N + - N- . Still, we may expect that some structure could be detected using differential technique. This hope finds support in the results of calculations
[56]
the behavior of reflectivity at
(for
N + - N- = 1016 cm -3) of
some fixed frequency from within
the resonance range as a function of the surface potential width,
w
oscillatory
).
The rather
reflectivity than
measured at a fixed frequency It is natural
has
monotonous
been
found
character.
The
to
(or SCL have
(20600 cm -I ) are shown in Fig.
to relate each successive peak
an
pulsations
to an emergence
43. of
one more level in the Stark well. This conclusion is oozu~oborated by estimates of the strength of the corresponding potential well. Experimentally,
the
ERL pulsations have been observed on a CdSe
crystal subjected to small irradiation
that
produced,
(and increasing) primarily,
exposures of electron
alteration
charge. The variation of R is presented in Fig. 44.
of
surface
270
A.E. Cherednichenko and V.A. Kiselev
R 0.4
0.2 I
I
0.1
0.2
I
0.3 )llZvllz $ 1w
Fig. 43. Pulsations of the reflectance in CdS at a fixed frequency with varying ~s (calculated).
R
0.6-
0.¢ 0.~ 0.2I
I
I
I
I
5
I
I
1 I
I~1
fd'fO
I
I
ZO JO
I
4O
Fig. 44. Variations of the reflectance at a fixed frequency in a CdSe crystal with increasing electron irradiation dose at T=4.2K.
Excitonsat SemiconductorSurfaces
271
A feasibility for the exoiton to be localized by the space-charge field settles one of the long-standing problems of excitonic spectroscopy. Quite often, reflectance spectra exhibit strongly enhanced structure for the An= I- exciton, including the spike and an unusually high value of Rmi n, whereas the structure for the Bn= I - exciton retains its usual appearance. This seems strange as the intrinsic properties of the A- and B- excitons are very similar. Differences
in the ERL for these excitons were observed
on virgin high-quality CdS surfaces [12,18] oriented both along and normal to the OdS optical axis. We have observed these differing
ERLs
upon
small
doses
of
electron
irradiation
(Figs.
9,10).
R
Ot
An=l 8n=I
/
/
/
b
An-1
,/~ Bn,,,t
F
0.1I I I I I I I
I
I
I
I~,I
6?9 "
I
i
I
670
I
669
~, n m Fig. 45. ERL of CdS (a) and CdSe crystals (b) after small dose of electron irradiation at T=4.2K.
272
A.E. Cherednichenko and V.A. Kiselev
Characteristic low-fluence
spectra
electron
can be made
is much more sensitive
has
localization
on
the
and
CdSe
are given
crystals in Fig.
conditions
45.
From
the
than that of the
consider the effect the damping in the
reflectance
spectrum
which
of the exciton by the SOL field
the calculation
following
that the ERL of the An= I- exciton
to the surface
Bn= I - exciton. To explain this behavior, r0
OdS
bombardment
figure a conclusion
bulk
of
are presented
in Fig.
case of N + - N- = 5-1016 cm -3 and
corresponds
[78]. The results
46 for the above
w = 31 n m .
to of
considered
With tO= I cm -I the
ERL displays anomalously high Rma x and R m in values and the presence of the left spike, with P = 2 and 5 cm -I there is no spike and Rma x and Rmi n are less. At F = 10 cm -I ERL responds no
transition
that
layer
were
the B- exciton
exoiton.
and 20 cm -I the
to the increased damping in a classical manner,
Hence,
present.
is damped
the
first
to the A- exciton and, the spectra in Fig.
So,
several
lineshape
e.g.,
the
it
is
times
in Fig.
third one
natural
stronger 46 should
to
suppose
than
to the B- exciton
R
0.6 0.4
0
2 =lC
5
10
the A-
correspond
45 get explanation.
O.Z
as if
?.0 ~-~
Fig. 46. Modification of the ERL of a CdS crystal with increasing the exciton damping PC in the bulk.
and
Excitons at Semiconductor Surfaces
273
The same reasoning may be used to interpret a drastic decrease of
R mn i
with
rise
Pevtsov et al. increased
of
temperature
from
2
to
30 K
observed
[27], - see Pig. 14 (at higher temperatures
in
a
usual
way,
same
as
in
Fig.
46).
by
R mn i
Analogous
explanation may be given to a non-monotonic dependence of Rmi n
on
excitation intensity [114].
F. The effect of SCL on the phase of reflected light So far, we have been coefficient R
, which
concerned only with
is an
the reflectivity
intensity ratio
of reflected and
incident light: R = Ir/al 2 Yet,
r/a
is
a
complex
quantity
(6.8)
characterized,
alongside
the
module, with a phase: r/a = Ir/a i e(~ where ~ has
the
sense
of
a phase
,
shift
(6.9)
of
the reflected wave
relative to the incident wave. Frequency dependence of 9, as well as
of
E,
resonance
exhibits and,
in
peculiarities essence,
in
should
the
also
be
region
of
exciton
sensitive
to
the
transition layer at the semiconductor surface. A "non-classical" phase behavior has been reported in a number of papers
[115-120,
31]. The frequency dependence of ~ in the vicinity of the exciton transition has been found to deviate markedly from the calculation that neglected both the transition layer and spatial dispersion, as seen in Fig. 47 (curve c). This dependence is either S-shaped and
shows
resonance
a
step-wise
[116]
(Fig.
change 47,
by
curve
2~ a)
in or
the
region
N-shaped
of
exciton
and
has
a
pronounced negative dip in the short waves [117] (Fig. 47, curve b). The non-classical phase variation can be adequately reproduced within a theory that considers
the uniform dead layer
(DL) and
274
A.E. Cheredniohenko and V.A. Kiselev
2~
0 =
b
Fig. 47. Types of the frequency dependence of the phase of reflected light near exciton resonance. a) non-classical S-shaped; b) non-classical N-shaped; c) classical; b and c may be derived from a if the damping F 0 is incresed.
•
I 0
5OO
I 1000 W ~ nnt
Fig. 48. Areas (I-III) of characteristic ERL behavior. Curves I to 5 correspond to an appearance of the first and subsequent spikes. + Thin ~ l i n e s correspond to constant values of N -N-(~IO'~).
Excitons at Semiconductor Surfaces
[116-119, 31]. The type of frequency dependence
spatial dispersion to
be
expected,
275
S
or N
, is governed
by
the
exciton
damping
[117,119]. An uncertainty remained, however, concerning the extent to which the DL in the spirit of the
Hopfield and Thomas model
essential
the
in
Regrettably,
accounting on
for
other m o d e l s
lacking, nevertheless,
observed
such
an
phase
[12] is
peculiarities.
exhaustive
analysis
is
the question raised may be answered right
now: the uniform DL serves a good approximation of real transition layers
as
regards
the phase,
though
this
model
should
not
be
considered as ultimate. Our results are as follows: i) with OdS crystals having N + - N- > 1016 cm -3 , which exhibit ERL rotation at
strong
bending
of
the
bands
(Sect.
6G)
and,
consequently,
possess a sharp enough DL boundary, even at small w values, sufficient for the ERL enhancement to begin,
- see Fig.
just
37,
the
S-type phase will be observed (Fig. 47, curve a). ii) In the ease of N + - N- < 1015 cm -3 when the real DL boundary is diffused and no ERL rotation occurs
(Sect. 6G) a model incorporating only the
SCL gives
differing
results
dependence - Fig. such
crystals
behavior, mechanisms considered,
not 47,
curve c. Therefore,
(not yet
their of
essentially
available)
interpretation the
iii)
exciton If,
reveal
would the
the
classical
should experiments on a non-classical
require
transition
initially,
from
that
layer
phase
is
phase
intrinsic
formation B-shaped
be (Fig.
47,curve a), then by introducing greater bulk damping ro into the SOL
model
classical
one phase
obtains (curves
the b,c).
N-shaped iiii)
and, The
subsequently,
latter
event
the
occurs
simultaneously with the disappearance of the ERL enhancement
(see
Fig. 46). At r 0 = 1 or 2 cm -I the phase is S-shaped, at r 0 near to 5 cm -1, when Rmi n turns zero, a line tangent to the phase curve at = ~L becomes vertical signifying the transition to N-type. These developments reproduce the
excitonic
Brewster
effect for normal
incidence described earlier on assumption of a uniform DL
[120]
and explain completely the results concerning the dependence of Rmi n on temperature and also the temperature dependence of ~ [27].
276
A.E. Cherednichenko and V.A. Kiselev
G.
Exciton reflectance So
far we
dealt
with
thin space charge and
II
wider
of
layers
with
w
that
at
effect
(SOLs). plane
values
these are represented noteworthy peculiar
the
the F s - w
layers,
important
in the case of wide SCLs the
in Fig.
by Areas
exciton
reflectance
Those are represented
up
larger
on
to III,
w
38.
103
Now we
nm.
On
IV and V
lower
N+
by Areas
shall
the
N-
consider
48).
values
It is
are
since just the shallow and wide potential wells to this N + - N- range produce a nontrivial
I
F s - w plane
(see Fig.
-
of
more
that are
ERL structure.
Fig. 49 shows ERLs, typical of a wide SCL, calculated for the N+-N - values of 1016 (a), 3-1015 (b), 1015cm -3 and ~ = 10 -4 eV by the method presented above. 49a and Fig. The
main
39) the ERL displays
reflectivity
peak
whereas
the minimum
a maximum,
as well
shifting
is similar
(see Fig.
to
the
surface. less the
sharp
than 10 nm) exciton
effect.
(~
is large
calculated
102
nm)
The ERL rotation
coefficient
enough
calculated
the
ERL
values
rotation. the
The
pulsations
pattern
of ERL behavior
of
curves
the
appearance
in
( ) 1016
makes
possible
Area, right
this
in
Fig. 6E
to Area
III
3 .....
spike which
reflectance peak moving to long wavelengths.
is of
interference
the reflection in the
curve will have an
Sect.
2,
layer
wavelength
the
in
I,
In the
the boundary
with
shown
corresponds
al.
near
at some fixed frequency
discussed
of a successive
of
The resultant
is
et
thickness.
cm -3)
a period corresponding
curve
this
This
of exoitons
(the spread
or measured
character with
and so on.
being the layer of a heavy
can be characterized
region of the exciton resonance. oscillating
into
to the characteristic
which
and
turns
by Evangelisti
dissociation
inner boundary
in relation
(see Fig.
wavelengths
spectrum
dead layer of variable
caused by field-induced
has a rather
long
to long wavelengths, one,
6) for a uniform
When N + - N-
to
of the initial
our case the DL is of different nature, damping
cm L~
a rotation when w is increased.
shifts
disappears picture
For N + - N- ) 1016
to one cycle of 50
(at
are
in Pig. is
small
seen).
9s
This
48. Each
related
then develops
to
an
into a
Excitons at Semiconductor Surfaces
277
c
b 0.6
0.4
w-6OOnm: to
-130Nn 900
0 ZOO
260
-"
~0520
|/I
I-~'"
20600
%..,~11,-.-I
20500
I-. 2000
20600
20560
20600
U,4 Grit"1---,,Fig. 49. Theoretical ERLs for the A(n=1)-exoiton in a CdS crystal calculated using a Shot~ky barrier model for different values of w and three values of N - N- (see text).
In contrast to published results [18]
(see Fig. 6) the amplitude
of oscillations dies out with distance from the origin of the F s w
plane. At the boundary of Area V (see a wavy curve in Fig. 48)
it is 1/10 of
the
non-uniformity
of
initial value. the SCL
field.
This
effect
However,
is related
to the
in experiment
it has
been found even stronger than our theory predicts. The ERL rotation was observed on a number of semiconductors GaAs [25,108,109], InP [17], CdS [107,19,121] was
applied,
using
[17,25,121,108,109], irradiatied
with
or
semitransparent the
electrons
metal
semiconductor [19]
(see
:
. Either the voltage surface
Sect.
4B).
electrode (CdS)
was
With
the
metal/OdS structure [121] it was possible to observe just a little
278
A.E. Cherednichenko and V.A. Kiselev
0./~ 0.3 0.2 0.1 0 B
I
o
I
0.5
,,I
1.0
1
1.5
2.0
Fig. 50. Variation of the reflectivity coefficient R of a CdS crystal at a fixed frequency in the v i c i n i t ~ of resonance of the A(n=1)- exciton (20600 cm ").
more than one ERL reversal
and the peak-to-peak
amplitude
of ERL
decayed noticably faster than it follows from our calculations. least
two factors may be responsible
of the resonant
structure
of the density of states electric
field,
use
of
the
band
the
symmetrical
arises
from
in reflectance.
One is the asymmetry
distribution which appears under
see Sect.
6B.
It is to be recalled
Lorentzian large
parametrization.
spreading
At
for the rapid disappearance
of
the
strong
that we made Asymmetry
levels
of
of
excited
states under applied field and their merging into the continuum of states.
With
increasing
field
the
continuum
encroaches
band of the exciton ground state as well,
see Fig.
physical
getting
DL
can
transparent.
The
not
the excitonic structure less conspicuous. The
second
structures
cause
of (see,
considered
as
(unstructured)
at
the the
more
hig~
and
fields
more makes
in the density of states of deeper layers
This conclusion is confirmed by the experimental
data on absorption occurrence
be
absorption
upon
34. Hence,
[121]. of
impact
the
rapid
ionization
e.g.,[63,53]),
decay of
of
the
ERL
the
exciton
the process
largely
may in
be
an
barrier
dependent
on
Excitons at Semiconductor Surfaces
279
the physical nature of a metal-semiconductor contact and on the intensity of incident light. Essentially different ERL behavior is obtained values,
around 1015
corresponding
cm -3,
as seen
in Fig.
at low N + - N-
49o.
In Fig.
48 the
area is Area IV . At large enough w , same as for
high N + - N-
levels,
after another,
there appears
a sequence of spikes,
represented by curves I , 2, 3,
... in
one
Area IV.
But, in contrast to the former case, the spikes may accumulate in the reflectance spectrum giving rise to a multi-spike structure. With further increase in the value of w the spikes are vanishing in order of appearance. Area IV encompasses ERLs which display no rotation but are characterized by the shift of the main peak to long
wavelengths
and
rotation does not
by
reduced
peak-to-peak
take place because
amplitudes.
The
the function r(z) has
the
form of a gradually sloping tail whose characteristic dimension by far exceeds kex c (kexc ~ 10 nm), i.e. the field is nearly uniform in this case. The shift of the ERL maximum to long wavelengths is a
consequence
of
a
field-induced
lowering
of
the
resonant
frequency. The boundary separating Areas III (rotation) and IV (multi-spike structure)
is not
transition between illustrated by spectra
is
the
In
two
w
intermediate of
spectra
structure further
the types
reflectance
the
increasing
sharp.
to
to
the
region
behavior
in Fig. red
long
ERL of
49b. ~L
wavelengths
a
is Seen
which
smooth
observed, in
these
moves
and
with
gradually
disappears. When crossing o v e r from Area III to IV the role of the damping
P(z) gets appreciably reduced whilst that of the lowering
of
resonant
the
frequency
A~ O(z)
near
the
surface
becomes
dominant. This has a bearing on the structure produced which may be
interpreted
quantization structure
in
of
excitons)
arises
dispersion is
terms
of in
interference the SOL
irrespective
taken into account,
of the
of
of polaritons a
(or of
semiconductor.
whether
or
two cases
not
The
spatial
differing only
quantitatively. As regions of shallower and wider potential wells are entered (Area IV), the excitons of progressively longer wavelengths get involved in quantization, so that the additional
280
A.E. Cherednichenko and V.A. Kiselev
structure Fig.
becomes more closely spaced and moves nearer
The
ERL
region
features
have
structure may,
pertaining
been
to Area
observed
observed
across
in a number
spectra
the
of
observed
transition
CdS
crystals
. Often,
the main
multiple
reflectance
IV
and
experimentally. of
in the first approximation,
light
on
CdSe
SCL
macroscopic
intermediate [88].
be attributed
layer a
assumed
two-spike
spikes
peak
region
These
spectra
(see Fig.
49b)
have
fitted
Evolution
of
the
ERL
typical
of
Au-CdS
barrier structures with a
reverse
bias.
excitons
Representative
are seen
in Fig.
(Figs.
[29]),
been
a
IV
can
the
increased
magnitude, On
the
(uniform)
estimated at
to dissociate
as
still
agreement cm -3,
hand,
when
for
observed
the
the SGL
more.
The
field
to
the
is non-uniform
lineshape
the
theoretical
see Fig.
490.
The shift
alteration
calculation
the
A-
and
B-
feature
is
peak shifts
amplitude crystal,
to about
to a value calculated for w = 2000 nm. and the
= 2000 nm.
The
above value of N + - Ndisappearance
also adequately described.
of
the
for by as
its
is insufficient
SOL
of
a
observed
for
N +-
~IO-4cm (f ~ I).
the ERL
of the reflectivity
at a 4V bias amounts
on
6B).
confined
with
wavelengths 4V
that
can not be accounted
within
(see Sect.
GaAs
the field may be an order of magnitude higher,
In addition, smeared
field
on
distinctive
F < 104 V/cm or f < O.1,
the exciton
other
thickness
the
This behavior
the
electrode under
peak-to-peak
with
diminishing.
of
theory
be
semitransparent
Their major
besides
similar to ours.
to
wavelengths,
be
indicating
obtained
using
The main reflectance
long
also
typical
the absence of the ERL rotation. lineshape
the
other factors
well
Area
may
of
In
by a structure
spectra
have
12,13)
to interference
12 and
modifications
51.
multi-spike
structure
Reflectance
been
intermediate
are accompanied
(see Fig.
field.
the
transparent.
assumed wide SOLs and a calculation procedure
of
(see
The
samples
that a theory is needed that would consider the
to ~L
17a).
N-
will
be
is
in
=
1015
peak
to long
40 cm -I which
is close
Assuming the one actually
of the structure
band bending obtains
w
in reflectance
is
Excitons at Semiconductor Surfaces
281
/
A B
2~
~ U : 0 V
I0 20
Fig. 51. ERLs for the A(n=l)and B(n=1)- exoitons as functions of the reverse bias applied via a semitransparent Au-film.
10-
m
Io
1o
~88.0 ~8~.0 x,nm The theory and experiment ~iffer in that the latter provides the evidence of the interference structure located between the A- and B- excitons (see Fig. 51, U = O and 1V). We regard this as one more argument in favor of the necessity to take into account other factors besides the macroscopic field near the surface. On the other hand, the experimental spectra display no multi-spike
282
A.E. Cherednichenko and V.A, Kiselev
structure.
This
fact
is
readily
exPlained
by
larger
exciton
damping to be expected in barrier structures compared to the value used in the calculation (kLF0 = 0.8 cm-1), large damping
being an
immediate cause of the spike dis~ppearance. An important and somewhat surprising conclusion to be drawn from the discussion in this section is that a more pure crystal do not necessarily possess
a more
pronounced
structure
in
the
exciton
reflectance. Let N A be either fixed or small compared to N D , so that the impurity content of an n-type material is dominated by the latter and let N D be lowered. bending
near
the
surface
If there occurs a fixed band
(which
is
typical
of
A3-B 5
semiconductors), then with N D lowering w will increase as (ND NA )-I/2 and F s will decrease in inverse proportion to the same factor.
The
increased
damping w
of
the
may happen
resonant
reflectivity
due
to
the
to outweigh the effect of the reduced
F s. Indeed, as long as F s > F I reduction in the value of F s does not lead to the enhancement of the resonant structure because the excitons in this case do not reach the surface. For example,
in
GaAs Ps does not equal F I until N D - N A is lowered to ~1013 cm -3. One has
to be
aware
yet
that
in measuring
the reflectivity
coefficient the effect of light on the band bending is inavoidable and
unless
the
light
intensity
is
kept
sufficiently
low
the
conclusions made might be in error. In conclusion we note that the exciton spectroscopy can be of great
value
in
understanding
certain
surface
properties
of
semiconductors though the problem of the exciton behavior at the surface looks very complicated. We think that future progress in the issue is intimately connected with the progress in the novel c~-ystal-growth and
surface-treatment
techniques
which
are
being
developed nowadays intensively. The
authors
wish
to
thank
Professor
B.
V.
Novikov
for
encouragement and support of the work and thank the colleages and associates for useful discussions and assistance.
Excitons at Semiconductor Surfaces
283
R E F E R E N C E S
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
14.
15. 16. 17. 18. 19. 20. 21. 22.
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284
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Excitons at Semiconductor Surfaces
285
49. E. Conwell, Solid State Commun., !4, 915 (1974). 50. I. Balslev, S ~ ~ _ a _ ~ e l C_omin__un., 3_9a 359 (1981). 51. J. Biellmann, M. Grosmann, S. Nikltine, Polaritons, in Proe. of the I st Res. Conf. on Structure of M a ~ e r ~ "Taormina, I~aiy ~T~9~, p. 183. 52. W. Ekardt, K. Losch, D. Bimberg,__P_h~s. Rev. .B, _2_0, 3303 (I 979 ). 53. J. Lagois, E. Wagner, W. Bludau, K. Losoh, P~s_. Rev. B, 18, 4325 (1978). -. . . . . . . . . . . . 54. J. Lagois, P_k4ys. Rev. B, 23, 5511 (1981). 55. V. A. Kiselev,-FT~.. Tverd. Tela (Leningrad), 20, 2173 (1978) [English transl.: Soy. Phys. - S o l i d State, 20, 1255 (1978)]. 56. V. A. Kiselev, F i z . - - T v ~ d . - T e i ~ - ( T . ~ n L ~ d ) ? 21, I069 (1979) [English transl.-Sov. P ~ s . - S o i z d State, 2_!, 62T (1979)]. 57. M. A. Lampert, P_b,ys. Rev. L_e_~_~._, 1, 50 (1958). 58. E. I. Rashba, Fiz. T~H~. Polu_~r., 8, 1241 (1974) [English transl.: Soy. P ~ s . ~' Semloond., 8,_807 (1975)]. 59. E. I. Rashba, G. E. Gurgenishvili, Fiz. Tverd. Tela (Leningrad), 4, 1029 (1962) [English transl. : Soy. Phys. Solid State, 4, 759 (1962)]. 60. D. G. Thomas, J. J. Hopfield, __Ph2ts_- _R_ev. 128, 2135 (1962). 61 A. S. Batyrev, B. V. Novikov, A. E. Oherednichenko, Fiz. Tverd. Tela (Leningrad), 23, 2982 (1981 ) [English transl. : Soy. Ph,ys. - Solid State, _2_3, 1739 (1981)]. 62 G. V. Benemanskaya, B. V. Novikov, A. E. Cheredniohenko, Fiz. Tverd. Tela (Leningrad),_-20, 1803 (1978) [English transi.: Soy. P h~s_. - Solid State, 20, 1042 (1978)]. 63. V. G. Lysel~6~ V. B. T~ofeev, Fiz. Tverd. Tela (Leningrad), I_8_, 1030 1976. 64. V. V. Travnikov, Pis'ma Zh. Eksp. Teor. Fiz., 40, 278 (1984) [English transl.: Soy. P h y s . - JETP T . ~ . , 4_O,-1060 (1984)]. 65. B. V. Novikov, G. V. Benemanskaya, A. Vestkhoff, A. E. Cherednichenko, Fiz. Tverd. Tela (Leningrad), _1_7, 2186 (1975) [English transl.: Soy. P~!~is_. - Solid State L 17L 1448 (1975)]66. V. V. Travnikov, _Pzs-_m~_Zh. ~_s~.-Te_o_r~.-T~z.? 42,_357 (1985) [English transl. : S o v . - P ~ s . - ~ P - T , ett..-42_~_ 440 (1985) ]. 67. R. P. Holmstrom, J. ~ o w s k I ~ H. O. Gatos, Surf. Sci. Lett., 1OO, 467 (1980). 68. J. Lagois, B. Fisher, in Surface Polaritons, V.M.Agranovieh and D.L.Mills, eds., North-Hoii~a~d , Amsterdam (1982), p.69. 69. V. I. Sugakov, Fiz. Tverd. Tela (Leningrad), 1_4_, 1977 (1972). 70. Y. Lozovik, V. Nishanov~ Ti~..--Tverd. Tela [Leningrad), !8, 3267 (1976). 71. R. Del Sole, A. Selloni, P__h~_s-__Bev_-__B, 3_0, 883 (1984). 72 P. E. Best, P_h,ys. Rev. B_, !2_, 5790 (1975). 73. G. J. Lapeyre, J. Anderson, P_hys. Rev. Lett., 3_5~ 117 (1975). 74. A. I. Bobrysheva, S. I. Beryl, V. T. Zhyukov, S. A. Moskalenko, E. P. Pokatilov, P_h~_s_._S_tatus Solidi (b_l, 115, ~53 (I983 ). 75. R. Del Sole, E. Tosatti, Solid State 0ommun.j 22j_307 (1977). 76. V. N. Ermakov, D. V. Korbu~ya/~7 V. G. T,itovchenko, O. Yu. Mikityuk, V. V. Nitsovich,__P_~s____Status Solidi B, _1_2_5, 815 (1984).
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Excitons at Semiconductor Surfaces
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So!.~b_.), 8S, 797 (1978). 106. J. W. A l l e n , Na.tur__e, 187,_51 107. 108. 109. 110. 111. 112. 113. 114
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(1980 ). 115.
I. Filinski, T. Skettrup, Solid State Commun., _11, 1651 (1972). 116. L. E. Solov'ev, A. V. Babinskii, Pis'ma Zh. Eksp. Teor. Fiz., 23, 291 (1976) [English transl.: Sov. Phys. - J E T P T . ~ . , 23, 2~3 (1976)]. -117. A. V. Komarov, S. M. Ryabchenko, M. I. Strashnikova, Zh. Eksp. Teo_r. Fiz., 74, 251 (1978) [English transl.: Soy. Ph~s_-.
U_J&--'T_P_, 47, 128 (1 c~78) ] . 118. S. B. Moskovskii, L. E. Solov'ev, Zh. Eksp_ Teor. Fiz., 8_6,
1419 (1984) [English transl.: Soy. Phys. - JETP, _59, 831 (1984)]. 119. ~. Gourdon, P. Lavallard, P ~ s . Rev. B L 3 1 L 6 6 5 4 (1985). 120. A. V. Pevtsov, A. V. Sel'kin~-Zh.-~sp. Teor- Fiz., 83, 516 (1982) [English transl. : Soy. P~_8?_---_-_~_-P_-,_5_6?-282 (~982) ]. 121. B. V. Novikov, A. B. Pavlov, V. G. Talalaev: Fiz. Tverd. Tela (Leningrad), 23, 1014 (1981) [English transl.: Soy. P hzs_.- Solid State, 2__3, 587 (1981)].
0079-6816/91 $0.00 + .50 Copyright @ 1991 Pergamon Press plc
SURFACE ASPECTS OF EXCITONS IN SEMICONDUCTORS A.E. CHEREDNICHENKO and V.A. KISELEV Leningrad State University, Solid State Department, Institute of Physics, 198904, Staryi Petergoff, Leningrad, U.S.S.R., and, A.F. Ioffe Physico- Technical Institute, Academy of Sciences of the U.S.S.R., 19402 I, Leningrad, U.S.S.R. Abstract This review considers the factors which influence the exciton behavior at the semiconductor surface and, consequently, are revealed through excitonic optical spectra. The basic knowledge of excitons in semiconductors is first presented. The experimental exciton reflectance spectra of OdS and CdSe crystals are discussed followed by an analysis of the results which make use of the near-surface "dead-layer" model. In particular, the effect of the electron irradiation on the exciton reflectance lineshape and exciton photoluminescence is considered with reference to the physical factors which control optical spectra. Various mechanisms of localization of excitons in a semiconductor surface region are presented. Relevant experiments on low-temperature photoluminescence and reflectance are discussed. Finally, the effect on the exciton reflectance of the electric field of the space-charge layer considered in detail.
Contents
Page
1.
Introduction
181
2.
Wannier-Mott excitons. Energy considerations
182
3.
Excitons: Mechanical, Coulomb, real (polaritons)
186
4.
Exciton reflectance spectra
191
A.
191
Thomas-Hopfield near-surface "dead" layer and the exciton reflecfivity
179
180
A.E. Cherednichenko and V.A. Kiselev
5.
B.
The effect of keV-electron irradiation on exciton reflectance spectra
198
C.
Interpretational problems
205
D.
Interference effects in thin crystals. Size quantization of excitons
211
E.
Exciton transition layers and a formal model
215
Crystal defects and localization of excitons
218
A.
Bound excitons
218
B.
The effect of keV-electron irradiation on the exciton photoluminescence spectra
220
C.
Surface and near-surface excitons
225
D.
Localization of excitons in rectangular potential wells
229
E.
Localization of excitons in defect-rich surface layers
237
E
Localization of excitons by fluctuations of a near-surface
6. A.
random potential
241
Band bending and its relevance to excitons and exeiton spectra
244
Band bending in semiconductors
245
B.
The effect of a uniform electric field on excitons
248
C.
The effect of a non-uniformelectric field on excitons
254
D.
Exciton relectivity in the ease of thin SCLs
257
E.
Quasilocalization of excitons in the SCL field and the effect of damping
267
E
The effect of SCL on the phase of the reflected light
273
G.
Exciton reflectance in the case of wide SCLs
276
References
283
Abbreviations
ABCs AWs DL EB ERL ETL SCL
Additional boundary conditions Additional waves Dead layer Electron bombardment Exciton reflectance lineshape ~xciton transition layer Space charge layer
Excitons at Semiconductor Surfaces
181
1.Introduction In surface physics, the most sensitive probe is the electron, be it in Auger spectroscopy, in low energy electron diffraction, or in photoelectron spectroscopy. However, the photon also provides information on surface phenomena, though its penetration into the bulk of the material usually exceeds that of the electrons. Optical techniques are, for instance, ellipsometry, spectroscopy of the vibrational structure of physisorbed chemisorbed molecules, spectroscopy of surface polaritons.
IR and
In this review we intend to discuss one more optical technique which can be used to investigate surface (near-surface) layers of semiconductors. The purpose of the paper is to present an exoiton as a probe in the studies of surface (near-surface) properties of real semiconductor crystals, either as grown or subjected to various surface-sensitive treatments (e.g., electron irradiation). We will conoentr~ate mainly on A2B 6 - compounds (CdS, OdSe) and summarize the present state of understanding of the problem based on
our
experimental
and
theoretical
studies
and
also
on
the
results obtained by other researchers specializing in the field of the exoiton behavior at a semiconductor surface. We shall show that some of the data on the exciton reflectance spectra
may
serve
a
useful
guide
to
understanding
surface
properties of real semiconductors. In particular, the overall shape of the exciton (polariton) reflectance signal appears to be a sensitive function of the presence below a semiconductor surface of the exciton-free layer (the so-called "dead" layer). The width of this layer can be varied from several tens to thousands of ~gstroms by a number of techniques (e.g., by keV-eleotron bombardment). A theoretical lineshape analysis of the reflectance will be given which evidence the role of the exciton-free layer and space charge layer (SOL) in the formation of the exciton
182
A.E. Cherednichenko and V.A. Kiselev
reflectance lineshape (ERL). Studies
of
the
so-called
bound
exciton
luminescence
and
its
modification under electron bombardment will be presented in Sect. 5 and various physical mechanisms of the exciton localization at a semiconductor surface will be discussed. Finally, in Sect. 6 the effect on the exciton reflectance of the SOL will be considered in detail. We also note that surface aspects of exciton photoconductivity phenomenon are not considered in this review. A detailed account of
a
surface
sensitivity
of
photoconductivity
spectra
is
in
preparation. 2. Wannier-Mott Excitons. Energy Considerations An
exciton
is
the energy
quantum of
an electronic
propagating in insulating or semiconductor crystals.
excitation In case the
energy band model is well applicable, it can also be regarded as a composite particle hole
in
the
of an electron
valence
band
bound
in the conduction band and a to
each
other
by
the
Coulomb
attraction between them. If the electron and hole are so far apart that
the atomic
structure
of
the
crystal
can be
ignored
their
mutual potential can be written as v = -e2/~
,
(2.1)
where p is the electron-hole separation and ~ is the macroscopic (long
wavelength)
dielectric
constant
generally
taken
as
the
square of the refractive index. Such a composite particle, which we
call
Wannier-Mott
exciton
(see
[1,2])
may
analogy with the hydrogen atom. The potential Coulomb potential
reduced by
the dielectric
be
visualized
(2.1)
by
is just the
constant,
and there
will be bound states of the exciton system having total energies lower than the bottom of the conduction band. By analogy with the hydrogen (n=1)
atom
state
the binding (the
energy
energy level
of
the
referred
exciton to
the
in
the ground
bottom
conduction band) is given by the m o d i f i e d R y d b e r g formula:
of
the
Excltorl lit 6ernk~nductor Surfaces
183
4
~e ~1 = - ~
= (~/=o)'s-2"~1~
,
(2.2)
where ~ is the reduced mass of the exoiton ~-1 = m;1 + ~ 1 , formed from
the
effective
masses
m e , mh
of
the
electron
and
hole,
respectively, m 0 is the free electron mass and IEIH i = 13.6 eV is the ionization potential of the free hydrogen atom. Pot a simple case of isotropio non-degenerate baud extrema at the zone center the kinetic energy of the carrier is E k = ~2k2/2m = where m is the effective mass (me for the electron and m h for the hole) and Phk is the momentum at the extremum. The resulting spectrum for the exciton energy consists of the sum of the internal energy and the exciton center-of-mass kinetic energy ~e 4
Q¢K)
=
2~2~2n2 +
~2K2
2.
'
¢2.3)
where I = m e + m h, h~ is the momentum of the center of mass and n = 1,2,3 .... o0. £n is measured from the band edge EG. Thus, the
Fig. 1. Energy of the exoiton bands for Wazluier Mot t excitons vs waveveotor K. The d~mhed line represents the energ~ vs waveveotor dispersion curve 0~ ~ o o u p l e d photon.
184
A.E. Cherednichenko and V.A. Kiselev
exciton spectrum consists of a series of parabolic bands below E G which merge into a continuum at high energies, The
binding
energy
of
the n=1
exciton
electron-hole at the band edge given by
see Fig. I.
relative
to
the
free
(2.2), often called the
"exciton Rydberg", can also be expressed in terms of the "exciton Bohr radius" given by
Rex c = ~ 2 s / ( ~ e 2 )
= (~/mo)-l.
8-R,I:I
,
(2.4)
where RH = 5.29"10 -9 om is the radius of the first Bohr orbit for the free hydrogen atom. (2.2)
and
(2.4)
Note that the factors ~/m 0 and 8 entering
reduce
the
binding
energy
and
correspondingly
increase the radius by a factor of 102 or even 103 as compared with the hydrogen atom. As a result, typical values of the exciton Rydberg
in A2B6-compounds
are
of
the
order
of
10 meV
and
the
exciton radius is about a few tens of ~ugstroms. By virtue of its large
size
the
Wannier-Mott
exciton
is
extremely
polarizable.
Consequently the applied electric field has a pronounced effect on exoitons. In what follows we shall show that the exciton spectra are
very
sensitive
to
the
surface
electric
field
which
is
an
scheme
is
an
essential point to be emphasized here. As
shown
infinity
of
transitions
in
Fig.
I
the
overlapping create
exciton continua;
excitons
of
energy
level
nevertheless,
well-defined
its
energy
optical and
can
therefore be sharp. This a consequence of momentum conservation. In
the
case
of
direct
transitions,
i.e.,
vertical
on
the E(K)
diagram, a photon of energy ~L0 can only create an exoiton with K = nt0/c, where n is the refractive cm -I so that ~2K2/2M ~ IO-SeV.
index.
For ~
~ eV, o/c
~ 105
Thus the only exoitons which can be directly created by a photon have
a
well-defined
negligible.
energy,
Transitions
electron-hole pairs,
and
involving
their
kinetic
the
creation
on the other hand,
give
energy of
is free
continuous spectra
for E > E G, since momentum conservation only puts one condition on k e and kh, leaving the relative motion of the electron and hole free to be specified by energy conservation. For the exciton, the
Exdtons at Semiconductor Surfaces
185
requirement that the electron and hole sts~V together eliminates this degree of freedom. The existence of sharp optical transitions in a crystal is evidence for the existence of exoitons. It is difficult to produce excitons in sufficient concentrations to observe directly the transitions amor~ the exoiton levels, but it is possible to observe the transitions between the valence band edge and an exoiton level. A series of sharp lines occurring on the long wavelength side of the absoz-ption edge and merging into the continuum has been first observed in cuprous oxide Ou20 (see, e.g., [3]). The spacing of the exoiton levels has been found to be in surprisingly good agreement with the Rydberg formula for n > 2. Later, large-radius Wanniez--Mott exoitons were revealed in a great number of semiconductors (for the A2B6-compounds see, e.g. [4]). It is evident that it the energy surfaces are not isotropio and non-degenerate the exoiton problem is considerably more complicated. A sketch of the energy band structure of A2B6-oompounds (wurtzite) is shown in Fig. 2. In wurtzite-type crystals the conduction band is simple, whereas the valence band consists of the three close-lying bands F 9, r 7, and r 7 which arise due to crystalline field and spin-orbit splitting effects. The valence bands are termed A, B, O, starting from the uppermost valence band, respectively. An appropriate optical excitation yields A-, B-, and O- exoiton series.
Fig. 2. A sketch of the energy bands in wurtzite-type crystals.
B
C
r, x(ooo)
186
A.E. Cherednichenl(o and V.A. Kiselev
Note, that the A-exciton is only allowed for light polarized perpendicular to the hexagonal axis ElO (~ is the electric field vector of the light and o is the unit vector along the hexagonal axis (c-axis)). A free exoiton at liquid helium temperatures has a velocity N of about 106 cm/s, so for an impurity concentration N ~ IO150m -3 its lifetime before capture iS expected to be (Nov)-I ~ I0-9s. Thus the exciton is a short lived excited state of the crystal observed at low temperatures near the band edge (below ~5000 ~ for OdS and ~7000 ~ for CdSe) so that virtually the only way of studing it is by optical spectra, v~Z., transmission, refleetlvity and luminescence. (We note that the exeiton photoconductivity phenomenon is b~¥ond the scope of the present review), To conclude, we note two more points important from the experimental point Of view. (a) Optical experiments are essentially performed at loW temperatures (moStly at 4.2K) insofaras the exciton binding energies are very small which results in an easy thermal dissociation of exoitons. (b) The exciton absorption coefficient in the cr~stal with allowed direct transitions may be of the Order of ~IO 5 om -I . This has two consequenoies. (i) Ex0itons thus excited are essentially generated in the near-stu~f~ce region of a cr~stal. (ii) Small penet~mtion depth of the exoiton does not allow transmission studies, hence, exciton reflectance ~easurements need to be carried out instead.
3, Exoitons! Mechanical, Ooulomb, Real (Polaritons) Figure I corresponds from an experimental point of V i m to a low-resolution spectroscopy, whe~ the intersection of the photon dispersion curve and the eXciton band is given by a point. In the case of a high resolution this approach would be inappropriate, especially for the ground (n = I) states of dipole-active exoitons [5] which are characterized by a large oscillator strength. More accurately the exciton states in the region of their intersection with the photon dlspe2sion should be considered as in Fig. 3.
Excitons at SemiconductorSurfaces
~
b
I I ! !
/
187
o
I wL~--" 7
/ k
~
k
Fig. 3. Exciton dispersion curves. (a)- mechanical approximation, (b)- Coulomb approximation which allows for the longitudinal field, (c)- with allowance for longitudinal and transverse fields.
Figure 3a corresponds to the approximation mentioned above, when the longitudinal electric field, produced due to the exciton wave propagation is not taken into account and the interaction with the transverse electromagnetic field is negleQted. Within that approximation the exoitons are referred to as mechanical [6], therefore, the approximation is termed mechanical. The inclusion of the longitudinal electric field arising already in a more simple system of the Lorentz oscillators [7], when longitudinal vibrations (exoitons) are excited, leads to increase in the energy of these excitations by the value h(~ L - ~0 ) ~ ~ L T (Fig. 3b). The longitudinal exciton, that is the exciton with P~K, P being the vector of polarization, is refem~ed to as the Coloumb exoiton [6], so that the approximation is a Coulomb one. The exoitons with the polarization PIK that do not interact with the longitudinal field
188
are
A.E. Cherednichenko and V.A. Kiselev
still
mechanical
in
this
case.
Lastly,
an
account
of
the
interaction with the transverse electromagnetic field brings about modification
of
yielding real The
above
the
transverse
excitons
[6]
(photoexcitons
considerations
vibrations by Born
exciton
were
applied
dispersion
(Fig.
3c),
[8], polaritons to
crystal
[10], and to excitons by Pekar
[9]).
lattice
[8], Hopfield
[9] and Agranovich [ 1 1 ] . Figure
3 can
also
dielectric
theory.
associated
with
be
The
the
treated
from
the point
of view
of
the
excitation of a dipole-active exciton polarization
of
the
medium
P(R,~)
is
which
satisfies the following equation [6,8,12]: B(~° where B = ~O/M, is
the resonant
+ q2(~))
= _~ 2 ( R , ~ )
~ = (2%)-1~O~LTSO , q2= _ B - I ( ~ frequency
the mechanical
~(~,~)
of
the mechanical
exoiton damping;
,
(3.1)
_ ~2 _ ¢ ~ p )
exoiton
at
a n d ~0
K = O; F i s
s 0 is the background dielectric
constant which contains contributions from all interactions except the exciton in question. A right-hand part of the electric field may be mechanical
field vector, treated as
exciton,
connected with
the driving
since
for
the
(3.1) incorporates
the exciton wave.
force with respect
latter
we
have
E
=
derivation of (3.1) was suggested by Pekar [8]. Equation a material
equation
for
the medium which
Maxwell's set of equations for fields.
is
additional
The
to
the
O.
The
(3.1) is to
the
For a homogeneous medium
one can make use of the Fourier transform and write down, in the case of an isotropic medium, the relation of the following type: 1
~(k,~)
= ~--~[s(E,~)
where the dielectric function ~(k,~) ~(k,~)
(3.2)
- s012(1~,~) is of the form:
= ~£O((fl) - 4'Fu30/((o2 - (,o2
- B k 2 + ~,~r)
,
(3.3)
For the mechanical excitons ~. = O, P # O, and, as it follows from (3.2),
~(~,~)
= ~,
or
Excitons at Semiconductor Surfaces
~(k)
~ (~0 + t ak2/(2M) -
(F/2
189
,
(3.4)
which corresponds to Fig. 3a. For the Coulomb exciton there is only a longitudinal component of the electric field E~k. Since the displacement field B = 80 ~ + 4~7 is transverse we have case D = O. From (3.2) follows 8(k,~) = O, or ~(k)
~ ~0 + %_,T + '%k2/(aM) -
61/2
,
in this
(3.5)
which corresponds to the longitudinal exciton in Fig. 3b. Lastly, for the real excitons (polaritons) there exists only a transverse component of the field EIK, for which the Maxwell equations are related to D by = (02k2/~2)E Solving
(3.6)
together with
(3.6)
(3.2) we arrive at the dispersion
relation: 8(k,~) = c2k2/602
.
(3.7)
It is convenient to solve the equation for k or n (n = ck/~ is the index of refraction). Resulting is the Pekar's which defines the polariton dispersion:
nl,2(~)
I = {60 + 2-
[Q ± (a2 + b ) 1 / 2 ] } 1 / 2
Mc 2 where G = ~°O~ 2 (~2 _ ~02 + ¢~F) - 60;
equation
,
[8]
(3.8)
Mc280~LT b = 8
t~02
These relations define the two polariton branches in Fig. 3c. The polariton approach will be utilized below. In some cases, however, the mechanical exciton approach appears to be more convenient for discussion. Such is the case of the exciton localization in potential wells, which we intend to treat at some length below. As to the calculation of the crystal optical properties, it will be
190
A.E. Cherednichenko and V.A. Kiselev
performed making allowance for the photon-exciton coupling,
i.e.,
within the framework of the polariton theory. In case the exciton mass M is not infinite the mechanical exciton band has dispersion and ~ depends not only on ~ but on k as well. The dependence of on k is referred to as spatial dispersion. Actually this implies a possibility for the excitation to propagate through the crystal in a purely mechanical way when no macroscopic [6,12].
This
also
implies
that
reference point of a medium
the
fields are involved
electric
is not
polarization
in
completely specified by
a
the
field at the same point. The situation is represented by the first member in (3.1) which contains the second derivatives with respect to coordinates.
Therefore a solution to the problem requires an
introduction of the boundary conditions. They are additional with respect
to the Maxwell's
theory and,
hence,
are referred to as
additional boundary conditions (ABCs). Additional boundary conditions have been intensively studied and were a subject of much discussion.
The efforts of the theorists
were essentially concentrated on three main problems: ABOs
appropriate
from a phemenological
point
(i) Type of
of view
(we note,
that even the necessity for introduction of the ABCs was sometimes under
question),
(ii)
The
comparison with experiment, making
use
of
various
attempts
to
specify
the
AB0s
from
(iii) The attempts to derive the ABOs
microscopic
models
of
the
surface.
The
results of the theoretical efforts in these fields were discussed in several reviews and monographs (see, e.g., [6,8,13,14]). The ABO can be given in the form of a linear relation [6]
c
(P + 7 ~ 0
c
dz
)lz:o
: (~
+ ~ ~o ~z
)lz=o
,
(3.9)
where 7, ~, and ~ are functions of frequency (usually ~ = ~ = O is supposed).
Specification of the ABC
is dictated by the physical
properties
of a medium
and,
and
surface
hence,
the ABC
is not
universal. To solve the problem one should specify the properties of the medium and its surface.
Excitons at Semiconductor Surfaces
191
Most common is the ABO introduced by Pekar [8]
#(z
= o) = o
(3.1o)
.
I n a number of papers the calculations of the exoiton reflectance lineshapes were compared for several different ABCs with the experimental spectra. Patella et al. [15] have concluded that the theory based only on the ABC approximation gives a poor fit to the experiment.
4. Exoiton Reflectance Spectra A. Thomas-Hopfield near-surface refleotivity
"dead" layer and
the
exoiton
A theory of exoitons in confined media based on the boundary conditions, the ABOs included, suggests that the exciton state (the exciton binding energy, oscillator strength, damping, etc.) is unperturbed up to the very boundary or perturbed in a very thin layer. The effect exerted on exoitons by the latter is, to a fair approximation, usually neglected. Such an approach, though, may prove inappropriate insofaras, in reality, there exists a near-surface layer where excitons are drastically perturbed. The part of the layer where exoitons are totally damped is referred to as the exciton-free layer or "dead" layer (DL). The concept of the DL was first introduced b y H o p f i e l d a n d Thomas [12]. 0onsider a simple hydrogenio exciton in its ground state. The interaction between the exoiton and its image results in a potential energy of the exoiton ( in a z-plane)
v(z)
1
=~.--
8-1
(--g~)
~
('
Rexo3
.--~--.
,
(4.1)
where E B is the exciton binding energy. Since 8 is greater than !, the force is repulsive. If all the effects of the surface could be
192
A.E. Cherednichenko and V.A. Kiselev
represented by a potential U(z) for the exciton, U(z) would have to be sufficiently repulsive to cause a free exciton to be totally reflected from an effective barrier a finite distance inside the surface. Thomas and Hopfield were first to replace the potential U(z) by an infinite potential barrier, as indicated in Fig. 4. The layer
1
is
referred
to
as
a
Thomas-Hopfield
dead
layer
for
excitons. We note that
the binding energy of the ground exciton vs the
distance z of its center of mass from the surface has been first calculated in
[16] by taking into account the influence of the
image
on
approaching
the
surface. The main conclusion arrived at by the authors was
forces
both
the
electron
and
hole
that
the exoiton binding energy was found to equal the bulk effective exoiton Rydberg only far away from the surface. For z ~ 1.5 Bohr radius, exciton ionization occurs. Oorrespondingly one has in this model
a
layer
decreasing
where
from
its
the bulk
exciton value
approximation depicted in Fig.
to
polarizability zero.
The
is
gradually
infinite
barrier
4 by a dashed line suggests
that
the Pekar's boundary condition P = 0 apply at and to the left of such a barrier.
ZL,
Fig. 4. A formal adiabatic potential (solid line) and the infinite barrier approximation (dashed line).
Excitons at Semiconductor Surfaces
Actually,
193
three spatial regions should be considered,
as applied
to excitons (see Fig. 5): z < 0 (I), 0 < z < 1 (II), and z > 1 (IiI). The first two are characterized by classical indices of refraction n = I and n O = ~ 0 ' respectively. The usual Maxwell boundary conditions plus the P=O boundary condition determine the connection between the second and third regions.
n-1 10
k~
R2 0
1,
z
Fig. 5. A schematic diagram of the normal-incidence exciton reflectacne problem with a dead layer of thickness i. I O, R I , R2 are incident and, respectively, reflected beams.
The exciton reflectance lineshape
(ERL) is primarily determined
by the phase delay between the two interfering beams R I and R 2. The first analysis of the experimental reflectivity measurements using a spatial dispersion approach was carried out by Hopfield and Thomas [121 who pointed out several anomalies in the reflectance spectra to be discussed below. The DL of Thomas and Hopfield, thus depicted is clearly of intrinsic origin and bears no connection with the particular conditions of the surface, taken as an ideal geometry boundary. In general, there will be other contributions to the DL to be discussed further. Whatever the origin of the DL may be, it allows, within a hard wall approximation, treatment of its effect on the reflectance in terms of interference between the light
194
A.E. Cherednichenko and V.A. Kiselev
beams reflected at the vacuum - semiconductor
(z = O) and at the
DL - bulk (z = l) boundaries, respectively. The hard wall DL model has been generalized to arbitrary values of 1 not related to Rex c by Evangelisti et al
[17,18] and it can
be depicted as in Fig. 6.
0.61-
m
-
o ,
,
,
,
I
i
I
I
l
I I
l
I I
g I
I
I i
Fig. 6. Calculated exciton lineshape of the normal-incidence reflectance for the A (n=l) exoiton in CdS. The DL depth 1 was used as a parameter. 1,nm: a- O; b- 4.7; c- 7.1; d- 9.5; e- 11.8; f- 14.2; g- 19.O; h- 26.1; i- 37.9; j- 52.1; k- 71.1; l- 80.6. The positionsof the transverse and longitudinal exciton frequencies are shown by arrows. [18].
Excitons at Semiconductor Surfaces
195
The DL is supposed to be uniform and is characterized by the parameter 1 and a real non-resonant index of refraction, equal, for instance,
to the background index of refraction of a crystal
n O . The crystal bulk is characterized by the dielectric function (3.3). The incident light I 0 generates
in the crystal
with the wave numbers k I and k 2 defined by (3.8). incidence reflectance is given in the form [18] P12 + P23
e~e
R
The
normal
,2 I
1 + /"12r'23e ~'e
two waves
I
,
(4.2)
where ~12 is a constant, ~23 is a function of a resonant character which defines the ERL, and the round-trip phase delay (or the phase angle) e is e = 4'11~_01/~,
,
where k is the wavelength of light in vacuo.
(4.3)
The value of R is
seen to be a periodical function of 1 with a period k/(2n O) which is of the order of 102nm for semiconductors. A reversal of the ERL for the An= I exoiton in OdS is shown in Fig. 6 [18]. With increasing 1 the reflectance minima give place to maxima (and u~ce uersa) and a reversal of the lineshape occurs. With further increase of 1 the maximum is shifted towards long wavelengths. A prominent feature of the ERL is the refleotivity peak at the frequency ~L - the so-called "spike". It is nominated, for the sake of brevity, the "right" spike as referred to the spectral position of the reflectivity minimum. The emergence of the right-spike structure in the case of a homogeneous DL model is accompanied by the decrease of the main reflectivity maximum. The spike problem has been treated in detail by Benemanskaya et al. [ 1 9 ] who attacked the problem of the exoiton reflectance spectra modification under keV-electron bombardment
(see Sect. 4B)
The authors made use of the approach, developed in [12,18,20] and calculated the ERLs in OdS using the two-layer model depicted in
196
Fig.
A.E. Cherednichenko and V.A. Kiselev
5 with
spatial
dispersion
taken
into account.
In addition,
the exciton damping parameter F was introduced and its effect on the ERL was investigated in a wide range of variations o~ 1. The calculation
was
carried
out
for
the
An=l-exciton
in p-geometry
(~,Io, klc) or normal incidence. The following formula was used to calculate
the
reflectivity
coefficient
(with
allowance
for
the
Pekar's ABO [8]) •
2
1 -n o
¢2knol no+n
/ e
R =
(4.4)
,
1 +
e ~2kno 1 I +no
no+n*
where n o = W ~oI' n = (nln2 + ~ol)/(nl + n2)' ~o1 is the background dielectric constant for light with Eic, k is the wave vector of the exciton,
i is the depth of the dead layer,
and n 1
and n 2 are the refractive indices of the transverse waves. For the computer
calculations
the
values of the parameters:
authors
of
[19]
used
the
following
the longitudinal - transverse splitting
~LT = 2 meV, Eoi = 8.3, M = 0.9 m O, and 0~0 = 2.5524 eV. The DL depth 1 was varied from 0 to 1000~ and the damping F from 0 to 1 meV.
Fig.
7
shows
the
calculated
reflectance
spectra
for
a
negligible increase of l, with ~F = 10 -5 eV corresponding to that value
of
experiment at ~L
the
damping
which
(see below).
yields
Note,
the
(the spike) appears at 1 ~ 60~,
2Rex c. At
layer values
best
agreement
with
that a subsidiary reflectance
1 > 150~ and
peak
i.e. at the value close to small
rtF ( < 10 -4 eV)
results were found to be close to the data of
the
[18] shown in Fig.
6. The inversion of the ERL occurs at 1 ~ 200~, and the return to the initial lineshape at 1 ~ 850~. The results make it possible to estimate
the
values
of
1
and
I~.
A
comparison
between
these
calculations and experiment will be done in Sect. 4B. Alterations of the DL depth observed to occur under application of the external
field,
surface-sensitive
keV-electron techniques
will
irradiation be
shown
to
and result
other from
the
Excitons at Semiconductor Sudaces
197
changes in the near-surface electric field value and also from the emergence of the defect-rich near-surface layer, which is evidenced by the studies of low-temperature exciton photoluminescence spectra and, in particular, of their modification under electron irradiation (Sect. 5B).
I
0.6
0.¢
•
~'\
0.~
Fig. 7. Calculated ERLs for the A (n=1) exciton in CdS as a function of the DL depth 1
Note,
on
the other hand,
that
the outlined model
suggests
a
completely transparent homogeneous DL, though it is evident that field-induced changes in the value of the exciton damping P should be taken into account. Large values of P yield an exciton-free layer which, in fact, is responsible for the observed alterations of the lineshapes under electric field. It is also evident that the layer can not be derived straightforward from the DL in the spirit of Hopfield and Thomas [12] by merely increasing i since the latter is of intrinsic origin and is due to increase in ~0 but not in r as in the former case. On the other hand, it is clear that the exoiton-free layer caused by the abrupt increase in the damping value should likewise result in the rotation of the ERL
198
A.E. Cherednichenko and V.A. Kiselev
when 1
is varied.
Sect.
6 where
field
is
This
allowance
made.
The
issue for
latter
shall the
be
considered at
inhomogeneous
brings
about
the
length
character of diffusion
of
in the the
boundary of the exciton-free layer which may be so strong that the ERL rotation will not proceed at all. In what follows it will be shown that the electric field may lead to localization of excitons near a semiconductor surface.
B. The
effect
of
keV-electron
irradiation
on
exciton
reflectance spectra The
exciton
reflectance
spectra
of
semiconductors
with
the
direct allowed transitions appear to be drastically influenced by the surface. The ERLs are markedly affected by the surface states, near-surface defects, and, especially, electric fields. It is well kwown that strong variations in the lineshapes can be found among different materials and, in a given material, among various levels of the hydrogen-like excitonic series. Quite often, a given line may
look
rather
different
semiconductor material.
from
This
sample
is
true,
to
sample
of
in particular,
the
same
of A2B 6 -
compounds in which the ERLs may vary depending on the conditions for crystal growth and generally uncontrollable surface conditions in a given specimen. The for
electron bombardment chapping
the
(EB) has proved a suitable
reflectance
spectra.
In
discussed in this section keV-electrons were
the
technique
experiments
taken advantage of
(the penetration depth into the material being of a few hundreds of ~ngstroms).
The
(CdS [19,21], CdSe oil-free
vacuum
experiments [22], ZnSe
using
the
were
performed
on A2B 6
crystals
[23]). The EB was performed in an cryostat
permitting
optical
and
electrical measurements at 4.2K. Irradiation-induced changes in the exciton spectra vs electron fluence were registered ~ s~tU without warm up. Near-to-normal-incidence reflectance spectra were measured from a plane containing the C-axis for light polarized both parallel and perpendicular to it. By the effect on the ERL
Excitons at Semiconductor Sudace$
199
the electron irradiation can be classified into a low-dose (1013-1015) e/cm 2 and a high-dose (9 1016) e/om 2 one. The low-dose EB produced reversible changes in the ERL which are discussed below. Fig. 8 shows the changes of the exciton reflectance spectra of a 0dS crystal for the An= I- exciton that occurred under low-fluence electron irradiation with 3 keV electrons. The as-grown crystal is seen to reveal the right-spike structure (curve I), whereas after EB the left-spike structure appears instead (curve 2). (Note that the left spike is the reflectance peak at ~L which is situated to the left from the reflectance minimum (also see Fig. 7). Further exposure to electrons results in a reversal ("rotation") of the ERL.
R
a~ a~ ?i 8. 8. Experimental exciton reflectance spectra of CdS at T=4.2K vs 3-keV electron irradiation dose (a low-fluence case).
0,I
200
A.E. Cherednichenko and V.A. Kiselev
R
A
O_
Ill,
0.2! 0
8"=1
,
I
,
,
,
2 I
~1
I
I
I
I
I
~o wl.
2.550
2.570
2.650
2.570
E,eV
Fig. 9. Experimental exciton reflectance spectra of CdS at T=4.2K vs electron irraditaion dose [19].
Figure 9 shows the modification of the exciton reflectance
for
the An= 1-, Bn= 1-, and An= 2- excitons in CdS [19]. (Note that according to the sketch of the energy bands presented in Fig. 2 along with the An= I- excitons, a more short-wavelength structure associated with the B-band and excited states of the excitons may arise). The initial ERL of the An=1-exoiton also features the
ExcilOnsat Semiconductor Surfaces
~,n'rn 680.0 6"/8.0 676.0 67#.0
A..I
670.0 668.0
201
666.0
8n=2
O#
An=2 O~ 0.2!
0.31-
Ill
A.
0.~
0.~ b, 0.2
4 1+?00 1#'/~ f#?80 ¢ t#820
1#900 ~ I#9#0" 1#g80 0251
Pig. 10. Experimental ERL of 0dSe vs dose of EB [22].
202
AE. Cherednichenko and V.A. Kiselev
right-spike structure whereas following low-fluenoe electron irradiation the left spike appears. This is accompanied by the enhancement oZ the main reflectance maximum. Similar effects are readily revealed in the spectra of 0dSe crystals
(Fig. 10, curve
2). It is pertinent to note that these findings are peculiar of the A-exciton. The low-fluence EB has only a slight effect on the B-exciton (compare curves I and 2). The reason for this difference is discussed in Sect. 6). The analysis of the exciton reflectance in CdS was performed in [19] using the DL model (see Sect. 4A). As seen from Fig. 9 the initial spectrum features the coefficient at the main maximum
right spike, the reflectivity is 0.42. A comparison with the
calculated spectra yields for the Afl=S-exciton a DL value liA SO~. After a low-dose EB (curve 2) the right spike is converted to the left one and the main reflectivity maximum is markedly increased spectra
(by approximately
O.1).
According
to
the
theoretical
(Fig. 7), the ERL in this case corresponds to a DL liA
60~, which is close to the intrinsic DL thickness (~2Rex c) in this compound. This was the smallest DL thickness that the authors could obtain experimentally in CdS (for CdSe, see Sect. 4C). With further increase in the EB dose, the ERL tends to return to the initial state (curve 3), i.e., the depth of the DL begins to increase. Curves 4 and 5 reveal a gradual "rotation" for all of the exciton resonances indicating that
of the ERL the DL is
increased. A comparison with the theoretical spectra yields liA 160~, ~ = 10 -4 eV and lIA ~ 200~, and ~ = 5-10 -4 eV for curves 4 and 5, respectively. With increasing electron
irradiation
dose
the
spike
evolves
towards a maximum and the ERL has an inverted appearance (see ctu~ve 5 in Fig. 9, curve 4 in Fig. 10). Note that curve 6 in Fig. 9 agrees in shape with the initial spectrtum (which corresponds to a phase shift of about 2~, see Fig. 6) though the reflectivity is markedly damped. This implies that the high-fluence EB increases both the DL depth and the value of the damping (~U ~ 5.10-4eV). It appears that the boundary between the DL and the crystal bulk can no longer be represented by a plane which results in an additional
Excitons at Semiconductor Surfaces
smearing
of
the
spectrum.
(Note,
that
the
203
shortcomings
of
the
DL-model have been touched upon at the end of Sect. 4A and will be considered
in
what
follows).
The
changes
in
the
reflectance
induced by the EB persisted for hours provided the sample was not heated.
Annealing
of
a
moderately
irradiated
sample
at
room
temperature with an exposure to air usually recovers the initial lineshapes.
The authors of
[19,221 concluded that the effect of
the EB was reversible. Following high-dose EB, when the total dose reached 1018 or more an overall decrease of the exciton structure was revealed. This effect was partially reversible. An incomplete of
the
high-fluence
recovery
EB
spectrum which
was was
observed
following
assumed
to
be
the
repeated
due
to
the
subthreshold-irradiation-induced defect creation. Similar reversal of
the
ERL
under
EB
was
observed
for
the
excited
states
of
excitons each featuring its characteristic phase of rotation. The analysis of the ERL and its modification under EB in CdS single
crystals
has
been
performed
by
Benemanskaya
et
al.[19]
within the framework of the DL-model (Sect. 4A) with low values of the exciton damping F, spatial dispersion having been taken into account.
The experimental
lineshapes were
compared with
the the
calculated ones. The changes of the spectra under low-fluenoe EB were interpreted as a result of a decrease of the DL depth to the value of about
60 ~
(Fig. 9, curve 2) which
is
typical of
the
intrinsic DL with the ERL featuring the left spike.
It is evident
that
the
reversible
reflectance
and
spectra
irreversible
are
of
changes
different
of
origin.
The
exciton
former
are
associated with oxygen desorption and recharging of surface and near-surface
centers under low-fluence EB resulting in the band
bending decrease. In the initial state there occurs at the surface of the CdS crystal a field corresponding to
the depletion.
The
transition I ~ 2 is corresponds to the decrease of the electric field,
which
takes
place,
apparently,
as
a
result
of
the
electon-irradiation-induced oxygen desorption (direct evidence of the band bending
decrease under low-dose EB has been obtained in
[22] for the case of 0dSe crystals). This means that curve 2, on the DL model, corresponds to an "ideal" surface, when the DL is
204
A.E. Cherednichenko and V.A. Kiselev
equal to the intrinsic DL of Thomas and Hopfield. No such state, however,
was observed on natural
crystal surfaces.
with further increasing of the EB dose an
appearance
of
the
initial
ERL.
We note
that
(see curve 3) the ERL has
Only
then
corresponding to an increase in 1 relative
do
the
changes
to the initial value
occur in the ERL (see curves 4,5). Thus, successive effect of the EB
is first
depth.
The
to reduce
and
then,
latter case p~oceeds
gradually,
to
from either
increses
the
the DL
electric
field
increase or, as it will be shown in Sect. 5, from the formation of the defect-rich near-surface layers. The increase in the amount of irradiation results
leads
in
the
to
the
storage
modification
of
of
the
the
electric ERL.
field
Note,
which
that
the
experimental data on the effect produced by EB on the ERL of CdS crystals result
of
at
SOK
the
reported by
Ratsch
[24] were
electron-beam-induced filling
interpreted as of
surface
a
states
with electrons from the bulk, producing a negative surface charge and a depletion layer below the surface. Thus,
that part of the
effect is reversible which is due to the inhomogeneous field and Franz-Keldysh effect at the surface
(Sect.
6). Annealing at room
temperature in atmosphere surroundings recovers the spectra. The irreversible decrease of the exciton reflectance following a repeated
high-dose
long-lived defects,
EB,
is
e.g.,
more complex defects
predominantly
sulfur
(see Sect.
associated
with
the
(selenium) vacancies Vs(Vse)
or
5.E). To account for the overall
decrease of the reflectance resulting from the repeated high-dose electron
irradiation
the
authors
oZ
[19]
had
to
use
in
calculations of the reflectance a somewhat enlarged value for the bulk
damping
near-surface
parameter layer
is
r.
The
formation
evidenced
by
of
the
the
defect-rich
analysis
of
the
the
main
photoluminescence spectra (see Sect. 5E). Additional
features
on
the
long-wavelength
reflectivity maximum may be observed
side
of
(see, Figs 9,10). The lines
labelled I s and I se appear to be very sensitive to the EB. These features are also revealed in the photoluminesoenoe
spectra and
will be discussed below. In conclusion we wish to point out that the DL-model with hard
Excitons at Semiconductor Surfaces
205
edges enables a qualitative description of the features in the ERL under EB. A more adequate description which makes allowance for the effect of the inhomogeneous electric field on the exciton resulting in its field dissociation and quasilooalization at the surface will be given in Sect. 6. Still that seems to be only an approximate model,
since
the electron irradiation has a complex
effect on semiconductors and a non-trivial electron energy loss distribution should result in complicated field and defect distributions in irradiated specimens.
O. Interpretational problems As noted, Hopfield and Thomas [12] were the first to introduce a near-surface exciton dead-layer (DL) model. The original variant of the DL, namely, the one with hard edges could explain several features in the exoiton reflectance lineshapes (ERLs). Starting from mid-1970s much effort was given to investigation of surface aspects of the exciton behavior in semiconductors with a particular emphasis on the study of the ERL in various semiconductor crystals. The spectra were explored under application of the electric field, following etching, electron and ion irradiation, additional illumination. Studies of the ERLs and their alterations under various external stimuli revealed several important features. Some of them, such as the occurrence of a maximum at the frequency corresponding to the longitudinal exciton ~L (the so-called spike) and its spectral stability have been interpreted within the framework of the intrinsic DL model whereas a reversal of the ERL has been explained within the extrinsic DL with a varying depth [18] (see, Sect. 4A). The extrinsic DL observed to occur in A3B5-compounds has been treated in [17,25] as arising due to the exoiton ionization in nea~-surfaoe electric fields. This issue as applied to A2B6-compounds shall be discussed in detail in Sect. 6. It appeared, however, that a number of experimental data could not be understood within the DL-model: the occurrence of the
206
A.E. Cherednichenko and V.A. Kiselev
"left"
spike
pulsation,
(see Sect.
stretching
multy-spike
4B), of
structure
exciton reflectance
the
ERL
(see Sect.
accounted for by the DL-model. the
attention
to
those
along
6).
enhancement
the
These
frequency
features
and
scale,
can not
be
The aim of this section is to draw
features
in
the
ERLs
of
CdS
and
CdSe
crystals that are anomalous from a traditional point of view. A marked sensitivity of the excitonic spectra to various surface treatments
indicate
that
physical
conditions
at
a
semiconductor
surface dominate the ERLs. Possible mechanisms of the influence of a
surface
on
excitons
are
discussed
in Sect.
5C.
A
comparison
between the calculated ERLs and experiment allows in some cases to gain
information
about
near-surface
exciton
transition
layers
(ETLs),- see Sect. 4E. Several
authors
Patella et al. with
the
definite
attention
to
the
anomalies
in
the
ERLs.
[15] compared the experimental ERLs in CdS crystals
theoretical choice
experiment Close
paid
ones
between
for the
different ABCs
DL-depths
was
made.
The
and
ABCs.
fit
with
No the
was rather poor for the large values of the DL-depth.
correlation
between
the
magnitude
of
the
spike
and
the
magnitude of the main reflectivity peak Rma x has not been revealed which is at variance with the DL-model. calculated
ERLs
and
experimental
ones
Discrepancies were
also
between the
pointed
out
by
Brozer et a1.[26]. Of
particular
interest
is
indicated by several
authors.
of
only
a homogeneous
damping ~ ,
DL
for
a
complex
The left spike arises very
(see, e.g.,
small
values
with minimal
the intrinsic DL
spike
in the model
of
the
exciton
values
(1 ~ 2Rexc).
of
in CdS crystals could only be the DL depths
or more
only
the
structure
right could
equal to zero [22]. Hence,
spike be
corresponding
For the CdSe crystal
DL should be about 10 nm [28], but, as Fig. left-spike
a
[12,27]). The left-spike ERL (Figs. 7,8)
observed by Benemanskaya et al.[19]
nm
of
about lO-5eV, which is too low a value relative to the
commonly used revealed
structure
is
11 shows,
available,
obtained
for
1
for DL of 5
though ~
to
the intrinsic
4 nm
a
slight
setting
F
the left-spike ERL in CdSe crystals can
not be reproduced within the frames of the homogeneous DL model.
Excitons at SemiconductorSudaces
207
0.2
Wo
wL
Fig. 11. Calculated ERLs for the A(n=1 ) exciton in OdSe as a function of the DL depth i. 1,nm: (I)- O; (2)- 4; (3)- 11; (4)- 15; (5)- 30. P = O. Fig. 12 shows the fine structure of the spike observed in the exciton reflectance of OdSe at T=4.2K by Batyrev et al. [29]. The spike consists of the two maxima near ~L; a more complicated structure was reported by Davydova et al.[30] on OdSe crystals with "shifted" exoiton spectra (Fig. 13). In Sect. 5D we shall discuss the nature of the shifted exoiton spectra in CdSe crystals and give comparison between the experimental and calculated lineshapes. In addition to the anomalies in the region of ~L' 0dS and CdSe crystals show various features on the long-wavelength side of the main reflectance maximum (see, e.g., Figs. 9,10). Prominent features seen in the figures result from exoiton localization at the surface (near-surface) centers which is also evidenced by the studies of the photoluminesoenoe spectra (see Sect. 5).
208
A.E. Cherednichenko and V.A. Kiselev
0.5
/
O,4 {z
O.3
/ 0.2
R 0.4
0.1
0.3
-0
/, b
Fig. 12. Experimental exciton reflectance spectra in the vicinity of the A (n=1) exciton in OdSe at T=4.2K [29]. (a)- initial spectrum; (b) - after a 3-keV low - fluence electron irradiation A complex structure o f the spike is seen.
0.1~
o.1
0
!
i
!
~t700 14720 74740
4
It is pertinent to note that exciton spectra are measured at low temperatures where the exciton damping due to temperature is insignificant. With temperature rise, one should expect a spectral shift, line broadening, and other temperature effects. As for the EP~L, a number of authors (see, e.g.,[23]) reported a reversal of the lineshape upon heating from T=4.2K to 77K which has been ascribed to a thermally induced increase in the DL depth value.
Excitonsat SemiconductorSurfaces
209
R ¢u~. un,.
Fig. 13. Experimental exoiton reflectance spectra in the vicinity of the A (n=1) exoiton in CdSe at T=4.2K [30].
I 682.0
R,, n m I 678.0
I
F, rt~V
I
~
t
0,8
t tA 0.*
"
/~°'2"' / I" /.I/.,.,'-~ the ERL of OdS vs temperature [27]. A solid curve represents a theoretical dependence on I~.
/RmLn'°l°
t
O.a
8or, K
210
A.E. Cherednichenko and V.A. Kiselev
Pevtsov et ai.[27] studied the exciton reflectance in CdS single crystals in the temperature range 2K
the
reflectivity
signal
were
observed,
especially
spectral region of a minimal reflectivity value 2 m n i The magnitude of the minimal reflectivity R m n i zero at T ~ 3OK increased. model
of
To
(plane " R mni
account
for
the
dropped to nearly
- T") and afterwards monotonically
this
Hopfield-Thomas
in
(Fig. 14).
[12]
observation has
been
the
homogeneous
utilized.
A
DL
distinct
step-wise change in the reflected wave phase by 2~ at about 3OK [31] allowed to ascribe this finding to the Brewster effect 0). Note,
(R =
that the values of the exciton damping r derived from
the experiments were
two orders
of magnitude
as large as those
obtained with making allowance for the exciton-phonon scattering. In Sect. 6 an alternative explanation for the observed anomalous effect is suggested. Permogorov et al.
[32] examined the exciton reflectance spectra
of CdS
crystals which
random
illumination
induced
the
spike
temperature for
a
initially revealed no
of in
a
sample
the
long
at
liquid
spectrum,
which
period
of
time
spike
structure.
helium
temperature
persisted
The
A
at
occurrence
this
of
the
illumination-induced spike has been ascribed as due to initiation or alteration of the SCL. Profound changes in the ERLs of several CdS samples have been reported
in
[33]
following
illumination
with
k
~
51OO~.
Preliminary illumination of a sample was performed at T = 4 . 2 K u s i n g a 250W halogen lamp. The light beam passed through a monochromator and
a
heat
filter.
ERLs
were
taken
with
increasing
amount
of
illumination (time of illumination). The intensity of the probing light was low enough not to affect the ERL. A pattern of
illumination-induced changes of
the lineshape of
the An=1-exoiton is given in Fig. 15 for a number of expositions (in minutes).
Initial spectrum is seen to have no peculiarities,
whereas after exposure to a 5-min illumination an enhancement of the
reflectance
exposition
time
was
revealed.
the
marked
With
changes
further in
the
increase
of
the
magnitude
of
the
reflectance as well as in the lineshape were observed and at the
Excitons at Semiconductor Surfaces
same
time
the
left
spike
appeared.
As
a
211
result
of
a
30-min
exposure a reversal of the ERL to the initial shape was observed. Alterations
of
the ERLs
with
increasing amount
of
illumination
will be discussed at some length in Sect. 6.
R
0.4
20
30m£~
O.Z
Fig. 15. Evolution of ERL of OdS with illumination time. T=4.2K.
D. Interference effects in thin
crystals.
Size
quantization
of excitons The
discussions
in
the
previous
sections
have
indicated
the
usefulness of the refleotivity measurements in bulk crystals for estimation of the magnitudes of the dead layer the
interference
features
(DL) which govern
in the exciton reflectance
lineshapes
212
A.E. Cherednichenko and V.A. Kiselev
(ERLs). This section shortly discusses a more complicated case of interference
features observed
to occur
in the vicinity of
the
exciton resonance transition in thin crystals. An important contribution to the exciton optical response comes from the so-called additional or anomalous waves
(AWs)
[6,12]. We
note, that inherent in the concept of AW's is the problem of the additional boundary conditions (ABO's) which was discussed at some length in Sect. been
3. The
demonstrated
by
existence
a
number
of additional of
light waves has
techniques
(see.
e.g.,
the
reviews [34,35]). The direct verification of the theory of the AW has been performed by Kiselev et al. experimental
and
theoretical
[36-38] as a result of the
studies
of
the
reflectivity
and
transmission in thin CdS and 0dSe crystals. The
computer and experimental results
for the reflectivity
in
the neighborhood of exciton A in a CdS platelet 0.34~ thick are compared in Fig. for
the
16. For
two-wave
the calculations,
interference
in
a
Pekar's
slab
were
formulas
used.
[8]
The
ABC
employed was that of Pekar eliminating the exeiton polarization at the boundaries of the medium. The computation involved a set of parameters common for different thicknesses but different damping terms.
The
calculations
following
the
ABC
scheme
enabled
authors to reproduce the essential features of the spectra, the side
one-period Fabri-Perot and
the
two-period
high-energy side of ~L" Fig. 16 shows that on interference
feature
(FP) (FP
interference plus
AW)
the low-energy
can
be
observed.
on
the
the ~z.
low-energy
interference
on
the
side of e L a distinct This
is
due
to
the
FP
interference of excitons from the lower polariton branch (see Fig. 3c).
On
the high-energy
side
of ~L
two
series
of
interference
peaks occcur. The first one corresponds to the FP interference of exciton polaritons from the upper branch; the peaks in this series are well separated. Another series superimposed on the first one consists
of much
closer
spaced peaks
and
is due
to
the mutual
interference of exciton polaritons from both of the branches. Thus, optical
the
AWs
spectra
produce of
thin
several crystal
interesting platelets.
features The
in
the
interference
Excitons at SemiconductorSudaces
213
structureadditional to the one that should be observed with M = is a direct proof of the fact that the AWs do exist. Studies of interference due to AWs provide evidence for proximity of the ABO to
that
of Pekar
for
low-energy
waves arise as a result of
exeitons
[37,38].
Additional
excitation of polaritons. Nonetheless,
it seems also useful to consider the problem within the mechanical approximation [39]. The mechanical exciton is depicted by (3.1) at E=O.
In
the
case
of
a
plane-parallel
conditions should be imposed (e.g., boundaries
platelet
the
boundary
the Pekar's ABO P=O) on both
(let them be z = 0 and z = a). In other words, a thin
plane-parallel semiconductor platelet constitutes a quantum well for excitons which is approximated by a rectangular potential well with
infinitely high
walls.
It
is
evident
that
the mechanical
exciton band (3.4) is size-quantized in the direction z normal to the crystal plane and the solutions of (3.1) along z would depend on z as
sin(N%z/a), where N = I, 2, 3 ....
part
Pig.
of
17
rectangular well
which
corresponds
is shown with
to
In the right - hand
real
values
of
k=k z
a
the levels spaced by the values
(P~V~/a)2/(2M) as referred to the bottom of the well. These levels are
represented
on
the
mechauical
exciton
dispersion
(dashed
curve) by points, or, more strictly, the levels and corresponding points indicate the bottoms of the 2D subbands in the x,y - plane. A much more complicated problem arises when the exciton - photon coupling is
to be
taken into account
[39] as
in the case when
transmission and reflection coefficients of a crystal platelet are to be derived (see Fig. 16). The exciton levels in a rectangular well can be shown [39] to govern the interference structure which is
the
result
of
both
the
FP
interference
within
the
entire
frequency range explored and the interference between the AWs for ~ ~L" A theoretical account of the size quantization of excitons fits fairly well the features of the experimental spectrum 16).
In
what
follows,
we
shall
discuss
the
(Pig.
long-wavelength
structure at ~ < ~L which is due to the exciton localization in a quantum well. We note, that the levels of the mechanical excitons in the well do not coincide with corresponding features optical specti~m [40].
in the
214
A.E. Cherednichenko and V.A. Kiselev
Rtheo,
Rez~r
- 0.6
0.# 0.6 0.2 0.4 0 0.2 0
486.4
486.0
#85.2
485.6
#84.8
~8~. ~ A, , n m
Fig. 16. Reflectivity from a CdS crystal (0.34 ~un thick) in the region of the A(n=1) exciton. The upper curve is experimental (in arbitrary units) taken at T=I.6K, the lower curve is theoretical.
I
O" z~'Ct ~z
z =O
k--link
k=Rek
Fig. 17. Schematic representation of the optical response of excitons localized in: rectangulsr well ( a ) ; surface 6-potential (b).
Excitons at Semiconductor surfaces
215
This observation is accounted for by Fig. 17. The points in Pig. 17a [40] o o ~ e s p o n d to mechanical exoitons and the arrows give the positions of the expected features in the optical spectrum. Such a spectral shift arising due to the polariton effect enables the observation of the exoiton quantization in much thicker platelets than in those where quantization of carriers, i.e., of electrons and/or holes is possible.
E. Exoiton transition layers and a formal model In sections
4A and 4B we have
demonstrated
that
the simplest
model to calculate the ERL is that of the exoiton-free layer, supplemented by the ABO (usually of the Pekar's type). Yet, there exists evidence that more sophisticated descriptions of the excitonic optical response need to be incorporated. In factj we should also consider more complicated models of the near-surface transition layers and, consequently, of the exoiton transition layers (ETLs). The ETL may be of intrinsic or extrinsic origin. Intrinsic mechanisms are those stemming from the internal structure of excitons and are due to (i) the interaction of electrons and holes with their mirror images at the semiconductor surface
(see Sect. 4A) and (ii) the effect of the surface-induced
confinement
of
the
electron
and
hole
motions.
A
number
of
theorists (see, e.g., [12,16,41-47]) considered the details of intrinsic mechanisms of formation of the intrinsic ETL. They found that the mechanisms yield the exoiton-free layer of about ~Rex c. Extrinsic ETLs for excitons arise due to external, with respect to excitons, mechanisms and they are motivated by various surface (near-surface) imperfections (see Sect. 5) including the electric field (see Sect. 6). (Note, that, contr-ary to the intrinsic ETL, the extrinsic one may have macroscopic dimensions, well exceeding the exciton wavelength ~/nex o ~ 102nm). Theoretical aspects of the perturbation of exciton states near the surface were discussed in a number of papers (see, e.g., [48-54]). Most drastically excitons are perturbed by the electric field of the SOL. This issue will be
216
A.E, Cherednichenko and V.A. Kiselev
discussed in detail in Sect. 6. Whatever the origin of the ETL may be, the ERL can be calculated by formally introducing a near-surface potential for excitons. A method of calculating the exoiton reflectance in the case of an arbitrary potential has been put forward by Kiselev [55]. Assuming that the adiabatic approximation (see, e.g., [42]) holds, the elastic scattering of an exciton by a considered. A smooth adiabatic potential
surface potential was was approximated by a
step curve. The limitations on the size of the steps were as follows: the height of the step is small compared with the energy of the exciton being scattered or the width small compared with the exciton wavelength. On the other hand, the number of steps must
be
finite.
The
problem
is
thus reduced
to the problem
of
calculating the optical response of a layered medium where the potential is not dependent, within each layer, on the coordinate which enables the usage of the crystal optics of a thin slab with spatial dispersion [6,8]. Note, that it is also possible to formulate
the problem within the mechanical
exciton quantization
approximation [39,40]. Principally, one should arrive at the same results insofaras the phenomenology of the layered medium takes into account
both the above-barrier
[39] and below-barrier
[40]
solutions for the mechanical exciton. A number of different model potentials V(z) are presented in the left-hand part of Fig. 18 (see curves 1-17). The results of calculations of the reflectivity R are shown against corresponding numbers on the right. The spectra were calculated for the A(n=1) exoiton [12,18] in OdS (curves 1-16) and OdSe [37] (cuzwe 17). The analysis of the calculated spectra shows that the reversal of the ERL results from the effect of the repulsive potential experienced by longwavelength exoitons with kinetic energies from within the interval of 0 - h~LT. The reversal is exemplified by oases 1,3, and 5 for the potentials in the form of the steps of different widths. Note, that the height of the step (see 2-4) does not affect the reversal. The reversal is seen to occur also for the potentials featuring tails (6-8) and for the linear potential
(9,10).
Exc#ons at Semiconductor Surfaces
217
¢~,,6z|
Tig. 18. A number of potential s V and re 1 evant calculated refleotivity coefficients R in the vicinity of the A(n=l) exciton in OdS (I-16) and OdBe
{~7).
i~
~Y
Y]:., "-'-""
I Z
0,2
218
A.E. Cherednichenko and V.A. Kiselev
Inspection of curves 2-4 shows that the spike (in the spirit of Hopfield and Thomas [12]) occurs only for a sufficiently high step. In other words, the potential must have a hard core which is inpermeable
even for the high-energy excitons
is enough transparent to the light wave). The spike, however, appears for curves
(the surface layer
11-16
which
represent
near-surface potential wells for excitons. By insignificantly varying the depth and shape of the well (11-13) prominent spikes can be observed. Curves 14-16 show the optical response of the surface mechanical exoiton, i.e., the exoiton having the bound state in a fairly deep well. The spectra are seen to reveal the additional maximum which shifts to longer wavelengths with increasing binding energy (for the details, see [40]). The outlined method for calculating the exciton optical response has been applied by Kiselev
[56] to the Shottky barrier case at
low temperatures which is considered in detail in Sect. 6. To conclude, we note, that when finding the exoiton optical response one solves the direct problem of the exciton scattering only. The inverse problem, as is well-known, can not be solved uniquely, that is, only a family of potentials can be found. An appropriate potential can be chosen proceeding from physical mechanisms and from comparison between the calculated exciton reflectance spectra and experimental ones.
5. Crystal Defects and Localization of Exoitons A. Bound excitons Exeitons have a tendency
to forfn bound exciton complexes with
defects and these are also 0teated by the optical excitation of the crystals. This section describes shortly the properties of the bound W a n n i e ~ M o t t excitons. Lampert [57] pointed out the analogy between the hydrogen molecule and an exciton interacting with a shallow neutral donor (aoceptor) which can, like the exciton, be described by the hydrogenic model, Just as the hydrogen atoms bind
Excitons at Semiconductor Surfaces
219
to each other in the H 2 molecule, so should the exciton be bound to the neutral impurity. Since a bound exciton complex is made up of an impurity atom (or lattice defect) to which an intrinsic (free) exciton is bound and since the intrinsic exciton is a property of the crystalline host lattice, a question naturally arises concerning the relative intensities of absorption lines that derive from these two types of excitons. This is especially so when the bound exciton lines are observed to be nearly as intense as the intrinsic exciton lines, a frequent observation in both the absorption - compounds. The
theory
of
and
"impurity"
reflectivity or
defect
spectra
absorption
of
the
A2B 6
intensities
in
semiconductors has been developed by Rashba [58]. He found that if the absorption transition is direct and if the discrete level associated with the impurity approaches the conduction band, the intensity of the absorption line increases. The explanation offered for this intensity behavior is that the optical exoitatign is not localized in the impurity but encompasses a number of neighboring lattice points of the host crystal. Hence, in the absorption process, light is absorbed by the entire region of the crystal consisting of the impurity and its surroundings. As a result of an attack on the particular problem of excitons which are weakly bound to localized impurities , Rashba and Gurgenishvily [ 5 9 ] derived the following relation between the oscillator strength of the bound exciton ~ and the oscillator strength of the intrinsic exciton ~exc' using the effective-mass approximation Ig = (Eo/IEgll3/2~exo where
E0 =
(2~2/m)
(%/00)2/3,
Eg
is
the binding
(5.1) energy of
the
exoiton to the impurity, m is the effective mass of the intrinsic exoiton, and ~0 is the volume of the unit cell. Note, that the value of ~ , when substituting the parameters for the bound exoiton lines derived experimentally, may exceed ~exc by more than four orders of magnitude (the effect of the "giant" oscillator
220
strength).
A.E. Cherednichenko and V.A. Kiselev
Assuming
that
the
line
intensities
are
equal
to
a
simple product of density and oscillator strength (In ~ Nd'f~) and assuming an impurity concentration of N d ~1017cm-3~and intrisic exciton concentration of Nexc ~ 1022 one arrives at the conclusion that bound exciton intensities can be within an order of magnitude of those of the intrinsic exoiton lines, hence, they are observable in transmission and reflectance. We note, that the bound exoiton dominate the emission spectra. The emission results from the
disintegration
(radiative decay)
complexes
and
the
occurs
on
of the bound exoiton
long-wavelength
side
of
the
free
exciton transition as narrow (having half-widths of ~10-4eV) lines yielding information on the associated defects. Another important conclusion which can be drawn from equation (5.1) is that, as the intrinsic exciton becomes more tightly bound to the associated center, the oscillator strength (and, hence, the intensity of the exoiton-complex line should decrease and u~ce versa). The extensive studies of Thomas and Hopfield [60] on CdS have resulted in the development of a theory of bound exoitons. The theory is based on the band symmetry properties associated with the wurtzite structure of this compound. Several bound exciton complexes are postulated and observed experimentally. These are the exoitons bound to neutral and ionized donors. The transitions were designated as 12 and 13, respectively. The exoitons bound to neutral aooeptors, designated as 11 are also observed.
B. The effect of
keV- electron
irradiation
on
the
exciton
photoluminescenoe spectra Exciton photoluminesoence spectra are also dependent upon surface conditions of a sample. This section briefly reviews the effect on the exoiton photoluminesoenoe spectra of the electron irradiation of OdS and OdSe crystals [61,62]. Irradiation induced changes in the spectra were registered ~n s~tu at 4.2K as a function of fluenoe (or dose) of incident keV-electrons. Figure 19 shows the dependences on the irradiation dose [62]
Excitonsat SemiconductorSurfaces
I
221
ziO 20 10
1
,6~.
F~.
19. Intensities
o~
the emission lines in OdS at T=4.2K vs eleotron dose. 1- I2B; 2- I2A; 3- 11; 4- A(n=l); 5- 13 .
~'~
~+.,.--o-----I~. _+,+Z~"
-~2
~ ~~i~e.~.e..~e~ 0.6F\5
e
~i
0 0.5
4
~e~3
40 50 6O 70 8O
T
.............Iz
Pig. 20. Transmission speotra in the region of the 12bound exoiton in 0dSe exposed to inoreasing doses of 3.5 keV electron irradiation. 1initial speotrum; 2 - after 2"1015e/om2; 3- 8"1017e/om2.
681.0 ' 680.5
6B0.0
679.5 ~,,rmt
222
A.E. Cherednichenko and V.A. Kiselev
of various exciton emission lines typical ¢f CdS. With low-fluence irradiation (D ~ 1015 e/cm 2) an overall increase in the luminescence intensity was observed for the I2A-, I2B-,
Ii-, 13-
bound exciton emission lines and for the An= I- exciton emission line
as
well.
crystals
Similar
changes
occurring
seemingly results,
dose
dependences
were
found
for
0dSe
[61]. These findings indicate that the exciton emission due in
to
under a
low-fluence
surface
particular,
exoiton non-radiative
band
in
the
escape
electron
bending
irradiation
decrease.
decrease
of
probability.
a
The
The
are
latter
field-dependent in
the
amount of band bending is also evidenced by our studies of
the
reflectance spectra and surface photovoltage
reduction
[22]. Note, that the
transmission in the region of the I2-bound exciton line in OdSe crystals
shown
insignificantly,
in
Fig.
hence,
20
(curve
[61]
2)
the low-fluence EB
is
does not
changed
practically
produce defects. This conclusion is also supported by a reversible character of the exciton spectra changes. After electron irradiation to a fluence of above 1016 e/cm 2 the emission
lines
intensities
show
different
behavior
(Fig.
19),
which is indicative of their different origin. The most pronounced increase, about 1.5 order of magnitude, was observed to occur for the I2-1ines corresponding to excitons bound to neutral donors. A corresponding
increase
absorption
the
in
is
region
observed of
for
these
486.1-487.1
nm.
lines The
in
CdS
electron
in
beam
induced enhancement of the I2-feature in CdSe is seen in Fig. 20. These observations
indicate an increase
in the
intrinsic defect
(neutral donor) content in the near-surface region after EB. Figure 19 shows a sharp
drop
in the intensity of
the I3-1ine
(the Is-line, see below) which is due to the exciton bound to the charged
(ionized) donor. The anticorrelation between the 12- and
13- line behavior (also see [60,63,64]) and a reversible character of the changes show that a neutralization of ionized donors occurs under EB. At the same time the intensity of the It-line, which is due to the exciton, bound to a neutral acceptor is less dependent on the dose,
probably
because
a
major
part
of
acceptors
in
n-type
Excitons at Semiconductor Surfaces
223
semiconductors is charged and, hence, not influenced by EB. With increasing irradiation dose the effects due to long-lived defects become dominant which is evidenced by an overall bound exciton emission intensity drop (Fig. 19), an increase in the absorption in the region of the I2-I2"-lines (Fig. 20), the emergence of the electron irradiation induced Ie-lines (see below). Electron irradiation as well as other treatments were found to strongly affect the broad-line structure observed to occur in CdS crystals in the spectral range of 486.1-486.5nm. Initially this feature has been ascribed to the emission of the exciton bound to the ionized donor (the so-called 13- line). The reason for this assignment was the observed anticorrelation between the intensities of the 12 - and 13-1ines. Typical of this feature is its dependence on surface-sensitive electron irradiation, excitation light intensity. This surface-sensitive structure was labelled I s [65]. Travnikov [66] studied the exciton spectra from as-cleaved and air-exposed surfaces of OdS and found two lines in the spectral o -line (A = 486.4nm) may vary in region under consideration. The Isl intensity whereas the I°2-1ine may as well change the spectral position within the spectral range 486.4-486.2nm. One of the lines was ascribed to surface defects involving oxygen adsorbed at the surface, while the other one was claimed to originate from the space charge layer. The observed disappearance of the Is-feature under low-dose electron irradiation, resulting in the oxygen desorption [67] points to the fact that oxygen is involved in the centers yielding the Is-feature. A surface-sensitive Is-feature was likewise revealed in the reflectance and photolumin.escence spectI~a of 0dSe crystals,- see Figs. I0,2!, The Is-line appears in t~e initial spectrum as broad line with the maximum at k = 679.9nm, slightly varyir~ its spectral position from sample to sample within 0.1-0.2nm, With minimal amount of e l @ c t r ~ i~r~iation a short-wavelength shift (Fig. 21, OU~-'ve 2) ~ d subSeQuent quenohin~g of the Is-line (curve 3) were revealed. These observations indicate that the I s - feature is most likely associated with the excitons bound to ionized
224
A.E. Cherednichenko and V.A. Kiselev
15
Fig. 21. Exciton photoluminescence spectra of CdSe at 4.2K. (1)- initial spectrum, (2 4)- after exposure to increasing doses
(1.1016; 5.1016; 3.1017e/cm2)
of 2 keV electron irradiation.
680
679 j,nm
donor-type centers at the surface. The Is-line in CdS is seemingly a surface counterpart of the I3-type line in the bulk. It features a
large
half-width
electric
which
may
be
field value affecting
assigned
this
to
variation
exciton-impurity
in
the
complex at
the surface. The authors of ref.
[61 ] claimed that the Is-line in
CdSe
field-dependent
may
be
associated
with
a
localization
of
excitons in near-surface quantum wells. As
the
electron
irradiation
dose
is
raised
a
new
feature
designated as I e s arises in the spectrum. (i) This new feature is observed both in luminescence and reflectance spectra of CdSe (Figs.
10,21 )
and
CdS
(Fig.
9)
single
crystals
following
Excitons at Semiconductor Surfaces
high-fluence
EB.
In fairly
thin
samples
it
225
can be
observed
in
transmission as well (Pig. 20). (ii) Contrast to the Is-feature, the spectral position of this line is practically fixed and makes 485.9 and 679.6nm in CdS and CdSe, respectively. (iii) Following a repeated high-fluence EB, some samples reveal a reversal (rotation) of the ERL corresponding to this transition (see Sect. 5E). These findings suggest that long-lived defects produced by electron irradiation yield the I~-lines. The defects are located in the near-surface region whose width is governed by the electron irradiation dose and incident electron energy. A reversal of the ERL corresponding to the I~-feature proceeds much in the same manner as in the case of the intrinsic exciton although in the former case the centers responsible for the I~-feature are located in the near-surface region and are absent in the bulk (Sect. 5E).
C
Surface and near-surface excitons
Surface states and resonances for excitons are predicted by both dielectric and microscopic theories [6]. For ideal surfaces and interfaces, specific long-wavelength solutions, with the electric field localized within thin layers adjacent to the surface, can be obtained using a macroscopic dielectric treatment provided the dielectric function and the ABCs are specified (both, however, should be derived using microscopic models). The results of experimental studies of this type reviewed, for instance, in [68].
of
excitations
have
been
Conclusions of microscopic theories depend on the assumptions about the electronic structure of the surface. For example, in the case of molecular crystals the surface-site-shift model has been proposed [69] which leads to the exciton localization within one or two subsurface monolayers characterized by exciton resonance frequencies shifted with respect to the bulk value. The theory explains the fine structure observed in the reflectivity spectra of antracene. For interfaces, the excitons may be attracted to the surface by the image potential [70] provided the dielectric
226
A.E. Cherednichenko and V.A. Kiselev
constant of the overlayer is higher than that of the medium. Of
particular
carriers are
interest
is
the
case
in surface electronic
when
one
or
both
states or bands,
of
the
formed,
for
instance, from dangling-bond states [71]. Experimental aspects of the problem have been discussed in [72,73]. Theoretical aspects of 2D
surface
have
been
excitons
excitons
and,
discussed
in
with
surface
in particular, [71,74,75].
impurity
their optical
The
centers
response
interaction and
of
initiation
these of
2D
exciton - impurity complexes have been discussed in [76]. This
short
effort
of
review
shows
(for more,
see
[77])
that
those engaged in the problem of surface
the
main
excitons has
been concentrated on the mechanisms of surface excitons formation for
the
case
of
ideal
crystals.
New mechanisms
of
the
exciton
localization at the surface and within near-surface regions arise if we take into account the interaction of excitons with defects and impurities in semiconductors are
listed
below.
Note,
that
[78]. These
the
"defect" mechanisms
best-developed
model
for
the
interaction of excitons and near-surface charged imperfections is based on the Stark effect in an inhomogeneous electric field. The latter,
under normal
circumtanoes,
surface of semiconductors. excitons
are
localized
is always
present
below
the
It is important that quite frequently within
extended
layers
(102-
103nm)
exceeding in value those typical of the ideal surfaces by orders of
magnitude.
These
layer
thio~2qesses
are
close
to
polariton
wavelengths near resonance and also to the penetration depth of light in a crystal. Hence, exoiton spectra appear to be sensitive functions of near-surface exciton transition layers are
strongly
dependent
on
the
physical
(ETLs) which
conditions
at
the
defects
and
semiconductor near-surface layers. Actually,
the
interaction
of
excitons
with
impurities is always present both in the bulk and at the surface of semiconductors. However,
the defects and impurities themselves
may acquire new properties near the surface, and that may be the reason
for
the modified
or even new mechanisms
of
localization
(see [78)]. Some variants, discussed below, are displayed in Fig. 22.
Excitons at Semiconductor Surfaces
227
W w0
3,4r ~@@
Fig. 22. A s c h e m e for "defe c t" mechanisms of the excit o n localization near semiconductor surfaces (see text).
• 1
~
I@ @@ @@@ @@ @
@@@ •
@@ @@ •
• • @@@
..-- ".. @ .-.-~... .
;@41@ D @
I.
It
is well
known
that
defects
and
impurities
yield under
optical excitation various excitonic complexes whose radii are of the order of IO nm or more (see Sect. 5A). It is natural to assume that
within
magnitude
surface
new
layers
excitonic
with
thicknesses
complexes
may
of
arise
this with
frequencies other than in the crystal bulk. Moreover, may
be
even
thicker
if
the
effect
on
the
order
of
resonance the layers
complexes
of
the
near-surface electric field (see point 4 of our list) is strong. This influence is twofold. Firstly, it may change charge states of imperfections, and, secondly, to
the
Stark
responsible
effect.
for
the
In
the
binding
it may shift the energy levels due other of
limiting
excitons
are
case
the
centers
localized
on
the
surface of a semiconductor,- see, e.g., [66]. Provided
the
complexes
are
independent
(see variant
I)
their
contribution to the dielectric constant of the medium may be taken into
account
by
adding
oscillator
terms.
These
terms
should
include the Rashba enhancement of the oscillator strength [58,59] and inhomogeneous dependences of the parameters of the complexes along
the
z-axis
normal
optical properties of
to
the
surface.
Calculations
of
the
the media characterized by the dielectric
constant varying in one dimension may be performed exploiting the approach outlined in Sect. 4E [55].
228
A.E. Cherednichenko and V.A. Kiselev
2. Another mechanism of localization may result from an increase in
the
imperfection
qualitatively
concentration
new feature appears
that an exciton-defect thickness
of defects.
a nonstoichiometric surface
band
resonance resonance point
(or exciton-impurity)
of the localization
the distribution may
term
layer be
typical
oscillator
may be not of great
A
is so high The
this layer may coincide with
a
shown
into
free in Fig.
However,
the
importance
the
The exciton-defect
theory
exciton, 22,
term should be z-dependent
strength.
surface.
band is created.
(see the next point). of
as
the
layer in this case should depend on Actually
incorporated
frequency,
I this
towards
if the concentration
but
effects
2.
include of
for this band,
adding
for
variant
and
by
a
a
lower
Just
as
in
the enhanced
spatial
dispersion
and formally case 2
may be reduced to case I. 3. A semiconductor the surface,
crystal may have a nonstoichiometric
for example,
layer at
a layer enriched with Cd in the case of
CdS. Also real crystals may be characterized by chemically reacted layers
or/and
layers
one-dimensional gap
of
normal
the material The some
the surface
[55,79,29].
continuum deformation well
for
vanishing
exciton may
be
excitons into
described,
case
3
states,
discrete
importance
of
interaction layer
to
the
refer
may
crystal.
the
direction
be
treated
frequency,
~O'
to modifications the
variant
that
in
the energy
3
in
it will This
Fig.
cause
and
situation
near the
of
22. a
wave
by
of
deformation
of
a The
quantum
functions has
been
[29]. for near-surface
it also provides due
gap
eigenstates,
the
predominantly
to the localization
resonance
see
significant
is the charge of imperfections. 3 and 5, but
-
so
the bulk
lead
concerns
with
in, e.g., ref.
4. Of great
the
I and 2 which
states
of
in a graded
of
(a
which may change
semiconductor
variation
to cases
exciton
solutions
This may also
graded-gap
introducing Contrary
solid
is expected)
resulting
to the surface.
excitons.
bound
of
disorder
localization
of excitons
It is true for variants a specific
electric
field
(SCL). The latter acts on excitons
additional within
the
from I to
mechanism space
of
charge
through the Stark effect
Excitons at Semiconductor Surfaces
229
(see Sect. 6). The coordinate dependence of the field may be subdivided into regular
(averaged over fluctuations)
and irregular contributions
In Sect. 6 we shall show that the macroscopic electric field may cause exciton localization in the SOL. Note, that the width of the SCL, w, depends on the surface charge, Qs' and on the net space charge density, N + - N-. magnitudes.
The
highest
It may vary in a very broad range of sensitivity
should be observed for w ~ ~exc
of
the
exciton
reflectance
~ I0-I02nm where kexc
is some
characteristic wavelength of the exciton. Another mechanism may also be operating, when only one of the carriers, comprising the exciton is l o c a l i z e d b y the field with a subsequent
attraction
of
the
other
carrier.
This
claimed to be revealed experimentally (see, e.g.,
mechanism
is
[80]), but its
consideration is beyond the scope of the present review. 5. At last we wish to mention highly disordered surface layers (amorphous, sputtered layers of solid solutions, etc.) that may be represented
by
random
spatial
oscillators,
- see variant
5
and
energy
in Fig.
22.
distributions
This
variant will
of be
illustrated in what follows.
D. Localization of excitons in rectangular potential wells To account for some features in the ERLs of semiconductors appeared
appropriate
to
make
use
of
the
model
it
involving
near-surface potential wells for excitons with fairly sharp edges. It is evident
that the diffusion of the well's
exceed the exciton wavelength (~102nm). the exciton reflectance lineshape described
in
circumventing
[55] the
(see,
difficult
edge should not
A method for calculating
(ERL) for a given potential was
Sect. problem
4E). of
This
method
finding
the
permits
levels
of
mechanical excitons in the well and then taking into account the photon-exciton interaction, and it makes it possible to directly calculate the optical spectra of exoitons.
230
A.E. Cherednichenko and V.A. Kise ev
0
2
#
6
8
N 10 tZ
-/I /"
1 I
0
-5
T
.
111 e
1 0
250
2
$
6
2,rim
8
I0 I2 "~$ 02 0
iz '
R
Fig. 23. A scheme showing the fine structure o f t h e EP~ o f a CdS crystal (on the right) arising due to the potential well with hard edges (on the left).
Consider, well,
for
the
shown in Fig.
sake 23.
of
simplicity,
We note,
the
that the
right-angle
shaped
particular case of a
well with infinite walls has been presented in Sect. 4D. Here, inner wall of the well
is of finite height,
levels
finite.
in
the
well
is
These
levels
hence, are
the
the number of plotted
in
the
left-hand part of Fig. 23. From the figure we note that there are resonant
states
quantum-mechanical Bound spectral
and
above
the
well
which
are
dominated
by
interference of excitons above the well.
resonant
states
both
lead
density of states of mechanical
to
oscillations
exoitons.
in
The maxima
the in
the density of states are indicated on the dispersion curve by the dots plotted
in the middle part of the figure.
They indicate
positions of the bottoms of N = 1,2... 2D-subbands plane). Such a plot is
made
by
analogy
with
the
(in the surface
the
case
of
an
Excitorl~ at Semiconductor Surfaces
231
O.5
0.~ 0.5 0.2 0.1
f
I
0
0
0
w o
wj.
w
Fig. 24. The calculated ERL of a CdS crystal for different parameters of the rectangular near-surface potential well of Fig. 23 (see text).
infinite-wall well (see Fig. 17a) and, is, It has been pointed out in Sect. 4D that polariton effect the singularities in density of states do not coincide with corresponding to them in the optical
therefore, qualitative. in the presence of the the mechanical exciton the spectral features spectra, e.g., exciton
232
A.E. Cherednichenko and V.A. Kiselev
J /
!!
P~Jf
~03
k
Fig. 25. Schematic illustration of the origin of the spike features due to the near-surface potential well with hard edges (see text).
reflectance spectra
[40]. The singularities are shown in Fig.
to be related by means of upward and downward arrows
17
(see Fig.
17a), i.e., by means of projecting the dots onto the two polariton branches. Similar plot was made in Fig. 23. The right-hand part of the figure shows the fine structure of the calculated reflectance spectrum which is seen to be dominated by the exciton levels in the well, as well as by the above-well resonant states. The width of the well w was taken to be 250 nm, the depth of the well being given
as
the resonance
frequency
shift
A~O=
-5cm-~
Henceforth,
where necessary (see, e.g., Sect. 6), these two parameters will be given as put in parentheses without (250,-5), denoting the width and respectively. The spectra calculated for several shown
in
Fig.
24.
By
varying
w
one
units of dimensions, the depth of the different arrives
values at
the
e.g., well,
of w are following
Excitons at Semiconductor Surfaces
conclusions interference
(i)
With
features
the
increase
marked
in
1,2,3...
the begin
233
value to
of
appear.
w
the They
originate from the above-well resonant states which are subsequently captured by the well with increasing w. ( i i ) The interference structure shifts to long wavelengths away from the main reflectivity peak. (iii) The structure is spread out and, eventually, disappears with a new short-wavelength feature arising in the region of the spike. Such a modification of the spectra are easy to follow in Fig. 17 and Fig. 23. In fact, similar interference features can likewise be obtaind with a right-angled step (see, e.g., Fig. 18, variant 17) and this circumstance need to be taken into account when performing the analysis of experimental data. Still, the models involving potential wells quite often give a more close fit to the features revealed in the experimental spectra. Quite often, one may observe a long-wavelength shift of the main refleotivity maximum and minimum, stretching of the spectrum along the frequency scale, and appearance of a more pronounced spike as compared with that normally observed in the region of ~L (see Fig. 24 and text below). These observations are readily explained within the framework of the polariton approach, see Fig. 25. Lowering of the exciton resonance frequency in the near-surface layer (~OV ~ ~OS ) brings about the spectral shift of the polariton branches below the surface. As a result, the polaritons of the upper branch in this layer (shown by horizontal smrows) fall into the region of the longitudinal-transverse splitting for bulk polaritons, hence, they are reflected from the bulk of the crystal back to the surface. Actually,
the crystal bulk serves as a metal
mirror for near-surface exciton polaritons. As regards the origin of the wells in question, we note, that, at least, two mechanisms should be considered. The first one is due to a random field of charged centers in the SCL, where a sharp drop in the frequency ~0 below the surface is associated with the fluctuation part of the electric field of charged defects (donors) of high concentration. This case is exemplified by the spectra of Fig. 12 [29]. Relevant theoretical treatments are now in progress,
234
A.E. Cherednichenko and V.A. Kiselev
and a detailed
account
of the problem
is beyond
the scope
of the
present review. Another mechanism crystals [81].
The
crystal
experimental
is
spectrum
shown
in
features
compared
with
experimental about
a a
stretched along 13.4om -I
exciton
part
normal also
spectral
note,
that
the frequency
6.4cm -1
(which
shift
of
the
properties,
perturbation
is
of
spectra
as
the
that
of the surface.
a whole
Another
spectra
of the ERLs are value
used
of
the
[37]).
(A)
is is
associated a
X-ray analysis
with
result
of
Both of the shifts are assumed
due to one and the same reason.
are
to a commonly the
(A~L)
stretching
The
IOcm -I , as
[37].
"widths"
splitting of 7.7 cm -1
while
a.
a
curve a, is shifted by
compared
less
in
such
curve
experimental
The
of
about
position
the
scale.
26(I),
shift
of CdSe
observed
spectrum
of Fig.
short-wavelength
in some
solution
reflectance
and 25.5cm -1 , respectively,
of
The
to the occurrence
layer of a solid
the upper
longitudinal-transverse bulk
due
spectrum shown in Fig. 26(II),
49cm -I . We
value
arises
of a near-surface
indicates
a
to be
that the
crystals in question were heavily doped with S to a concentration of 1019-I020cm -3 . Actually, the samples represented solid solutions
of
CdSe1_xS x
Corresponding 1.5%
for
of
second
one
the
extrapolating
involving extended and
can
a fairly tail,
IIb,
deep well with
only
be
by
the
the of
order
first S
emploing
sharp
27. The upper
to calculate (Note,
6).
of
I%.
spectrum
was
accounted
regards for
edges shown in Fig. 27.
by
the
a
and
estimated
by
spectra and the model
that
the
potential
supplemented
(I) and lower of Fig.
see curve
feature
long-wavelength a
ERLs
are
is seemingly due to
by the inhomogeneous
incorporating
by an
(2) curves of 26,
theoretical
The tail of the potential
exerted on excitons As
edges
the ERLs
field which yields a multy-spike
Sect.
of
for
percentage
achieved
respectively.
the Stark effect electric
be
see Fig.
given as unshifted).
also,
(the
being
0.3%
agreement between the experimental
ones
the figure were used Ib
x
x were
the data of [82]).
A qualitative calculated
with
values
deep
surface
in the ERL feature, well
with
(see,
it
can
sharp
Ex¢itons at Semiconductor Surfaces
235
T R OL
0.#0.3- / 0.20.10 ! , I~720 , R
I
I
I
I
14700
I
14720
1"[ b
0.5-
0,#0.~- /
m
0.2o,1-
u~,cm 1 0
i
I
1#?~0
I
I
I
!
1#760 1~780
ua~c,m-t i
~
i
I~700
I
I
I
1#720
Pig. 26. (a) Experimental exciton reflectance spectra of OdSe(1-x)S(x) in the vicinity of the A(n=1) -exciton at 4.2K (see text); (b) Theoretical spectra calculated using near-surface potentials shown in Pig. 27. I corresponds to x =0.3% (see potential I in Pig. 27); II corresponds to x =1.5% (potential 2 in Pig. 27).
236
A.E. Cherednichenko and V.A. Kiselev
t" •,~ r ~ o , 4 --- .
.
.
crl, t"l
F .
"1
L,_.
0 -4 -'1 %.
0
Fig. 27. A resonant exoiton frequency shift A~ and exciton damping P vs coordinate z. (I) corresponds to curve Ib and (2) corresponds to curve IIb (see Fig. 26).
-8
-/2
z,~m
-16 I
I
I
I
0.2 0.4
0
In
i
view
of
I
0.6
I
I
these
(one-dimensional)
I
0.8 1.0
studies
near-surface
it
seems
natural
to
assume
a
fluctuation of the composition due
to a layer of a solid solution (only a one-dimensional fluctuation is
capable
estimates,
of
producing
sharp
a characteristic
be of about ~102nm.
features).
According
to
our
linear size of the fluctuation should
Sulfur content at the surface should be less
than in the interior, which yields a potential well rather that a step.
There
is
also
an
additional
that,
from the point
excitons
in
CdS(Se)
solid
does not seem to proceed. that,
depending
surface excitons.
on
fluctuation
the may
which
of view of solid solution physics,
these findings appear to be rather surprising, of
evidence
[81].
supports our suggestions Note,
experimental
solutions
Probably, growth arise
with
low
sulfur
content
the effect of surface is such
conditions,
which
since localization
allows
a for
one-dimensional localization
of
Excitons at Semiconductor Surfaces
237
E. Localization of excitons in defect-rich surface layers In
the
previous
section
the
exciton
localization
in
the
near-surface potential wells has been treated. Another important mechanism
of
localization,
U~Z.,
localization
field
the
space
layer
(SOL)
of
detail
in Beet.
charge
6. In the present
by
shall
the
be
electric
considered
in
section we consider one more
type of localization which is due to the binding of exeitons by near-surface defects. Several
types
of
bound
excitons
associated
with
different
centers can be excited in a semiconductor bulk (see Sect. 5A) but we
will
concentrate
herein
on
those
which
near-surface layer (Fig. 22, variants 1,2). impurities) occur
may exist
predominantly
in as-grown
as
a
result
are
peculiar
of
a
Such centers (defects,
(untreated)
samples,
of
treatments,
various
but
they e.g.,
etching, electron or ion irradiation. In Sect. 5B we have outlined that the electron irradiation of CdS and OdSe or~jstals to a dose of
IO17e/cm 2
spectra
yields
(taken ~
irreversible
modificaton
of
the
exciton
s~t~ at 4.2K) resulting from the electron-beam
stimulated formation of long-lived defects.
This
issue has been
treated in detail by Batyrev et al. [83] and this section reviews briefly the results of these studies. In
Sect.
5B
we
have
shown
that
after
a
prolonged
electron
irradiation of CdS and OdSe crystals some new features, designated e and 12* arise in the reflectance and emission spectra (see as I s Figs. 9.10,21].
0dSe samples, may be classified, with respect to
their photolumineseence spectra, into two groups; group I concerns "perfect" or~Jstals while group II include "defect" crystals. Fig. 28 shows
the evolution of
the exciton reflectance
spectra of a
defect CdSe crystal with increasing the irradiation dose. Photolumineseence spectra of a "perfect" CdSe crystal before and after
prolonged
electron-beam
irradiation
induced
line
are Ie s
is
presented seen
to
in shift
Fig.
29.
towards
The long
wavelengths by about O.Inm (curves 2,3). The donor-bound exciton, induced by the EB (see Fig. 20) is modified into a broad line (Ak ~0.6 nm) labelled I2s which also shifts towards long wavelengths.
A.E. Cherednichenko and V.A. Kiselev
238
IS
R~2
G
I2s l 1~
0.4
I2s
jv
I I 0/3 _I I I -I I I 0.2
b
J/
2
j~
I
I .I II
L-- I
6
\
6
0.~
G81
6~
681
680
~nm Fig. 28. Exciton reflectance curves of CdSe (2-61~exposed to increasing doses of electron irradiation ( x10 "e/cm). a: (1)-initial; (2)-I; (3)-2; (4)-8; (5)-20; (6)-60. b: the magnified spectra.
Excitons at Semiconductor Surfaces
239
~l Izs
Iz
I i}
Fig. 29. Exciton pho toluminesoence spec tra of a "perfect" CdSe sample (1)-initial spectrum, (2,3) - after exposure to increasing doses of electron irradiation, (4)-initial spectrum of a "defect" crystal (see text ).
x 10
~
J
j ......
,
" ";'"
, x3 '........-2",.;
°
\
i
i
681
i
i
i
i
679
i"--.,/.i
~,nm
Eventually, the I2s-line (~ ~ 680.65nm) becomes a predominant feature of the photoluminescence spetrum. Also seen is the Ie-line (~ u 679.7nm). The observed changes were found to be irreversible. s Note that the I s - and I2S- features can be revealed in the optical spectra of as-grown samples. Curve to curve 3, indicating that a prolonged yields modification of "perfect" (group I) (group II) ones. The ERL of a "defect" CdSe sample versus
4 is almost identical electron irradiation crystals to "defect" the electron dose is
shown in Fig. 28. Initially, a distinct structure associated with An=l-, I2S-, and I s- excitons is observed. Electron irradiation brings about the "rotation" of the ERL of the An= I- exoiton, which is indicative of an initiation of the exciton-free layer.
240
A.E. Cherednichenko and V.A. Kiselev
It
oi.;
° Fig. 30. Calculated ERLs of CdSe vs the width of the near-surface layer with an additional resonance, a 20nm; b - 50nm; c - 100nm; d lO00nm.
14710' "/4720
The reflectivity are
of
a
14730
e transitions features associated with 12S- and I s-
typical
dispersion
"rotation".
Furthermore,
the feature
associated
with
the
accompanied by an enhancement The
reflectivity
irradiation centers
dose
(defects)
that a defect-rich
features
point
character
and
also
reveal
the
curves 4-6 indicate a strong quenching of
to
capable
intrinsic
An= I- exciton
which
is
of the I2S- feature. and
the
their fact
of binding
reversal
that
the
excitons.
with
EB
increasing
produces
From
layer arises below the surface
local
this we infer
(the penetration
depth of keV electrons
is about ~102nm) whose width is governed by
the
energy
incident
Contrary excitons
to
electron the
and
conventional
the
electron
dead-layer
irradiation
(see
Sect.
excited at the centers of the near-surface
close to the surface and are essentially absent Relevant
calculations
in this case. The model
indicate
is considered
the
that the ERL can also be reversed
in question suggests to be
4A),
layer do exist
in the bulk.
the existence
layer of a finite width within and beyond of which constant
dose.
equal
to
of the
the dielectric
the background
dielectric
Excitons at Semiconductor Surfaces
241
constant ~0(~)" Within the layer, the dielectric constant is contributed by an oscillator term of the type: ~/(~I~0 + 47), where ~I is the excitation frequency of the e exciton (e.g., I2S and Is). The calculated 30, reproduce the effect of rotation of the in the width of the layer. The rotation is
relevant near-surface spectra shown in Fig. ERL with the increase observed for both the
intrinsic exciton and additional resonance outlined above. As regards the origin of the electron-beam-induced defects, latter seem to involve
the anion vacancies
Vse(V S)
the
and the I2s
line is a counterpart of the bulk 12 line. The I e s - and I2sfeatures show similar dependence on the irradiation dose and, in heavily treated samples they rather than local centers.
F. Localization of exoitons
are
by
associated
fluctuations
with
of
a
defect
bands
near-surface
random potential Low-temperature photoluminescence spectra also demonstrate the feasibility of surface localization of excitons (carriers) by fluctuations of a random potential. There is a variety of reasons for the occurrence of a random potential and corresponding in the density of states (see, Fig. 22, variant 5).
tails
In Sect. 5B we have indicated that the exciton photoluminescence spectra of CdS and CdSe crystals were modified by electron bombardment (EB). This section shortly reports a study of a broad band observed in the emission spectra of some CdSe crystals which was attributed to the luminescence of excitons localized at surface fluctuations of the potential (or charge) [84]. Figure 31 shows the photoluminescence spectrum of one of OdSe samples at T=4.2K typical of those reported in [84]. In the short-wavelength part of the spectrum (on the right of the figure) there
are
observed
luminescence
features
due
to
free
excitons
An=l, surface exciton I s (see Sect. 5B), and a strong line 12 due to exoitons bound to neutral donors. In the long-wavelength part of the spectrum ( on the left of the figure ) a very broad band
242
A.E. Cherednichenko and V.A, Kiselev
Zs
1
A .=Z
/jlAI:/ "i ')
~"=' (,=~)
/~
,l
/ilz~ ~. /:i~!):
/
21 ~,
[, I"
!.~x30
r 679 ~n.t
Fig. 31. Photoluminescence spectrum of a CdSe sample at T=4.2K (see ref.[84]). 1) initial spectrtun; 2),3) after low-dose electron irradiation (see text).
labelled X is revealed and against its background one can see oneand two-phonon replicas of the An= I- exciton. Several new ~eatures were observed for the new band. I ) The band
is wide with a halfwidth of at
least 20 meV.
The
spectral position of the band maximum varied from sample to samle within 6830-6850 ~. The intensity of the band was anticorrelated with the intensity of the exciton photoluminescence. 2)
The
band
X was
extremely
sensitive
to
surface
treatments.
Excitons at Semiconductor Surfaces
243
Figure 31 shows the photoluminescence spectra before (curve i) and after 1 sec ( ~2 1015e/cm 2) and 3 sec (~ 6 1015e/cm 2) of 3.5 keV electron
irradiation.
It
is
clear
from
the
figure
that
the
electron irradiation resulted in a marked quenching of the band and
in
a
characteristic
wavelengths.
The
shift
observed
of
its
quenching
maximum was
toward
shorter
accompanied
considerable increase in the exciton luminescence
by
intensity
a
(see
the right-hand part of the figure). 3) The band X depended on temperature in a complex mam~cr.
It
markedly shifted toward long wavelengths and reduced in intensity disappearing at about 20K [84]. These findings
(for more, see [84,85]) suggest that the X band
results from the radiative recombination of excitons localized at fluctuations of the random potential relief at the semiconductor surface.
The
fluctuations centers
fluctuations of
the
(defects,
result,
charge
due
impurities)
most
to
a
likely, random
from
spatial
distribution
of
in the surface layer. The presence
of these centers may give rise to wells
in the potential relief
and these wells may have localized (exciton) states similar to the exciton states in solid solutions of semiconductors. Small Sect.
electron 4B)
initially
and
doses also
reduce
reduce
localized
the
surface
charge
states
electric
fluctuations,
disappear.
This
efficiency of other luminescent channels, ~ Z . , and bound excitons, 31).
The
field
so
temperature
the
increases
the
those due to free
which is evidenced experimentally
influence of
(see
that
can be attributed
(see ~'ig. to thermal
delocalization of excitons. This results in a gradual quenching of the
band,
starting
from
its
short-wavelength
edge,
and
by
a
relative shift of its maximum toward longer wavelengths. To conclude, we note that surface fluctuations of the potential do not necessarily result in localization of excitons as a whole but
only
one
of
the
carriers
in
an
exciton
can
be
localized
instead. As an alternative to the model
in question we can also
consider
electrons
recombination
with
holes
of
localized
at
surface centers. This model is also popular and ft~ther studies are needed to identify the origin of the broad emission bands.
244
A.E. Cherednichenko and V.A. Kiselev
6. Band Bending and Its Relevance to Excitons and Exciton Spectra In
this
section
we
shall
discuss
extrinsic mechanisms by which
one
of
the
most
obvious
the exciton transition layer (ETL)
is formed, namely, the interaction of excitons with a non-uniform electric field of a near-surface space charge layer the near-surface and
modifies
drastically.
electric
the
field that perturbs
exciton
This
is
reflectance
particularly
compounds
With
the
direct
absorption
coefficient
at
the
of
(ERL)
the
most
semiconductor
transitions,
exoiton
It is
the exciton state
lineshape
true
allowed
(SCL).
resonance
where
the
very
high
is
(I04-I05cm -I ) so that excitons are excited close to the surface. Hence,
the
aim
of
this
section
is
to
consider
the
issue
in
detail and also to demonstrate the sensitivity of the ETL to the changes
in the SOL.
Many anomalies
of
the
exciton reflectance
spectra can be explained by this mechanism. The influence of the SOL
may
be
extremely
varied
depending
on
the
electric
field
strength at the surface F s and the field penetration depth into a crystal w which is related to the screening length. For calculations of the reflectance
the approach developed in
[55] has been employed (see Sect. 4E). The SCL was devided into a set of thin Pekar's slabs [8]. By means of this approximation the problem is reduced to the calculation of the optical properties of a layered medium. The effect of the potential is to increment the exciton resonant frequency in each layer by an a m o u n t e q u a l to the height of the corresponding step. The fact that within each layer the potential
is
independent
of
the
coordinate
simplifies
the
problem considerably, since for each layer it becomes possible to apply the crystal optics of a slab with spatial dispersion. At the inner
boundaries
conditions
between
(ABCs)
require
the the
slabs
the
continuity
additional of
the
boundary excitonic
polarization and its normal derivatives. The set of ABCs and Maxwell boundary conditions at all boundaries, inner and outer, form
the
set
of
equations
that
are
to be
solved.
Results
of
calculations shall be compared with data by other authors as well
Excitons at Semiconductor Surfaces
as with our own results on the
245
effect of preliminary illumination
with light, of electron bombardment and externally applied field on the exciton reflectance.
Characteristic
features of the ERLs
and their response to external stimuli may provide an estimate of SCL parameters: F s, w and the charge density e(ND-NA).
A. Band bending in semiconductors Bending of energy bands in the near-surface region is considered to
be
one
of
the
most
important
features
of
semiconductor
surfaces. By depleting the near-surface layer of majority carriers the band bending builds a potential barrier for them (the Shottky barrier) devices
which based
plays on
a key
role
in operation
metal-semiconductor
or
of
metal
semiconductor -
insulator
-
semiconductor structures. The band bending near a free surface is the result of a screening of the field by the surface charge. This charge
is held by surface
electronic
states.
A great
effort
is
devoted at present to achieving an understanding of the mechanism of Shottky barrier formation on an atomic level,
i.e. to clarify
the physical nature and energy structure of the surface states. The
surface
charge
density
determines
the
electric
field
strength at the surface F s. Given the space charge density value near a semiconductor surface,
p = e(N + - N-),
concentrations
of
positively
and negatively
respectively,
one
obtains,
coordinate
dependence
of
the
using field F
the and
N + and N- being charged
Poisson of
the
centers,
equation,
a
electrostatic
potential ~ . For a depletion layer of a n-type semiconductor we have 4%e + F(z) =--~-(N -N-)
(z-
~(z) =
w),
(F(O) = F s, F(z > w) = O ) , (6.1)
2~e ~ (N+ - N-) (z - w) 2 ,
where g is the static dielectric constant.
(6.2)
246
A.E. Cherednichenko and V.A. Kiselev
Q
bIFI C
W
Fig. 32. Distribution of the charge density p (a), of the electric field F (b), and electrostatic potential @ (c) assuming complete depletion.
W These
equations
Z
will
be
reflectivity coefficient. -
N-)
is
constant
(in
used
(6.1)
equilibrium
(N+
bending of free carriers The
approximation.
last
calculate
the
exciton
,and (6.2) apply if (N+ -
N-)
is
a
difference
aoceptor concentrations, N D and only one type of donors is present
the temperature is low enough,
depletion
to
Equations
between the donor and respectively), if, then,
excitons.
below
NA, and
so that a contribution to the band
( ~ kT) will be too small to affect
condition
defines
Variations
with
the z
so-called of
the
the
complete
space
charge
density p as well as of the quantities F and 9, calculated in this approximation,
are
plotted
in
Fig.
32.
This
model
is
able
to
account for many anomalies in the exciton reflectance spectra that could not be explained in the framework of traditional approaches (see Sect. level,
4) and, as well,
modifications
(illumination, variations,
of
electron
applied
field,
to explain, at least on a qualitative the
ERL
under
external
bombardment,
etching,
oxidation
others).
and
influence temperature
On
the
model
depicted by (6.1) any two parameters out of available three: F s, N + - N- and w may be chosen to describe these modifications, w being the depth to which the field penetrates the crystal width of the SOL).
It is clear
that external
factors
(or the
(including
Excltonsat SemiconductorSurfaces
247
probing light in optical investigations) may affect not only the three parameters mentioned but also the function F(z) (as well as p(z) and ~(z)). For instance, high concentrations of the electron-hole pairs generated in SOL by intrinsic light may disturb the constancy of p(z) across SOL in such a way that the holes are accumulated near the surface and electrons near the inner SOL boundary (see, e.g., [86]). A non-uniform charging of centers across the SOL will proceed and subsequent charge redistribution will be acting so as to flatten the energy bands near the surface. Note, that the authors of refs. [87,88] used other expressions for P(z) We have also tried to modify their model by adopting F ~ (z - w)~ where n = 2, 3, or more, but no improvement on the results was found. Direct observation of flattening of the energy bands by light is possible utilizing exoitonio spectroscopy (see [89] and further). Therefore, the light intensity has to be kept very low for the photogeneration of pairs to be negligible. This is a proper way in probing with light. In other kinds of experiment the light intensity should by sufficiently high to permit the amount of band bending to be contr~lled and the model depicted by (6.1) may be used only in a qualitative consideration of the effect (Sect. 6D). Note, that illumination with light from within an appropriately chosen spectral range enables one to charge the surface states in a controllable way [89]. The model in question still applies in this case but an appropriate change in the parameter F s is required. Excitonic spectroscopy is most suocessfull at low temperatures. Besides allowing to ignore free carriers while seeking to establish the form of the barrier, low temperatures lead to long relaxation times, such that certain non-equilibrium processes (externally provoked band bending, e.g., under illumination with light [89], depletion [90], charging of a near-surface layer by irradiation with electrons [19], charging of surface states [89,33] and others) get "frozen-in" and can persist for long enough to permit measurements of optical characteristics of the exoiton at any given band bending value. The effects of this kind
248
A.E. Cherednichenko and VoA.Kiselev
include the dependence of ERL on whether a crystal is being cooled when exposed to light or in the dark. To get reproducible results, it is extremely important to take account of such effects. One
more
bending
property
of
the
of
energy
the
B0L
bands
that
is
the
may
be
observed
quantization
of
at
strong
the
carrier
movement normal to the surface resulting in the formation of a 2D exciton
(see, e.g.,E80,91]).
A detailed account of the problem of
2D excitons is beyond the scope of the present review.
B. The effect of a uniform electric field on excitons Prior field
to a
of
useful
treatment
BCL
to
on
the
consider
assumptions
of
the
effect
exciton the
to be made
and
case
of
in Beet.
of
its a
a non-uniform
optical
uniform
field
of
an
proximation)
exciton
as
it
because
is the
6C will allow us to substitute
uniform field for the non-uniform one. One assumption movement
electric
properties
a
whole
is
slow
a
is that the
(adiabatic
ap-
and the other is that field variations over distances
comparable to the exciton radius are small
(the approximation of a
point
of small
increments
in
field does not mean small absolute values of the field, which,
in
fact,
exciton). may
Note,
be
Investigations
that the assumption
very of
strong
the
effect
near on
a
semiconductor
excitons
of
have been initially prompted by observations on
excitons
e.g.,J33)
by
Gross
and later,
et
al.
the electron-hole
interaction
electric
(the
field
"solid-state"
as
by results
far
electric
field
of the Stark effect
as
the
mid-50s
(see,
of the research on the effect of
on
the absorption
Franz-Keldysh
investigations
back
the
surface.
were
effect
carried out,
of light
in the
[92,931).
These
at least
for some
time, in isolation from traditional calculations of the electronic states of the h~vd~ogen atom in the electric field
(see references
in [94,95]) that can equally be applied to a theory of an exciton. Out only
of a large number of works those
energies
that
close
deal
with
in this
theories
to the ground
state
field we
applicable
in
shall the
energy of an exciton
review
range
of
and,
in
Excitons at Semiconductor Sudaces
249
addition, cover the case of strong electric fields, up to such that ionize a hydrogen-like system in the ground state. We note, that the ranges of the Coulomb interaction and field magnitudes outlined above rule out the usage of the perturbation theory. The problem was found to permit analytical treatment if either some special form of potential distribution is used instead of the actual one (see Fig. 33) [96-98] or a limit is imposed on the magnitude of the field [99,100]. The most reliable results for a wide
range
of
field
values
have
been
obtained
by
numerical
integration of the Schrodinger equation [101-103, 94]. Characteristic scales of energy, length and field strength are defined in this model as follows [103]. The exciton binding energy in the ground state has the form (2.2) and the exciton radius is given by (2.4). The field required to ionize a system is found by equating an increment in electrostatic energy binding energy:
over
the
exciton
radius
F I = IEll/(eRexc) = (~/mo)2. g-3Fi H
to
,
hydrogen-like the electron the
exciton
(6.3)
where FIH = 2.57"109 V/cm is the ionization field for the ground state of the hydrogen atom. Factors in (2.2), (2.4), and (6.3) containing the reduced mass ~ and the static dielectric permeability diminish
IEll and F I by several orders of magnitude
and increase Rex c compared to the CorTesponding quantities for the free hydrogen atom so that fields as high as F I (and even stronger) seem quite feasible in semiconductors. Some examples are given in Table I (for more see [103,102]). Major results obtained for the hydrogen-like system in an electric field may be summed up as follows. In weak fields, F/F I ~ f << I, the exoiton level is shifted to long wavelengths (the quadratic Stark effect ) by (9/8)f 2 in units of the exciton Rydberg: IEl(f = 0)I. This shift is due to some broadening of the Coulomb well under applied field, see Fig. 33 [103]. The field creates a potential barrier, such that the energy needed to pass it over would have dissociated the exciton. As the field is increased, the top of the barrier lowers
250
A.E. Cherednichenko and V.A. Kiselev
Fig. 33. A potential of the electron hole interaction in the presence of an extez*nally applied uniform electric field (solid curve) and without field (dashed curve).
TABLE
1
Rexc, nm
IEII, meV
F I, V/cm
GaAs
13.6
4.2
) -103
InP
12.5
4.9
4"103
OdSe
5 •I
16
3.104
OdS
2.8
28
1 • 105
Excitons at Semiconductor Surfaces
faster
than
the
ground
state
level
and
at
251
f
=
fc
~
0.13
a
classical transition of the barrier becomes possible (if the Stark shift is neglected,
then fc = 1/8 [1021). The finite probability
of the exciton dissociation (both tunnel and classical) the
exciton
fulfilled,
level. i.e.,
When
if
f
a
condition
<<
fc
the
for
density
deep
broadens
tunnelling
of
states
in
is the
resulting band is described by the Lorentzian distribution with a halfwidth (in units of IEll)
r/2
=
(6.4)
(81f) exp [-4/(3f)]
Despite the fact that the resonance happens to occur above the barrier for f > fc the
shift of the
level to long wavelengths is
nearly quadratic up to field strengths of about f ~ 0.25
[1031,
and the density of states distribution still retains the maximum, which can be followed up to f ~ I [101-103] (Pig. 34), i.e., up to the field value at which the bandwidth exceeds the exoiton binding energy (P =fEll for f ~ 0.75 [94]). The occurrence of the maximum in the spectrum of unbound states is due to quantum interference that an electron wave experience when passing over the well [102] (this effect for the case of a rectangular well has been discussed in Sect.
5).
It seems that few of those engaged in research on
excitons are aware of the fact that they often deal, in fact, with the unbound state at the above-barrier resonance rather than with the bound exciton state. The situation when in the semiconductor near surface region f ~ fc ~ 0.13 is met rather frequently. this
field value
the
shift
to
long wavelengths
and
broadening
amounts to a merely ~ 0.O2.1Eli and O.003.1EII, respectively, example, for the An= I exciton in 0dS
At (for
the respective values are
0.6 and 0.09 meV). As the field f ~ 0.4 is reached, the energy of the resonant exciton state assumes a value of ~ 1.1.E1(f = O) and with further field increase the resonance shifts in reverse, i.e., towards short wavelengths, - see Fig. 35 [103]. Hence, the maximum possible value
of
macroscopic field
the exciton resonance can produce
exoiton binding energy.
frequency
shift
that a
amounts to about one tenth of the
252
A.E. Cherednichenko and V.A. Kiselev
Eg.-ILll
Lg
Fig. 34. Profiles of the'density of states ~2(0) (relative units) of a hydrogen-like system for different values of the uniform electric field [101].
E,(,f)/IE,(O)I -0.2.
-0.6 -'1.0 O.q
0.5
2.0
Fig. 35. Variation with eleotrio field f of the ground state energ~y of a hydrogen-like system relative to the unperturbed oontlnuum [103]; dashed curve - perturbation theory.
Excitons at SemiconductorSurfaces
253
For the excited exciton states with the quantum number n above unity the exciton dissociation is possible at substantially lower fields, e.g., at fc(n) = fc(1)/n4 [102]. Therefore, at f ~ 0.2 essentially no peaks remain in the density of states spectrum except the one with n = I. This peak becomes broader and merges into the continuum, loosing its symmetry, longer be described by the Lorentz formula. At
stronger
fields
the 0oulomb
that
interaction gets
is,
it
can
no
progressively
less effective and for f ~ 10 the density of states calculations are but little influenced by an account of this interaction [103]. Most convenient for treating the effect of a non-uniform field on excitons happen to be the expressions, derived for the hydrogen atom using the results of ref° [94]. The calculations include strong fields and resonant energies, with the energy and broadening as major parameters of resonances. The field and energy values can be given in units of F I and IEII, respectively, of atomic units. To
solve
the
Schr8dinger
equation
for
the
instead
hydrogen
atom
subjected to the electric field the variables were separated by employing special "quadratic" pax~abolic coordinates and the problem thus reduced to solving a system of two differential equations. Numerical integration of the equations was performed using the method of series reexpansion. If a condition is imposed on one of the two equations that the wave function be finite at zero and decay exponentially to infinity, then a discrete spectrum of
eigenvalues
is
obtained
(with
the
energy
E and
field F
as
parameters). In our calculations, though, the demand of exponential decay was replaced by that of the wavefunction be zero at some point sufficiently remote from the origin of coordinate system. The other equation which depicts the transition through a barrier leads, in fact, to a problem on quasi-stationary states. Its solution was made to matah with an asymptote of the wavefunction also in a enough remote point, which yielded the asymptote parameters, the amplitude and phase, as functions of the energy E and field F. By analogy with the scattering problems, from the analysis of dependences on energy of the phase and
254
A.E. Cherednichenko and V.A. Kiselev
amplitude value.
the
The
EI
and
r
values
calculations
were
can
be
deduced
carried
out
for
along
every
the
field
lines
of
Breit-Wigner theory. The values of E I were deduced under condition that the squared asymptotic divergent
wave
is present)
parametrization phase
shift
Generally
amplitude
of
in
the
the
the P values
dependence
proximity
speaking,
parametrization
and
the
be minimum
on
of
obtained using
of
the
of
the
asymptotic state.
Breit-Wigner
is open to question.
of the ground state band [101-103]
only the
quasi-stationary
applicability
at high fields
were
energy
the
(i.e.,
An asymmetry
is indicative of a necessity to
go beyond the bounds of this theory.
However,
the probability
of
exciton dissociation for the noticable asymmetry of the band is so great,
that
the
bandshape
effects considered.
is
not
of
much
importance
Still, we intend to discuss,
for
the
on a qualitative
level, deviations from the Lorentzian lineshape evident in the ERL rotation above
(Sect.
deal
6G).
To conclude,
exclusively
calculations
to
Generalization
follow
we note
with
simple
will
be
of the results
that
the works
electronic
restricted
cited
bands.
to
to cover non-uniform
Our
this
case.
fields
can by
performed using the method discussed in Sect. 4.
C. The effect of a non-uniform electric field on excitons As
indicated
semiconductor considered
in
above,
the
may
strong
be
the previous
electric
field
enough
section.
to
within cause
all
Inhomogeneity
makes the density of states distribution
the of
SCL
the
of
a
effects
this
field
(Fig. 35) vary from point
to point along z. The effect on excitons of a non-uniform electric field near
the
crystal
surface
Gribnikov and Rashba
[104].
that
pulled
an
exciton
interaction result,
of
is
the
induced
was
first
In particular, into
the
dipole
treated
by
they have demonstrated
high-field
moment
in a paper
with
the quadratic Stark effect takes place,
region
this
due
field.
to
As
a
i.e., the resonant
frequency of the exciton near the surface gets lower. In ref.[IO4] the
influence
of
this
effect
on
the
exciton
diffusion
was
Excitons at Semiconductor Surfaces
analysed.
The analisys was later continued in
255
[105], emphasizing
the role the near-surface field plays in exciton dissociation. The possibility that the near-surface field might pull in and ionize the exciton was also suggested by Allen [106]. In the most definite manner the influence of the space-charge field on the exciton states in semiconductors was established in experiments o n reflectance and electroreflectance. For some crystals interference of light has been observed in the region of the exciton resonance. An additional reflection of light from the inner SOL boundary where an abrupt change of the dielectric permeability occurs was observed on CdS crystals at 273K [107] and 4.2K [121] as well as on GaAs [25,108,109] and InP [17] at liquid helium temperatures. At low temperature this interference could be observed without modulating the applied voltage, as a rotation of the ERL following alteration of the contact bias or under electron bombardment of the surface (see Sect. 4B). At room temperatures when the exciton peak in reflectance does not appear, voltage modulation is indispensible. (Analogous rotation can also be caused by an "excitonless" Franz-Keldysh effect in the SOL of a semiconductor [110], present review).
but
The ERL rotation was
this issue obtained
is beyond
the scope of
in terms of a theory
the
[17, 18]
which simulated the high-field region (F a F I ) by a uniform dead layer [12] of varying thickness (Sect. 4A). A m o r e adequate description based on adoption of the actual variation of the electric field with coordinate z (Sect. 6A) and assuming a gradual variation of the resonant frequency and of exciton damping with Z was proposed in [56,111], the calculation method has been given in Sect. 4E). In these studies an account was taken of the fact that the layer free of excitons is formed near the surface due to high probability of the field dissociation of the exciton (and not to an abrupt change of the exciton resonant frequency ~0 [17,18] ). Besides providing a reprodution of the effect of the ERLrotation, these calculations have also predicted or confirmed a variety of effects in the exciton reflectance spectra (see below). Other models were proposed to describe the exciton behavior
256
A.E. Cherednichenko and V.A. Kiselev
the near-surface field. In ref. [53] the field dissociation of the exciton
was
assumed
to depend
exponentially
on
coordinate.
The
probability of the field dissociation was found to decrease with flattening
of
the bands
near
the
surface.
Lagois
[54] using
a
three-layer approximation has derived the exciton potential, again relating Balslev
this
to a
strong near-surface
[87] used the exponential
screening
length,
to
describe
field.
Schultheis
term exp(-z/d),
the
coordinate
and
d - being the
dependence
of
the
field strength. Intrinsic mechanisms of the transition layer formation were
taken into consideration and variational method was
used to specify the adiabatic potential. Eleotrodynamic equations including
those
for
the
exciton
polarization
have
been
solved
numerically. The calculation has confirmed the possibility of the ERL
enhancement
and,
as
well,
Besides,
the
with the
variation
of
alternative
calculations
the near-surface
mechanism
performed
in
of
field
spike
[87]
formation.
yielded
structure typical of very wide and shallow wells
[111]
the
ERL
(see Sect. 6G),
the result to be expected in view of the assumptions made: d ~ 400 nm
and the field at
Sect.
6B).
near-surface
The
increase
excitons
essentially affect was
the surface not by
in
the
such
justifiably neglected
value
fields
the ERL, hence, [87].
in excess of 0 . 3 F I of
damping
is not
yet
(see
of
the
sufficient
to
to a first approximation,
it
Schultheis and Lagois
[88]
took
account of both the resonant frequency shift in the field, varying with coordinate as
(z - w) 2 and the damping, which,
was assumed exponential. The way in which the
as in [53],
ERL is affected by
the parameters of the SOL in GaAs crystals has been considered. Lowering of the exciton resonant frequency near the surface due to the Stark effect may lead to localization of the exciton as a whole in the potential well thus formed [78,29] (manifestations of the
exciton
localization
in
the
photoluminescence
spectra
have
been discussed i n s e c t . 5). Apart from field effects an impact ionization of excitons near the
surface
structures
can (see,
be e.g.,
scope of the review.
observed [53])
due but
to these
current effects
flow are
in
contact
beyond
the
Excitons at Semiconductor Surfaces
D.
257
Exoiton refleotivity in the case of thin SOLs In
this
section
calculations
the
results
will
for the An= I exciton
in a
[56,33]. The calculation procedure, proceeds the
from defining
exciton
resonant
presented
of
ERL
CdS crystal with a SOL
that has been outlined above
the functions
frequency
be
shift
A~o(Z) and
and r(z)
damping,
Outside the SOL, i.e., at z ~ w , one has A~ 0 = 0
which are
respectively.
and P = F O, the
latter being the exciton damping in the bulk of the crystal. If a field variation near the surface F(z) is known (see Sect. 6A) then the functions A~o(Z) and r(z) may be readily determined (see Sect. 6B). For a linear field variation two
values
of
w
the
resonant
(6.1), N D - N A = I016cm -3 and
frequency
shift
and
damping
as
functions of z are shown in Fig. 36 [56]. It is seen from this figure that A~o(Z)
forms a potential well
and r(z) increases towards the surface producing a physically dead layer
for
excitons.
For
larger
surface
fields
Ps
(which
is
proportional to w) the curves for A~ 0 and P are shifted as a whole along
the
surface
z-axis.
(Fig.
36b)
Eventually, where
A~ 0
a
layer may
and
r
are
adiabatic conditions are not warranted.
develop
not
adjacent
definable
However,
and
the the
this region is
inaccessible to the exciton and various attempts to determine A~ 0 and
r
there
have
had
little
effect
on
the
result
of
ERL
calculations. In calculating the ERL the following parameter values were used [38]: ~0 = 20590.2 cm -1 ~LT 15 cm -I ~0 9.5, M 0.78 mo, 9.3 [112], where ~0 is the resonant frequency of the An=1-exciton , ~LT' the longitudinal-transverse splitting, dielectric permeability in a region of translational optical
axis
mass O
of
, m O,
the
electron
the free
moving
~0' the background resonance, M, the at
electron mass,
a
normal
and ~,
to
the
the static
dielectric permeability. For the given parameter values the result of ERL calculation is defined by any two of the three SOL parameters: F s,
the SOL
thickness w,
and
the centers
the surface field
concentration N + - N-
258
A.E. Cherednichenko and V.A, Kiselev
r c~ t i
Z00 a
b
Z,~m
° V,.7°., -10 ~ I~
I
I"%1
I
-20 -,.Jx.L/ AUJO ~rlz "1
Fig. 36. Damping and resonant frequency shift A(0 of the exciton in the Shottky barrier on a CdS crystal; a -IV, b - 2.8V. which
defines
the
space
charge
density.
ERLs for a variety of combinations
of
We
have
(N +- N-,
calculated w) values,
the some
examples are given in Fig. 37 (for N D - N A equal to 2-1017cm-3(a), 5.1016cm-3(b), and 1 . 1 0 16cm-3(c)), more will be presented in Sect. 6G. With respect to their peculiar features all ERLs may be divided into types plane, Fig.
I, II, III, and IV. If mapped on the
38 by
bold
curves.
origin of coordinates (N+ - N-). certain of
(F s- w)
each ERL type is confined to a well-defined area shown in
Of
the
Thin
("rays")
sets
straight
lines
coming
out
of
the
correspond to different values of
of ERLs
in Fig.
37
each
corresponds
(Fs, w) points on a particular ray which means
to
constancy
(N+ - N-). The numbers at each curve indicate the values of w
in nanometers. fine structure,
Area I in Fig.
38 contains
spectra
exhibiting no
the examples are the spectra for w = 0.12
in the
first row, w = 10, 27 in the second and w = 20, 50 and 70 third row in Fig. spike
37. Area
II
contains
the ERLs with
in the the left
(see Sect. 4 ) i.e., the one to the left of the reflectance
minimum
( for the chosen direction of wavelength increase).
These
Excitons at Semiconductor Surfaces
259
a
ioI ? 0.6
0.4
Iz
0,2 0
0.~
Fig. 37. 0alculated ERLs for the A(n=l) - exciton in a CdS crystal calculated using a model of the Shottky barrier for different w values and at three values of N D - N A (see text).
are exemplified
in Fig.
the first row and
37 by the lineshapes
for w = 17,
18
in
w = 30, 31 in the second one.
It is important to note that the left spike is accounted for in our calculations assuming reasonable values for the bulk damping, say, ~
= 0.Scm -I ~ 0.1 meV
(the value used in
computing
all
of
260
A.E. Cherednichenko and V.A. Kiselev
F0 / 0.5_
05
0
,*5
5O
75 -~
w, nrn
Fig. 38. Contour plots of constant Rma x value (thin curves) and areas
(I, II,
III) of characteristic ERL behavior on
the (Fs-W) plane. Straight lines correspond to constant values of N +- N- in units of 1017 cm-3; ~F O = 10 -4 eV.
the
spectra
in
measurements
Fig.
of
37).
the
Close
forbidden
value
has
exciton
been
obtained
linewidth
from
[12],
and
longitudinal exciton linewidth [27] at liquid helium temperatures. On
the other hand,
the
spike
of
such a
shape
explain within the uniform dead-layer model substantially [19,22].
lower
values
In addition,
(~ 10-5eV)
in our model
of
damping r increases towards the surface very
high
values,
for
instance,
190
for the
respectively,
in Fig.
37.
The
difficult
the
exciton
to
damping
transition layer
the
(see Fig. 36) and reaches cm -1
linesh~pes with w = 18 nm in the first row and second,
is
[12] as it requires
and
30
cm -1
for
w = 31 nm in the
observation
that
for
the
Excitons at Semiconductor Surfaces
left spike to appear the damping should
261
not at all be too
low
attests
the mechanism under consideration as more realistic
than
the one
discussed
in
[12].
In particular,
a serious
discrepancy
between the values obtained from experiments on reflectance and on luminescence Area
III
[19] gets an explanation. in the F s- w plane plane
right spike
contains
the ERLs with
the
(see w = 20, 22 in the first, w = 33, 40 in the second
and w = 75, 80 and 90 in the third row of Fig. 37)
as well as the
"reversed" ERL. The effect of the ERL reversal (or rotation) demonstrated in Fig. 39 for N + - N- = 1016 cm -3. The effect
is is
seen to start with an appearance of the right spike and follows in its major features the predictions of the uniform dead layer model [18], though detailed agreement
is lacking. Area III, as well as
area IV, featuring small F s and large w values shall be discussed at length in Sect. 6G. Here we will concentrate on areas I and II, i.e.,
restrict
ourselves
to
fields F s around 104 V/cm
the and
cases
of
w
w ~ 25 nm
~
10 nm
with
with
the
F s around 105
V/cm, in conformity with this section's title. Besides
producing
the left
and right
spikes,
the near-surface
field changes reflectivity coefficients at the ERL extrema and Rmin)
stretches
the spectrum over greater
Fig. 37. These changes may be mapped on the F s-
w
drawing
values
curves,
parameters. constant
corresponding
As an illustration,
reflectivity values
to
(Rma x
energy range, s e e
constant
we show in Pig.
plane too, by of
ERL
38 contours of
at Rma x with a spacing between
the
curves ARma x = 0.05 . It is seen that Rm~ x as a function of F s and w has a peak value of 0.82 at F s = 0.25.105 V/cm and w = 27 nm, that is, Note,
the ERL is enhanced at these values
that at the values of F s and w close
of the parameters. to the above
quoted
there also occurs peak in R m n i (see Pig. 37). In the next section we demonstrate that the most likely cause of the ERL enhancement is the quasilocalization
of
the exciton at a near-surface
Stark
well. The presence
in the spectrum of
the anomalies mentioned above
(enhancement, left and right spikes) sets the following limits on SCL parameter values (for CdS):
262
A.E. Cherednichenko and V.A. Kiselev
Ri
~_
0.3 0.2 0.1 0 L~550
!i -
20600 20550
f
2060O oJ ~ C1~ t
Fig. 39. Reflectance of a CdS crystal in the vicinity of the A(n=1) -exciton for different values of the surface potential ~s V: a)- O, b)- 0.065, c)- 0.1, d)- 0.2, e)- 0.5, f)- 0.75.
104 <~ F s ~< 105 V/cm , 10 <~ w ~< 102 nm , 1016 ~< N + - N- ~< 5-I017cm -3
(6.5)
Many of the results of experimental investigations on exciton reflectance discussed in Sect. 4 can be understood in terms of the effect of SCL. For instance, two very similar F~Ls for w = 70 and 80 mm in the third row of Fig. 37, the latter featuring the spike and the former having none, may be compared to the spectra from the paper by Permogorov et al. [32]. In that stud~ the spectrum with the spike has been obtained umder intense illumination of the sample, which could, for example, produce charging of surface states [113] or cause a slow (frozen-in at low temperatures) recharging of the centers in the SOL. Below, we consider at some length the effect of preliminary illumination described in [33J. Short exposure to light also makes w somewhat larger but changes
Excitons at SemiconductorSurfaces
263
of the ERL will be different in cha1~aoter and similar to those in the first row of Fig. 37. It is natural to suppose that these different kinds of behavior arise from differences between the (ND - N A) values in the crpstals under study. Evangelisti et al. [18] and, especially, Patella et al. [15] stressed that the intensity of the spike is not rigidly related to that of the main reflectivity peak Rma x as the uniform DL model would have implied [12,18]. This subSect has been already discussed in Seat. 4, and a new mechanism of spike formation should be suggested, namely, the one that assumes the existence of a potential well for the exciton near the surface. In the model of the transition layer for the exciton considered here the above well apRears as the S~ark well produced by the nea~-surface field (see Pig. 36). Indeed, as seen f ~ m Pig. 37, with variation of the SOL pa1~ameters the relationship between Rma x and spike intensity might be most diverse. AS indicated in Sect. 4, by subjecting the surface to electron bombardment a varied and complex influence is exerted on the near-surface region and, consequently, on the exciton reflectance, It seems, though, that in some cases chauges in the ERL are q u i t e definite in character and ma~v be related to o h ~ e s in SOL parameters. So, after small (undestructive) doses of electron irradiation there is observed a conversion of the right spike into the left one and back, the ERL enhancement and at greater doses the ERL rotation is revealed (see Pigs. 9,10). The values of Rma x and Rmi n may also be changed by varying temperature (see Pig. 14). it is of special interest that as the temperature is raised from 2 to 30 K the value of R m i n may drop to practically zero, as reported by Pevtsov et al. [27]. This is contrary to the result of the calculations assuming an increase in PO provided no transition layer at the surface is taken into account, Bearing in mind that the SOL may undergo changes with temperature variation the outlined behavior is no longer surprising. For example, for w = 70 and 50 nm, in the third row in Pig. 37, the value of R m i n drops frc~n 4.6% down to 0.24% while the lineshape remains essentially the same. Another, more definite
264
A.E. Cherednichenko and V.A. Kiselev
interpretation of such behavior will be put forward in
Sect. 6E
where the dependence of the reflectance spectrum on the damping in the bulk PO is considered. There was investigated the influence on reflectance spectra [33] of preliminary exposure Fig.
to light with k ~ 51Ohm
(see Sect.
40,
15). Reversibility and reproducibility of the results prove
that
the major
various
effect
electronic
parameters.
These
of
states
illumination was with
experiments
that
resultant have
given
of
recharging
changes
in
evidence
SOL
of
a
quasilocalization of the 3D exciton revealed through reflectance spectra. Modifications of the ERLs with increasing amount of preliminary illumination Fig.
40.
The
(time of illumination in minutes) illumination-induced
changes
are displayed in
persisted
at
liquid
helium temperature for long enough, so that in the course of the reflectivity measurements, with the use of a probing light of very low
intensity,
the
SOL
configuration
remained
essentially
unchanged. As seen in the figure, the experimental spectra display all the major features
that appear in the calculated spectra of
Fig. 37, namely, the ERL enhancement followed by the emergence of the left spike, with an accompanying decrease (Rmin)
and by stretching of
the ERL along
(increase) of Rma x
the frequency scale.
Then the ERL is again enhanced and the reflectance damped. Such a complicated behavior of the ERL under illumination indicates that the corresponding trajectory in the
F s- w plane (Fig. 38) must be
very intricate as it should pass twice the area of high values of Rma x at exposures of 5 and 15 minutes. Modifications of the ERL at fairly low exposures to light
(0.5
and 10 rain) correspond best to a set of lineshapes seen in the upper row in Fig. 37 for which N +- N- = 2.1017 cm -3 and w = O, 12, 17 n m , though, of course, the trajectory might be, as well, both more steep (like the one for N + - N- = 5"1017 om -3 and more sloping
(as for F s ~ 105 V/cm).
trajectory starts
from Area
I,
Still,
it is certain that this
then passes
through a region of
fairly high Rma x values and finally enters region II where F s 105 V/cm. From here variations of SOL assume different character.
Excitons at Semiconductor Surfaces
R
0.3
F fO~n
265
f 2O
o.2[
/
p
Fig. 40. Modifications of the ERL of a CdS crystal at T=4.2K caused by preliminary illumination (of. Fig. 15).
One of the most likely reasons for the growth of F s and w at low doses of illumination is charging of surface states [113]. This seems
to be
minutes,
also
the reason
times of relaxation
for fairly
long,
to the initial
of
state.
the order
of
A 10 minute
exposure completely fills the surface states. Longer exposures lead to flattening of the bands, with the band bending region receding towards the surface. Flattening is effected by photogenerated pairs of carriers when these get separated in the near-surface field and driven toward the low field region (electrons) or the high field region (holes). The electrons will be captured by ionized donors thus decreasing the space charge density in the low-field region while the holes will be captured
266
A.E. Cherednichenko and V.A. Kiselev
R 0.5
0"3
n
0.2-
V 0.1
--
0 Fig. 41. Modifications of the ERL in the vicinity of the A(n=1)- exciton induced by various doses of 3.5 keV electron irradiation.
by charged aeeeptors, surface.
The holes,
increasing the space charge density near the in addition,
might be captured on the levels
of the 2D quantum well on the surface, acceptors
are
completely
filled.
As
both during and after the
calculations
show,
10 5 V/cm and w of the order of tens of nanometers series of two-dimensional To
establish
circumstances self-consistent
the is
a
subbands separated by
spatial
distribution
difficult
task
that
of
manner and requires knowledge
Therefore,
a
calculated
on the model
comparision
of
the
assuming
may turn out to be inadequate.
the
must
Nevertheless,
Fs ~
there exists a
~ 10 -2 eV. field be
in
solved
these in
a
of many parameters.
experimental a linear
at
ERLs
decrease
with of
those
the field
it is clear that the
Excitons at Semiconductor Surfaces
exposures average,
of
interesting control,
15,
hence
20
the
that
and
30 min
repeated
in
some
make
the
enhancement
267
field weaker of
experiments,
the
where
on
ERL.
situation
is
evade
this enhancement is accompanied by transformation,
that the left spike gives place to the right one,
the
It
such
see Fig.
40,
which implies that the descending part of the trajectory on the F s- w
plane transverse Area II in Fig. 38.
Very much doses
of
irradiation
crystals. this
similar modifications with
These are shown
case
the
of
the ERL result
electrons
of
the
surface
(for the same sample)
experimental
ERLs
compare
from of
in Fig.
best
to
a
small CdS
41. set
In of
lineshapes calculated for N + - N-= 5-1016 cm -3. It is evident that these lineshapes belong to Areas II and III of the F s - w plane as they
display
demonstrates
all that
the
typical
the surface
anomalies. is being
The
ERL
modification
charged negatively when
subjected to electron irradiation.
E.
Quasilocalization of excitons in the SCL field and the effect
of damping In the preceeding Section we have shown that for thin
SCLs ( w
~< 102 nm for CdS) the reflectivity in the vicinity of the An= 1exciton
might
be
expected
to
enhance.
An
anomalously
high
reflectivity may occur both in as-grown crystals and after some external
factor
has
been
applied.
The
reflectivity has been actually observed,
enhancement
of
the
some examples have been
given in Sect. 4, more will be discussed below. The effect under consideration was
found to be related to a feasibility for the
exciton to be localized by the SOL [78]. Fig. 42a shows grafs of A(~o(Z) and r(z) for N + - N- = 5.1016 cm -3 and w = 31 nm. A corresponding ERL in Fig. 37 is strongly enhanced and features the left spike. We note Fig.
that the lower part of the curve A~00(z) in
42b may be approximated with a triangular and show in this
figure two lowermost levels of the calculated using a known formula:
triangular well
gO
and £1'
268
A.E. Cherednichenko and V.A. Kiselev
r , cm -t
20~
a b
gl
f
/
0
/
-10 -2D
!
0
10 ,~U.)o~Cm-t
20
30
Z,~
Fig. 42. (a) The resonant energy shift A~ and damping F(z) for the first ERL of Fig. 46. (b) A~(z) for the potential approximated with a triangular. C O and C I are the bound states.
[
3~e
e~ =
Fs
3 ] 2/3
(~ + ~)
(6.6)
,
2v' 2M
where F s is not the electric field at the surface but a quantity defined as
~ o (z) i Fs : B ~-~ Iz=O
(6.7)
As seen in the figure, with the parameters as indicated,
the gO
level occurs somewhat below the edge of the real potential well. In
other
words,
an
enhanced
ERL
points
to
the
presence
of
potential well, sufficient to localize the exoiton. Simulation of a real potential with a triangular one gives, course, Yet,
only a crude estimate of the level position
a more
the magnitude
accurate of
calculation
P(z)
near
seems unnecessary.
the surface,
this
a of
in the well. To
level,
judge by due
to a
Excitons at Semiconductor Surfaces
269
high probability of the field dissociation of the exciton, spreads over the energy
interval
that exceeds
the binding energy of
the
exciton in the well. We have, in fact, dealt with the ERL enhancement due a single level. At concentrations N +- N- greater than 5.1016 cm -3 the enhancement looses in strength (see Fig. 38) as the potential well gets narrower while its depth is limited to a value ~ O.IIEII
(or
about
the
3
meV
contrary, of
for
CdS).
For
lower
values
of
N +-
N-,
on
the wells become wider and more levels for localization
excitons
may
arise.
At N +- N- = 5"1016
cm -3
the
levels
are
separated by more than I meV (see Fig. 42) and at N +- N - = I016cm -3 the separation between the levels is estimated at a few tens of a millielectronvolt. ERL
enhancement,
as
seen
With several levels in the well
in
Fig.
37,
is
less
pronounced
arises at larger w values. At still lower N + - N- values cm -3)
the levels
in the well
form a continuous
the and
( ~ 1015
spectrum and the
effect of enhancement is no longer observed. A
conclusion
to
be
drawn,
that
is
very
important
for
interpreting the ERL anomalies observed,
is that because of large
r
does
values
the
near-surface
detectable structure
field
not
produce
of the main reflectivity peak whatever
any the
value of N + - N- . Still, we may expect that some structure could be detected using differential technique. This hope finds support in the results of calculations
[56]
the behavior of reflectivity at
(for
N + - N- = 1016 cm -3) of
some fixed frequency from within
the resonance range as a function of the surface potential width,
w
oscillatory
).
The rather
reflectivity than
measured at a fixed frequency It is natural
has
monotonous
been
found
character.
The
to
(or SCL have
(20600 cm -I ) are shown in Fig.
to relate each successive peak
an
pulsations
to an emergence
43. of
one more level in the Stark well. This conclusion is oozu~oborated by estimates of the strength of the corresponding potential well. Experimentally,
the
ERL pulsations have been observed on a CdSe
crystal subjected to small irradiation
that
produced,
(and increasing) primarily,
exposures of electron
alteration
charge. The variation of R is presented in Fig. 44.
of
surface
270
A.E. Cherednichenko and V.A. Kiselev
R 0.4
0.2 I
I
0.1
0.2
I
0.3 )llZvllz $ 1w
Fig. 43. Pulsations of the reflectance in CdS at a fixed frequency with varying ~s (calculated).
R
0.6-
0.¢ 0.~ 0.2I
I
I
I
I
5
I
I
1 I
I~1
fd'fO
I
I
ZO JO
I
4O
Fig. 44. Variations of the reflectance at a fixed frequency in a CdSe crystal with increasing electron irradiation dose at T=4.2K.
Excitonsat SemiconductorSurfaces
271
A feasibility for the exoiton to be localized by the space-charge field settles one of the long-standing problems of excitonic spectroscopy. Quite often, reflectance spectra exhibit strongly enhanced structure for the An= I- exciton, including the spike and an unusually high value of Rmi n, whereas the structure for the Bn= I - exciton retains its usual appearance. This seems strange as the intrinsic properties of the A- and B- excitons are very similar. Differences
in the ERL for these excitons were observed
on virgin high-quality CdS surfaces [12,18] oriented both along and normal to the OdS optical axis. We have observed these differing
ERLs
upon
small
doses
of
electron
irradiation
(Figs.
9,10).
R
Ot
An=l 8n=I
/
/
/
b
An-1
,/~ Bn,,,t
F
0.1I I I I I I I
I
I
I
I~,I
6?9 "
I
i
I
670
I
669
~, n m Fig. 45. ERL of CdS (a) and CdSe crystals (b) after small dose of electron irradiation at T=4.2K.
272
A.E. Cherednichenko and V.A. Kiselev
Characteristic low-fluence
spectra
electron
can be made
is much more sensitive
has
localization
on
the
and
CdSe
are given
crystals in Fig.
conditions
45.
From
the
than that of the
consider the effect the damping in the
reflectance
spectrum
which
of the exciton by the SOL field
the calculation
following
that the ERL of the An= I- exciton
to the surface
Bn= I - exciton. To explain this behavior, r0
OdS
bombardment
figure a conclusion
bulk
of
are presented
in Fig.
case of N + - N- = 5-1016 cm -3 and
corresponds
[78]. The results
46 for the above
w = 31 n m .
to of
considered
With tO= I cm -I the
ERL displays anomalously high Rma x and R m in values and the presence of the left spike, with P = 2 and 5 cm -I there is no spike and Rma x and Rmi n are less. At F = 10 cm -I ERL responds no
transition
that
layer
were
the B- exciton
exoiton.
and 20 cm -I the
to the increased damping in a classical manner,
Hence,
present.
is damped
the
first
to the A- exciton and, the spectra in Fig.
So,
several
lineshape
e.g.,
the
it
is
times
in Fig.
third one
natural
stronger 46 should
to
suppose
than
to the B- exciton
R
0.6 0.4
0
2 =lC
5
10
the A-
correspond
45 get explanation.
O.Z
as if
?.0 ~-~
Fig. 46. Modification of the ERL of a CdS crystal with increasing the exciton damping PC in the bulk.
and
Excitons at Semiconductor Surfaces
273
The same reasoning may be used to interpret a drastic decrease of
R mn i
with
rise
Pevtsov et al. increased
of
temperature
from
2
to
30 K
observed
[27], - see Pig. 14 (at higher temperatures
in
a
usual
way,
same
as
in
Fig.
46).
by
R mn i
Analogous
explanation may be given to a non-monotonic dependence of Rmi n
on
excitation intensity [114].
F. The effect of SCL on the phase of reflected light So far, we have been coefficient R
, which
concerned only with
is an
the reflectivity
intensity ratio
of reflected and
incident light: R = Ir/al 2 Yet,
r/a
is
a
complex
quantity
(6.8)
characterized,
alongside
the
module, with a phase: r/a = Ir/a i e(~ where ~ has
the
sense
of
a phase
,
shift
(6.9)
of
the reflected wave
relative to the incident wave. Frequency dependence of 9, as well as
of
E,
resonance
exhibits and,
in
peculiarities essence,
in
should
the
also
be
region
of
exciton
sensitive
to
the
transition layer at the semiconductor surface. A "non-classical" phase behavior has been reported in a number of papers
[115-120,
31]. The frequency dependence of ~ in the vicinity of the exciton transition has been found to deviate markedly from the calculation that neglected both the transition layer and spatial dispersion, as seen in Fig. 47 (curve c). This dependence is either S-shaped and
shows
resonance
a
step-wise
[116]
(Fig.
change 47,
by
curve
2~ a)
in or
the
region
N-shaped
of
exciton
and
has
a
pronounced negative dip in the short waves [117] (Fig. 47, curve b). The non-classical phase variation can be adequately reproduced within a theory that considers
the uniform dead layer
(DL) and
274
A.E. Cheredniohenko and V.A. Kiselev
2~
0 =
b
Fig. 47. Types of the frequency dependence of the phase of reflected light near exciton resonance. a) non-classical S-shaped; b) non-classical N-shaped; c) classical; b and c may be derived from a if the damping F 0 is incresed.
•
I 0
5OO
I 1000 W ~ nnt
Fig. 48. Areas (I-III) of characteristic ERL behavior. Curves I to 5 correspond to an appearance of the first and subsequent spikes. + Thin ~ l i n e s correspond to constant values of N -N-(~IO'~).
Excitons at Semiconductor Surfaces
[116-119, 31]. The type of frequency dependence
spatial dispersion to
be
expected,
275
S
or N
, is governed
by
the
exciton
damping
[117,119]. An uncertainty remained, however, concerning the extent to which the DL in the spirit of the
Hopfield and Thomas model
essential
the
in
Regrettably,
accounting on
for
other m o d e l s
lacking, nevertheless,
observed
such
an
phase
[12] is
peculiarities.
exhaustive
analysis
is
the question raised may be answered right
now: the uniform DL serves a good approximation of real transition layers
as
regards
the phase,
though
this
model
should
not
be
considered as ultimate. Our results are as follows: i) with OdS crystals having N + - N- > 1016 cm -3 , which exhibit ERL rotation at
strong
bending
of
the
bands
(Sect.
6G)
and,
consequently,
possess a sharp enough DL boundary, even at small w values, sufficient for the ERL enhancement to begin,
- see Fig.
just
37,
the
S-type phase will be observed (Fig. 47, curve a). ii) In the ease of N + - N- < 1015 cm -3 when the real DL boundary is diffused and no ERL rotation occurs
(Sect. 6G) a model incorporating only the
SCL gives
differing
results
dependence - Fig. such
crystals
behavior, mechanisms considered,
not 47,
curve c. Therefore,
(not yet
their of
essentially
available)
interpretation the
iii)
exciton If,
reveal
would the
the
classical
should experiments on a non-classical
require
transition
initially,
from
that
layer
phase
is
phase
intrinsic
formation B-shaped
be (Fig.
47,curve a), then by introducing greater bulk damping ro into the SOL
model
classical
one phase
obtains (curves
the b,c).
N-shaped iiii)
and, The
subsequently,
latter
event
the
occurs
simultaneously with the disappearance of the ERL enhancement
(see
Fig. 46). At r 0 = 1 or 2 cm -I the phase is S-shaped, at r 0 near to 5 cm -1, when Rmi n turns zero, a line tangent to the phase curve at = ~L becomes vertical signifying the transition to N-type. These developments reproduce the
excitonic
Brewster
effect for normal
incidence described earlier on assumption of a uniform DL
[120]
and explain completely the results concerning the dependence of Rmi n on temperature and also the temperature dependence of ~ [27].
276
A.E. Cherednichenko and V.A. Kiselev
G.
Exciton reflectance So
far we
dealt
with
thin space charge and
II
wider
of
layers
with
w
that
at
effect
(SOLs). plane
values
these are represented noteworthy peculiar
the
the F s - w
layers,
important
in the case of wide SCLs the
in Fig.
by Areas
exciton
reflectance
Those are represented
up
larger
on
to III,
w
38.
103
Now we
nm.
On
IV and V
lower
N+
by Areas
shall
the
N-
consider
48).
values
It is
are
since just the shallow and wide potential wells to this N + - N- range produce a nontrivial
I
F s - w plane
(see Fig.
-
of
more
that are
ERL structure.
Fig. 49 shows ERLs, typical of a wide SCL, calculated for the N+-N - values of 1016 (a), 3-1015 (b), 1015cm -3 and ~ = 10 -4 eV by the method presented above. 49a and Fig. The
main
39) the ERL displays
reflectivity
peak
whereas
the minimum
a maximum,
as well
shifting
is similar
(see Fig.
to
the
surface. less the
sharp
than 10 nm) exciton
effect.
(~
is large
calculated
102
nm)
The ERL rotation
coefficient
enough
calculated
the
ERL
values
rotation. the
The
pulsations
pattern
of ERL behavior
of
curves
the
appearance
in
( ) 1016
makes
possible
Area, right
this
in
Fig. 6E
to Area
III
3 .....
spike which
reflectance peak moving to long wavelengths.
is of
interference
the reflection in the
curve will have an
Sect.
2,
layer
wavelength
the
in
I,
In the
the boundary
with
shown
corresponds
al.
near
at some fixed frequency
discussed
of a successive
of
The resultant
is
et
thickness.
cm -3)
a period corresponding
curve
this
This
of exoitons
(the spread
or measured
character with
and so on.
being the layer of a heavy
can be characterized
region of the exciton resonance. oscillating
into
to the characteristic
which
and
turns
by Evangelisti
dissociation
inner boundary
in relation
(see Fig.
wavelengths
spectrum
dead layer of variable
caused by field-induced
has a rather
long
to long wavelengths, one,
6) for a uniform
When N + - N-
to
of the initial
our case the DL is of different nature, damping
cm L~
a rotation when w is increased.
shifts
disappears picture
For N + - N- ) 1016
to one cycle of 50
(at
are
in Pig. is
small
seen).
9s
This
48. Each
related
then develops
to
an
into a
Excitons at Semiconductor Surfaces
277
c
b 0.6
0.4
w-6OOnm: to
-130Nn 900
0 ZOO
260
-"
~0520
|/I
I-~'"
20600
%..,~11,-.-I
20500
I-. 2000
20600
20560
20600
U,4 Grit"1---,,Fig. 49. Theoretical ERLs for the A(n=1)-exoiton in a CdS crystal calculated using a Shot~ky barrier model for different values of w and three values of N - N- (see text).
In contrast to published results [18]
(see Fig. 6) the amplitude
of oscillations dies out with distance from the origin of the F s w
plane. At the boundary of Area V (see a wavy curve in Fig. 48)
it is 1/10 of
the
non-uniformity
of
initial value. the SCL
field.
This
effect
However,
is related
to the
in experiment
it has
been found even stronger than our theory predicts. The ERL rotation was observed on a number of semiconductors GaAs [25,108,109], InP [17], CdS [107,19,121] was
applied,
using
[17,25,121,108,109], irradiatied
with
or
semitransparent the
electrons
metal
semiconductor [19]
(see
:
. Either the voltage surface
Sect.
4B).
electrode (CdS)
was
With
the
metal/OdS structure [121] it was possible to observe just a little
278
A.E. Cherednichenko and V.A. Kiselev
0./~ 0.3 0.2 0.1 0 B
I
o
I
0.5
,,I
1.0
1
1.5
2.0
Fig. 50. Variation of the reflectivity coefficient R of a CdS crystal at a fixed frequency in the v i c i n i t ~ of resonance of the A(n=1)- exciton (20600 cm ").
more than one ERL reversal
and the peak-to-peak
amplitude
of ERL
decayed noticably faster than it follows from our calculations. least
two factors may be responsible
of the resonant
structure
of the density of states electric
field,
use
of
the
band
the
symmetrical
arises
from
in reflectance.
One is the asymmetry
distribution which appears under
see Sect.
6B.
It is to be recalled
Lorentzian large
parametrization.
spreading
At
for the rapid disappearance
of
the
strong
that we made Asymmetry
levels
of
of
excited
states under applied field and their merging into the continuum of states.
With
increasing
field
the
continuum
encroaches
band of the exciton ground state as well,
see Fig.
physical
getting
DL
can
transparent.
The
not
the excitonic structure less conspicuous. The
second
structures
cause
of (see,
considered
as
(unstructured)
at
the the
more
hig~
and
fields
more makes
in the density of states of deeper layers
This conclusion is confirmed by the experimental
data on absorption occurrence
be
absorption
upon
34. Hence,
[121]. of
impact
the
rapid
ionization
e.g.,[63,53]),
decay of
of
the
ERL
the
exciton
the process
largely
may in
be
an
barrier
dependent
on
Excitons at Semiconductor Surfaces
279
the physical nature of a metal-semiconductor contact and on the intensity of incident light. Essentially different ERL behavior is obtained values,
around 1015
corresponding
cm -3,
as seen
in Fig.
at low N + - N-
49o.
In Fig.
48 the
area is Area IV . At large enough w , same as for
high N + - N-
levels,
after another,
there appears
a sequence of spikes,
represented by curves I , 2, 3,
... in
one
Area IV.
But, in contrast to the former case, the spikes may accumulate in the reflectance spectrum giving rise to a multi-spike structure. With further increase in the value of w the spikes are vanishing in order of appearance. Area IV encompasses ERLs which display no rotation but are characterized by the shift of the main peak to long
wavelengths
and
rotation does not
by
reduced
peak-to-peak
take place because
amplitudes.
The
the function r(z) has
the
form of a gradually sloping tail whose characteristic dimension by far exceeds kex c (kexc ~ 10 nm), i.e. the field is nearly uniform in this case. The shift of the ERL maximum to long wavelengths is a
consequence
of
a
field-induced
lowering
of
the
resonant
frequency. The boundary separating Areas III (rotation) and IV (multi-spike structure)
is not
transition between illustrated by spectra
is
the
In
two
w
intermediate of
spectra
structure further
the types
reflectance
the
increasing
sharp.
to
to
the
region
behavior
in Fig. red
long
ERL of
49b. ~L
wavelengths
a
is Seen
which
smooth
observed, in
these
moves
and
with
gradually
disappears. When crossing o v e r from Area III to IV the role of the damping
P(z) gets appreciably reduced whilst that of the lowering
of
resonant
the
frequency
A~ O(z)
near
the
surface
becomes
dominant. This has a bearing on the structure produced which may be
interpreted
quantization structure
in
of
excitons)
arises
dispersion is
terms
of in
interference the SOL
irrespective
taken into account,
of the
of
of polaritons a
(or of
semiconductor.
whether
or
two cases
not
The
spatial
differing only
quantitatively. As regions of shallower and wider potential wells are entered (Area IV), the excitons of progressively longer wavelengths get involved in quantization, so that the additional
280
A.E. Cherednichenko and V.A. Kiselev
structure Fig.
becomes more closely spaced and moves nearer
The
ERL
region
features
have
structure may,
pertaining
been
to Area
observed
observed
across
in a number
spectra
the
of
observed
transition
CdS
crystals
. Often,
the main
multiple
reflectance
IV
and
experimentally. of
in the first approximation,
light
on
CdSe
SCL
macroscopic
intermediate [88].
be attributed
layer a
assumed
two-spike
spikes
peak
region
These
spectra
(see Fig.
49b)
have
fitted
Evolution
of
the
ERL
typical
of
Au-CdS
barrier structures with a
reverse
bias.
excitons
Representative
are seen
in Fig.
(Figs.
[29]),
been
a
IV
can
the
increased
magnitude, On
the
(uniform)
estimated at
to dissociate
as
still
agreement cm -3,
hand,
when
for
observed
the
the SGL
more.
The
field
to
the
is non-uniform
lineshape
the
theoretical
see Fig.
490.
The shift
alteration
calculation
the
A-
and
B-
feature
is
peak shifts
amplitude crystal,
to about
to a value calculated for w = 2000 nm. and the
= 2000 nm.
The
above value of N + - Ndisappearance
also adequately described.
of
the
for by as
its
is insufficient
SOL
of
a
observed
for
N +-
~IO-4cm (f ~ I).
the ERL
of the reflectivity
at a 4V bias amounts
on
6B).
confined
with
wavelengths 4V
that
can not be accounted
within
(see Sect.
GaAs
the field may be an order of magnitude higher,
In addition, smeared
field
on
distinctive
F < 104 V/cm or f < O.1,
the exciton
other
thickness
the
This behavior
the
electrode under
peak-to-peak
with
diminishing.
of
theory
be
semitransparent
Their major
besides
similar to ours.
to
wavelengths,
be
indicating
obtained
using
The main reflectance
long
also
typical
the absence of the ERL rotation. lineshape
the
other factors
well
Area
may
of
In
by a structure
spectra
have
12,13)
to interference
12 and
modifications
51.
multi-spike
structure
Reflectance
been
intermediate
are accompanied
(see Fig.
field.
the
transparent.
assumed wide SOLs and a calculation procedure
of
(see
The
samples
that a theory is needed that would consider the
to ~L
17a).
N-
will
be
is
in
=
1015
peak
to long
40 cm -I which
is close
Assuming the one actually
of the structure
band bending obtains
w
in reflectance
is
Excitons at Semiconductor Surfaces
281
/
A B
2~
~ U : 0 V
I0 20
Fig. 51. ERLs for the A(n=l)and B(n=1)- exoitons as functions of the reverse bias applied via a semitransparent Au-film.
10-
m
Io
1o
~88.0 ~8~.0 x,nm The theory and experiment ~iffer in that the latter provides the evidence of the interference structure located between the A- and B- excitons (see Fig. 51, U = O and 1V). We regard this as one more argument in favor of the necessity to take into account other factors besides the macroscopic field near the surface. On the other hand, the experimental spectra display no multi-spike
282
A.E. Cherednichenko and V.A, Kiselev
structure.
This
fact
is
readily
exPlained
by
larger
exciton
damping to be expected in barrier structures compared to the value used in the calculation (kLF0 = 0.8 cm-1), large damping
being an
immediate cause of the spike dis~ppearance. An important and somewhat surprising conclusion to be drawn from the discussion in this section is that a more pure crystal do not necessarily possess
a more
pronounced
structure
in
the
exciton
reflectance. Let N A be either fixed or small compared to N D , so that the impurity content of an n-type material is dominated by the latter and let N D be lowered. bending
near
the
surface
If there occurs a fixed band
(which
is
typical
of
A3-B 5
semiconductors), then with N D lowering w will increase as (ND NA )-I/2 and F s will decrease in inverse proportion to the same factor.
The
increased
damping w
of
the
may happen
resonant
reflectivity
due
to
the
to outweigh the effect of the reduced
F s. Indeed, as long as F s > F I reduction in the value of F s does not lead to the enhancement of the resonant structure because the excitons in this case do not reach the surface. For example,
in
GaAs Ps does not equal F I until N D - N A is lowered to ~1013 cm -3. One has
to be
aware
yet
that
in measuring
the reflectivity
coefficient the effect of light on the band bending is inavoidable and
unless
the
light
intensity
is
kept
sufficiently
low
the
conclusions made might be in error. In conclusion we note that the exciton spectroscopy can be of great
value
in
understanding
certain
surface
properties
of
semiconductors though the problem of the exciton behavior at the surface looks very complicated. We think that future progress in the issue is intimately connected with the progress in the novel c~-ystal-growth and
surface-treatment
techniques
which
are
being
developed nowadays intensively. The
authors
wish
to
thank
Professor
B.
V.
Novikov
for
encouragement and support of the work and thank the colleages and associates for useful discussions and assistance.
Excitons at Semiconductor Surfaces
283
R E F E R E N C E S
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
14.
15. 16. 17. 18. 19. 20. 21. 22.
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