- Email: [email protected]
Polarization of porous silicon photoluminescence: alignment and built-in anisotropy
",i ,'
'l?
ELSEVIER
Thin Solid Films 276 (1996) 120-122
Polarization of porous silicon photoluminescence: alignment and built-in anisotropy D. Kovalev "*, M. Ben Chorin ", J. Diener ", F. Koch ", A. Kux b, A1.L. Efros c, M. Rosen c, N.A. Gippius d, S.G. Tikhodeev d a Technische Universiti~tManchen, Physik-Department El6, D85747 Garching, Germany b Laboratoire de Spectrometric Phydque, Universite Joseph Fourier de Grenoble, BP8Z 38402 Saint Martin d'Heres, France c Beam Theory Section, Naval Research Laboratory, Washington, DC 20375, USA d General Physics Institute, Russian Academy of Sciences, Vavilova Str. 38, Moscow 117333, Russia
Abstract We report the observation of the anisotropy of the polarization properties of the porous Si photoluminescence. In the edge excitation geometry (exciting light incident on a cleaved edge of the sample) the luminescence polarization is aligned mainly in the [ I00] direction normal to the surface. The effect is described within the framework of a dielectric model in which porous Si is considered as an aggregate of slightly deformed, elongated and/or flattened, dielectric elliptical Si nanocrystals with preferred orientation normal to the surface. Keywords: Luminescence; Anisotropy; Dielectric properties; Silicon
1. Introduction The strong photoluminescence (PL) of porous Si has stimulated great interest in its optical properties. An early model of this material as a system of isolated quantum wires, proposed in the first papers [ 1,2], is probably far from reality. Recent transmission electron microscopy (TEM) [3] has shown that it is a conglomerate of nanometer-sized and randomly shaped Si erystallites. Ti~e wide distribution of shapes and sizes of these particles, together with their random distribution of orientations leads to the conclusion that porous Si should be an isotropic optical system. We demonstrate here that on the contrary an anisotropy of the optical properties does exist, and that it is related to the geometry of the nanocrystals formed in the anodization process. The linear polarization memory effect is well known for the direct bandgap bulk semiconductors [4] or for the system of spherical CdSe nanocrystals with randomly oriented hexagonal axes [5 ]. However, the huge value of the degree of linear polarization in porous Si (up to 30%) at the conditions of more than 1 eV energy loss for the excited carriers [6] shows that the effect cannot be understood in these traditional terms. In this paper we show that the unusual behaviour of the porous Si polarization can be related to the non-sphericity of the nanocrystals in porous Si. * Corresponding author. 0040-6090/96/$15.00 © 1996 Elsevier Science S.A. All rights reserved SSD! 0040-6090 (95) 08074-0
2. Experimental Microporous Si layers were prepared from p-type (100), B-doped substrates with typical resistivity of 5 l'~ cm. The electrochemical etching was performed with a current density of 30 mA cm- z. The etching solution was a 1:1 by voh~me mixture of hydrofluoric acid (49 wt.% in water) and ethanol. The layer thickness was 20 I~m. All the samples were fabricated in the dark. The PL was excited using 442 nm unpolarized radiation from a He-Cd laser which was polarized by a glass polarizer. The light was deflected using a small mirror and was incident onto the sample normal to the investigated surface. The polarization of the detected light was determined by a second polarizer, placed between the lenses. A depolarizer was placed at the entrance slit of the monochromator, in order to avoid problems with the internal polarization properties of the monochromator grating. Two different excitation geometries were used. In the normal geometry, the exciting beam had normal incidence and the polarization vector lay in the sample surface. For edge excitation, the light was incident on a cleaved edge of the sample. The position of the 10 I~m laser spot was controlled with the optical microscope. The cleaved edge was a (100) plane, with one special direction, namely the [ 100] axis along the direction of anodization. The polarization of the exciting
121
D. Kovalev et al. /Thin Solid Fihns 276 (1996) 120-122
light lies in this plane, and its angle with respect to the anodization direction can be tuned. All PL spectra were normalized for the spectral response of the optical system. The degree of the linear polarization of the PL was measured as P = Iil - 11/Ill W l . t , here III is the intensity of the PL polarized parallel to that of the exciting light and 1. is the intensity of the PL polarized in a perpendicular direction.
3. Experimental results Fig. I depicts the PL spectrum (dotted line) and the degree of the linear polarization of PL (solid line) for the normal excitation geometry. The PL is linearly polarized throughout the spectral range investigated with a polarization level that varied from 0.17 at the blue edge of the PL band to 0.05 at the red one. The angular dependence of the PL intensity, for the two perpendicular polarization directions of the exciting light in the normal geometry, is shown in the inset of Fig. 1. The 180° periodicity of the PL intensity is very apparent. The amplitude of the oscillations corresponds to p = 0.1. When the direction of the exciting light polarization is shifted by 90 ° in the plane of the sample (dotted line), the PL angular 0.4
....
distribution also exhibits the equivalent shift. The degree of polarization does not depend on the polarization orientation of the exciting light in respect to the crystalline axes in the surface plane. The low-energy part of the low-temperature PL spectrum is shown in Fig. 2. As readily seen, the value of p monotonically vanishes with the decrease of the detection energy and reaches zero exactly at the energy of the bulk Si fundamental bandgap. The infrared luminescence band is found to be completely unpolarized. Quite different behaviour of the polarization properties of porous Si PL is observed when the exciting light enters the sample in the edge geometry (Fig. 3). We have found that excitation with the light polarized in the [ 100] direction of the surface normal leads to the significant increase of the p values with respect to the plane geometry. The values of p are nearly 0.2 for all of the spectral range investigated (Fig. 3(a)). On the contrary, the excitation with the light polarized perpendicular to the [ 100] surface normal gives rise to negative values of p over nearly all spectral range except the high-frequency part of the PL spectrum (Fig. 3(b)). This discrepancy indicates that PL is preferentially polarized along [ 100] direction normal to the surface. The parameters of this anisotropy depend strongly on the
.--10.4
L (a)
ii:I\ / %
-= 0.3
.~
-
-f
It
0.3 ~-
"-s
[10Ol ~0.4
,/
',,
t_.llee
- 0.3 ~
=_.. o
0.1
0.1
,
,J
0.2 ~"
•==
L
,._?,,
,
i 1.6
,
, 1.8
,
, 2.0
,
~ 2.2
/00 "
0.1,40~
Energy, eV Fig. 1. PL (dotted line) and linear polarization spectra (,,;olid line) of p-type porous Si under linearly polarized excitation. Normal incidence on the (100) plane of the sample is used. Inset: the angular polariz~,ti=on dependence of the porous Si PL. For the solid line the direction of the electric field of the excited light coincides with zero degrees of the detection. For the dotted line the distribution in the same detection coordinates for the 90 ° shifted polarization of the exciting light.
T=6K Eex.=2.8eV
.,.
..................
...
tO .,..
14
0.06
tO
5" "13 rm
..
..-
.. ......... .,
0.04
.. "
:..:"::'"
==_.
""".. ~. ":..... (
n°
0.3-
0.00 1.2
1.4
1.6
1.6
2.0
(b)
"
•~ 0.2.
•
2.0
2.2
....
[1001
.-...,
0.3 •
.
0.2
#J
"13 o ----L 0.1 N
• •'.. •'.
0.1
0.0
~ o.oc _J 11,,-0.1 . . . .
14 1.0
1.8
O :3 .~
01
v
0.02
0.8
0.0 1.6
Energy (eV)
m 0.08
0.1 -o
"".",
o, 00 • 1.4
--
2.2
Energy, eV Fig. 2. PL (dotted line) and linear polarization spectra (solid line ) of porous Si at T= 6 K.
1'6
1'8 Energy (eV)
20
22
Fig. 3. PL (dotted line) and linear polarization spectra under linearly polarized excitation. Normal incidence on the cleaved edge of the sample is used. For the solid line the direction of the electric field of the exciting light coincides with the (lO0) direction. For the dashed line the electric field is perpendicular to the (100) direction.
122
D. Kovalev et al. / Thin Solid Fibns 276 (1996) 120-122
1.1 |
'['00l--
"-- 1.0
E~
~. /.,' ~
It
o.9
•
,o.t7
• ,,
1.1
..,,
1.0
70
0.9
",
"...,..,,:.
"
it..
..
~.
•
""
0.7
/
i
,o
,,,.... • -.
sS
:~ "
'
-100
-. •
I
I
I
:" I
I
0 100 200 Detection Polarization Angle a d (degree)
'
o., 0.7
I
300
Fig. 4. Angular polarization dependence of the porous Si PL under normal incidence of the exciting light on the cleaved edge of the sample. The angles between the electric field of the excited light and the (100) direction are 00, 150, 30°, 450, 60°, 75* and 90* respectively.
angle between the surface normal and the direction of the electric field of the exciting light. The angular distributions of the PL intensity at different angles between the electric field of the exciting light and the [ 100] direction is shown in Fig. 4. Contrary to the planar geometry (inset of Fig. 1) the PL angular distribution does not follow the polarization direction of the exciting light. There is a pronounced direction of the optical anisotropy that coincides with the [ 100] crystalline direction normal to the surface. The level of polarization is determined from Fig. 3 and depends on the angle between the polarization of the exciting light and the [ 100] direction. The maximum value p=0.2 is achieved for the E parallel to the [ 100] surface normal and the minimum p ffi0.08 for the E perpendicular to this axis. We mention here also the small phase shift of the angular distributions which is result of the two effects: the in-plane alignment-type behaviour and PL polarization anisotropy in the [ 100] direction.
model was proposed by Lavallard and Suris [9] ). We note that p reaches zero exactly at the energy of the Si bandgap (1.17 eV). The unpolarized luminescence of the IR band is consistent with the deep level nature of the luminescing centres responsible for this band. For these radiative transitions the excitation step ( trapping of photogenerated carriers) is not relevant to the polarization of the exciting light. The deep-level centre is highly localized and exists on the surface of the dielectric object where the local electric field are on an atomic scale and determines by neighbouring surface atoms. The high level of the atomic disorder on the surface of the crystallites leads to the complete randomization of the emitted photons polarization. The isotropic angular distribution of the p values in the surface plane (inset of Fig. 1) assume the random distribution of the nanocrystallites shapes and orientations in the surface plane. However, the observed dependence of p (Fig. 4) and pinning of the polarization direction of the emitting light.in the [ 100] growth direction suggest the existence of the preferential alignment of the long crystalline axes in the [ 100] direction, i.e. the microscopical internal anisotropy of the [ 100] p-type porous Si layer. In the effective medium approximation this anisotropy should create the difference in the refractive index for light polarized parallel and perpendicular to the porous layer. This is in a good agreement with the previous observation of the macroscopical anisotropy in the reflection of the polarized light from the porous Si layer [101.
Acknowledgements DK acknowledges the support of the Alexander yon Humboldt Foundation. JD is sponsored by Siemens AG.
4. Discussion References We assume that porous Si consists of dielectric ellipsoids embedded in a dielectric medium. The wavelength of the exciting light is much longer than typical dimensions of the ellipsoid axis. In the electrostatic model description [7] the internal electric field will be reduced by the depolarization field. The decrease in the component of the electric field along the long axis of ellipsoids is less than along the small axis. Thus assuming isotropy of the interband dipole matrix element, the probability of the optical absorption is larger for the crystallites with their long axes aligned in the direction of the electric field of the excited light. The strength of the coupling of the internal dipole with the electric field of emitt,~d photon is affected by the dielectric asymmetry of the ocrystallites as well. As a result the emission will be preferentially polarized in the direction of the excited light (detailed calculations are shown in Ref. [ 8] and a similar
[1] L.T. Canham, Appl. Phys. Lett., 57 (1990) 1046. [2] V. Lehmann and U. G6sele, Appl. Phys. Letr, 58 ( 1991 ) 856. [3] A.G. Cull/s, L.T. Canham and O.D. Dosser, Mater. Res. Soc. Syrup. Proc., 256 (1992) 7. [41 F. Meier and B.P. Zakharchenya (eds.), Optical orientation, Modern Problems in Condensed Matter Science, Vol. 8, North-Holland, Amsterdam, 1984. [51 M.G. Bawendi, P.J. Carroll, W.L. Wilson and L.E. Brus, J. Chem. Phys., 96 (1992) 946. [6] A.V. Andrianov, D.I. Kovalev, N.N, Zinov'ev and I.D. Yaroshetskii, JETP Lett., 58 (1993) 427. [71 L.D. Landau, E.M. Lifshitz and L.P. Pitaevskii, Electrodynamic.~ of Continuous Media, 2nd edn., Pergamon Press, Oxford, 1984. [ 8 ] D. Kovalev, M. Ben-Chorin, J. Diener, F. Koch, AI.L. Efros, M. Rosen, N.A. Gippius and S.G. Tikhodeev, Appl. Phys. Lett., 67 (1995) 1585. [9] P. Lavailard and R.A. Suris, SolidState Commun., 95 (1995) 267. [1o] P. Basmaji, V.S. Bagnato, V. Grivickas, G.I. Surdutovich and R. Vitlina, Thin Solid Films, 223 (1993) 131.
'l?
ELSEVIER
Thin Solid Films 276 (1996) 120-122
Polarization of porous silicon photoluminescence: alignment and built-in anisotropy D. Kovalev "*, M. Ben Chorin ", J. Diener ", F. Koch ", A. Kux b, A1.L. Efros c, M. Rosen c, N.A. Gippius d, S.G. Tikhodeev d a Technische Universiti~tManchen, Physik-Department El6, D85747 Garching, Germany b Laboratoire de Spectrometric Phydque, Universite Joseph Fourier de Grenoble, BP8Z 38402 Saint Martin d'Heres, France c Beam Theory Section, Naval Research Laboratory, Washington, DC 20375, USA d General Physics Institute, Russian Academy of Sciences, Vavilova Str. 38, Moscow 117333, Russia
Abstract We report the observation of the anisotropy of the polarization properties of the porous Si photoluminescence. In the edge excitation geometry (exciting light incident on a cleaved edge of the sample) the luminescence polarization is aligned mainly in the [ I00] direction normal to the surface. The effect is described within the framework of a dielectric model in which porous Si is considered as an aggregate of slightly deformed, elongated and/or flattened, dielectric elliptical Si nanocrystals with preferred orientation normal to the surface. Keywords: Luminescence; Anisotropy; Dielectric properties; Silicon
1. Introduction The strong photoluminescence (PL) of porous Si has stimulated great interest in its optical properties. An early model of this material as a system of isolated quantum wires, proposed in the first papers [ 1,2], is probably far from reality. Recent transmission electron microscopy (TEM) [3] has shown that it is a conglomerate of nanometer-sized and randomly shaped Si erystallites. Ti~e wide distribution of shapes and sizes of these particles, together with their random distribution of orientations leads to the conclusion that porous Si should be an isotropic optical system. We demonstrate here that on the contrary an anisotropy of the optical properties does exist, and that it is related to the geometry of the nanocrystals formed in the anodization process. The linear polarization memory effect is well known for the direct bandgap bulk semiconductors [4] or for the system of spherical CdSe nanocrystals with randomly oriented hexagonal axes [5 ]. However, the huge value of the degree of linear polarization in porous Si (up to 30%) at the conditions of more than 1 eV energy loss for the excited carriers [6] shows that the effect cannot be understood in these traditional terms. In this paper we show that the unusual behaviour of the porous Si polarization can be related to the non-sphericity of the nanocrystals in porous Si. * Corresponding author. 0040-6090/96/$15.00 © 1996 Elsevier Science S.A. All rights reserved SSD! 0040-6090 (95) 08074-0
2. Experimental Microporous Si layers were prepared from p-type (100), B-doped substrates with typical resistivity of 5 l'~ cm. The electrochemical etching was performed with a current density of 30 mA cm- z. The etching solution was a 1:1 by voh~me mixture of hydrofluoric acid (49 wt.% in water) and ethanol. The layer thickness was 20 I~m. All the samples were fabricated in the dark. The PL was excited using 442 nm unpolarized radiation from a He-Cd laser which was polarized by a glass polarizer. The light was deflected using a small mirror and was incident onto the sample normal to the investigated surface. The polarization of the detected light was determined by a second polarizer, placed between the lenses. A depolarizer was placed at the entrance slit of the monochromator, in order to avoid problems with the internal polarization properties of the monochromator grating. Two different excitation geometries were used. In the normal geometry, the exciting beam had normal incidence and the polarization vector lay in the sample surface. For edge excitation, the light was incident on a cleaved edge of the sample. The position of the 10 I~m laser spot was controlled with the optical microscope. The cleaved edge was a (100) plane, with one special direction, namely the [ 100] axis along the direction of anodization. The polarization of the exciting
121
D. Kovalev et al. /Thin Solid Fihns 276 (1996) 120-122
light lies in this plane, and its angle with respect to the anodization direction can be tuned. All PL spectra were normalized for the spectral response of the optical system. The degree of the linear polarization of the PL was measured as P = Iil - 11/Ill W l . t , here III is the intensity of the PL polarized parallel to that of the exciting light and 1. is the intensity of the PL polarized in a perpendicular direction.
3. Experimental results Fig. I depicts the PL spectrum (dotted line) and the degree of the linear polarization of PL (solid line) for the normal excitation geometry. The PL is linearly polarized throughout the spectral range investigated with a polarization level that varied from 0.17 at the blue edge of the PL band to 0.05 at the red one. The angular dependence of the PL intensity, for the two perpendicular polarization directions of the exciting light in the normal geometry, is shown in the inset of Fig. 1. The 180° periodicity of the PL intensity is very apparent. The amplitude of the oscillations corresponds to p = 0.1. When the direction of the exciting light polarization is shifted by 90 ° in the plane of the sample (dotted line), the PL angular 0.4
....
distribution also exhibits the equivalent shift. The degree of polarization does not depend on the polarization orientation of the exciting light in respect to the crystalline axes in the surface plane. The low-energy part of the low-temperature PL spectrum is shown in Fig. 2. As readily seen, the value of p monotonically vanishes with the decrease of the detection energy and reaches zero exactly at the energy of the bulk Si fundamental bandgap. The infrared luminescence band is found to be completely unpolarized. Quite different behaviour of the polarization properties of porous Si PL is observed when the exciting light enters the sample in the edge geometry (Fig. 3). We have found that excitation with the light polarized in the [ 100] direction of the surface normal leads to the significant increase of the p values with respect to the plane geometry. The values of p are nearly 0.2 for all of the spectral range investigated (Fig. 3(a)). On the contrary, the excitation with the light polarized perpendicular to the [ 100] surface normal gives rise to negative values of p over nearly all spectral range except the high-frequency part of the PL spectrum (Fig. 3(b)). This discrepancy indicates that PL is preferentially polarized along [ 100] direction normal to the surface. The parameters of this anisotropy depend strongly on the
.--10.4
L (a)
ii:I\ / %
-= 0.3
.~
-
-f
It
0.3 ~-
"-s
[10Ol ~0.4
,/
',,
t_.llee
- 0.3 ~
=_.. o
0.1
0.1
,
,J
0.2 ~"
•==
L
,._?,,
,
i 1.6
,
, 1.8
,
, 2.0
,
~ 2.2
/00 "
0.1,40~
Energy, eV Fig. 1. PL (dotted line) and linear polarization spectra (,,;olid line) of p-type porous Si under linearly polarized excitation. Normal incidence on the (100) plane of the sample is used. Inset: the angular polariz~,ti=on dependence of the porous Si PL. For the solid line the direction of the electric field of the excited light coincides with zero degrees of the detection. For the dotted line the distribution in the same detection coordinates for the 90 ° shifted polarization of the exciting light.
T=6K Eex.=2.8eV
.,.
..................
...
tO .,..
14
0.06
tO
5" "13 rm
..
..-
.. ......... .,
0.04
.. "
:..:"::'"
==_.
""".. ~. ":..... (
n°
0.3-
0.00 1.2
1.4
1.6
1.6
2.0
(b)
"
•~ 0.2.
•
2.0
2.2
....
[1001
.-...,
0.3 •
.
0.2
#J
"13 o ----L 0.1 N
• •'.. •'.
0.1
0.0
~ o.oc _J 11,,-0.1 . . . .
14 1.0
1.8
O :3 .~
01
v
0.02
0.8
0.0 1.6
Energy (eV)
m 0.08
0.1 -o
"".",
o, 00 • 1.4
--
2.2
Energy, eV Fig. 2. PL (dotted line) and linear polarization spectra (solid line ) of porous Si at T= 6 K.
1'6
1'8 Energy (eV)
20
22
Fig. 3. PL (dotted line) and linear polarization spectra under linearly polarized excitation. Normal incidence on the cleaved edge of the sample is used. For the solid line the direction of the electric field of the exciting light coincides with the (lO0) direction. For the dashed line the electric field is perpendicular to the (100) direction.
122
D. Kovalev et al. / Thin Solid Fibns 276 (1996) 120-122
1.1 |
'['00l--
"-- 1.0
E~
~. /.,' ~
It
o.9
•
,o.t7
• ,,
1.1
..,,
1.0
70
0.9
",
"...,..,,:.
"
it..
..
~.
•
""
0.7
/
i
,o
,,,.... • -.
sS
:~ "
'
-100
-. •
I
I
I
:" I
I
0 100 200 Detection Polarization Angle a d (degree)
'
o., 0.7
I
300
Fig. 4. Angular polarization dependence of the porous Si PL under normal incidence of the exciting light on the cleaved edge of the sample. The angles between the electric field of the excited light and the (100) direction are 00, 150, 30°, 450, 60°, 75* and 90* respectively.
angle between the surface normal and the direction of the electric field of the exciting light. The angular distributions of the PL intensity at different angles between the electric field of the exciting light and the [ 100] direction is shown in Fig. 4. Contrary to the planar geometry (inset of Fig. 1) the PL angular distribution does not follow the polarization direction of the exciting light. There is a pronounced direction of the optical anisotropy that coincides with the [ 100] crystalline direction normal to the surface. The level of polarization is determined from Fig. 3 and depends on the angle between the polarization of the exciting light and the [ 100] direction. The maximum value p=0.2 is achieved for the E parallel to the [ 100] surface normal and the minimum p ffi0.08 for the E perpendicular to this axis. We mention here also the small phase shift of the angular distributions which is result of the two effects: the in-plane alignment-type behaviour and PL polarization anisotropy in the [ 100] direction.
model was proposed by Lavallard and Suris [9] ). We note that p reaches zero exactly at the energy of the Si bandgap (1.17 eV). The unpolarized luminescence of the IR band is consistent with the deep level nature of the luminescing centres responsible for this band. For these radiative transitions the excitation step ( trapping of photogenerated carriers) is not relevant to the polarization of the exciting light. The deep-level centre is highly localized and exists on the surface of the dielectric object where the local electric field are on an atomic scale and determines by neighbouring surface atoms. The high level of the atomic disorder on the surface of the crystallites leads to the complete randomization of the emitted photons polarization. The isotropic angular distribution of the p values in the surface plane (inset of Fig. 1) assume the random distribution of the nanocrystallites shapes and orientations in the surface plane. However, the observed dependence of p (Fig. 4) and pinning of the polarization direction of the emitting light.in the [ 100] growth direction suggest the existence of the preferential alignment of the long crystalline axes in the [ 100] direction, i.e. the microscopical internal anisotropy of the [ 100] p-type porous Si layer. In the effective medium approximation this anisotropy should create the difference in the refractive index for light polarized parallel and perpendicular to the porous layer. This is in a good agreement with the previous observation of the macroscopical anisotropy in the reflection of the polarized light from the porous Si layer [101.
Acknowledgements DK acknowledges the support of the Alexander yon Humboldt Foundation. JD is sponsored by Siemens AG.
4. Discussion References We assume that porous Si consists of dielectric ellipsoids embedded in a dielectric medium. The wavelength of the exciting light is much longer than typical dimensions of the ellipsoid axis. In the electrostatic model description [7] the internal electric field will be reduced by the depolarization field. The decrease in the component of the electric field along the long axis of ellipsoids is less than along the small axis. Thus assuming isotropy of the interband dipole matrix element, the probability of the optical absorption is larger for the crystallites with their long axes aligned in the direction of the electric field of the excited light. The strength of the coupling of the internal dipole with the electric field of emitt,~d photon is affected by the dielectric asymmetry of the ocrystallites as well. As a result the emission will be preferentially polarized in the direction of the excited light (detailed calculations are shown in Ref. [ 8] and a similar
[1] L.T. Canham, Appl. Phys. Lett., 57 (1990) 1046. [2] V. Lehmann and U. G6sele, Appl. Phys. Letr, 58 ( 1991 ) 856. [3] A.G. Cull/s, L.T. Canham and O.D. Dosser, Mater. Res. Soc. Syrup. Proc., 256 (1992) 7. [41 F. Meier and B.P. Zakharchenya (eds.), Optical orientation, Modern Problems in Condensed Matter Science, Vol. 8, North-Holland, Amsterdam, 1984. [51 M.G. Bawendi, P.J. Carroll, W.L. Wilson and L.E. Brus, J. Chem. Phys., 96 (1992) 946. [6] A.V. Andrianov, D.I. Kovalev, N.N, Zinov'ev and I.D. Yaroshetskii, JETP Lett., 58 (1993) 427. [71 L.D. Landau, E.M. Lifshitz and L.P. Pitaevskii, Electrodynamic.~ of Continuous Media, 2nd edn., Pergamon Press, Oxford, 1984. [ 8 ] D. Kovalev, M. Ben-Chorin, J. Diener, F. Koch, AI.L. Efros, M. Rosen, N.A. Gippius and S.G. Tikhodeev, Appl. Phys. Lett., 67 (1995) 1585. [9] P. Lavailard and R.A. Suris, SolidState Commun., 95 (1995) 267. [1o] P. Basmaji, V.S. Bagnato, V. Grivickas, G.I. Surdutovich and R. Vitlina, Thin Solid Films, 223 (1993) 131.