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Silicon needles in porous silicon
ELSEVIER
Thin Solid Films 276 (1996) 293-295
Silicon needles in porous silicon P. Lavallard a, R.A. Suris b Groupe de Physique des Solides, CNRS, Universitd Paris 6 et Paris 7, Paris, France b A.F. loffe, Technical hzstitute, St. Petersburg 194021, Russia
Abstract
A model is presented which explains the high degree of linear polarization of luminescence which was observed in porous silicon under non-resonant excitation. Porous silicon is supposed to be composed of elongated nanocrystals. We show that because of the anisotropy of the depolarizing field in silicon needles polarized light excites preferentially those nanocrystals which emit light with the same polarization Keywords: Silicon, Luminescence; Nanostructures; Anisotropy
1. Introduction
Linear polarization of the photoluminescence of porous silicon was observed by excitation with linearly polarized light [ 1 ]. The degree of linear polarization (DLP) is isotropic in the (100) plane and is as large as 0.275 in the luminescence line of porous silicon oxidized at 1 200 °(2. It equals 0.060.1 in the main line of as-prepared porous silicon. For both lines the DLP is almost independent of the excitation energy and keeps a high value even when the difference between the energies of the exciting light and photoluminescence photons is more than 1 eV. A high degree of linear polarization is indeed observed in the luminescence of semiconductors in the case of hot luminescence, alignment of excitons [2] or preferential population of the valleys in an indirect semiconductor [ 3 ]. However such DLPs are observed only for resonant excitation. The usual models of optical pumping cannot explain the polarization of luminescence in porous silicon.
2. Model
One other situation is found when the absorption coefficient of each nanocrystal (NC) is anisotropic. The morphology of porous silicon has indeed been studied by scanning tunnelling microscopy (STM) and transmission electron microscopy (TEM). STM measurements show that filament-like structures with a lateral extension of 20-30 nm are observed to run both parallel and perpendicular to the surface [4 ]. From TEM measurements it was concluded that quantum wires of the required sizes (3 nm) are present throughout layers that give visible luminescence [ 5 ]. Elsevier Science S.A. SSD! 0040-6090 (95) 08100-3
Let us consider the probability of optical transition in a NC. It is proportional to the square of the matrix element xEi,, between initial and final states; x is the material dipole operator and Ei,t the electric field creation operator inside the luminescent material. Anisotropy of the absorption coefficient can be obtained either by anisotropy ofx or by a tensorial relationship between the internal and external fields. I. The anisotropy of x was considered by Andrianov et al. [ 1] to explain their results. They postulated that the Si NCs are quasi-one-dimensional ( 1D) and that the absorption is much stronger for a quasi-ID NC whose long dimension is parallel to the polarization of the exciting light than for a NC which is oriented perpendicularly. This explanation cannot be discarded for resonant excitation but one must note that the observation of a high DLP is very unlikely for non-resonant excitation since several transitions with different selections rules in polarized light can participate to the absorption. 2. As the NC size is much less than the wavelength of light the description of the electromagnetic fields can be made as in electrostatics theory. Because of the different optical dielectric constants inside and outside NCs a tensorial relationship between internal and external fields is obtained in an elongated Si NC embedded in a dielectric matrix. Polarization charges appear on the surfaces when NCs are excited by light. They create a depolarizing field which reduces the amplitude of the electromagnetic field. The reduction is smallest along the elongation direction of the NCs. As a result the more the angle between the axis of the NC and the field direction is small, the more the excitation is large. As the excited NCs also emit light preferably with polarization along their great axes, the
294
P. Lavallard, R.A. Suris / Thin Solid Films 276 (1996)
emitted light from the collection of NCs is mainly polarized along the polarization direction of the exciting light.
293-295
0.5~.~---,
-
.
,
•
•
-
,
•
•
-
,
•
•
•
0.4
3. Calculation
0,3
Let us define the direction of the NC axis in polar coordinates by the angle 0 with the z axis and the angle ~pbetween the x axis and the projection of the axis on the (x, y) plane. For the sake of simplicity we assume a complete isotropy of the NC in the plane orthogonal to the long axis. We call k~ (/it) the ratio of the longitudinal (transverse) electric fields inside and outside a NC. The number of electron-hole pairs in a NC created by absorption of light polarized along the z axis is proportional to the square of the interaction energy
0.2
0.1
0
0
0.2
0.4
0.6
0.8
b/a F i g . 1. D e g r e e o f p o l a r i z a t i o n o f p h o t o l u m i n e s c e n c e anisotropy,
b/a
as a function of NC
is t h e r a t i o o f s e m i - m i n o r a n d s e m i - m a j o r
axes. The calcu-
l a t i o n is d o n e f o r e l / e ~ = 8.5 X E i n t"
N = A ( ~ c o s 2 0 + ~ s i n 2 0) The total number of electron-hole pairs created by absorption in the sample is found easily by integrating over 0. This number is proportional to (t~ + 2 ~ ) . The absorption due to a collection of anisotropic NCs is not proportional to the quantity of matter but depends on the shape of the NCs. The intensity of luminescence emitted by one NC in one polarization direction is proportional to NE2,t. The external electric fields along the z :and x directions are easily found by taking into account the contributions of the transverse and longitudinal internal electric fields in one NC. The intensities of luminescence emitted by one NC and polarized along the z and x axes are:
l~.ffiBN(~cos 2 0 + ~ s i n 2 O) Ix = BN(k~sin =0cos = ~p+ ~ ( i - sin = 0cos 2 ~p)) where B is a proportionality constant. The integration along the 0 and tp angles is easily done. The degree of polarization of light emitted by a collection of random NCs is:
p /_/,,
9(k 2 - 1 )
where e=
b2 1 -a--~
is the eccentricity of the spheroid. The optical dielectric constants e~and Ecare equal to the square of the refractive indices in the internal and external medium respectively. The same DLP is obtained for prolate or oblate spheroids. The DLP of luminescence varies weakly with the excitation wavelength through the refractive index dispersion in both media. The numerical calculation is performed for the parameters of porous silicon. Experimentally E~ is found equal to 1.77 for a porous layer with porosity of 74% [7]. Taking this value and ¢i -" 15 [ 8] for silicon we have drawn in Fig. 1 the DLP of PL as a function of the ratio b/a. The DLP of the main line in oxidized porous silicon, 0.275 is obtained tbr NCs in which the major axis is five times the minor axis (b/ a = 0.2.). The DLP of the luminescence line in as-prepared porous silicon, 0.1, is obtained for a smaller anisotropy of the NCs (b/a-0.5).
2
=I~+i'~=23k~"+ 34k2 + 3 4. Conclusion
where
k=k_, kl One can relate k to the depolarizing factor n [6] which depends on the NC shape: k--
2 + (ei/ec- l ) ( 1 - n ) 2 + 2 ( e J e ~ - 1)n
Let us assume that the NCs are spheroids with semi-major axis a and semi-minor axis b. One obtains:
l-e2(
l+e
2e)
We have presented a very general model which can be applied to any heterogeneous medium. We show that a high DLP of luminescence has to be observed when small elongated crystals (smaller than the wavelength of light) of a luminescent material are embedded in a medium which has a different refractive index. The DLP of luminescence of porous silicon is very large because the refractive indices of the Si nanocrystals and the mean medium are very different. In agreement with TEM and STM measurements vveconclude that luminescence arises from elongated NCs. From the experimental value of the DLP of luminescence we deduce the NC anisotropy. It is larger for smaller NCs which emit light in the blue region of the visible spectrum.
P. Lavallard, R.A. Suris / Thin Solid Fihns 276 0996) 293-295
5. Note added in proof During the E - M R S conference one other communication was pre:~ented about the polarization of porous silicon luminescence [ 9 ] . The authors propose the same model to explain their experimental results. The expeiiments done in the edge excitation geometry allowed them to conclude there is a preferred orientation of the elongated Si NCs normal to the surface of the sample.
References [ ! ] A.V. Andrianov, D.I. Kovaiev and I.D. Yaroshetskii,Phys.-Solid State, 35(10) (1993) 1323.
295
[2] For a review, see F. Meier and B.P. Zakharchenya (eds.), Optical Pumping, North Holland, Amsterdam, 1984. [3] P. Lavallard, R. Bichard and B. Sapovai, Solid State Commun., 17 (1975) 1275. [4] V.A. Karavanskii, M.A. Kachalov, A.P. Maslov, Yu.N. Petrov, V.N. Seleznev and A.O. Shuvalov, JETP Lett., 57 (1993) 239. [5] A.G. Cullis and L.T. Canham, Nature, 353 ( 1991 ) 335. [6] L.D. Landau and E.M. Lifshiiz, Electrodynamics of Continuo,,s Media, Pergamon Press, Oxford, 1962. [7] I. Sagnes, A. Haiimaoui, G. Vincent and P.A. Badoz, Appl. Phys. Lett., 62 (1993) 1155. [8] O. Madelung (ed.), Data in Science and Technology, Semiconductors Group IV Elements and IlI-V Compounds, Spnnger-Vedag, Berlin, 1991. [9] Kovalev et al., Polarization of porous silicon photoluminesce~ice: alignment and built-in anisotropy, Thin Solid Fihns 276 (1996) 120.
Thin Solid Films 276 (1996) 293-295
Silicon needles in porous silicon P. Lavallard a, R.A. Suris b Groupe de Physique des Solides, CNRS, Universitd Paris 6 et Paris 7, Paris, France b A.F. loffe, Technical hzstitute, St. Petersburg 194021, Russia
Abstract
A model is presented which explains the high degree of linear polarization of luminescence which was observed in porous silicon under non-resonant excitation. Porous silicon is supposed to be composed of elongated nanocrystals. We show that because of the anisotropy of the depolarizing field in silicon needles polarized light excites preferentially those nanocrystals which emit light with the same polarization Keywords: Silicon, Luminescence; Nanostructures; Anisotropy
1. Introduction
Linear polarization of the photoluminescence of porous silicon was observed by excitation with linearly polarized light [ 1 ]. The degree of linear polarization (DLP) is isotropic in the (100) plane and is as large as 0.275 in the luminescence line of porous silicon oxidized at 1 200 °(2. It equals 0.060.1 in the main line of as-prepared porous silicon. For both lines the DLP is almost independent of the excitation energy and keeps a high value even when the difference between the energies of the exciting light and photoluminescence photons is more than 1 eV. A high degree of linear polarization is indeed observed in the luminescence of semiconductors in the case of hot luminescence, alignment of excitons [2] or preferential population of the valleys in an indirect semiconductor [ 3 ]. However such DLPs are observed only for resonant excitation. The usual models of optical pumping cannot explain the polarization of luminescence in porous silicon.
2. Model
One other situation is found when the absorption coefficient of each nanocrystal (NC) is anisotropic. The morphology of porous silicon has indeed been studied by scanning tunnelling microscopy (STM) and transmission electron microscopy (TEM). STM measurements show that filament-like structures with a lateral extension of 20-30 nm are observed to run both parallel and perpendicular to the surface [4 ]. From TEM measurements it was concluded that quantum wires of the required sizes (3 nm) are present throughout layers that give visible luminescence [ 5 ]. Elsevier Science S.A. SSD! 0040-6090 (95) 08100-3
Let us consider the probability of optical transition in a NC. It is proportional to the square of the matrix element xEi,, between initial and final states; x is the material dipole operator and Ei,t the electric field creation operator inside the luminescent material. Anisotropy of the absorption coefficient can be obtained either by anisotropy ofx or by a tensorial relationship between the internal and external fields. I. The anisotropy of x was considered by Andrianov et al. [ 1] to explain their results. They postulated that the Si NCs are quasi-one-dimensional ( 1D) and that the absorption is much stronger for a quasi-ID NC whose long dimension is parallel to the polarization of the exciting light than for a NC which is oriented perpendicularly. This explanation cannot be discarded for resonant excitation but one must note that the observation of a high DLP is very unlikely for non-resonant excitation since several transitions with different selections rules in polarized light can participate to the absorption. 2. As the NC size is much less than the wavelength of light the description of the electromagnetic fields can be made as in electrostatics theory. Because of the different optical dielectric constants inside and outside NCs a tensorial relationship between internal and external fields is obtained in an elongated Si NC embedded in a dielectric matrix. Polarization charges appear on the surfaces when NCs are excited by light. They create a depolarizing field which reduces the amplitude of the electromagnetic field. The reduction is smallest along the elongation direction of the NCs. As a result the more the angle between the axis of the NC and the field direction is small, the more the excitation is large. As the excited NCs also emit light preferably with polarization along their great axes, the
294
P. Lavallard, R.A. Suris / Thin Solid Films 276 (1996)
emitted light from the collection of NCs is mainly polarized along the polarization direction of the exciting light.
293-295
0.5~.~---,
-
.
,
•
•
-
,
•
•
-
,
•
•
•
0.4
3. Calculation
0,3
Let us define the direction of the NC axis in polar coordinates by the angle 0 with the z axis and the angle ~pbetween the x axis and the projection of the axis on the (x, y) plane. For the sake of simplicity we assume a complete isotropy of the NC in the plane orthogonal to the long axis. We call k~ (/it) the ratio of the longitudinal (transverse) electric fields inside and outside a NC. The number of electron-hole pairs in a NC created by absorption of light polarized along the z axis is proportional to the square of the interaction energy
0.2
0.1
0
0
0.2
0.4
0.6
0.8
b/a F i g . 1. D e g r e e o f p o l a r i z a t i o n o f p h o t o l u m i n e s c e n c e anisotropy,
b/a
as a function of NC
is t h e r a t i o o f s e m i - m i n o r a n d s e m i - m a j o r
axes. The calcu-
l a t i o n is d o n e f o r e l / e ~ = 8.5 X E i n t"
N = A ( ~ c o s 2 0 + ~ s i n 2 0) The total number of electron-hole pairs created by absorption in the sample is found easily by integrating over 0. This number is proportional to (t~ + 2 ~ ) . The absorption due to a collection of anisotropic NCs is not proportional to the quantity of matter but depends on the shape of the NCs. The intensity of luminescence emitted by one NC in one polarization direction is proportional to NE2,t. The external electric fields along the z :and x directions are easily found by taking into account the contributions of the transverse and longitudinal internal electric fields in one NC. The intensities of luminescence emitted by one NC and polarized along the z and x axes are:
l~.ffiBN(~cos 2 0 + ~ s i n 2 O) Ix = BN(k~sin =0cos = ~p+ ~ ( i - sin = 0cos 2 ~p)) where B is a proportionality constant. The integration along the 0 and tp angles is easily done. The degree of polarization of light emitted by a collection of random NCs is:
p /_/,,
9(k 2 - 1 )
where e=
b2 1 -a--~
is the eccentricity of the spheroid. The optical dielectric constants e~and Ecare equal to the square of the refractive indices in the internal and external medium respectively. The same DLP is obtained for prolate or oblate spheroids. The DLP of luminescence varies weakly with the excitation wavelength through the refractive index dispersion in both media. The numerical calculation is performed for the parameters of porous silicon. Experimentally E~ is found equal to 1.77 for a porous layer with porosity of 74% [7]. Taking this value and ¢i -" 15 [ 8] for silicon we have drawn in Fig. 1 the DLP of PL as a function of the ratio b/a. The DLP of the main line in oxidized porous silicon, 0.275 is obtained tbr NCs in which the major axis is five times the minor axis (b/ a = 0.2.). The DLP of the luminescence line in as-prepared porous silicon, 0.1, is obtained for a smaller anisotropy of the NCs (b/a-0.5).
2
=I~+i'~=23k~"+ 34k2 + 3 4. Conclusion
where
k=k_, kl One can relate k to the depolarizing factor n [6] which depends on the NC shape: k--
2 + (ei/ec- l ) ( 1 - n ) 2 + 2 ( e J e ~ - 1)n
Let us assume that the NCs are spheroids with semi-major axis a and semi-minor axis b. One obtains:
l-e2(
l+e
2e)
We have presented a very general model which can be applied to any heterogeneous medium. We show that a high DLP of luminescence has to be observed when small elongated crystals (smaller than the wavelength of light) of a luminescent material are embedded in a medium which has a different refractive index. The DLP of luminescence of porous silicon is very large because the refractive indices of the Si nanocrystals and the mean medium are very different. In agreement with TEM and STM measurements vveconclude that luminescence arises from elongated NCs. From the experimental value of the DLP of luminescence we deduce the NC anisotropy. It is larger for smaller NCs which emit light in the blue region of the visible spectrum.
P. Lavallard, R.A. Suris / Thin Solid Fihns 276 0996) 293-295
5. Note added in proof During the E - M R S conference one other communication was pre:~ented about the polarization of porous silicon luminescence [ 9 ] . The authors propose the same model to explain their experimental results. The expeiiments done in the edge excitation geometry allowed them to conclude there is a preferred orientation of the elongated Si NCs normal to the surface of the sample.
References [ ! ] A.V. Andrianov, D.I. Kovaiev and I.D. Yaroshetskii,Phys.-Solid State, 35(10) (1993) 1323.
295
[2] For a review, see F. Meier and B.P. Zakharchenya (eds.), Optical Pumping, North Holland, Amsterdam, 1984. [3] P. Lavallard, R. Bichard and B. Sapovai, Solid State Commun., 17 (1975) 1275. [4] V.A. Karavanskii, M.A. Kachalov, A.P. Maslov, Yu.N. Petrov, V.N. Seleznev and A.O. Shuvalov, JETP Lett., 57 (1993) 239. [5] A.G. Cullis and L.T. Canham, Nature, 353 ( 1991 ) 335. [6] L.D. Landau and E.M. Lifshiiz, Electrodynamics of Continuo,,s Media, Pergamon Press, Oxford, 1962. [7] I. Sagnes, A. Haiimaoui, G. Vincent and P.A. Badoz, Appl. Phys. Lett., 62 (1993) 1155. [8] O. Madelung (ed.), Data in Science and Technology, Semiconductors Group IV Elements and IlI-V Compounds, Spnnger-Vedag, Berlin, 1991. [9] Kovalev et al., Polarization of porous silicon photoluminesce~ice: alignment and built-in anisotropy, Thin Solid Fihns 276 (1996) 120.