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Appl. Phys. B 74, 671 ­ 675 (2002) DOI: 10.1007/s00340-002-0916-6

Applied Physics B
Lasers and Optics

t.v. dolgova1 v.g. avramenko1 a.a. nikulin1 g. marowsky2 a.f. pudonin3 a.a. fedyanin1 o.a. aktsipetrov1,

Second-harmonic spectroscopy of electronic structure of Si/SiO2 multiple quantum wells
1 2 3



Department of Physics, Moscow State University, 119992 Moscow, Russia Laser-Laboratorium GÆttingen e.V., 37077 GÆttingen, Germany Lebedev Physical Institute, Russian Academy of Science, 177924 Moscow, Russia

Received: 16 October 2001/Revised version: 16 April 2002 Published online: 6 June 2002 · © Springer-Verlag 2002
ABSTRACT

Size effects in the resonant nonlinear optical response of amorphous Si/SiO2 multiple quantum wells (MQW) are studied by second-harmonic generation (SHG) spectroscopy in a spectral interval of second-harmonic photon energies from 2.5to3.4 eV. The sensitivity of SHG spectroscopy to thicknessdependent electronic structure (sub-band energy position and density of states line shape) of MQW is demonstrated. A monotonic red shift of central energies of SHG resonances by ° 120 meV upon increase of the well thickness from 2.5 to 10 Ais observed. This is interpreted as a size dependence of the position of singularities in the combined density of states for a 2D gas of electrons moving in an effective potential well. It is shown that, for agreement with experiment, the simplest (rectangular) shape of the well should be modified in order to take into account the lattice-potential distortion at the interfaces. 73.20.Dx; 78.66.-w; 42.65 Ky
Introduction

PACS

1

One of the limitations in using crystalline silicon in optical devices is its indirect band gap and, therefore, small luminescence efficiency. However, the incorporation of perfect silicon processing into the photonics industry is of great importance, and much effort has been made in this field [1]. Recently, a greatly enhanced quantum efficiency has been observed in porous silicon. The quantisation effect coming from confinement of electron motion in one or several directions is suggested as the origin of strong visible photoluminescence of porous silicon constituted from nanocrystals ­ quantum wires (1D) and quantum dots (0D) ­ depending on porosity [2]. In this way, the study of 2D Si quantum wells (QW) is quite important. The only possibility of fabricating quantum wells from crystalline Si is to separate silicon layers by germanium or SiGe alloys. Si/Ge superlattices become stable, since the mechanical strain produced by the lattice mismatch is minimal. However, a small difference in band gaps of Si and Ge weakens the quantisation effect in Si/Ge superlattices [3]. On the contrary, amorphous silicon (a-Si) does not require lattice matching, and QW fabricated from a-Si can be separated
Fax: +7-095/939-1104, E-mail: aktsip@shg.ru

by materials with a quite large band gap, even by dielectrics such as silicon oxide [4, 5]. Additionally, due to the direct effective band gap [6], the photoluminescence efficiency of a-Si is larger, which makes a-Si QW very promising for optoelectronic applications. As QW mostly consist of interfaces, second-harmonic generation (SHG), which is an intrinsically interface-sensitive technique, is expected to be a particularly suitable probe for these structures [7]. In centrosymmetric media the quadratic polarisation vanishes in the dipole approximation and is determined by the next-order (quadrupole) term of the multipole expansion [8]. Therefore for QW with a substrate fabricated from a centrosymmetric material the QW-to-substrate signals ratio becomes significantly better for SHG than for conventional linear optical techniques, such as spectroscopic ellipsometry [9] and photoreflection [10]. In this paper ultra-thin amorphous Si/SiO2 multiple quantum wells (MQW) are studied by SHG spectroscopy. The observed resonant features in the SHG response are attributed to direct inter-sub-band electron transitions in Si quantum wells, whose band structure (combined density of states (DOS)) is strongly modified by the quantisation effect. The observed second-harmonic (SH) intensity spectra are shown to be well fitted by a 2D line shape of the quadratic susceptibility corresponding to a 2D singularity (critical point) in the combined DOS. The dependence of the resonant energy on the well thickness in the interval from 2.5å to 10 å is found to be more flat than that calculated for rectangular wells. This effect is interpreted as an influence of the lattice-potential distortion in the Si/SiO2 interface region.
2 Experimental

Si/SiO2 MQW that are multilayered structures with alternation of amorphous silicon and silicon dioxide layers (Fig. 1) are grown using RF sputtering of silicon and fused-quartz targets in argon-plasma discharge in a vacuum chamber with a residual pressure of 7.5 â 10-4 Torr. The deposition rates are 60 å per minute for the silicon and 25 å per minute for the dioxide. The optically polished chemically clean Si (100) wafer is the substrate. To avoid absorption of ions the substrate is biased under a negative potential of 5 ­ 10 V during the sputtering procedure. The temperature of the substrate is kept below 70 . The SiO2 layer thickness is fixed at 11 å and the silicon thickness is varied with


672

Applied Physics B ­ Lasers and Optics

SHG azimuthal anisotropy of Si/SiO2 MQW for several SH photon energies. Curves are fits of I (2) () =|C0 + ei C1 cos |2 to experimental data. Bottom: SHG anisotropy of Si (100) substrate for 2h = 3.9eV. Vertical line denotes angle with only an isotropic SHG contribution
FIGURE 2

observed for all other MQW samples. The same shape of the anisotropic dependences has been reported for similar Si/SiO2 MQW [11, 12]. The dependences differ from the subFIGURE 1

Spatial structure (a) and energy bands (b) of a Si/SiO2 MQW

sample

the sample. Four samples with silicon thickness d of 2.5å, 5.5å, 7.5å and 10 å are fabricated. The number of periods is several tens (30­70) and is varied with the sample to keep the total silicon thickness a constant. SHG spectroscopy is performed using a tunable nanosecond parametric generator/amplifier laser system (SpectraPhysics MOPO 710) as a source of p-polarised fundamental radiation. The operating interval of wavelengths is 745 nm to 1000 nm with 4ns pulse duration and approximately 10 mJ/pulse. The reflected p-polarised SH wave is spectrally separated by a set of filters (9mm of BG39) and detected by a photomultiplier tube and an electronic peak-hold detector. The reference channel is used to normalise the SH intensity spectrum over the laser fluence and the spectral sensitivity of the optical detection system. A slightly wedged z -cut quartz plate is used as a source of the reference SH signal and the detection system is identical to the system in the sample channel. Figure 2 shows azimuthal anisotropic dependences of SH intensity measured for the sample with the thinnest wells for several SH photon energies. The curves possess one- or two-fold symmetry and change strongly with the spectrum. Analogous symmetry of the SHG azimuthal dependences is

FIGURE 3 SH intensity spectra of the MQW sample with 2.5 å thick a-Si wells (filled circles) and of a Si (100) substrate (open circles). The curves show a fit of a single 2D SHG resonance (1) with = 1.36 eV to the part of the MQW spectrum in the interval of 2h from 2.5 to 3.1eV (solid curve) and a fit of the Lorentz SHG resonance (2) with central energy Si = 3.26 eV and broadening Si = 0.10 eV to the substrate SHG spectrum (dashed curve)


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et al. Second-harmonic spectroscopy of electronic structure of Si/SiO2 multiple quantum wells

673

strate SHG anisotropy (Fig. 2, bottom), which shows a fourfold symmetry standard for the Si (100) face. The strong spectral dependence of SHG azimuthal anisotropy of MQW comes from the pronounced spectral dependence of the relative phase between isotropic C0 and anisotropic C1 ei cos components of the SH field. The anisotropy observed in structures made from amorphous material can be caused by the sputtering geometry, which has a mirror symmetry m z where z is perpendicular to the surface. All SHG spectra presented below are measured in an intermediate azimuthal position containing only an isotropic contribution (marked in Fig. 2 by a vertical line). The intermediate point of the anisotropic dependences of MQW coincides with the intermediate point for the Si (100) substrate, which can contribute to the resulting SHG spectra as well. Figure 3 represents the SH intensity spectrum of a MQW sample with the thinnest wells (solid circles) and the substrate SHG spectrum (open circles). The SHG spectrum of MQW differs significantly from that of the substrate. The MQW

spectrum has a pronounced narrow dip near the resonant energy of the substrate spectrum. The SH intensity spectra for the four MQW samples with different well thicknesses are shown in Fig. 4. The spectra have similar resonant features. At the low-energy edge of the spectra a monotonic red shift of the interval of the SH intensity growth is observed upon increase of the QW thickness from 2.5 to 10 å. The intensity dip at the blue edge of the spectra becomes broader as the QW thickness increases, but its spectral position remains the same for all MQW samples.
3 3.1 Theory and discussion SHG spectra shape

To describe the SH intensity spectra of Si/SiO2 MQW the following assumptions are made. SHG sources are supposed to be localised in the silicon wells and the 2 SH polarisation is considered to be dipolar: PQW (2) = (2) (2) QW E() E(), with QW denoting the effective quadratic susceptibility of QW. Multiple reflections in the structure are neglected, and the SH field from MQW is written as (2) (2) E 2 G QW (2) PQW (2) G 0 PQW (2), where the factor G QW characterises the propagation of the SH wave in the structure, including effects of the superposition and the retardation of the SH fields from different QW. For simplicity, it is replaced by the spectral-independent factor G 0 . The resonant features in the SHG spectra of MQW are expected to be related to direct optical transitions between the first valence and first conduction sub-bands of a 2D electron gas confined in each a-Si QW. The value of the effective energy band gap in the bulk of both amorphous and hydrogenated amorphous silicon falls in the interval from 0.95 eV to 1.65 eV depending on the preparation method [13]. Therefore, the resonance of the combined DOS is also located in this energy interval. Thus it is reasonable to associate the observed SHG resonance with a resonance at the energy of fundamental photons. Suppose there is a parabolic dispersion law of electrons in conduction (c) and valence (v) sub-bands E c ( p) = E c +| p|2/(2m c ) and E v ( p) = E v -| p|2/(2 mv ) ­where p is a 2D quasi-momentum and m c,v is the effective mass (Fig. 1b). (2) Then the line shape of QW in the vicinity of the resonant energy = E c - E v (critical point) is a step-like function [14]:

(2) QW



-ln( - h - ih ); i - ln(h - + ih );

<; > ,

(1)

FIGURE 4 SH intensity spectra for Si/SiO2 MQW with several thicknesses of a-Si QW. The curves represent fits of (3) to the experimental spectra. Vertical lines show the difference in central energies of 2D SHG resonances for the MQW samples with the thickest and the thinnest wells

where the constant characterises the broadening of the resonance. For a 2D critical point the central energy corres(2) ponds to the maximum of the function Im(QW /). The solid line in Fig. 3 shows the modelled spectral (2) dependence |QW (2)|2 for the 2D line shape from (1) with = 1.36 eV. The SHG spectrum has a step-like shape and saturates at energies above 3eV. In the spectral range above (2) 3eV the calculated |QW (2)|2 spectrum deviates from the experimental data. The independence of the spectral position of the SH intensity dip on the well thickness suggests an idea that this dip results from the destructive interference of the resonant contribution to SHG from MQW with that from the Si (001) substrate. The latter resonance, peaked at


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Applied Physics B ­ Lasers and Optics

approximately 3.25 eV, is sufficiently shifted from the spectral position of the bulk E 0 / E 1 resonance at 3.4eV. This allows the description of the SHG substrate contribution (2 by a single surface-susceptibility tensor, Si ) , which gen(2 erates the dipole-type nonlinear polarisation, PSi ) (2) = (2 Si ) E() E(). The spectral profile of the SHG resonance from the silicon-substrate surface is supposed to have the Lorentz line shape (excitonic critical point):

(2) Si

monotonically at 60 meV as the QW thickness increases from 2.5å to 10 å and approaches approximately 1.28 eV for bulk a-Si.
3.2 Size effect

1/( - Si + iSi ),

(2)

with central energy Si and the broadening Si . The resulting SH field can be expressed as
E
(2)



(2) QW

+ ei

(2) Si

.

(3)

In the studied MQW structures the electron tunnelling between neighbouring quantum wells can be neglected. Therefore, the problem concerning the energy spectrum of an electron moving in the z direction reduces to that for a single Si quantum well in a SiO2 host. As a starting point in interpreting the dependence (d ) we consider the ( energy - E 1c,v) of the ground state of a particle with a mass m c,v moving in a rectangular 1D potential well of a width d (Fig. 6):
U
()

The relative phase, , between the two resonant SHG contributions is the adjustable parameter which takes into account the retardation of the substrate SH wave with respect to the SH field from the MQW sample. The SH intensity spectra for all MQW samples are fitted by (3) with a spectral-independent relative phase as an addi(2 tional adjustable parameter. The resonant parameters of Si ) ­ Si and Si ­ were first extracted from the fit of the SH intensity substrate spectrum by (2) (dashed line in Fig. 3) and then are kept fixed in the approximation of the MQW spectra. The results of the least-square approximation are shown in Fig. 3 by (2) solid curves. The broadening of QW is approximately 0.09 eV for all MQW samples. The phase changes monotonically from 3.5rad for d = 2.5å to 3.0rad for d = 10 å. The dependence of the extracted central energies on the silicon well thickness d is shown in Fig. 5. The resonant energy decreases

(z , d ) =

-V , |z | d/2 0, |z | > d/2

,
SiO2 g

(4)
-E
a-Si g

where = c,v and Vc + Vv = E thickness dependence we have

. Then for the (5)

( ( (d ) = Vc + Vv + E 1c) (d ) + E 1v) (d ).

However, the numerical solution of the one-electron SchrÆdinger equation with the potential given by (4) at any reasonable values of m c,v and V yields a dependence (d ) that is essentially steeper than that obtained from the experiment (inset in Fig. 5). This indicates that the expression for the model potential describing the confinement of an electron within a quantum well should take into account additional physical factors that can flatten the dependence (d ). One of these factors is the electrostatic interaction of an electron with its inhomogeneous environment (Fig. 6). For a three-layer dielectric medium the image potential Wim (z , d ) describing this interaction takes the form [15]:
() Wim (z , d ) =

F (z , d )/Si , - F (z , d )/SiO2 ,

|z | d/2 |z | > d/2

,

(6)

sign (m ) e2 Si - SiO2 F (z , d ) = â 4 Si + SiO2 1 1 , â + d2 d2 (z + 2 ) + 2 (z - 2 ) + 2

The dependence of the resonant energy of the SHG resonance in Si/SiO2 MQW on a-Si well thickness d obtained from the approximation of SH intensity spectra by (3) (filled circles). Solid line shows the model dependence (d ) obtained for a rectangular well with the interface -function distortion WS given by (7). Inset: the same thickness dependences are shown in comparison with model dependences (d ) calculated for a rectangular well (solid curve) and for a rectangular well with image-potential corrections Wim (dashed line)
FIGURE 5

where = v,c and Si and SiO2 are the dielectric constants of Si and SiO2 , respectively; the `softening' parameter is introduced phenomenologically to take into account a nonzero thickness of the interface regions. In (6) only the nearest image sources are taken into account. As before, the dependence (d ) is described by (5), with the only difference () that - E 1 is now the ground-state energy for the potential () () U (z , d ) + Wim (z , d ). The numerical solution of the corresponding SchrÆdinger equation with Si = 11.7, SiO2 = 3.7 and = 1å results only in a slight correction to the dependence (d ) for the rectangular well (inset in Fig. 5). This is caused by the fact that quantitatively the image-potential effect is reduced by the factors 1/ 4Si,SiO2 1 in the exc, pression for Wimv (z , d ).


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et al. Second-harmonic spectroscopy of electronic structure of Si/SiO2 multiple quantum wells

675

sition interface layers between Si and SiO2 is comparable with the QW thickness d . Second, the thickness and structure of these interface layers may depend on d , whereas we treat the parameters of the interface perturbation Q and as independent of d . Thus comparison of the experimentally observed size effect with that calculated for the model potential taken in the form of a rectangular well with delta-function interface perturbations is meaningful only on a qualitative level.
4 Conclusions

In conclusion, the resonant enhancement in the SHG spectra of amorphous Si/SiO2 multiple quantum wells is observed for SH photon energies from 2.5 to 3.4eV. The spectral features revealed are associated with electron transitions between 2D electronic sub-bands in a-Si QW. The central energy of SHG resonance obtained using a 2D line shape of the effective quadratic susceptibility of QW corresponds to the lowest-lying interband transition energy in these QW. A monotonic dependence of the SHG resonant energy on the well thickness in the interval from 2.5å to 10 å is observed. The size effect is described within the rectangular-well model with delta-function interface perturbations.
ACKNOWLEDGEMENTS The authors are pleased to acknowledge S. Soria for helpful discussions and D. Schuhmacher for experimental assistance. This work was supported by the Russian Foundation for Basic Research (RFBR) and Deutsche Forschungsgemeinschaft (DFG): DFG Grant No. 436 RUS 113/640/0-1, RFBR Grant Nos. 01-02-04018, 01-02-16746 and 01-02-17524, Special Grants for Leading Russian Science Schools Nos. 00-15-96555 and 00-15-96558 and INTAS Grant No. YSF-2001/1-160.

FIGURE 6

Model potentials U

(c,v)

(c, ( (z , d ), Wim v) (z , d ) and Ws

c,v)

(z , d )

The modification of the lattice potential at the interface, in comparison to that in the bulk (see Fig. 6), is another factor that affects the dependence (d ) but is ignored by using the model rectangular potential U(z ). To take into consideration the potential distortion within interface regions, we introduce Ws (z , d ), a simple model correction to the effective potential taken as Dirac delta-functions localised at the quantum-well boundaries (Fig. 6):
Ws(c,v) (z , d ) = sign (m ) âQ z- d d + + z+ - 2 2 ,

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1 D.J. Lockwood (Ed.): Light Emission in Silicon, Vol. 49 of Semiconductors and Semimetals (Academic, Orlando 1997) 2 W. Theiú: Surf. Sci. Rep. 29, 91 (1997) 3 T.P. Pearsall, J. Bevk, L.C. Feldmann, J.M. Bonar, J.P. Mannaerts, A. Ourmazd: Phys. Rev. Lett. 58, 729 (1987); T.P. Pearsall, J.M. Vandenberg, R. Hull, J.M. Bonar: Phys. Rev. Lett. 63, 2104 (1989) 4 D.J. Lockwood, Z.H. Lu, J.-M. Baribeau: Phys. Rev. Lett. 76, 539 (1996) 5 Y. Kanemitsu, S. Okamoto: Phys. Rev. B 56, R15 561 (1997) 6 J.D. Joannopoulos, G. Lucovsky (Eds.): The Physics of Hydrogenated Amorphous Silicon, Vols. I and II (Springer, New York 1983) 7 G. LÝpke: Surf. Sci. Rep. 35, 75 (1999) 8 T.F. Heinz: In Nonlinear Surface Electromagnetic Phenomena, ed. by H.-E. Ponath, G.I. Stegeman (Elsevier, Amsterdam 1991) 9 D.E. Aspnes, A.A. Studna: Phys. Rev. B 27, 985 (1983) 10 M. Cardona: Modulation Spectroscopy, Suppl. 11 of Solid State Physics, ed. by F. Seitz, D. Turnbull, H. Ehrenreich (Academic, New York 1969) Chapt. 2 11 O.A. Aktsipetrov, V.N. Golovkina, A.I. Zayats, T.V. Murzina, A.A. Nikulin, A.A. Fedyanin: Phys. Dokl. 40, 12 (1995) 12 O.A. Aktsipetrov, A.N. Rubtsov, A.V. Zayats, W. de Jong, C.W. van Hasselt, M.A.C. Devillers, T. Rasing, E.D. Mishina: Sov. Phys. JETP 82, 668 (1996) [Zh. Eksp. Teor. Fiz. 109, 1240 (1996)] 13 D.R. McKenzie, N. Savvides, R.C. McPhedran, L.C. Botten, R.P. Netterfield: J. Phys. C 16, 4933 (1983); R.H. Klazes, M.H.L.M. van den Broek, J. Bezemer, S. Radelaar: Philos. Mag. B 45, 377 (1982) 14 The line shapes of (2) depending on the dimensionality of critical points are supposed to be the same as of (1) , which can be found, for instance, in [10] 15 O.A. Aktsipetrov, P.V. Elyutin, E.V. Malinnikova, E.D. Mishina, A.N. Rubtsov, W. de Jong, C.W. van Hasselt, M.A.C. Devillers, T. Rasing: Phys. Dokl. 42, 340 (1997) 16 K.T. Queeney, M.K. Weldon, J.P. Chang, Y.J. Chabal, A.B. Gurevich, J. Sapjeta, R.L. Opila: J. Appl. Phys. 87, 1322 (2000)

(7)

where Q and are parameters of the model. It is reasonable to expect that, by the order of magnitude, Q / 1eV, 1å. The dependence (d ) is given by (5) with the energies ( E 1c,v) calculated numerically for the potential U () (z , d ) + Ws() (z , d ). In the fitting of (d ) to the experimental data the values of m c,v and Vc,v are chosen fixed, whereas the quantities Q and are treated as adjustable parameters. This yields a Q / = 6.1eV and = 0.5å for E g-Si = 1.28 eV The fitting results shown in Fig. 5 provide much better agreement with the experiment, which indicates that apparently the observed thickness dependence of is significantly influenced by the properties of the Si/SiO2 interface regions. However, this model still remains oversimplified in comparison with the properties of real Si/SiO2 interfaces [4, 16] at least for two reasons. First, the effective thickness of the tran-