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PHYSICAL REVIEW B 66, 033305 2002

Optical second-harmonic interferometric spectroscopy of Si,, 111... - SiO2 interface in the vicinity of E 2 critical points
T. V. Dolgova, A. A. Fedyanin, and O. A. Aktsipetrov*
Department of Physics, Moscow State University, 119992 Moscow, Russia

G. Marowsky
Ё Ё Laser-Laboratorium Gottingen e.V., D-37077 Gottingen, Germany Received 4 January 2002; published 17 July 2002 The direct electron transitions at the Si( 111) -SiO2 interface are studied by combined second-harmonic SH intensity and phase spectroscopy with the SH photon energy range from 3.5 eV to 5 eV. A strong resonant behavior of the quadratic optical response is observed in the vicinity of the E 2 ( X , ) critical points CP's of the silicon band structure. Good representation of the experimental SH intensity and phase spectra is provided by a line shape of the quadratic susceptibility calculated both with excitonic and two-dimensional 2D types of E 2 ( X , ) CP's. The line-shape dependence of resonance energies, namely, repulsion of nearby E 2 ( X , ) resonances in the case of a 2D CP in comparison with the excitonic ones, is observed. DOI: 10.1103/PhysRevB.66.033305 PACS number s : 71.20. b, 73.20. r, 73.20.At, 78.68. m

Studies of the electronic response of semiconductor surfaces and interfaces are of interest and importance for both fundamental and technological reasons. The prevalent techniques are variants of the spectroscopy of the linear dielectric function, e.g., photoreflectance and spectroscopic ellipsometry.1 The resonances in spectra reflect critical behavior Van Hove singularities of combined density of states. The parameters of the band-structure critical point CP --central energy, broadening, and dimensionality--can be deduced from the optical spectra. Second-harmonic generation SHG spectroscopy,2 which is intrinsically sensitive to interfaces,3 successfully complements well-established optical methods. The resonances of the second-order susceptibility, (2) , are due to direct electron transitions and therefore contain information about electron spectra depending on morphological, structural, and symmetry features of the bulk and the surface. For example, symmetry variation of a reconstructed surface,4 surface strain5 and its removal during H termination,6 and interface7 and surface8-states spectra can be studied by SHG spectroscopy. Most authors interpret SHG spectroscopy data within a Lorentz excitonic CP (2) line shape9 because a large number of adjustable parameters makes doubtful determination of the CP types of nearby resonances based upon only the second-harmonic SH intensity spectrum. The (2) line shapes are quite different for various CP dimensionalities and influence the resonance parameters significantly, mainly the central energies. One of the ways to define resonance parameters more accurately would be to supplement the SH intensity spectrum with the SH phase as an additional independent characteristic of the SH field. The latter is essentially a complex value due to the complex (2) and Green'sfunction corrections near resonances. Single-wavelength SHG interferometry is conventionally used for determination of the (2) phase and the absolute direction of molecules on liquid surfaces,10 and for separation of SHG contributions from thin films and the substrates.11 Another application of SH phase measurements
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is a homodyne mixing technique that improves the signal-tonoise ratio for the surface SHG studies.12 The use of external and internal homodynes for dc-electric-field-induced SHG allows the measurement of the in-plane spatial distribution of the electric-field vector with micron resolution,13 or the visualization of weak nonlinear contributions.14 Further developments of SHG interferometry are frequency-domain interferometric SHG spectroscopy,15 which explores the broad bandwidth of femtosecond laser pulses, and hyper-Rayleigh scattering interferometry,16 which uses the correlation of fluctuations in linear and nonlinear optical properties of thin inhomogeneous films. SHG interferometric spectroscopy unites amplitude intensity SHG spectroscopy with SHG interferometry. It deals with SH intensity and phase spectra which are two independent combinations of resonance parameters. The difference of SHG interferometric spectroscopy from conventional SHG interferometry is the proper choice of the phasereference sample. A quartz plate10 or a poled polymer film11 cannot be used because of strong spectral dependence of SH intensity and phase due to Maker fringers and the (2) resonances in the experimental spectral range. In this paper the spectral dependences of the phase and amplitude of the SH field generated at the Si( 111) -SiO2 interface are measured in the spectral vicinity of the silicon E 2 critical points. The resonant contributions of E 2 ( ), E 2 ( X ), and E 1 CP's to (2) are extracted taking into account the complex Green's-function corrections along with the complex (2) . The setup of the SHG interferometric spectroscopy is shown in Fig. 1 a . The output of a tunable ns-parametric generator/amplifier Spectra-Physics MOPO 710 operating from 490 nm to 690 nm is directed onto the sample at an angle of incidence of 45° . The SHG signal is selected by filters and detected by a photomultiplier tube. To normalize the SH intensity spectrum over the laser fluence and the spectral sensitivity of the optical detection system, a SH intensity reference channel is used with a slightly wedged z-cut
©2002 The American Physical Society

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FIG. 1. a Experimental setup of SHG interferometric spectroscopy. b Raw SHG interferograms for different SH photon energies. Solid curves are the dependencies given by Eq. 1 . c SH phase spectra of the indium tin oxide reference.

quartz plate and with a detection system identical to the one in the sample channel. The phase- reference sample i is thin enough to avoid Maker fringes in the SHG response , ii is opduring tuning of the fundamental wavelength tically inactive for conservation of the polarization state of the fundamental radiation, and iii has no resonance features in the tuning regions of both the fundamental and SH waves. A 30-nm-thick indium tin oxide ITO film coated on the 1-mm-thick plate of fused quartz is chosen as a phasereference sample. An SHG interferogram is obtained by translating the reference along the laser beam by changing the distance l between the reference and the sample. Figure 1 b shows typical SHG interference patterns measured for different SH energies. A spectral dependence of the interferogram period is obvious. The result of the interference of the SH waves from the reference and the sample is the detected SH inten2 E 2 2: sity I ( c /8 ) E r II
2 r

FIG. 2. SH intensity panel a and phase panel b spectra of the Si( 111) -SiO2 interface. The inset: sketch of a typical SH phase spectrum of a multiresonant system with central energies 1 , 2 , and 3 .

I

2

2

I

2 r

I

2

cos

2l L

r

,

1

where L ( ) (2 n ) 1 is the interferogram period with arg( E 2 ) , and n n 2 n describing air dispersion, 2 1 indicates the laser cor arg( E r ) arg( R 2 ) , and herence. The deviation of from unity is found to be small and is neglected. R 2 is the Fresnel factor of the SH wave reflection from the sample. 2 The position-dependent phase shift 2 l / L between E r 2 and E comes from the different refractive indices of air for the fundamental and SH waves. The position-independent shift originates from complex surface, (2), S , and bulk quadrupole, (2), BQ , quadratic susceptibilities as well as from Green's-function corrections G 2 :17 E
2

/E

2

G

2

(2)

G

2

(2)

,

2
2 (2)

where subscripts and are related to components of E parallel and normal to the surface, respectively. (2) and

are the linear combinations of the (2), S and (2), BQ components with nonresonant coefficients depending only on the fundamental wave vector. The SH phase spectrum of the ITO 2 film, arg( E r ) , is measured using a 1-mm-thick backsideimmersed y-cut quartz as a sample since quartz surface generates the SH wave with constant phase in the whole used spectral region Fig. 1 c . The ITO SH phase spectrum is found to be monotonic and is taken into account as normalization curve for silicon SH phase spectra. The samples are natively oxidized p-type Si 111 wafers with resistivity of 10 cm. SHG interferograms are measured at the maximum of the azimuthal SHG rotational anisotropy for the p-in, p-out polarization combination of the fundamental and SH waves. The fit of the SHG interference pattern for every wavelength by Eq. 1 with , I 2 , and L as adjustable parameters yields their spectral dependences. The I 2 spectrum measured directly with a fine resolution in SH photon energy is combined with I 2 (2 ) obtained from the interferograms. The I 2 spectrum Fig. 2 a has a pronounced peak, centered approximately at 4.4 eV, with a shoulder below 4 eV. The phase , shown in Fig. 2 b , increases approximately by 0.9 rad within the interval from 4.3 eV to 4.7 eV. Some nonmonotonic features are seen below 4.3 eV. The experimental spectra denote that at least four resonances have to be considered. Two resonances localized near 4.3 eV and 4.5 eV account for the forked peak in the SH intensity spectrum and corresponding phase growth in the center of the spectrum. The increasing SH intensity at the

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BRIEF REPORTS TABLE I. SH resonance energies in eV extracted from Si 111 SH intensity and combined spectra fitted by different line shapes in comparison with results of the excitonic fit to Si 001 the SH intensity spectrum in Ref. 9. Reference values for CP energies deduced using linear techniques in Refs. 20 ­ 22 and from nonlocal pseudopotential calculations Ref. 23 are extrapolated to 300 K by Eq. 3b of Ref. 20. E 2( X ) E 2( ) Excitonic fit to SH intensity Combined excitonic fit Combined 2D fit Si 001 SH intensity a Ellipsometry b Electroreflectance ER c ER and ellipsometry d Theory e
a

PHYSICAL REVIEW B 66, 033305 2002

E

1

4.28 4.44 4.31 4.47 4.24 4.60 4.34-4.39 4.27 4.51 4.28 4.54 4.27 4.57 4.41 4.57

5.1-5.3 5.18 5.27 5.31 5.38 5.50

3.9 3.78 3.86 3.60-3.68

Reference Reference c Reference d Reference e Reference
b

9. 20. 21. 22. 23.

edges of the spectrum and the inflections of the SH phase spectrum show the influence of two additional resonances that should be included into the model to achieve a reasonable agreement. The inset in Fig. 2 shows schematically the phase spectrum for the case of superposition of several nearby resonances. A single resonance causes phase growth, while superposition of several resonances modifies the phase spectrum strongly, producing curve inflections. One can consider the SHG spectra as the means of derivation of the (2) resonance parameters, mainly the central energies. The spectral dependence of (2) ) is superposition ( of two-photon resonances related to several CP's and spectral background B:
(2)

FIG. 3. Solid lines are the result of the combined fit to SH intensity top panels and phase bottom panels experimental spectra with excitonic and 2D line shapes. Arrows indicate the resonance energies. Thin lines: fit of SH intensity spectra with excitonic line shapes and corresponding reconstructed phase spectra.

2

B
m

f m exp i

m

2

m

i

m

n

,

3

where is or , m 1 . . . 4 , m is a multiple of /2, and n takes the values 1, 1/2, 0, and 1/2 depending on the CP type.1 The exponent n 0 denotes the function ln(2 m i m) symbolically and relates to the two-dimensional 2D CP. The spectral profiles of (2) and (2) are assumed to be the same though they could have, in principle, different spectral dependences because of the presence of the surface.18 The oscillator strengths f m( ) are supposed to be real numbers. For the present experiment, a slight spectral dependence of B taking into account the resonances at fundamental energies below 1.5 eV Ref. 8 is neglected. First, only the I 2 spectrum is fitted by Eq. 2 with Lorentz (2) line shapes used traditionally in SH intensity spectroscopy9 i.e., n 1 in Eq. 3 . The curve calculated for resonance energies listed in the first row of Table I thin line in Fig. 3 fits the experimental I 2 data well. The resonances at 4.28 eV and 4.44 eV can be attributed to E 2 ( X ) and E 2 ( ) CP's.19 The

latter is significant in SHG spectra whereas it is weakly exhibited in linear investigations.20 The resonance at approximately 5.2 eV out of range corresponds more likely to the E 1 CP. The spectral position of the low-energy resonance is uncertain: the fit is found to be reachable with any central energy less than 3.9 eV. The phase spectrum calculated with the intensity fit parameters thin line in Fig. 3 shows much worse agreement with the experimental data. Thus the set of parameters deduced from the intensity fit does not appear to be valid and the SH intensity spectrum is not enough for an analysis of the (2) spectra. The obvious way to overcome the situation is to use a wider spectral range, but practically it is the primary experimental difficulty. We propose another way, which is to fit the SH intensity spectrum simultaneously with the spectral dependence of the SH phase as an additional independent combination of resonance parameters. The simultaneous least-squares fit to both spectra with the appropriate weights for each spectrum solid lines in Fig. 3, left panels demonstrates a good agreement. The total chi is reduced by a factor of 5 in comsquared for I 2 and parison with that calculated for the intensity fit parameters. The results of the intensity and combined fits are compared in the first two rows of Table I. The central energies of the E 2 ( X , ) resonances extracted from the combined fit are shifted from the intensity fit ones. Then, the combined fit is achievable with a certain and unique position of the 3.78-eV resonance and the out-of-range 5.18-eV resonance. The resonance energy of 3.78 eV has no analog in linear spectral investigations of silicon. The nearest known resonances are E 0 / E 1 CP's at 3.4 eV and a E 0 CP at approximately 4 eV which is very weak in the linear response.22 Such a significant energy difference can be caused by dis-

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tinction of the (2) line shapes from CP types used in linear spectroscopy, for instance, by Lautenschlager et al.20 In order to test the model dependence of resonance parameters we used 2D line shapes for E 2 ( X ), E 2 ( ) , and E 1 CP's following Ref. 20, and the excitonic line shape for the low-energy resonance. The curves demonstrate again very good agreement with the experiment Fig. 3, right panels . Chi squared is even less than that for the excitonic model. The new resonance parameters differ from the excitonic model fit and the resonances repel each other. This emphasizes the importance of the model choice in the (2) spectral properties problem. The central energy of the left SHG resonance is certain for the new fit and fixed at 3.86 eV. This means that the resonance originally differs from ones observed in linear spectra. The SHG resonance at similar energy was recently observed by Erley and Daum9 at the oxidized Si 001 surface and was attributed to electron transitions at the silicon-dioxide interface. The same explanation can be applied here for the silicon 111 . Table I shows a comparison of the SHG spectra resonance parameters with parameters known from linear spectroscopy20,22,21 and theoretical calculations23 of the silicon band structure. The diversity of resonance energies in the literature does not allow us to differentiate between an excitonic and a 2D CP model for SHG spectra. The definite choice of the CP type from the combined SHG spectra requires precise measurements. The measurements of SHG spectra for various azimuthal positions of the sample possi-

bly leading to understanding the role of surface in (2) lineshape modification are also in further development. In conclusion, the spectroscopic modification of SH phase measurements, namely, combined SH intensity and phase spectroscopy, is developed. Resonant contributions of nearby E 2 ( X ) and E 2 ( ) critical points of Si 111 to quadratic susceptibility are resolved. Their magnitudes are comparable with each other in contrast to the linear case. The resonance parameters are certain and unique only if they are deduced from both the SH intensity and phase spectra, and they are shown to be strongly dependent on the chosen CP types. Moreover, the combination of the SH intensity and phase spectra is found to be the way to examine out-of-range resonances. At the blue edge of the SHG spectrum the E 1 outof-range resonance is obtained. An additional resonance, much wider than the others and located at 3.86 eV, is observed. It has no equivalent in linear spectroscopy investigations but its spectral position is close to the one reported in Ref. 9 for the SH intensity spectrum of a Si 001 wafer. Authors are pleased to acknowledge D. Schuhmacher for experimental assistance. This work was supported by the Russian Foundation for Basic Research RFBR and the Deutsche Forschungsgemeinschaft DFG : DFG Grant No. 436 RUS 113/640/0-1, RFBR Grant Nos. 01-02-04018, 0102-16746, and 01-02-17524, and INTAS Grant No. YSF2001/1-160.

*Web address: http://www.shg.ru
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