Investigation of the photoelastic effect in si at high values of the absorptivity

Investigation of the photoelastic effect in si at high values of the absorptivity

The birefrigence effect induced by uniaxial elastic deformation has been investigated in monocrystalline Si using theoretical and experimental technigues. To improve a measuring system sensitivity in the range of band absorption, polarization modulation of reffected emission was used. Deformation ch...

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Дата:1999
Автори: Boiko, I.I., Venger, Ye.F., Nikitenko, E.V., Serdega, B.K.
Формат: Стаття
Мова:English
Опубліковано: Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України 1999
Назва видання:Semiconductor Physics Quantum Electronics & Optoelectronics
Онлайн доступ:http://dspace.nbuv.gov.ua/handle/123456789/119860
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Цитувати:Investigation of the photoelastic effect in si at high values of the absorptivity / I.I. Boiko, Ye.F. Venger, E.V. Nikitenko, B.K. Serdega // Semiconductor Physics Quantum Electronics & Optoelectronics. — 1999. — Т. 2, № 2. — С. 54-58. — Бібліогр.: 8 назв. — англ.

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spelling irk-123456789-1198602017-06-11T03:02:57Z Investigation of the photoelastic effect in si at high values of the absorptivity Boiko, I.I. Venger, Ye.F. Nikitenko, E.V. Serdega, B.K. The birefrigence effect induced by uniaxial elastic deformation has been investigated in monocrystalline Si using theoretical and experimental technigues. To improve a measuring system sensitivity in the range of band absorption, polarization modulation of reffected emission was used. Deformation characteristics of this effect has been measured. The main results obtained for probing light wavelengths from the range of crystal transparency are in a good accordance with conclusions of previons studies. In the range of strong absorption, a considerable change of effect characteristic shapes was found, and their satisfactory accordance with results of theoretical estimations was also ascertained. 1999 Article Investigation of the photoelastic effect in si at high values of the absorptivity / I.I. Boiko, Ye.F. Venger, E.V. Nikitenko, B.K. Serdega // Semiconductor Physics Quantum Electronics & Optoelectronics. — 1999. — Т. 2, № 2. — С. 54-58. — Бібліогр.: 8 назв. — англ. 1560-8034 PACS 78.20.C http://dspace.nbuv.gov.ua/handle/123456789/119860 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
description The birefrigence effect induced by uniaxial elastic deformation has been investigated in monocrystalline Si using theoretical and experimental technigues. To improve a measuring system sensitivity in the range of band absorption, polarization modulation of reffected emission was used. Deformation characteristics of this effect has been measured. The main results obtained for probing light wavelengths from the range of crystal transparency are in a good accordance with conclusions of previons studies. In the range of strong absorption, a considerable change of effect characteristic shapes was found, and their satisfactory accordance with results of theoretical estimations was also ascertained.
format Article
author Boiko, I.I.
Venger, Ye.F.
Nikitenko, E.V.
Serdega, B.K.
spellingShingle Boiko, I.I.
Venger, Ye.F.
Nikitenko, E.V.
Serdega, B.K.
Investigation of the photoelastic effect in si at high values of the absorptivity
Semiconductor Physics Quantum Electronics & Optoelectronics
author_facet Boiko, I.I.
Venger, Ye.F.
Nikitenko, E.V.
Serdega, B.K.
author_sort Boiko, I.I.
title Investigation of the photoelastic effect in si at high values of the absorptivity
title_short Investigation of the photoelastic effect in si at high values of the absorptivity
title_full Investigation of the photoelastic effect in si at high values of the absorptivity
title_fullStr Investigation of the photoelastic effect in si at high values of the absorptivity
title_full_unstemmed Investigation of the photoelastic effect in si at high values of the absorptivity
title_sort investigation of the photoelastic effect in si at high values of the absorptivity
publisher Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
publishDate 1999
url http://dspace.nbuv.gov.ua/handle/123456789/119860
citation_txt Investigation of the photoelastic effect in si at high values of the absorptivity / I.I. Boiko, Ye.F. Venger, E.V. Nikitenko, B.K. Serdega // Semiconductor Physics Quantum Electronics & Optoelectronics. — 1999. — Т. 2, № 2. — С. 54-58. — Бібліогр.: 8 назв. — англ.
series Semiconductor Physics Quantum Electronics & Optoelectronics
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AT serdegabk investigationofthephotoelasticeffectinsiathighvaluesoftheabsorptivity
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fulltext 54 © 1999, Institute of Semiconductor Physics, National Academy of Sciences of Ukraine Semiconductor Physics, Quantum Electronics & Optoelectronics. 1999. V. 2, N 2. P. 54-58. 1. The effect of birefringence in semiconducting crys- tals caused by uniaxial elastic deformation (also known as photoelastic or piezo-optical effect) has been investi- gated comprehensively and in much detail. This phenom- enon, defined as the difference in the propagation veloc- ity of waves polarized in two mutually orthogonal direc- tions, is usually observed in anisotropic substances in the wavelength range corresponding to their transpar- ency. It may seem that in materials with a strong ab- sorption this effect can be observed only in an indirect way. For example, in [1] the components of the photo- elastic tensor in silicon were measured by the Bragg dif- fraction technique, while in [2, 3] the photoelastic con- stants in the same material were calculated using Ra- man scattering measurements for several wavelengths from the fundamental absorption range and for the en- ergy range 0.7 to 3.4 eV. It is known that at large absorption coefficients the reflected wave is formed in the crystal layer of a thick- ness comparable to the absorption length. This indicates a finite length of the wave penetration into the crystal and a certain role of the bulk interaction. If the dielectric constant of the crystal is anisotropic, at least within the absorption length, the polarization of the reflected wave will be changed in comparison to the incident one. For example, a linearly polarized incident beam, upon reflec- tion from the surface of such a crystal, in the general PACS 78.20.C Investigation of the photoelastic effect in Si at high values of the absorptivity I. I. Boiko, Ye. F. Venger, E. V. Nikitenko, B. K. Serdega Institute of Semiconductor Physics of NASU, 45, prospect Nauki , 252028 Kiev, Ukraine Phone (044) 265-4020; e-mail: bshl@polarget.semicond.kiev.ua Abstract. The birefrigence effect induced by uniaxial elastic deformation has been investigated in monoc- rystalline Si using theoretical and experimental technigues. To improve a measuring system sensitivity in the range of band absorption, polarization modulation of reffected emission was used. Deformation characteristics of this effect has been measured. The main results obtained for probing light wavelengths from the range of crystal transparency are in a good accordance with conclusions of previons studies. In the range of strong absorption, a considerable change of effect characteristic shapes was found, and their satisfactory accordance with results of theoretical estimations was also ascertained. Keywords: birefrigence, polarization, modulation, photovoltage, anisotropy, thermal stress. Paper received 10.03.99; revised manuscript received 01.06.99; accepted for publication 12.07.98. case becomes elliptical, if the polarization plane does not coincide with the axes of the optical indicatrix of the sam- ple. Therefore, by detecting a change in the polarization of a normally incident beam reflected from the crystal surface, it is possible to observe the photoelastic effect directly. However, in the range of fundamental absorption, the value of the polarization change, measured experi- mentally as the phase difference ϕ between the orthogo- nal components of the wave, is by many orders of mag- nitude lower than that in the conditions of transparen- cy. This is related to the fact that the length of the opti- cal path appearing in the formula ϕ ~ d⋅∆n is now de- termined by the light absorption length rather than the thickness of the sample. Therefore, to observe this ef- fect, one should use a technique having a high sensitivity with respect to the magnitude of the anisotropy. One of the techniques satisfying this requirement is based on the modulation of the probing beam polarization [4]. In the present paper, we study the birefringence in single crystals of Si in the fundamental absorption range. The anisotropy of the dielectric constant is induced by a uniaxial elastic deformation controlled by the external force. 2. Let us consider this phenomenon qualitatively. We assume that the dielectric properties of a semi-infinite crystal are determined by a diagonal tensor I. I. Boiko et al.: Investigation of the photoelastic effect in Si at high... 55SQO, 2(2), 1999 z y x ε ε ε ε 00 00 00 =) , and the geometry of the experiment is such that the elec- tric and magnetic components of the wave are presented by the vectors ( ) ( )0,, ,0,, yxyx HHEE =Η=Ε . In this case we assume ( )z∇=∇ ,0,0 . Then, Maxwell�s equations ( ) Ε⋅=Ε×∇×∇      Η⋅−=Ε Ε⋅=Η εω ∂ ∂ ∂ ∂ε ) ) 2 2 1 1 c tc rot tc rot (1) for waves z , iktie +−∞ΗΕ ω are reduced to the equations 02 ,0 2 2 2 2 2 =        −=        − yEy c kxEx c k εωεω (2) Here, ).,( , ,0 0 yxjjijjyk ykxkxcxk =′′⋅+′== === εεεε εεω The electric field and its gradient in the region z ≤ 0:    ⋅−⋅⋅=∇ ⋅+⋅= ⋅−⋅ ⋅−⋅ zik j zik jjz zik j zik jj eEeEikE eEeEE 0 2 0 10 0 2 0 1 (3) In the region 0≥z zik jjz zik jj eikEE eEE ⋅ ⋅ ⋅⋅=∇ ⋅= 3 3 (4) Here jE1 is the amplitude of the incident wave, so that for jE2 ( ) 321321 , EikEEikEEE ⋅=−=+ (5) The amplitude of the reflected wave jE2 expressed through that of the incident wave jE1 is ( )jjjj EE k k EE 21 0 21 −=+ (6) Thus, we obtain the two components of the wave: y y yy x x xx EEEE ε ε ε ε + − = + − = 1 1 , 1 1 1 2 1 2 (7) Let us separate the real and imaginary parts of the refrac- tive index: yyyyxxxx ninnninn ′′+′==′′+′== εε , (8) Substitution of them in Eq. (7) yields the phase of the reflected wave:         −      = y y x x E E arctg E E arctg 2 2 2 2 Re Im Re Im δ (9) Here jERe and jEIm are the real and imaginary parts of the j-th component of the wave field. Substituting (7) and (8) in (9), we obtain the final equation for the phase of the reflected wave: . 1 2 2 1 2 22 2                ′′−   ′′− ′′ − −                  ′′−   ′− ′′ = xx x yy y nn n arctg nn n arctgδ (10) We assume the refractive indices to be linear func- tions of the uniaxial stress X. They can be expressed as . , , , XcnnXcn nXcnnXcnn y yxx γγ −′′=′′−′= =′⋅+′′=′′⋅+′=′ (11) Here c is the photoelasticity constant for a given wave- length, and γ is the Poisson coefficient. Fig. 1 shows the dependence of the phase δ of the reflected wave on c and X, which are the most signifi- cant factors. One can clearly see an anisotropy, which in the linear approximation is related to the uniaxial stress. The magnitude of the photoelasticity effect in the mate- rial under study is also affected by the Brewster constant c. As we might expect, in the isotropic case (at X = 0) the polarization of light is determined only by the «metal- lic» character of reflection. This can be seen from the fact that every component of the wave changes its phase in the same manner. The difference of the wave phases appears only at Õ ≠ 0 (when the anisotropy is induced), which was detected by our measurements. 3. For experimental studies of the birefringence man- ifested in the reflected wave, we used a modified optical I. I. Boiko et al.: Investigation of the photoelastic effect in Si at high... 56 SQO, 2(2), 1999 polarization technique, in which the phase difference ϕ be- tween the orthogonal components of the wave was mea- sured after passing through a sample with the thickness d and anisotropy IInnn −=∆ ⊥ . Here II,nn⊥ are the re- fractive indices in the directions of the optical indicatrix axes, and λπϕ / 2 dn∆= . We had to elaborate this technique, since the experimental condition required a higher sensitiv- ity with respect to the magnitude of ∆n. Indeed, in the case of a strong absorption the length of the optical path is no longer determined by the crystal thickness d, as in the trans- parency case, but takes the values comparable to the ab- sorption length, which is much less than d. The elaboration of the technique consisted in using a polarization modulator, and in a modification of the optical scheme of measurements, so that the polarization state was analyzed and recorded in comparison to the reference signal [4]. The probe beam in this case was directed normally to the sample surface, in order to avoid the Fresnel distortion of the reflected beam. As a result, the detection capability of the technique with respect to birefringence was improved to the level of ∆ϕ = 2 π×10−6 (in terms of phase), which is approximately equivalent to one angular second. The magnitude of the birefringence was measured in the samples of single-crystalline silicon. The optical anisotropy was induced by a uniaxial elastic deformation induced by an external force applied in parallel to the illuminated surface. Rectangular samples with dimensions 10x5x3 mm3 were cut from a lightly doped (ρ = 200 Ohm.cm) n-type single-crystal ingot in such a way that the faces were parallel to the crys- tallographic planes {100}. The surfaces of the samples illu- minated by the probing light were subjected to two kinds of treatment: mechanical and chemical mirror polishing. During the sample preparation, a special attention was paid to the treatment of their faces to which the compres- sion forces were applied. Their surfaces should be not only parallel to each other, but also geometrically flat. This is necessary both to ensure the uniformity of deformation over the cross-section of the sample and to eliminate any bend- ing, which could cause a deflection of the reflected beam during the process of deformation. Furthermore, the deform- ing system was equipped with an independent source of the compressive force, to ensure the immobility of the entire system with respect to the beam. Since the orientation of the optical indicatrix of the sam- ple deformed by the uniaxial stress is known (it is deter- mined by the direction of stress), then, as shown by the analysis of the entire system based on the matrix optics approach [5], the optimal polarization of the incident beam is the linear one. In this case, the polarization analyser con- sisting of a polarization modulator and a linear polarizer was set (using the adjustment of the azimuthal angle) to record the circular polarization component transformed from the initially linear one due to the interaction of light with the sample. A germanium photodiode with an enlarged receiv- ing area was used as a photodetector. Its signal I in this case is characterized by the relation ( )tI ⋅⋅∆≈ ωδ sinsinsin 0 , where ∆ is the birefringence magnitude in the sample; 0δ is the modulation depth of the polarization modulator expressed through the value of the ellipticity angle; ω is the main fre- quency of the modulator. Note that at ∆ < 1 the magnitude of the circular component of the beam (proportional to the photodetector signal) is a measure of the sample anisotro- py, i.e. I ∼ ∆. A He-Ne laser LG-126 was used as a source of probing light. Its radiation with the wavelength I ∼ ∆ is weakly absorbed in silicon crystals at room temperature; α ≅ 1 ñm-1 (see [6]). It was used to compare the results of the modula- tion technique in the transparency region with those of oth- er authors. Radiation with the wavelength λ = 0.63 is ab- sorbed, according to [6], over the length of 2.25⋅10 -4, which corresponds, at the sample thickness of 3 mm, to the condi- tion of a semi-infinite media. The signal, amplified by a loc-in nanovoltmeter, was recorded by a plotter at various magni- tudes of the mechanical stress in the sample. The magnitude of the stress was measured using a silicon transformer based Fig. 1. A set of dependencies of the phase difference calculated by Eq.(10) for two orthogonal components of the polarized light in the presence of an anisotropy created by a mechanical stress X. The values used for calculations are: n = 3.8; c = 0.04; g = 0.3. -0.7 -0.5 -0.3 -0.1 0 0 400 800 1200 16000.001 0.002 0.003 0.004 U X I. I. Boiko et al.: Investigation of the photoelastic effect in Si at high... 57SQO, 2(2), 1999 on the transverse tenso-emf (see [7]), placed in the deform- ing unit in such a way that it was subjected to the same force. Taking into account the high degree of linearity of the transformer�s characteristics, this technique of measure- ments provided reproducible and reliable results. 4. First of all, we should note that, at a chosen doping level of the samples, the experimental results obtained are attributed to a mechanism of birefringence related to the deformation of the crystal lattice. This is the fact that should be taken into account when discussing the results of the polarization-modulation technique shown in Fig. 2. The de- pendence of the photodetector signal on the magnitude of the compressive force is plotted for measurements at the wavelength λ = 1.15 µm. It generally agrees with the results of birefringence studies reported by other authors (see, for example, [8]). Indeed, these oscillations of the signal, like those ob- served by other researchers, are related to a variation of the phase difference between the two orthogonal components by a magnitude multiple to 2π. (By the way, this formal at- tribute has motivated the introduction of the notion of fringe order into the mechanics of the elastic deformed body). At the same time, the curve shown in Fig. 2 displays such fea- tures as variation of the period and amplitude and the asym- metry of oscillations, which make it different from the similar dependencies reported in other papers. To understand these peculiarities, note that it was established, using this tech- nique, that crystals free of any intrinsic stresses are not available in practice. Therefore, even without any external stress, the polarization of radiation is changed under the action of intrinsic stresses already present in the crystal, which results in the appearance of the initial signal. This is the factor allowing the magnitude of birefringence to be measured within the first period of oscillations, which is an advantage of our technique. As for the variations of the period and amplitude, they can be explained by the same reason � namely, by sample nonuniformities which can be of at least two types. The first type of nonuniformities is the uncontrollable variation of the composition in the direction of light propagation and a nonuniform deformation of the crystal related to this. The second type is related to a different value of the refractive index near the sample surface in comparison to that in the bulk. Its main peculiarity consists in the fact that the defor- mation characteristic of the photoelasticity effect in the sur- face layer can differ from that in the uniform bulk, and may even have the opposite sign. The photoelasticity constant determined from Figure 1 is (ε⊥⊥⊥⊥⊥ -ε)/ ε = 6.8⋅10-6 , which in the units comparable to those employed in [8] differs only by 30 % from the value reported there for the wavelength used in this paper. When using the probing beam in the fundamental ab- sorption range, the deformation characteristics of birefrin- gence change dramatically. First of all, the signal becomes weaker by 3 to 4 orders of magnitude (at equal intensities of the incident radiation). Furthermore, the dependence of the phase of the reflected wave on the magnitude of the me- chanical stress changes qualitatively. This curve, shown in Figure 3 by a thick line, is juxtaposed with that theoretically calculated according to Eq.(10) under the assumption of the linear dependence of the anisotropy on the stress nx - ny = c ( 1 + γ )⋅X (thin line). Since the calculations have not explic- itly taken into account the dispersion of the refractivity, the above agreement between the theory and experiment can be considered as quite satisfactory. This is even more so in view of the fact that the photoelasticity constants in such a material as Si, where the energy of vertical band transition is located far from the minimum bandgap, have a weak disper- sion not only in the transparency range, but also in the case of a strong absorption of light. This could be easily verified by comparing the two practically identical deformation char- acteristics obtained in the reflection mode for the wave- lengths of 1.15 and 0.63 µm. It was unexpected to find that the phase dependences shown in Fig. 3 were nonmonotonic. This fact can be under- stood using analysis of Eq.(10) in which every orthogonal component of the wave consists, in its turn, of two terms. The first term is responsible for the surface reflection related to the presence of an interface between the two media, while the second one is attributed to the «metallic» reflection caused by a strong absorption. Each of these mechanisms in the case of an anisotropic tensor of the dielectric constant has a Fig. 2. An experimental dependence of the photodetector signal induced by radiation with the wavelength of 1.15 mm passed through a Si sample plotted as a function of the uniaxial stress. Fig. 3. The experimental (line 1) and calculated (line 2) dependences of the wave phase on the mechanical stress X. The theoretical curve is calculated using Eq.(10) for n = 3.8; g = 0.3; c = 0.001. P h o to d e te c to r si g n a l, a . u . Stress, tnf/cm2 Pressure, kgf/cm2 P h o to d e te c to r si g n a l, a . u . 4 8 12 16 1 2 0 0 5 10 10 20 40 0 80 120 140 0 40 20 -20 -40 -60 I. I. Boiko et al.: Investigation of the photoelastic effect in Si at high... 58 SQO, 2(2), 1999 different effect on the phase magnitude, resulting in the ap- pearance of the above-mentioned peculiarities. The investigated photoelasticity effect manifested in the reflected beam could be used as a technique to measure the optical and mechanical parameters of the crystal. How- ever, the presence of the features described above can be a strong factor hindering such measurements. Nevertheless, the considered effect is undoubtedly of great practical im- portance. It can be used both to determine the critical points of the band spectrum and to characterize the uniformity of semiconducting materials. Polarization analysis of the re- flected radiation can provide information about the anisot- ropy of the material, caused either by the growth conditions or by an external action, for example, in the case of an elastic deformation of a cubic crystal. Since the presence of intrin- sic mechanical stresses may be related to a gradient of the density of impurities or other defects of the crystal lattice, this technique is capable of both detecting and, if the cali- bration curve ∆ = f (x) is known, assessing them quantita- tively. Furthermore, the fact that the dispersion of the Brew- ster constant in the fundamental absorption range of Si is small makes it possible to detect a variation of the mechan- ical stress towards the crystal bulk by probing it with the light of a variable wavelength. We have considered here only one case of orientation of the incident wave and the deforming force. Studies of other orientations offer additional possibilities. In that case the photoelasticity effect manifested in the reflected beam in the fundamental absorption range will be characterized by a tensor whose components may have, in the vicinity of the critical points of the Brillouin zone, an especially strong dis- persion. References 1. D. K. Biegelsen, Photoelastic Tensor of Silicon and the Volume Dependence of the Average Gap // Phys. Rev. Lett. 32 (21), p.1196 (1974). 2. M. Grimsditch, M. Cardona, Absolute Cross-Section for Raman Scattering by Phonons in Silicon // Phys. Stat. Sol. 102 (1), p.155 (1980). 3. M. Chandrasekar, M. Grimsditch, M. Cardona, Piezobirefringence above the fundamental edge in Si // Phys. Rev. B 18 (8), p.4301 (1978). 4. B. K. Serdega. Sposob izmereniya dvoinogo lucheprelomleniya (A technique of birefringence measurements), Ukrainian Patent No. 19983A of 01.07.94 (in Russian). 5. A. Gerard, C. M. Burch. Vvedenie v matrichnuyu optiku (Intro- duction to matrix optics).- Moscow, Mir Publ., 1978 (Russian translation). 6. W. C. Dash, R. Newman, Intrinsic Optical Absorption in Single- Crystal Ge and Si at 77 K and 300 K // Phys. Rev. 99 (4), p.1151- 1155 (1955). 7. I. P. Zhad�ko, V. A. Romanov, B. K. Serdega // Pribory i tekhnika eksperimenta, No.5, pp.1151-1155 (1972) (in Russian). 8. C. W. Higgingbotham, M. Cardona, F. H. Pollak, Intrinsic Piezo- birefringence of Ge, Si, and GaAs // Phys. Rev. 184 (3), p.821-829 (1969).

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