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Resonant Brillouin scattering by acoustoelectrically amplified phonons in CdSe
I. Pfiys. Ch,
So/ids, 1976,Vol. 37, pp. 181-W.
PergamonPress. Printd in Gnat Britain
RESONANT BRILLOUIN SCATTERING BY ACOUSTOELECTRICALLY AMPLIFIED PHONONS IN Cc& K. YAM~OTO, K, ~ISAWA,~ H. SHI~IZU and K. ABE Dep~ment of Electronic Engineering, Faculty of En~n~~ng, Kobe University, Rokko, Nada, Kobe, Japan (Received 21 April 1975;accepted IS My 1975) Abstract-The dispersion of Brikouin scattering efficiency from the acoustoelectrically amplified phonon flux in CdSe has been measured in the range 1.37 to 1.65eV. The dispersion curve shows enhancement and cancellation similar to those of GaAs, CdS and 2110,and is in a goodagreementwith modifiedLoudontheoryincludingexciton effects. INTRODUCTION
Investigations of the resonant dispersion relation of Briliouin scattering efficiency from the acoustoeiectrically ~piifi~ phonon flux in GaAs[l], CdS [2-r] and ZnO[6] have recently been reported at optical wavelen~hs near the abso~tion edge. These results are caused mainIy by the piezobirefringence changing the sign near the absorption edge of the crystal, and ,many investigators have found resonant enhancement and cancellation of the Brillouin scattering cross-section near the fundamental absorption edge. The stress coefficient of the birefringence of CdSe was measured more recently by Reza and Babonas[7]. Their results showed that the sign of the piezobirefringence changes at l-56 eV. We report here a study of the effect of the resonant Brihouin scattering by the strong phonon flux in a CdSe single crystal in the t~nsp~ency region and near the fundament~ abso~tion edge. This method also gives useful information about the dispersion of the photoelastic constant of the crystal.
dispersion of the refractive index of CdSe[8] using the relations give by DixsonD].
Figure 1 shows the photon energy dependence of B~liouin scattering efficiency for the on-axis @8GHz phonon. The ratio of scattered light intensity to the transmitted light intensity, L/Z,, is proportional to the square of the photoelastic constant {PM+ (PN)ind},where Ph( is Pockels photoelastic constant and (P.&, is the indirect photoelastic constant given by (PM)= - e15rJI/eII[a], where eu, r5! and ll1 are the piezoelectric, Pockels electro-optics and dielectric constants, respectively. Since rst of CdSe is unknown, we cannot estimate the value of (Pnq)M.In the case of Cd& the value of (PM)bdis 18%[6] of the Pockels phot~lastic cons~t, By analogy with the results for CdS, the value of (PM& in CdSe may be assumed to be smali, so that the change in Z,i& for the photon energy originates mainly from P,.
EXPERIMENTALMETHOD
experimental arrangement was essentially similar to the one described in[l]. The phonon flux was produced by applying a short pulse of high voltage to electrodes soldered at the ends of a CdSe semiconducting sample, which was oriented perpendicular to the c axis. The incident light was obtained from a monochrometer illuminated by a Xenon lamp. The diffracted hght signals were detected by a photomuItiplier with a S-l photocathode, and led directly through a Keithley wideband amplifier into a FAR boxcar integrator to improve the signal-to-noise ratio. The output signal was displayed on a X-Y recorder. The incident light polarization was parallel to the c-axis of the CdSe sample. For each value of the wavelength of the incident light, the incident and scattering angles were calculated taking into account the The
bond
1.4 PHOTON
@resent address: Department of Electronics, Faculty of En~n~ering, Osaka diversity, Yamad~ami, Suita, Osaka, Japan.
to bond
1.5 ENERGY
1.6 (av)
Fii. 1. Dispersion curves of Brillouin scattering cross-section for 04GHz acoustic phonons. Solid curves were calculated by a modified Loudon theory taking into account the exe&on effects. Dotted curves were estimated by unm~ified Loudon theory.
181
182
K. YAMAMOTOetal.
The resonant contribution in the dispersion curve becomes significant when the Brillouin scattering efficiency commences its drop towards the minimum at a photon energy near 1.37eV. Close to the intrinsic absorption edge, the dispersion curve has a deep minimum at 1.53 eV, similar to the resonant cancellation observed in GaAs, CdS and ZnO. This value gives fairly good agreement with the previous experimentabresult [7]. The simplest and obvious interpretation of the deep minimum is that the scattering amplitude, represented by the photoelastic constant P,, passes through zero while undergoing a reversal of sign. At still higher energies, the scattering efficiency increases rapidly compared to the resonant enhancement. These features are explained in terms of resonant enhancement and cancellation as proposed in the case of Raman scattering[lO, 111.Such an analysis is based on the result of Loudon’s theory of light scattering[l2]. A relatively simple expression[2,4,13] is obtained for the spherical and parabolic band cases. Since the band structure of CdSe is similar to that of CdS, we apply the result for CdS [4] to the analysis of the present data. In this experimental configuration, dipole transitions between the conduction band and the A valence band are forbidden, but are allowed between the conduction band and the B, C valence bands. The values of the matrix elements of the deformation potential, 8, are &a f 0, 8,, f 0, B. f 0 and 8,, = &a = %c = 0, respectively[4]. We can ignore the contribution from the C band because of the large energy separation. Therefore the incident photons excite virtual electrons in the conduction band and virtual holes in the B valence band (or B excitons). The holes in the B valence band are scattered to the A valence band via the deformation potential matrix element &a, and subsequently the holes in the A valence band recombine with excited electrons in the conduction band to emit photons of polarization perpendicular to the c-axis. This mechanism suggests that we may use the optical energy gaps EgB = 1.76 eV for the incident light and EC+,= 1.73 eV for the scattered light, where these energies are estimated from the electroreflectance peaks in CdSe[l4], assuming the exciton binding energy R = 15.7 meV [IS]. Theoretical calculations of the scattering intensities were carried out by assuming that the intensity is proportional to lRi, - Rol, where R,, is the scattering amplitude of the resonant term associated with the band gaps EXA and EB6, and (-Ro) is the non-resonant term arising from more distant critical points in the band structure[4]. The dotted curve was calculated using Loudon’s expression for &, while the solid curve was obtained using modified Loudon theory by taking into account the exciton effect[4,16]. The modified Loudon theory is shown as follows[4];
where 00 is the photon angular frequency and p is the reduced mass. a0 = eh*/pe’ is the Bohr radius of the exciton respecting and R = pe4/2h2e2 the Rydberg constant of the exciton respecting. M’ also contains the matrix element of the deformation potential momentum according to the above description. We find better agreement between experiment and the modified Loudon theory including exciton effects than between the experiment and modified theory. By similar arguments, past studies of piezoreflection[l7] and thermoreflection [ 181in CdSe have shown that the polarization dependences of the exciton structures satisfy a quasicubic model for the structure of the valence bands. CONCLUSION
The resonant Brillouin scattering in a CdSe single crystal has been observed for the first time, and the results are in better agreement with modified Loudon theory than with unmodified theory. Near the absorption edge of CdSe exciton transition are important. The theory used here also accounts for the spectral character dependence on the photoelastic constant, PU, which is qualitatively reminiscent of that observed for II-VI and III-V semiconductors. Moreover, our result for the deep minimum in CdSe gives fairly good agreement with more recently experimental results obtained by Reza et al. [7]. Acknowledgements-The authors are most grateful to Dr. C. Hamaguchi and Mr. K. Ando for many helpful suggestions and contributions to the theoretical analysis. REFERENCES 1. Garrod D. K. and Bray R., Phys. Reu. B6, 1314(1972). 2. Gelbat U. and Many A., Phys..Left. 43A, 329 (1973). 3. Yamada M.. Ando K.. Hamaeuchi C. and Nakai J.. J. Phvs. . Sot. Japan’34, 1969(1973). 4. Ando K. and Hamaguchi C., Phys. Reo. 871 3876 (1975). 5. Berkowitz R. and Price D. H. R., Solid State Comm. 14, 195 (1974). 6. Hamaguchi C., Ando K., San’ya M. and Yamada M., Symp. on Microwaoe Acoustics, Lancaster (1974)To be published. 7. Reza A. A. and Babonas G. A., Sou. Phys. Solid Stale 16,909 (1974). 8. Parison R. B., Wardzynski W. and Yoffe A. D., Proc. Roy. Sot. 262A, 120 (1961). 9. Dixson R. W., I.E.E.E., J. Quantum Electronics QE-3, 85 (1%7). 10. Raiston J. M., Wadsack R. L. and Chang R. K., Phys. Rev. Lett. 25, 814 (1970). 11. Loudon R., Proc. Roy. Sot. 275A, 218 (1963). 12. Dament C. and Scott .I.F., SolidState Comm. 9,383 (1971). 13. Pine A. S., Phys. Rev. B5, 3003 (1972). 14. Cardona M.. Shaklee K. C. and Pollak F. H.. Phvs. . Rev. 54, 154 (1961). 15. Blossey D. F., Phys. Reu. B3, 1382(1971). 16. GangulyA. K. and Birman J. L., Phys. Rev. 162,806(1967). 17. Galvini A. and Cardona M., Phys. Rev. Bl, 672 (1970). 18. IlievM. andBalevaM., Phys. StatusSolidi (b)47, K87(1971).
So/ids, 1976,Vol. 37, pp. 181-W.
PergamonPress. Printd in Gnat Britain
RESONANT BRILLOUIN SCATTERING BY ACOUSTOELECTRICALLY AMPLIFIED PHONONS IN Cc& K. YAM~OTO, K, ~ISAWA,~ H. SHI~IZU and K. ABE Dep~ment of Electronic Engineering, Faculty of En~n~~ng, Kobe University, Rokko, Nada, Kobe, Japan (Received 21 April 1975;accepted IS My 1975) Abstract-The dispersion of Brikouin scattering efficiency from the acoustoelectrically amplified phonon flux in CdSe has been measured in the range 1.37 to 1.65eV. The dispersion curve shows enhancement and cancellation similar to those of GaAs, CdS and 2110,and is in a goodagreementwith modifiedLoudontheoryincludingexciton effects. INTRODUCTION
Investigations of the resonant dispersion relation of Briliouin scattering efficiency from the acoustoeiectrically ~piifi~ phonon flux in GaAs[l], CdS [2-r] and ZnO[6] have recently been reported at optical wavelen~hs near the abso~tion edge. These results are caused mainIy by the piezobirefringence changing the sign near the absorption edge of the crystal, and ,many investigators have found resonant enhancement and cancellation of the Brillouin scattering cross-section near the fundamental absorption edge. The stress coefficient of the birefringence of CdSe was measured more recently by Reza and Babonas[7]. Their results showed that the sign of the piezobirefringence changes at l-56 eV. We report here a study of the effect of the resonant Brihouin scattering by the strong phonon flux in a CdSe single crystal in the t~nsp~ency region and near the fundament~ abso~tion edge. This method also gives useful information about the dispersion of the photoelastic constant of the crystal.
dispersion of the refractive index of CdSe[8] using the relations give by DixsonD].
Figure 1 shows the photon energy dependence of B~liouin scattering efficiency for the on-axis @8GHz phonon. The ratio of scattered light intensity to the transmitted light intensity, L/Z,, is proportional to the square of the photoelastic constant {PM+ (PN)ind},where Ph( is Pockels photoelastic constant and (P.&, is the indirect photoelastic constant given by (PM)= - e15rJI/eII[a], where eu, r5! and ll1 are the piezoelectric, Pockels electro-optics and dielectric constants, respectively. Since rst of CdSe is unknown, we cannot estimate the value of (Pnq)M.In the case of Cd& the value of (PM)bdis 18%[6] of the Pockels phot~lastic cons~t, By analogy with the results for CdS, the value of (PM& in CdSe may be assumed to be smali, so that the change in Z,i& for the photon energy originates mainly from P,.
EXPERIMENTALMETHOD
experimental arrangement was essentially similar to the one described in[l]. The phonon flux was produced by applying a short pulse of high voltage to electrodes soldered at the ends of a CdSe semiconducting sample, which was oriented perpendicular to the c axis. The incident light was obtained from a monochrometer illuminated by a Xenon lamp. The diffracted hght signals were detected by a photomuItiplier with a S-l photocathode, and led directly through a Keithley wideband amplifier into a FAR boxcar integrator to improve the signal-to-noise ratio. The output signal was displayed on a X-Y recorder. The incident light polarization was parallel to the c-axis of the CdSe sample. For each value of the wavelength of the incident light, the incident and scattering angles were calculated taking into account the The
bond
1.4 PHOTON
@resent address: Department of Electronics, Faculty of En~n~ering, Osaka diversity, Yamad~ami, Suita, Osaka, Japan.
to bond
1.5 ENERGY
1.6 (av)
Fii. 1. Dispersion curves of Brillouin scattering cross-section for 04GHz acoustic phonons. Solid curves were calculated by a modified Loudon theory taking into account the exe&on effects. Dotted curves were estimated by unm~ified Loudon theory.
181
182
K. YAMAMOTOetal.
The resonant contribution in the dispersion curve becomes significant when the Brillouin scattering efficiency commences its drop towards the minimum at a photon energy near 1.37eV. Close to the intrinsic absorption edge, the dispersion curve has a deep minimum at 1.53 eV, similar to the resonant cancellation observed in GaAs, CdS and ZnO. This value gives fairly good agreement with the previous experimentabresult [7]. The simplest and obvious interpretation of the deep minimum is that the scattering amplitude, represented by the photoelastic constant P,, passes through zero while undergoing a reversal of sign. At still higher energies, the scattering efficiency increases rapidly compared to the resonant enhancement. These features are explained in terms of resonant enhancement and cancellation as proposed in the case of Raman scattering[lO, 111.Such an analysis is based on the result of Loudon’s theory of light scattering[l2]. A relatively simple expression[2,4,13] is obtained for the spherical and parabolic band cases. Since the band structure of CdSe is similar to that of CdS, we apply the result for CdS [4] to the analysis of the present data. In this experimental configuration, dipole transitions between the conduction band and the A valence band are forbidden, but are allowed between the conduction band and the B, C valence bands. The values of the matrix elements of the deformation potential, 8, are &a f 0, 8,, f 0, B. f 0 and 8,, = &a = %c = 0, respectively[4]. We can ignore the contribution from the C band because of the large energy separation. Therefore the incident photons excite virtual electrons in the conduction band and virtual holes in the B valence band (or B excitons). The holes in the B valence band are scattered to the A valence band via the deformation potential matrix element &a, and subsequently the holes in the A valence band recombine with excited electrons in the conduction band to emit photons of polarization perpendicular to the c-axis. This mechanism suggests that we may use the optical energy gaps EgB = 1.76 eV for the incident light and EC+,= 1.73 eV for the scattered light, where these energies are estimated from the electroreflectance peaks in CdSe[l4], assuming the exciton binding energy R = 15.7 meV [IS]. Theoretical calculations of the scattering intensities were carried out by assuming that the intensity is proportional to lRi, - Rol, where R,, is the scattering amplitude of the resonant term associated with the band gaps EXA and EB6, and (-Ro) is the non-resonant term arising from more distant critical points in the band structure[4]. The dotted curve was calculated using Loudon’s expression for &, while the solid curve was obtained using modified Loudon theory by taking into account the exciton effect[4,16]. The modified Loudon theory is shown as follows[4];
where 00 is the photon angular frequency and p is the reduced mass. a0 = eh*/pe’ is the Bohr radius of the exciton respecting and R = pe4/2h2e2 the Rydberg constant of the exciton respecting. M’ also contains the matrix element of the deformation potential momentum according to the above description. We find better agreement between experiment and the modified Loudon theory including exciton effects than between the experiment and modified theory. By similar arguments, past studies of piezoreflection[l7] and thermoreflection [ 181in CdSe have shown that the polarization dependences of the exciton structures satisfy a quasicubic model for the structure of the valence bands. CONCLUSION
The resonant Brillouin scattering in a CdSe single crystal has been observed for the first time, and the results are in better agreement with modified Loudon theory than with unmodified theory. Near the absorption edge of CdSe exciton transition are important. The theory used here also accounts for the spectral character dependence on the photoelastic constant, PU, which is qualitatively reminiscent of that observed for II-VI and III-V semiconductors. Moreover, our result for the deep minimum in CdSe gives fairly good agreement with more recently experimental results obtained by Reza et al. [7]. Acknowledgements-The authors are most grateful to Dr. C. Hamaguchi and Mr. K. Ando for many helpful suggestions and contributions to the theoretical analysis. REFERENCES 1. Garrod D. K. and Bray R., Phys. Reu. B6, 1314(1972). 2. Gelbat U. and Many A., Phys..Left. 43A, 329 (1973). 3. Yamada M.. Ando K.. Hamaeuchi C. and Nakai J.. J. Phvs. . Sot. Japan’34, 1969(1973). 4. Ando K. and Hamaguchi C., Phys. Reo. 871 3876 (1975). 5. Berkowitz R. and Price D. H. R., Solid State Comm. 14, 195 (1974). 6. Hamaguchi C., Ando K., San’ya M. and Yamada M., Symp. on Microwaoe Acoustics, Lancaster (1974)To be published. 7. Reza A. A. and Babonas G. A., Sou. Phys. Solid Stale 16,909 (1974). 8. Parison R. B., Wardzynski W. and Yoffe A. D., Proc. Roy. Sot. 262A, 120 (1961). 9. Dixson R. W., I.E.E.E., J. Quantum Electronics QE-3, 85 (1%7). 10. Raiston J. M., Wadsack R. L. and Chang R. K., Phys. Rev. Lett. 25, 814 (1970). 11. Loudon R., Proc. Roy. Sot. 275A, 218 (1963). 12. Dament C. and Scott .I.F., SolidState Comm. 9,383 (1971). 13. Pine A. S., Phys. Rev. B5, 3003 (1972). 14. Cardona M.. Shaklee K. C. and Pollak F. H.. Phvs. . Rev. 54, 154 (1961). 15. Blossey D. F., Phys. Reu. B3, 1382(1971). 16. GangulyA. K. and Birman J. L., Phys. Rev. 162,806(1967). 17. Galvini A. and Cardona M., Phys. Rev. Bl, 672 (1970). 18. IlievM. andBalevaM., Phys. StatusSolidi (b)47, K87(1971).