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Inelastic light scattering on coupled plasmon-LO phonon modes in high magnetic fields
Physica B 298 (2001) 216}220
Inelastic light scattering on coupled plasmon-LO phonon modes in high magnetic "elds A. Wysmolek , M. Potemski *, T. Slupinski Grenoble High Magnetic Field Laboratory, MPI/FKF-CNRS, 25 avenue des Martyrs, BP 166X, F-38042 Grenoble Cedex 9, France Institute of Experimental Physics, Warsaw University, Hoza 69, PL-00-681 Warszawa, Poland
Abstract We report on the results of high-magnetic-"eld (up to 28 T) inelastic light scattering measurements of coupled plasmon}phonon modes in metallic GaAs samples with electron concentrations ranging from 10 to 2;10 cm\. The zero-"eld Raman spectra of the investigated samples show peaks related to the transverse optical phonon (TO) as well as the coupled plasmon}phonon (below TO frequency) and (above LO frequency) modes. In the backscattering \ > Voigt geometry (kNB) in which the modes propagating across the "eld direction are probed, the magnetic "eld does not in#uence the TO resonance but leads to a splitting of both and branches. The experimental data are > \ approximated using the dielectric function formalism. For a highly doped sample, the interaction between coupled plasmon}phonon modes with the Bernstein mode is observed. 2001 Elsevier Science B.V. All rights reserved. Keywords: Raman scattering; Plasmon-LO phonon modes; Berstein modes; Dielectric function
1. Introduction Wave propagation in solids has been intensively studied in the 1960s with signi"cant interest focused on doped polar semiconductors in which the nowadays text-book e!ect of coupling between plasmons and optical phonon modes can be clearly observed as demonstrated, for example, by the pioneering work of Mooradian and Wright [1] on light scattering in n-type GaAs samples. The dielectric function theory applied to coupled electron} phonon excitations has interesting consequences in
* Corresponding author. Tel.: #33-476-88-7876; fax: #33476-85-5610. E-mail address: [email protected] (M. Potemski).
the limit of high magnetic "elds when the cyclotron frequency is tuned over all other undressed excitations present in the crystal [2]. We report, to our knowledge for the "rst time, on the results of high-magnetic-"eld (up to 28 T) inelastic light scattering measurements of coupled plasmon}phonon modes in metallic n-GaAs samples.
2. Experimental details The Raman scattering experiments have been performed on bulk n-type Czochralski grown GaAs samples with room temperature concentration in the range from 10 up to 2;10 cm\. A tunable Ti : sapphire laser operating in the range between
0921-4526/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 1 ) 0 0 3 0 5 - 2
A. Wysmolek et al. / Physica B 298 (2001) 216}220
820 and 860 nm, typically with 100 mW power have been used for the below band-gap excitation. A specially designed optical "bre system has been employed for the measurements in the magnetic "eld up to 28 T, supplied by a resistive magnet. The Raman spectra have been measured at liquid helium temperatures, in four di!erent con"gurations of the exciting and scattered light wavevectors with respect to the magnetic "eld direction.
3. Results and discussion The zero-"eld Raman scattering spectra, for three samples with di!erent electron concentrations, are presented in Fig. 1. As can be seen in this "gure, each sample shows a sharp peak associated with the transverse optical phonon mode (TO) observed at the energy of 33.7 meV characteristic for undoped material. None of the spectrum displays a Raman signal related to the longitudinal optical phonon mode (LO) of undoped GaAs. Instead, this mode, in our heavily doped samples, is replaced by coupled plasmon}phonon modes observed be\ low the TO frequency and observed above the > LO frequency. As expected, and frequencies \ >
Fig. 1. Raman scattering spectra of n-GaAs samples with di!erent electron concentrations, measured at ¹"4.2 K.
217
depend on electron concentration, n, or in other words on the plasma frequency "(4ne/ N mH) (where is the high-frequency dielectric constant and mH denotes the electron e!ective mass). Each and mode is characterised by \ > its own dispersion relation but in the limit of long wavevectors, k&0, the corresponding frequencies are given by the following formula [3]: 2 " #$[( #)!4 ] . ! N J N J N R Here, and are, respectively, the LO and TO J R phonon frequencies of an undoped material. The plasmon- or phonon-like character of the coupled modes changes with electron concentration. can be considered as a plasmon-like and as \ > a phonon-like mode at low electron concentrations, when ; . In contrast, is phonon-like N J \ wheras is plasmon-like when < , i.e., in > N J the high concentration limit. The plasmon- or phonon-like character of coupled modes can be clearly seen in the spectral broadening of the associated Raman scattering signals (see Fig. 1): phonon-like peaks are much narrower when compared to plasmon-like ones. The application of a magnetic "eld modi"es the measured Raman response, although the e!ect is remarkably dependent on the con"guration of the incident and scattered wavevectors (correspondingly, k and k ) with respect to the magnetic "eld (B) direction. In Fig. 2, the spectra measured at 28 T for three representative (k , k )-B con"gurations are shown and compared with the zero-"eld spectrum, for the sample with a high electron concentration. Except the e!ect of "eld-dependent spectral broadening, the coupled plasmon}phonon modes are little in#uenced by the magnetic "eld applied in the backscattering Faraday con"guration. This is to be expected for scattering experiments as they predominantly probe the longitudinal excitations which remain unchanged when propagating along the magnetic "eld. In contrast, the scattering signal from coupled plasmon}phonon modes shows more pronounced changes, both in spectral position and broadening, for the other (k , k )-B con"gurations which are illustrated schematically in Fig. 2. After showing in Fig. 2 the potential richness of scattering experiments on plasmon}phonon modes in magnetic "elds, we concentrate in what follows on
218
A. Wysmolek et al. / Physica B 298 (2001) 216}220
Fig. 2. Raman scattering spectra of coupled plasmon-LO phonon modes for the sample with n"1;10 cm\, measured for di!erent con"gurations (displayed schematically in the "gure) of the incident and scattered light wavevectors with respect to the magnetic "eld direction.
the results obtained for the backscattering Voigt con"guration. This con"guration favours the observation of "eld-sensitive modes propagating across the magnetic "eld direction. The experimental data obtained for two samples, with electron concentrations of 1.3;10 cm\ (sample (a)) and 2;10 cm\ (sample (b)) are considered. The Raman scattering spectra measured for these samples in the range of magnetic "eld up to 28 T are shown in Fig. 3. The corresponding diagrams of peak positions versus the magnetic "eld are presented in Fig. 4. Roughly speaking, the magnetic "eld behaviour of Raman scattering spectra for the case of sample (a) with high electron concentration can be understood in terms of the observation of hybrid cyclotron}plasmon}phonon modes shown with solid lines in Fig. 4 according to the following formula [4]: 2(! )"# A J $[(#)!4(#)] , A J A J N R
Fig. 3. The evolution of Raman scattering spectra measured for the backscattering Voigt con"guration in magnetic "elds up to 28 T for samples with n"1.3;10 cm\ (a) and n"2;10 cm\ (b). Spectra shown in (b) were measured with 2 T "eld step.
A. Wysmolek et al. / Physica B 298 (2001) 216}220
219
where "# and is the cyclotron freA A N ! quency. These modes can be derived, when searching for singularities of the dielectric function () given by [5,6]: 2 ()" P J # P J in the Voigt con"guration considered, where
Fig. 4. Magnetic "eld dependence of the plasmon-LO phonon modes for the sample with (a) n"1.3;10 cm\ and (b) 2;10 cm\. Di!erent symbols denote the data obtained using di!erent excitation energies. Dashed lines show single- and double- cyclotron energy. Solid and dotted lines represent the evolution of collective modes according to the predictions of the dielectric function formalism (see text).
(1)
! R ! N " 1# J P J ! ($ ) R A represent dielectric functions for the right and left circularly polarised waves. As seen in Figs. 3a and 4a, \ and > shift towards higher energies with increasing magnetic "eld. \ crosses the TO mode and eventually approaches the energy of the undressed LO phonon of undoped material. In the high "eld limit, \ is twice as narrow as at zero magnetic "eld which indicates the increase of its (LO) phonon-like character at high magnetic "elds. More subtle effects observed for sample (a) are the "eld dependent broadening and/or splitting of each \ and > mode. This can already be seen in the row data presented in Fig. 3a. The \ peak is relatively narrow in both limits of low and high "elds, but show appreciable broadening around 10}12 T. One may even presume that at these "elds, \ shows a rather nonmonotonic "eld dependence indicating some anticrossing e!ect. The anticrossing behaviour is more clearly pronounced for the > mode, but occurs at higher magnetic "elds, around &16 T. As illustrated in Fig. 4a, we think this anticrossing is a result of interaction between collective modes and single particle excitations (Bernstein modes) with frequency of 2 [7]. Since ! > mode shows more pronounced anticrossing than the \ , we deduce that Bernstein modes predominantly couple with plasmon-like and not with phonon-like modes. The consideration of only hybrid cyclotron} plasmon}phonon modes is not su$cient to interpret the magnetic "eld evolution of the Raman scattering spectra measured for sample (b) with relatively small electron concentration. This is particularly the case when investigating the behaviour of the upper branch of the plasmon}phonon
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A. Wysmolek et al. / Physica B 298 (2001) 216}220
peak. As can be seen in the raw data presented in Fig. 3b, the upper branch of the plasmon}phonon peak splits, in this sample, into two components at "elds around 15 T. The remaining component of this upper branch, which persists at high magnetic "elds, is a sharp peak approaching the LO phonon of undoped material from the high energy side. On the other hand the magnetic "eld evolution of the lower plasmon}phonon peak is quite similar to the case of sample (b) with higher electron concentration. This latter peak can be easily investigated at high magnetic "elds, when it crosses the TO line, shows narrowing and approaches the LO frequency from the low energy side. This peak is however more di$cult to investigate at low magnetic "elds, when it is much broader and in addition overlaps with a relatively strong luminescence signal. Nevertheless, employing di!erent excitation conditions we were able to trace the evolution of this peak over the whole range of magnetic "eld considered and this is shown in Fig. 4b. The solid line is the calculated "eld dependence of the \ mode and describes relatively well the ob served "eld evolution of lower plasmon}phonon peak in analogy to the case of sample (b). The signal associated with the > mode is probably also observed in the low concentration sample but only at low magnetic "eld, whereas it disappears from the spectrum at high "elds showing appreciable broadening. To account for the sharp peak which clearly develops with the magnetic "eld and originates from the upper plasmon}phonon branch, we have taken into consideration not only singularities but also zeros of the dielectric function given by Eq. (1). The energy positions corresponding to zeros of the dielectric function calculated for sample (b) are shown in Fig. 4b with dotted lines. As can be seen in this "gure, a possible candidate which may account for the observation of the sharp high "eld component of the upper phonon}plasmon branch is the mode which we called \ derived from one of > possible zeros of the dielectric function. Neverthe-
less, further work on the selection rules is needed to clarify the reason why only certain speci"c modes associated with zeros of the dielectric function are observed in light scattering experiments.
4. Conclusions Raman scattering experiments on heavily doped GaAs samples show the development of hybrid cyclotron}plasmon}phonon modes as well as certain other collective excitations related to zeros of the dielectric function. In the high "eld limit, the collective modes which resemble longitudinal optical phonon excitations propagating across the "eld direction have been observed. The interaction of coupled plasmon}phonon modes with Bernstein modes have also been observed. This interaction is stronger if it involves collective modes of plasmonlike character.
Acknowledgements A. Wysmolek gratefully acknowledges "nancial support from the Alexander von Humboldt Foundation.
References [1] A. Mooradian, G.B. Wright, Phys. Rev. Lett. 16 (1966) 999. [2] B. Lax, Proceedings of the International Conference on the Physics of Semiconductors, Kyoto, 1966, J. Phys. Soc. Japan 21 (1966) 165. [3] B.B. Varga, Phys. Rev. 137 (1965) A1896. [4] R. Kaplan, E.D. Palik, R.F. Wallis, S. Iwasa, E. Burstein, Y. Sawada, Phys. Rev. Lett. 18 (1967) 159. [5] S. Ivasa, in: E.D. Haidemenakis (Ed.), Physics of Solids in Intense Magnetic Fields, Plenum Press, New York, 1969, p. 126}135. [6] I. Yokota, Proceedings of the International Conference on the Physics of Semiconductors, Kyoto, 1966, J. Phys. Soc. Japan 21 (1966) 738. [7] C.K. Patel, R.E. Slusher, Phys. Rev. Lett. 21 (1968) 1563.
Inelastic light scattering on coupled plasmon-LO phonon modes in high magnetic "elds A. Wysmolek , M. Potemski *, T. Slupinski Grenoble High Magnetic Field Laboratory, MPI/FKF-CNRS, 25 avenue des Martyrs, BP 166X, F-38042 Grenoble Cedex 9, France Institute of Experimental Physics, Warsaw University, Hoza 69, PL-00-681 Warszawa, Poland
Abstract We report on the results of high-magnetic-"eld (up to 28 T) inelastic light scattering measurements of coupled plasmon}phonon modes in metallic GaAs samples with electron concentrations ranging from 10 to 2;10 cm\. The zero-"eld Raman spectra of the investigated samples show peaks related to the transverse optical phonon (TO) as well as the coupled plasmon}phonon (below TO frequency) and (above LO frequency) modes. In the backscattering \ > Voigt geometry (kNB) in which the modes propagating across the "eld direction are probed, the magnetic "eld does not in#uence the TO resonance but leads to a splitting of both and branches. The experimental data are > \ approximated using the dielectric function formalism. For a highly doped sample, the interaction between coupled plasmon}phonon modes with the Bernstein mode is observed. 2001 Elsevier Science B.V. All rights reserved. Keywords: Raman scattering; Plasmon-LO phonon modes; Berstein modes; Dielectric function
1. Introduction Wave propagation in solids has been intensively studied in the 1960s with signi"cant interest focused on doped polar semiconductors in which the nowadays text-book e!ect of coupling between plasmons and optical phonon modes can be clearly observed as demonstrated, for example, by the pioneering work of Mooradian and Wright [1] on light scattering in n-type GaAs samples. The dielectric function theory applied to coupled electron} phonon excitations has interesting consequences in
* Corresponding author. Tel.: #33-476-88-7876; fax: #33476-85-5610. E-mail address: [email protected] (M. Potemski).
the limit of high magnetic "elds when the cyclotron frequency is tuned over all other undressed excitations present in the crystal [2]. We report, to our knowledge for the "rst time, on the results of high-magnetic-"eld (up to 28 T) inelastic light scattering measurements of coupled plasmon}phonon modes in metallic n-GaAs samples.
2. Experimental details The Raman scattering experiments have been performed on bulk n-type Czochralski grown GaAs samples with room temperature concentration in the range from 10 up to 2;10 cm\. A tunable Ti : sapphire laser operating in the range between
0921-4526/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 1 ) 0 0 3 0 5 - 2
A. Wysmolek et al. / Physica B 298 (2001) 216}220
820 and 860 nm, typically with 100 mW power have been used for the below band-gap excitation. A specially designed optical "bre system has been employed for the measurements in the magnetic "eld up to 28 T, supplied by a resistive magnet. The Raman spectra have been measured at liquid helium temperatures, in four di!erent con"gurations of the exciting and scattered light wavevectors with respect to the magnetic "eld direction.
3. Results and discussion The zero-"eld Raman scattering spectra, for three samples with di!erent electron concentrations, are presented in Fig. 1. As can be seen in this "gure, each sample shows a sharp peak associated with the transverse optical phonon mode (TO) observed at the energy of 33.7 meV characteristic for undoped material. None of the spectrum displays a Raman signal related to the longitudinal optical phonon mode (LO) of undoped GaAs. Instead, this mode, in our heavily doped samples, is replaced by coupled plasmon}phonon modes observed be\ low the TO frequency and observed above the > LO frequency. As expected, and frequencies \ >
Fig. 1. Raman scattering spectra of n-GaAs samples with di!erent electron concentrations, measured at ¹"4.2 K.
217
depend on electron concentration, n, or in other words on the plasma frequency "(4ne/ N mH) (where is the high-frequency dielectric constant and mH denotes the electron e!ective mass). Each and mode is characterised by \ > its own dispersion relation but in the limit of long wavevectors, k&0, the corresponding frequencies are given by the following formula [3]: 2 " #$[( #)!4 ] . ! N J N J N R Here, and are, respectively, the LO and TO J R phonon frequencies of an undoped material. The plasmon- or phonon-like character of the coupled modes changes with electron concentration. can be considered as a plasmon-like and as \ > a phonon-like mode at low electron concentrations, when ; . In contrast, is phonon-like N J \ wheras is plasmon-like when < , i.e., in > N J the high concentration limit. The plasmon- or phonon-like character of coupled modes can be clearly seen in the spectral broadening of the associated Raman scattering signals (see Fig. 1): phonon-like peaks are much narrower when compared to plasmon-like ones. The application of a magnetic "eld modi"es the measured Raman response, although the e!ect is remarkably dependent on the con"guration of the incident and scattered wavevectors (correspondingly, k and k ) with respect to the magnetic "eld (B) direction. In Fig. 2, the spectra measured at 28 T for three representative (k , k )-B con"gurations are shown and compared with the zero-"eld spectrum, for the sample with a high electron concentration. Except the e!ect of "eld-dependent spectral broadening, the coupled plasmon}phonon modes are little in#uenced by the magnetic "eld applied in the backscattering Faraday con"guration. This is to be expected for scattering experiments as they predominantly probe the longitudinal excitations which remain unchanged when propagating along the magnetic "eld. In contrast, the scattering signal from coupled plasmon}phonon modes shows more pronounced changes, both in spectral position and broadening, for the other (k , k )-B con"gurations which are illustrated schematically in Fig. 2. After showing in Fig. 2 the potential richness of scattering experiments on plasmon}phonon modes in magnetic "elds, we concentrate in what follows on
218
A. Wysmolek et al. / Physica B 298 (2001) 216}220
Fig. 2. Raman scattering spectra of coupled plasmon-LO phonon modes for the sample with n"1;10 cm\, measured for di!erent con"gurations (displayed schematically in the "gure) of the incident and scattered light wavevectors with respect to the magnetic "eld direction.
the results obtained for the backscattering Voigt con"guration. This con"guration favours the observation of "eld-sensitive modes propagating across the magnetic "eld direction. The experimental data obtained for two samples, with electron concentrations of 1.3;10 cm\ (sample (a)) and 2;10 cm\ (sample (b)) are considered. The Raman scattering spectra measured for these samples in the range of magnetic "eld up to 28 T are shown in Fig. 3. The corresponding diagrams of peak positions versus the magnetic "eld are presented in Fig. 4. Roughly speaking, the magnetic "eld behaviour of Raman scattering spectra for the case of sample (a) with high electron concentration can be understood in terms of the observation of hybrid cyclotron}plasmon}phonon modes shown with solid lines in Fig. 4 according to the following formula [4]: 2(! )"# A J $[(#)!4(#)] , A J A J N R
Fig. 3. The evolution of Raman scattering spectra measured for the backscattering Voigt con"guration in magnetic "elds up to 28 T for samples with n"1.3;10 cm\ (a) and n"2;10 cm\ (b). Spectra shown in (b) were measured with 2 T "eld step.
A. Wysmolek et al. / Physica B 298 (2001) 216}220
219
where "# and is the cyclotron freA A N ! quency. These modes can be derived, when searching for singularities of the dielectric function () given by [5,6]: 2 ()" P J # P J in the Voigt con"guration considered, where
Fig. 4. Magnetic "eld dependence of the plasmon-LO phonon modes for the sample with (a) n"1.3;10 cm\ and (b) 2;10 cm\. Di!erent symbols denote the data obtained using di!erent excitation energies. Dashed lines show single- and double- cyclotron energy. Solid and dotted lines represent the evolution of collective modes according to the predictions of the dielectric function formalism (see text).
(1)
! R ! N " 1# J P J ! ($ ) R A represent dielectric functions for the right and left circularly polarised waves. As seen in Figs. 3a and 4a, \ and > shift towards higher energies with increasing magnetic "eld. \ crosses the TO mode and eventually approaches the energy of the undressed LO phonon of undoped material. In the high "eld limit, \ is twice as narrow as at zero magnetic "eld which indicates the increase of its (LO) phonon-like character at high magnetic "elds. More subtle effects observed for sample (a) are the "eld dependent broadening and/or splitting of each \ and > mode. This can already be seen in the row data presented in Fig. 3a. The \ peak is relatively narrow in both limits of low and high "elds, but show appreciable broadening around 10}12 T. One may even presume that at these "elds, \ shows a rather nonmonotonic "eld dependence indicating some anticrossing e!ect. The anticrossing behaviour is more clearly pronounced for the > mode, but occurs at higher magnetic "elds, around &16 T. As illustrated in Fig. 4a, we think this anticrossing is a result of interaction between collective modes and single particle excitations (Bernstein modes) with frequency of 2 [7]. Since ! > mode shows more pronounced anticrossing than the \ , we deduce that Bernstein modes predominantly couple with plasmon-like and not with phonon-like modes. The consideration of only hybrid cyclotron} plasmon}phonon modes is not su$cient to interpret the magnetic "eld evolution of the Raman scattering spectra measured for sample (b) with relatively small electron concentration. This is particularly the case when investigating the behaviour of the upper branch of the plasmon}phonon
220
A. Wysmolek et al. / Physica B 298 (2001) 216}220
peak. As can be seen in the raw data presented in Fig. 3b, the upper branch of the plasmon}phonon peak splits, in this sample, into two components at "elds around 15 T. The remaining component of this upper branch, which persists at high magnetic "elds, is a sharp peak approaching the LO phonon of undoped material from the high energy side. On the other hand the magnetic "eld evolution of the lower plasmon}phonon peak is quite similar to the case of sample (b) with higher electron concentration. This latter peak can be easily investigated at high magnetic "elds, when it crosses the TO line, shows narrowing and approaches the LO frequency from the low energy side. This peak is however more di$cult to investigate at low magnetic "elds, when it is much broader and in addition overlaps with a relatively strong luminescence signal. Nevertheless, employing di!erent excitation conditions we were able to trace the evolution of this peak over the whole range of magnetic "eld considered and this is shown in Fig. 4b. The solid line is the calculated "eld dependence of the \ mode and describes relatively well the ob served "eld evolution of lower plasmon}phonon peak in analogy to the case of sample (b). The signal associated with the > mode is probably also observed in the low concentration sample but only at low magnetic "eld, whereas it disappears from the spectrum at high "elds showing appreciable broadening. To account for the sharp peak which clearly develops with the magnetic "eld and originates from the upper plasmon}phonon branch, we have taken into consideration not only singularities but also zeros of the dielectric function given by Eq. (1). The energy positions corresponding to zeros of the dielectric function calculated for sample (b) are shown in Fig. 4b with dotted lines. As can be seen in this "gure, a possible candidate which may account for the observation of the sharp high "eld component of the upper phonon}plasmon branch is the mode which we called \ derived from one of > possible zeros of the dielectric function. Neverthe-
less, further work on the selection rules is needed to clarify the reason why only certain speci"c modes associated with zeros of the dielectric function are observed in light scattering experiments.
4. Conclusions Raman scattering experiments on heavily doped GaAs samples show the development of hybrid cyclotron}plasmon}phonon modes as well as certain other collective excitations related to zeros of the dielectric function. In the high "eld limit, the collective modes which resemble longitudinal optical phonon excitations propagating across the "eld direction have been observed. The interaction of coupled plasmon}phonon modes with Bernstein modes have also been observed. This interaction is stronger if it involves collective modes of plasmonlike character.
Acknowledgements A. Wysmolek gratefully acknowledges "nancial support from the Alexander von Humboldt Foundation.
References [1] A. Mooradian, G.B. Wright, Phys. Rev. Lett. 16 (1966) 999. [2] B. Lax, Proceedings of the International Conference on the Physics of Semiconductors, Kyoto, 1966, J. Phys. Soc. Japan 21 (1966) 165. [3] B.B. Varga, Phys. Rev. 137 (1965) A1896. [4] R. Kaplan, E.D. Palik, R.F. Wallis, S. Iwasa, E. Burstein, Y. Sawada, Phys. Rev. Lett. 18 (1967) 159. [5] S. Ivasa, in: E.D. Haidemenakis (Ed.), Physics of Solids in Intense Magnetic Fields, Plenum Press, New York, 1969, p. 126}135. [6] I. Yokota, Proceedings of the International Conference on the Physics of Semiconductors, Kyoto, 1966, J. Phys. Soc. Japan 21 (1966) 738. [7] C.K. Patel, R.E. Slusher, Phys. Rev. Lett. 21 (1968) 1563.