- Email: [email protected]
i-GaAs heterostructure
Solid State Communications 224 (2015) 21–23
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Experimental observation of interface-phonon-plasmon mode in n-GaAs/i-GaAs heterostructure V.A. Volodin a,b, M.P. Sinyukov a, B.R. Semyagin a, M.A. Putyato a, V.V. Preobrazhenskiy a a b
A.V. Rzhanov Institute of Semiconductor Physics, Siberian Division of the Russian Academy of Sciences, Lavrenteva 13, Novosibirsk 630090, Russia Novosibirsk State University, Pirogova street 2, 630090 Novosibirsk, Russia
art ic l e i nf o
a b s t r a c t
Article history: Received 1 October 2015 Accepted 13 October 2015 Available online 19 October 2015
Interface-phonon-plasmon mode was observed for the first time in n-GaAs/i-GaAs heterostructure with the use of micro-Raman spectroscopy. To study this mode the geometry of the scattering in which the wave vectors of the incident and scattered light lie in the plane of the heterostructure was used. It was found that the frequency of interface-phonon-plasmon mode is less than the frequency of “volume” phonon–plasmon mode in the same heterostructure, which was observed in the geometry of the scattering with the wave vectors of the incident and scattered light directed perpendicular to the plane of the heterostructure. The frequency of interface-phonon-plasmon mode was calculated using continual model with boundary conditions, there is good agreement with the experimentally observed one. & 2015 Elsevier Ltd. All rights reserved.
1. Introduction The study of dispersion of phonons and other quasiparticles near interfaces attracted the researchers for many years. The presence of interface means that there is a preferred direction, so the angular anisotropy of quasiparticle dispersion should be manifested in heterostructures [1]. Researchers began to study the dispersion of phonons in GaAs/AlAs superlattices (SLs) both theoretically [2] and experimentally [3] for more than 30 years ago, and since then it has been adequately studied [4–9]. Interface phonons in GaAs/AlAs heterostructures were experimentally found almost 30 years ago [10]. The manifestation of the interaction of collective excitations of the electron gas (plasmons) and phonons in Raman spectra of bulk polar semiconductors was observed almost half a century ago [11]. Since, the phonon–plasmon modes in doped GaAs/AlAs SLs are investigated in details both theoretically and experimentally [12–17]. It is surprising, but to date, remains "white spot" in the study of the anisotropy of the phonon-plasmon modes not in SLs, but in a simple heterostructuresemi-insulating GaAs substrates / doped GaAs film.
thickness 1 μm was grown on i-GaAs (001) substrate using solid source molecular-beam epitaxy (MBE). According to Hall measurements data the concentration of free electrons in the n-GaAs film was 1.45·1018 cm 3. The Raman spectrometer T64000 (Horiba Jobin Yvon) with micro-Raman setup (based on Olympus optical microscope) and the 514.5 nm line of an Ar þ laser were used. The spectral resolution was better than 2 cm 1. The radiation power reaching the sample was 1 mW, which is insufficient to cause significant heating of the sample. The spatial resolution is determined by the size of the laser beam at the focus and was equal to 1 μm. This allowed us to use the geometry of the scattering in which the wave vectors of the incident and scattered light lie in the plane of the heterostructure. In these experiments, the light was focused onto an interface of the heterostructures. The “normal” geometry in which directions of the wave-vector of light q was perpendicular to the heterostructure was also used. This approach is described in more details elsewhere [8,9].
3. Results and discussion 2. Experimental details The bulk semi-insulating GaAs (i-GaAs) and n-GaAs/i-GaAs heterostructure were studied using Raman scattering spectroscopy in the backscattering geometry. The Si doped n-GaAs film with E-mail address: [email protected] (V.A. Volodin). http://dx.doi.org/10.1016/j.ssc.2015.10.006 0038-1098/& 2015 Elsevier Ltd. All rights reserved.
Fig. 1 shows experimental spectra of bulk i-GaAs and n-GaAs film in a case where the wave vectors of the incident and the scattered photons are perpendicular to the plane of the samples – the geometry Z(XY)-Z. Here, in brackets axis correspond to directions of the polarization of the electric field vector of the incident and scattered electromagnetic waves, and out the brackets axis indicate the direction of the wave vector of the incident and
22
V.A. Volodin et al. / Solid State Communications 224 (2015) 21–23
scattered light. The axes X, Y, and Z correspond to crystallographic directions (100), (010) and (001). The Raman scattering tensors Dαβ for phonons with wave vector along Z axis and with different directions of atomic displacements (X, Y, or Z) are: 0 1 0 1 0 1 0 0 0 0 0 d 0 d 0 B C B C B C TOX @ 0 0 d A; TOY @ 0 0 0 A;LOZ @ d 0 0 A 0
d
0
d
0
0
0
0
0
where d is the additive differential polarizability of bonds. According to symmetry selection rules, the Raman intensity is proportional to EsαDαβEiβ, where Ei is the polarization vector of the incident light and Es is the polarization vector of the scattered light. The geometry Z(XY)-Z is thus allowed only for longitudinal optic (LO) phonons. This phonon is observed in the experimental spectra (curves 1 and 2 – 292 cm 1). Some week peculiarities are also observed. It is forbidden transverse optical (TO) phonon – 268 cm 1, and peculiarities due to many-phonon scattering, which are shown by arrows. In the spectrum of the n-GaAs film, in addition to LO and TO phonon peaks and many-phonon scattering features, there is broad peak at 450 cm 1 due to scattering on the phonon–plasmon modes. The contribution of the phonon–plasmon interaction in the Raman spectrum can be calculated using a simple model [18]. For the studied frequency range the dielectric permittivity of the doped crystal (taking into account the contribution of phonons and plasmons) is: ! ω2p ω2LO ω2 εdop ¼ ε1 2 ð1Þ ωTO ω2 iγ phon ω ω2 þ iγ p ω
volume concentration of charge carriers (in cm 3), e is elementary charge, m* is effective mass of charge carriers (0.063 of mass of free electron for GaAs), c is velocity of light in vacuum. When the dielectric permittivity is equal to zero, the spontaneous fluctuation vibrations of the media are appeared. Then, the frequencies of the phonon–plasmon modes (marked plus and minus in formula bellow) without taking into account the damping of plasmons and phonons are: 2 1 2 ð3Þ ω2LO þ ;LO ¼ ωp þ ω2LO 7 ω2p þ ω2LO 4ω2p ω2TO 1=2 2 According to the fluctuation–dissipation theorem [18] the contribution of phonon–plasmon modes to the Raman scattering is proportional to:
1 IðωÞ ðnðωÞ þ 1Þ Im ð4Þ
εdop
where ε1 is the dielectric permittivity at frequencies much larger than the phonon frequencies, for GaAs it is equal to 10.89, N is
where n(ɷ) is Bose–Einstein factor nðωÞ ¼ ð1=expðℏω=kTÞ 1Þ, k is Boltzmann constant, T is temperature in Kelvin's scale, ħ is Plank constant. Fig. 1 (curve 3) shows the spectrum calculated using the formula (4), the parameters ɷp, γphon and γp were obtained with approximation of calculated spectrum to the experimental one. The best agreement was obtained for ɷp, γphon and γp equal to 440, 3 and 130 cm 1, respectively. The concentration of free electrons obtained from the plasma frequency 440 cm 1 (Eq. (2)) is equal to 1.48 1018 cm 3, which agrees well with the data obtained from the Hall effect. Observed in the experimental spectrum (curve 2) LO and TO peaks associated with the presence of depleted layer on the surface of the n-GaAs film [19]. Fig. 2 shows the spectra obtained in the geometry of the scattering in which the wave vectors of the incident and scattered light lie in the plane of the heterostructure – X(ZY)-X geometry. Two areas of the heterostructure were studied. First, the laser beam is focused in the area of semi-insulating substrate far from the film (curve 1), second, it was focused on interface (curve 2). In this geometry allowed phonon is LO mode, for which the atoms are displaced along the X. The geometries Z(XY)-Z and X(ZY)-X are
Fig. 1. Raman spectra of bulk semi-insulating GaAs (curve 1) and n-GaAs/i-GaAs heterostructure (curve 2)-the wave vectors of the incident and the scattered photons are perpendicular to the plane of the samples; curve 3 is calculated spectrum of phonon–plasmon mode.
Fig. 2. Raman spectra of bulk semi-insulating GaAs (curve 1) and n-GaAs/i-GaAs heterostructure (curve 2) – the wave vectors of the incident and scattered light lie in the plane of the heterostructure; curve 3 is calculated spectrum of interfacephonon-plasmon mode.
where γphon and γp damping parameters for the phonon and plasmon, respectively. Plasmon frequency (in reversed centimeters) is: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4π Ne2 ð2Þ ωp ¼ ð2π cÞ2 m U ε1
V.A. Volodin et al. / Solid State Communications 224 (2015) 21–23
equal, so one can see that spectra for i-GaAs (curves 1 in Figs. 1 and 2) are totally coincide. One can see that in the spectrum obtained from interface area (curve 2 in Fig. 2) phonon–plasmon peak is shifted to approximately 100 cm 1, comparing with phonon–plasmon peak from “bulk” mode (curve 2 in Fig. 1). It should be noted, that the two spectra in Figs. 1 and 2 are obtained in different geometries but from the same sample, therefore, the concentration of free electrons is the same. So, we assume that the shift is due to the influence of the interface. To calculate the spectrum of interface-phonon-plasmon we used continual model with boundary conditions, developed for undoped heterostructures [10]. A solution is the waves propagating along the interface (along the X axis) with a wave vector (in our case 106 cm 1). As for the Z direction, these waves are the superposition of growing and decaying exponential, but they are only damped as z tends to plus infinity (air) and to minus infinity (substrate as a semi-infinite medium). The eigen-vectors and the eigen-frequencies are obtained from the boundary conditions: the tangential component of the electric field E and the normal components of the electric induction vector D at the boundaries need to be continuous. In our case, this leads to a system of four equations, because of two boundary conditions on two intrefaces – the air-doped GaAs film and doped GaS film – semi-insulating GaAs substrate. We used a simplified approach, believing that the waves are strongly damped along Z, and hence, each interface can be considered as independent. Then, according to boundary conditions [1], the frequency of the interface modes at the interface of two media (in our case the medium with permittivity εdop and media with permittivity εi ¼ ε1 ðω2LO ω2 =ω2TO ω2 iγ phon ωÞ) is obtained from the equation:
εdop þ εi ¼ 0: In our case the Eq. (5) is transformed to: ! ω2 ω2 ω2 1 ε1 2 LO 2 ω 2 p ¼ 0: ωTO ω i τ 2 ω þiγω
ð5Þ
ð6Þ
The frequencies of the interface-phonon-plasmon modes (marked plus and minus in formula bellow) without taking into account the damping of plasmons and phonons are: 2 1 2 ω2LO þ ;LO ¼ ωp =2 þ ω2LO 7 ω2p =2 þ ω2LO 2ω2p ω2TO 1=2 2 ð7Þ One should note the main difference from formula 3 (“volume” phonon-plasmon) p that ffiffiffi plasmon frequency now is less, for our case it is equal to 440/ 2 cm 1 calculated with such parameter spectrum is shown in Fig. 2 (curve 3). One can see a good agreement with experiment, despite the fact that in this region of the spectrum there are peculiarities associated with many-phonon scattering.
23
It is worth noting that in the experimental spectrum (curve 2 in Fig. 2) the intensity of peak from forbidden in this geometry TO phonon peak comparable with intensity of allowed LO phonon. Such a violation of the symmetry selection rules will be the subject of further research. Presumably, this may be a manifestation of the electro-optical mechanism of Raman scattering. The analysis of frequency interface phonon–plasmon modes, taking into account the influence of all interfaces also the subject of further research. To study the dispersion of these modes for different values of the angle between the wave vector of light and the normal to the structure must be used in structures with specially made wedges [20].
4. Conclusion So, interface-phonon-plasmon mode was observed with the use of micro-Raman spectroscopy in n-GaAs/i-GaAs heterostructure for the first time. The spectrum of interface-phononplasmon mode was calculated using continual model with boundary conditions, the good agreement with the experimentally observed spectra is observed. This approach can be used for contactless determination of the concentration of free charge carriers, if doped layers are under opaque layers, and conventional scattering geometry does not allow receiving the Raman signal from this layer.
References [1] Light scattering in Solids V, In: M. Cardona, G. Guntherodt (Eds.), Superlattices and Dther microstructures, Springer Verlag, Berlin, 1989, p.335. [2] E.P. Pokatilov, S.I. Beril, Phys. Status Solidi (b) 110 (1982) K75. [3] A.K. Sood, J. Menendez, M. Cardona, K. Ploog, Phys. Rev. Lett. 54 (1985) 2115. [4] Shang-Fen Ren, Hanyou Chu, Yia-Chung Chang, Phys. Rev. B 37 (1988) 8899. [5] M. Cardona, Superlattices Microstruct. 5 (1989) 27. [6] S.F. Ren, G. Qin, Solid State Commun. 121 (2002) 171. [7] V.A. Volodin, A.S. Kozhukhov, A.V. Latyshev, D.V. Shcheglov, JETP Lett. 95 (2012) 70. [8] V.A. Volodin, M.P. Sinyukov, JETP Lett. 99 (2014) 396. [9] V.A. Volodin, V.A. Sachkov, МP. Sinyukov, J. Exp. Theor. Phys. 120 (2015) 781. [10] M. Nakayama, M. Ishida, N. Sano, Phys. Rev. B 38 (1988) 6348. [11] A. Mooradian, G.B. Wright, Phys. Rev. Lett. 16 (1966) 999. [12] A. Diego Olego, A.C. Pinczuk, Gossard, W. Wiegmann, Phys. Rev. B 26 (1982) 7867. [13] B. Lou, Solid State Commun. 84 (1992) 685. [14] R. Merlin, C. Colvard, M.V. Klein, H. Morkoc, A.Y. Cho, A.C. Gossard, Appl. Phys. Lett. 36 (1980) 43. [15] V.A. Volodin, M.D. Efremov, V.A. Sachkov, J. Exp. Theor. Phys. 103 (2006) 646. [16] V.A. Volodin, JETP Lett. 89 (2009) 483. [17] V.A. Volodin, Phys. Solid State 53 (2011) 404. [18] Light Scattering in Solids IV, In: M. Cardona, G. Gunterodt (Eds.), Electronic Scattering, Spin Effects, SERC, and Morphic Effects, Springer-Verlag, Berlin, 1982, p. 382. [19] P.D. Wang, M.A. Foad, C.M. Sotomayor‐Torres, S. Thoms, M. Watt, R. Cheung, C. D.W. Wilkinson, S.P. Beaumont, J. Appl. Phys. 71 (1992) 3754. [20] A. Huber, T. Egeler, W. Ettmiiller, H. Rothfritz, G. Trankle, G. Abstreiter, Superlattices Microstruct. 9 (1991) 309.
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Experimental observation of interface-phonon-plasmon mode in n-GaAs/i-GaAs heterostructure V.A. Volodin a,b, M.P. Sinyukov a, B.R. Semyagin a, M.A. Putyato a, V.V. Preobrazhenskiy a a b
A.V. Rzhanov Institute of Semiconductor Physics, Siberian Division of the Russian Academy of Sciences, Lavrenteva 13, Novosibirsk 630090, Russia Novosibirsk State University, Pirogova street 2, 630090 Novosibirsk, Russia
art ic l e i nf o
a b s t r a c t
Article history: Received 1 October 2015 Accepted 13 October 2015 Available online 19 October 2015
Interface-phonon-plasmon mode was observed for the first time in n-GaAs/i-GaAs heterostructure with the use of micro-Raman spectroscopy. To study this mode the geometry of the scattering in which the wave vectors of the incident and scattered light lie in the plane of the heterostructure was used. It was found that the frequency of interface-phonon-plasmon mode is less than the frequency of “volume” phonon–plasmon mode in the same heterostructure, which was observed in the geometry of the scattering with the wave vectors of the incident and scattered light directed perpendicular to the plane of the heterostructure. The frequency of interface-phonon-plasmon mode was calculated using continual model with boundary conditions, there is good agreement with the experimentally observed one. & 2015 Elsevier Ltd. All rights reserved.
1. Introduction The study of dispersion of phonons and other quasiparticles near interfaces attracted the researchers for many years. The presence of interface means that there is a preferred direction, so the angular anisotropy of quasiparticle dispersion should be manifested in heterostructures [1]. Researchers began to study the dispersion of phonons in GaAs/AlAs superlattices (SLs) both theoretically [2] and experimentally [3] for more than 30 years ago, and since then it has been adequately studied [4–9]. Interface phonons in GaAs/AlAs heterostructures were experimentally found almost 30 years ago [10]. The manifestation of the interaction of collective excitations of the electron gas (plasmons) and phonons in Raman spectra of bulk polar semiconductors was observed almost half a century ago [11]. Since, the phonon–plasmon modes in doped GaAs/AlAs SLs are investigated in details both theoretically and experimentally [12–17]. It is surprising, but to date, remains "white spot" in the study of the anisotropy of the phonon-plasmon modes not in SLs, but in a simple heterostructuresemi-insulating GaAs substrates / doped GaAs film.
thickness 1 μm was grown on i-GaAs (001) substrate using solid source molecular-beam epitaxy (MBE). According to Hall measurements data the concentration of free electrons in the n-GaAs film was 1.45·1018 cm 3. The Raman spectrometer T64000 (Horiba Jobin Yvon) with micro-Raman setup (based on Olympus optical microscope) and the 514.5 nm line of an Ar þ laser were used. The spectral resolution was better than 2 cm 1. The radiation power reaching the sample was 1 mW, which is insufficient to cause significant heating of the sample. The spatial resolution is determined by the size of the laser beam at the focus and was equal to 1 μm. This allowed us to use the geometry of the scattering in which the wave vectors of the incident and scattered light lie in the plane of the heterostructure. In these experiments, the light was focused onto an interface of the heterostructures. The “normal” geometry in which directions of the wave-vector of light q was perpendicular to the heterostructure was also used. This approach is described in more details elsewhere [8,9].
3. Results and discussion 2. Experimental details The bulk semi-insulating GaAs (i-GaAs) and n-GaAs/i-GaAs heterostructure were studied using Raman scattering spectroscopy in the backscattering geometry. The Si doped n-GaAs film with E-mail address: [email protected] (V.A. Volodin). http://dx.doi.org/10.1016/j.ssc.2015.10.006 0038-1098/& 2015 Elsevier Ltd. All rights reserved.
Fig. 1 shows experimental spectra of bulk i-GaAs and n-GaAs film in a case where the wave vectors of the incident and the scattered photons are perpendicular to the plane of the samples – the geometry Z(XY)-Z. Here, in brackets axis correspond to directions of the polarization of the electric field vector of the incident and scattered electromagnetic waves, and out the brackets axis indicate the direction of the wave vector of the incident and
22
V.A. Volodin et al. / Solid State Communications 224 (2015) 21–23
scattered light. The axes X, Y, and Z correspond to crystallographic directions (100), (010) and (001). The Raman scattering tensors Dαβ for phonons with wave vector along Z axis and with different directions of atomic displacements (X, Y, or Z) are: 0 1 0 1 0 1 0 0 0 0 0 d 0 d 0 B C B C B C TOX @ 0 0 d A; TOY @ 0 0 0 A;LOZ @ d 0 0 A 0
d
0
d
0
0
0
0
0
where d is the additive differential polarizability of bonds. According to symmetry selection rules, the Raman intensity is proportional to EsαDαβEiβ, where Ei is the polarization vector of the incident light and Es is the polarization vector of the scattered light. The geometry Z(XY)-Z is thus allowed only for longitudinal optic (LO) phonons. This phonon is observed in the experimental spectra (curves 1 and 2 – 292 cm 1). Some week peculiarities are also observed. It is forbidden transverse optical (TO) phonon – 268 cm 1, and peculiarities due to many-phonon scattering, which are shown by arrows. In the spectrum of the n-GaAs film, in addition to LO and TO phonon peaks and many-phonon scattering features, there is broad peak at 450 cm 1 due to scattering on the phonon–plasmon modes. The contribution of the phonon–plasmon interaction in the Raman spectrum can be calculated using a simple model [18]. For the studied frequency range the dielectric permittivity of the doped crystal (taking into account the contribution of phonons and plasmons) is: ! ω2p ω2LO ω2 εdop ¼ ε1 2 ð1Þ ωTO ω2 iγ phon ω ω2 þ iγ p ω
volume concentration of charge carriers (in cm 3), e is elementary charge, m* is effective mass of charge carriers (0.063 of mass of free electron for GaAs), c is velocity of light in vacuum. When the dielectric permittivity is equal to zero, the spontaneous fluctuation vibrations of the media are appeared. Then, the frequencies of the phonon–plasmon modes (marked plus and minus in formula bellow) without taking into account the damping of plasmons and phonons are: 2 1 2 ð3Þ ω2LO þ ;LO ¼ ωp þ ω2LO 7 ω2p þ ω2LO 4ω2p ω2TO 1=2 2 According to the fluctuation–dissipation theorem [18] the contribution of phonon–plasmon modes to the Raman scattering is proportional to:
1 IðωÞ ðnðωÞ þ 1Þ Im ð4Þ
εdop
where ε1 is the dielectric permittivity at frequencies much larger than the phonon frequencies, for GaAs it is equal to 10.89, N is
where n(ɷ) is Bose–Einstein factor nðωÞ ¼ ð1=expðℏω=kTÞ 1Þ, k is Boltzmann constant, T is temperature in Kelvin's scale, ħ is Plank constant. Fig. 1 (curve 3) shows the spectrum calculated using the formula (4), the parameters ɷp, γphon and γp were obtained with approximation of calculated spectrum to the experimental one. The best agreement was obtained for ɷp, γphon and γp equal to 440, 3 and 130 cm 1, respectively. The concentration of free electrons obtained from the plasma frequency 440 cm 1 (Eq. (2)) is equal to 1.48 1018 cm 3, which agrees well with the data obtained from the Hall effect. Observed in the experimental spectrum (curve 2) LO and TO peaks associated with the presence of depleted layer on the surface of the n-GaAs film [19]. Fig. 2 shows the spectra obtained in the geometry of the scattering in which the wave vectors of the incident and scattered light lie in the plane of the heterostructure – X(ZY)-X geometry. Two areas of the heterostructure were studied. First, the laser beam is focused in the area of semi-insulating substrate far from the film (curve 1), second, it was focused on interface (curve 2). In this geometry allowed phonon is LO mode, for which the atoms are displaced along the X. The geometries Z(XY)-Z and X(ZY)-X are
Fig. 1. Raman spectra of bulk semi-insulating GaAs (curve 1) and n-GaAs/i-GaAs heterostructure (curve 2)-the wave vectors of the incident and the scattered photons are perpendicular to the plane of the samples; curve 3 is calculated spectrum of phonon–plasmon mode.
Fig. 2. Raman spectra of bulk semi-insulating GaAs (curve 1) and n-GaAs/i-GaAs heterostructure (curve 2) – the wave vectors of the incident and scattered light lie in the plane of the heterostructure; curve 3 is calculated spectrum of interfacephonon-plasmon mode.
where γphon and γp damping parameters for the phonon and plasmon, respectively. Plasmon frequency (in reversed centimeters) is: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4π Ne2 ð2Þ ωp ¼ ð2π cÞ2 m U ε1
V.A. Volodin et al. / Solid State Communications 224 (2015) 21–23
equal, so one can see that spectra for i-GaAs (curves 1 in Figs. 1 and 2) are totally coincide. One can see that in the spectrum obtained from interface area (curve 2 in Fig. 2) phonon–plasmon peak is shifted to approximately 100 cm 1, comparing with phonon–plasmon peak from “bulk” mode (curve 2 in Fig. 1). It should be noted, that the two spectra in Figs. 1 and 2 are obtained in different geometries but from the same sample, therefore, the concentration of free electrons is the same. So, we assume that the shift is due to the influence of the interface. To calculate the spectrum of interface-phonon-plasmon we used continual model with boundary conditions, developed for undoped heterostructures [10]. A solution is the waves propagating along the interface (along the X axis) with a wave vector (in our case 106 cm 1). As for the Z direction, these waves are the superposition of growing and decaying exponential, but they are only damped as z tends to plus infinity (air) and to minus infinity (substrate as a semi-infinite medium). The eigen-vectors and the eigen-frequencies are obtained from the boundary conditions: the tangential component of the electric field E and the normal components of the electric induction vector D at the boundaries need to be continuous. In our case, this leads to a system of four equations, because of two boundary conditions on two intrefaces – the air-doped GaAs film and doped GaS film – semi-insulating GaAs substrate. We used a simplified approach, believing that the waves are strongly damped along Z, and hence, each interface can be considered as independent. Then, according to boundary conditions [1], the frequency of the interface modes at the interface of two media (in our case the medium with permittivity εdop and media with permittivity εi ¼ ε1 ðω2LO ω2 =ω2TO ω2 iγ phon ωÞ) is obtained from the equation:
εdop þ εi ¼ 0: In our case the Eq. (5) is transformed to: ! ω2 ω2 ω2 1 ε1 2 LO 2 ω 2 p ¼ 0: ωTO ω i τ 2 ω þiγω
ð5Þ
ð6Þ
The frequencies of the interface-phonon-plasmon modes (marked plus and minus in formula bellow) without taking into account the damping of plasmons and phonons are: 2 1 2 ω2LO þ ;LO ¼ ωp =2 þ ω2LO 7 ω2p =2 þ ω2LO 2ω2p ω2TO 1=2 2 ð7Þ One should note the main difference from formula 3 (“volume” phonon-plasmon) p that ffiffiffi plasmon frequency now is less, for our case it is equal to 440/ 2 cm 1 calculated with such parameter spectrum is shown in Fig. 2 (curve 3). One can see a good agreement with experiment, despite the fact that in this region of the spectrum there are peculiarities associated with many-phonon scattering.
23
It is worth noting that in the experimental spectrum (curve 2 in Fig. 2) the intensity of peak from forbidden in this geometry TO phonon peak comparable with intensity of allowed LO phonon. Such a violation of the symmetry selection rules will be the subject of further research. Presumably, this may be a manifestation of the electro-optical mechanism of Raman scattering. The analysis of frequency interface phonon–plasmon modes, taking into account the influence of all interfaces also the subject of further research. To study the dispersion of these modes for different values of the angle between the wave vector of light and the normal to the structure must be used in structures with specially made wedges [20].
4. Conclusion So, interface-phonon-plasmon mode was observed with the use of micro-Raman spectroscopy in n-GaAs/i-GaAs heterostructure for the first time. The spectrum of interface-phononplasmon mode was calculated using continual model with boundary conditions, the good agreement with the experimentally observed spectra is observed. This approach can be used for contactless determination of the concentration of free charge carriers, if doped layers are under opaque layers, and conventional scattering geometry does not allow receiving the Raman signal from this layer.
References [1] Light scattering in Solids V, In: M. Cardona, G. Guntherodt (Eds.), Superlattices and Dther microstructures, Springer Verlag, Berlin, 1989, p.335. [2] E.P. Pokatilov, S.I. Beril, Phys. Status Solidi (b) 110 (1982) K75. [3] A.K. Sood, J. Menendez, M. Cardona, K. Ploog, Phys. Rev. Lett. 54 (1985) 2115. [4] Shang-Fen Ren, Hanyou Chu, Yia-Chung Chang, Phys. Rev. B 37 (1988) 8899. [5] M. Cardona, Superlattices Microstruct. 5 (1989) 27. [6] S.F. Ren, G. Qin, Solid State Commun. 121 (2002) 171. [7] V.A. Volodin, A.S. Kozhukhov, A.V. Latyshev, D.V. Shcheglov, JETP Lett. 95 (2012) 70. [8] V.A. Volodin, M.P. Sinyukov, JETP Lett. 99 (2014) 396. [9] V.A. Volodin, V.A. Sachkov, МP. Sinyukov, J. Exp. Theor. Phys. 120 (2015) 781. [10] M. Nakayama, M. Ishida, N. Sano, Phys. Rev. B 38 (1988) 6348. [11] A. Mooradian, G.B. Wright, Phys. Rev. Lett. 16 (1966) 999. [12] A. Diego Olego, A.C. Pinczuk, Gossard, W. Wiegmann, Phys. Rev. B 26 (1982) 7867. [13] B. Lou, Solid State Commun. 84 (1992) 685. [14] R. Merlin, C. Colvard, M.V. Klein, H. Morkoc, A.Y. Cho, A.C. Gossard, Appl. Phys. Lett. 36 (1980) 43. [15] V.A. Volodin, M.D. Efremov, V.A. Sachkov, J. Exp. Theor. Phys. 103 (2006) 646. [16] V.A. Volodin, JETP Lett. 89 (2009) 483. [17] V.A. Volodin, Phys. Solid State 53 (2011) 404. [18] Light Scattering in Solids IV, In: M. Cardona, G. Gunterodt (Eds.), Electronic Scattering, Spin Effects, SERC, and Morphic Effects, Springer-Verlag, Berlin, 1982, p. 382. [19] P.D. Wang, M.A. Foad, C.M. Sotomayor‐Torres, S. Thoms, M. Watt, R. Cheung, C. D.W. Wilkinson, S.P. Beaumont, J. Appl. Phys. 71 (1992) 3754. [20] A. Huber, T. Egeler, W. Ettmiiller, H. Rothfritz, G. Trankle, G. Abstreiter, Superlattices Microstruct. 9 (1991) 309.