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Confinement and amplification of terahertz acoustic phonons in cubic heterostructures
Physica B 316–317 (2002) 356–358
Confinement and amplification of terahertz acoustic phonons in cubic heterostructures S.M. Komirenkoa,*, K.W. Kima, V.A. Kochelapb, M.A. Stroscioc a
Department of Electrical and Computer Engineering, North Carolina State University, 301-A, EGRC, 1010 Main Campus, Raleigh, NC 27695-7911, USA b Institute of Semiconductor Physics, Kiev-28, 252650, Ukraine c US Army Research Office, Research Triangle Park, NC 27709-2211, USA
Abstract A general criterion for phonon confinement is derived in the model of elastically anisotropic (cubic) media. The results are applied to the calculation of the dispersion curves of the confined phonons in Si=Si1x Gex =Si and AlAs=GaAs=AlAs heterostructures. For these structures, we show that the lowest-order phonon branches behave differently from those in the model of isotropic media. We have found that confinement is strong in the terahertz frequency range. For p-Si=SiGe=Si and n-AlAs/GaAs/AlAs quantum well heterostructures, we have studied the effect of amplification of confined high-frequency phonons by the drift of low-dimensional carriers. Two electron–phonon interaction mechanisms were taken into account: interaction via the deformation potential (p-SiGe and n-AlGaAs) and the piezoelectric interaction (n-AlGaAs). It was found that an amplification coefficient of the order of 102 cm1 for the AlGaAs heterostructures and 103 cm1 for the SiGe heterostructures can be obtained in spectrally-separated narrow amplification bands. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Acoustic phonon confinement; Phonon amplification; Elastic anisotropy
Recently, it was suggested that confined highfrequency phonons can be amplified by the drift of two-dimensional electrons [1]. In Ref. [1] this effect was studied for the case of elastically isotropic media and for electron–phonon interaction via the deformation potential. In the present paper, we develop a criterion of phonon confinement in quantum wells (QWs) fabricated from the cubic materials grown in a direction of high symmetry. Then, we analyze the possibility of Cerenkov
*Corresponding author. Fax: +19195153027. E-mail address: [email protected] (S.M. Komirenko).
amplification of acoustic phonons via both deformation and piezoelectric potentials. Consider a QW of width 2d grown in the ½0 0 1 (z) direction. In such a heterostructure, two classes of the confined waves can be specified: shearhorizontal waves (SHW) and shear-vertical waves (SVW). The SHW are purely transverse with displacement vector ð0; uy ; 0Þ: The SVW have two projections of the displacement vector ðux ; 0; uz Þ: Since the waves with SH and SV polarization do not mix upon wave reflection from a heterointerface, one can show that the criterion for confinement of the SHW as well as their dispersion are the same as in the case of isotropic media. However, the parameter of anisotropy, D ¼ C11 C12
0921-4526/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 2 ) 0 0 5 0 6 - 9
S.M. Komirenko et al. / Physica B 316–317 (2002) 356–358 8 7
Si/Ge/Si
6 5
Ω
2C44 (Ci;j are the stiffness coefficients), enters the dispersion relation of the SVW and, consequently, modifies the criterion for the phonon confinement and the behavior of the long-wavelength modes. Our analysis shows that the frequencies o of all confined SVW with a wave number q obey the relation qVcA oopqVcB ; where the index A (B) denotes the material of the well (barrier), and 2ðC11 þ C44 Þ C11 C44 D Vc ¼ C11 þ C44 rðC11 C44 Þ2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D þ C44 C11 2C11 C44 D 2 1 2 D þ C44 C11 ð1Þ D 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi if DpDn C11 pCffiffiffiffiffiffiffiffiffiffiffiffi 44 ffi C11 ðC11 C44 Þ: Otherwise, Vc ¼ VT C44 =r; r is the mass density. For most semiconductor QWs, the relation DpDn is satisfied. The line qVc can, in general, fall into a sector S limited in the (o; q)-plane by the lines qVT0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 and qVT ; where VT ¼ ðC44 þ D=2Þ=r: Note that in a heterostructure with a relatively small elastic mismatch, such as for a AlGaAs-based QW, the sectors S B and S A can overlap. As a result, the isotropic elastic model cannot predict the existence of confined SVW because in this model each sector reduces to a line. In Fig. 1, we show the dispersion relation calculated in terms of the dimensionless quantities O ¼ od=VTA and Q ¼ qd for the four lowest SV phonon branches in a Si/Ge/Si QW. One can see that the behavior of the lowest symmetric branch is different from that predicted by the isotropic model. In elastically isotropic media the lowest symmetric mode is the first excited mode with finite O and Q onsets [2]. For elastically anisotropic media, we found that the modes of both symmetries can exist at Q-0: This is stipulated by the fact that the magnitude of a confined SVW propagating in the QW with phase velocity below that determined by Eq. (1) not only decays, but also oscillates in the barrier. For the lowest symmetric mode, the latter allows to satisfy the boundary conditions even at Q-0: Confinement of waves with phase velocities close to VcB is weak. However, the confinement is stronger for the
357
4 3 2 1 1
2
3
4
5
6
7
Q Fig. 1. Dispersions of the lowest four confined phonon branches. Symmetric (antisymmetric) SVW are shown by the thick solid (dashed) lines . Sector SB (SA ) is delimited by the thin dashed (solid) lines. In each sector, the line with slope Vc is shown as the dotted line.
modes with larger Q: For all the structures studied, the confinement of the lowest phonon branches is strong in the THz frequency range. As a result, these high-frequency phonons can interact strongly with the confined electron subsystem and be amplified the most efficiently. In general, the magnitude of amplification depends on three factors: carrier population factor, degree of the wave localization, and relative magnitude of longitudinal-like vibrations in the confined wave. All these factors can be changed by variation of materials composing the QW, doping type, concentration, and the temperature as well as by the width of the well. Taking the concentration of the carriers and drift velocity to be 1012 cm2 and 2:5VTA ; respectively, and varying the temperature in the interval 50–300 K for QWs with ( we have estimated the maximum d ¼ 50–100 A; amplification of SVW to be 1200–1900 cm1 at frequencies 140–280 GHz for Si=Si0:5 Ge0:5 =Si QWs; 3000–7000 cm1 at frequencies 114– 380 GHz for Si=Ge=Si QWs; and 225–280 cm1 at frequencies 106–213 GHz for AlAs=GaAs= AlAs QWs. These amplification magnitudes exceed those obtained in Ref. [1], indicating that the QW configuration considered in this paper is favorable for efficient phonon amplification. As
S.M. Komirenko et al. / Physica B 316–317 (2002) 356–358
358
Si/Ge/Si QW
1
T=50 K T=100 K T=200 K T=300 K
d = 2.5 nm _1
α0 = 3470 cm
0.8
α/α 0
ω0 /2π = 228 GHz 0.6
0.4
0.2
0
0
1
2
3
4
5
6
7
8
ω/ω 0 Fig. 2. Amplification coefficient vs. phonon frequency at different temperatures, T:
illustrated in Fig. 2, where we depict the amplification of two lowest symmetric SVW of a Si=Ge=Si QW as a function of phonon frequency, the interplay of the previously mentioned factors leads to the appearance of relatively narrow, spectrally separated amplification bands. Confinement of acoustic waves in a QW provides another possibility of phonon amplification. In cubic bulk crystals all directions of wave propagation in the ð0 0 1Þ-plane are not piezoactive. Confinement of the SHW gives rise to a dependence of the displacement on the z-coordinate and induces their piezoactivity. We have found that symmetric SHW do not interact with carriers
populating the lowest electron subband. Only antisymmetric SHW interact with carriers. They are always represented by the excited branches of the dispersion relations and, hence, are characterized by the onsets of the wavevectors and frequencies. Applying our results to a AlGaAs QW, we have found that although the frequencies of amplified SHW, 180–200 GHz; are close to those of SVW, the piezoelectric interaction leads to lower magnitudes of amplification: 77–110 cm1 : Our estimates, however, suggest that for cubic AlGaN-based heterostructures the amplification can exceed 200 cm1 at frequencies of about 500 GHz: Our findings suggest that the Cerenkov generator=amplifier of confined acoustic phonons via the drift of 2D electrons is an efficient DC-current induced source of coherent high-frequency phonons. The proper modeling and engineering of the high-intensity beams cannot be done without taking into account the features related to the elastic anisotropy of the media. This work was supported by AFOSR and US ARO. V.A.K. would like to acknowledge the support from ERO of US Army.
References [1] S.M. Komirenko, et al., Phys. Rev. B 62 (2000) 7459. [2] L. Wendler, V.G. Grigoryan, Surf. Sci. 206 (1988) 203.
Confinement and amplification of terahertz acoustic phonons in cubic heterostructures S.M. Komirenkoa,*, K.W. Kima, V.A. Kochelapb, M.A. Stroscioc a
Department of Electrical and Computer Engineering, North Carolina State University, 301-A, EGRC, 1010 Main Campus, Raleigh, NC 27695-7911, USA b Institute of Semiconductor Physics, Kiev-28, 252650, Ukraine c US Army Research Office, Research Triangle Park, NC 27709-2211, USA
Abstract A general criterion for phonon confinement is derived in the model of elastically anisotropic (cubic) media. The results are applied to the calculation of the dispersion curves of the confined phonons in Si=Si1x Gex =Si and AlAs=GaAs=AlAs heterostructures. For these structures, we show that the lowest-order phonon branches behave differently from those in the model of isotropic media. We have found that confinement is strong in the terahertz frequency range. For p-Si=SiGe=Si and n-AlAs/GaAs/AlAs quantum well heterostructures, we have studied the effect of amplification of confined high-frequency phonons by the drift of low-dimensional carriers. Two electron–phonon interaction mechanisms were taken into account: interaction via the deformation potential (p-SiGe and n-AlGaAs) and the piezoelectric interaction (n-AlGaAs). It was found that an amplification coefficient of the order of 102 cm1 for the AlGaAs heterostructures and 103 cm1 for the SiGe heterostructures can be obtained in spectrally-separated narrow amplification bands. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Acoustic phonon confinement; Phonon amplification; Elastic anisotropy
Recently, it was suggested that confined highfrequency phonons can be amplified by the drift of two-dimensional electrons [1]. In Ref. [1] this effect was studied for the case of elastically isotropic media and for electron–phonon interaction via the deformation potential. In the present paper, we develop a criterion of phonon confinement in quantum wells (QWs) fabricated from the cubic materials grown in a direction of high symmetry. Then, we analyze the possibility of Cerenkov
*Corresponding author. Fax: +19195153027. E-mail address: [email protected] (S.M. Komirenko).
amplification of acoustic phonons via both deformation and piezoelectric potentials. Consider a QW of width 2d grown in the ½0 0 1 (z) direction. In such a heterostructure, two classes of the confined waves can be specified: shearhorizontal waves (SHW) and shear-vertical waves (SVW). The SHW are purely transverse with displacement vector ð0; uy ; 0Þ: The SVW have two projections of the displacement vector ðux ; 0; uz Þ: Since the waves with SH and SV polarization do not mix upon wave reflection from a heterointerface, one can show that the criterion for confinement of the SHW as well as their dispersion are the same as in the case of isotropic media. However, the parameter of anisotropy, D ¼ C11 C12
0921-4526/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 2 ) 0 0 5 0 6 - 9
S.M. Komirenko et al. / Physica B 316–317 (2002) 356–358 8 7
Si/Ge/Si
6 5
Ω
2C44 (Ci;j are the stiffness coefficients), enters the dispersion relation of the SVW and, consequently, modifies the criterion for the phonon confinement and the behavior of the long-wavelength modes. Our analysis shows that the frequencies o of all confined SVW with a wave number q obey the relation qVcA oopqVcB ; where the index A (B) denotes the material of the well (barrier), and 2ðC11 þ C44 Þ C11 C44 D Vc ¼ C11 þ C44 rðC11 C44 Þ2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D þ C44 C11 2C11 C44 D 2 1 2 D þ C44 C11 ð1Þ D 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi if DpDn C11 pCffiffiffiffiffiffiffiffiffiffiffiffi 44 ffi C11 ðC11 C44 Þ: Otherwise, Vc ¼ VT C44 =r; r is the mass density. For most semiconductor QWs, the relation DpDn is satisfied. The line qVc can, in general, fall into a sector S limited in the (o; q)-plane by the lines qVT0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 and qVT ; where VT ¼ ðC44 þ D=2Þ=r: Note that in a heterostructure with a relatively small elastic mismatch, such as for a AlGaAs-based QW, the sectors S B and S A can overlap. As a result, the isotropic elastic model cannot predict the existence of confined SVW because in this model each sector reduces to a line. In Fig. 1, we show the dispersion relation calculated in terms of the dimensionless quantities O ¼ od=VTA and Q ¼ qd for the four lowest SV phonon branches in a Si/Ge/Si QW. One can see that the behavior of the lowest symmetric branch is different from that predicted by the isotropic model. In elastically isotropic media the lowest symmetric mode is the first excited mode with finite O and Q onsets [2]. For elastically anisotropic media, we found that the modes of both symmetries can exist at Q-0: This is stipulated by the fact that the magnitude of a confined SVW propagating in the QW with phase velocity below that determined by Eq. (1) not only decays, but also oscillates in the barrier. For the lowest symmetric mode, the latter allows to satisfy the boundary conditions even at Q-0: Confinement of waves with phase velocities close to VcB is weak. However, the confinement is stronger for the
357
4 3 2 1 1
2
3
4
5
6
7
Q Fig. 1. Dispersions of the lowest four confined phonon branches. Symmetric (antisymmetric) SVW are shown by the thick solid (dashed) lines . Sector SB (SA ) is delimited by the thin dashed (solid) lines. In each sector, the line with slope Vc is shown as the dotted line.
modes with larger Q: For all the structures studied, the confinement of the lowest phonon branches is strong in the THz frequency range. As a result, these high-frequency phonons can interact strongly with the confined electron subsystem and be amplified the most efficiently. In general, the magnitude of amplification depends on three factors: carrier population factor, degree of the wave localization, and relative magnitude of longitudinal-like vibrations in the confined wave. All these factors can be changed by variation of materials composing the QW, doping type, concentration, and the temperature as well as by the width of the well. Taking the concentration of the carriers and drift velocity to be 1012 cm2 and 2:5VTA ; respectively, and varying the temperature in the interval 50–300 K for QWs with ( we have estimated the maximum d ¼ 50–100 A; amplification of SVW to be 1200–1900 cm1 at frequencies 140–280 GHz for Si=Si0:5 Ge0:5 =Si QWs; 3000–7000 cm1 at frequencies 114– 380 GHz for Si=Ge=Si QWs; and 225–280 cm1 at frequencies 106–213 GHz for AlAs=GaAs= AlAs QWs. These amplification magnitudes exceed those obtained in Ref. [1], indicating that the QW configuration considered in this paper is favorable for efficient phonon amplification. As
S.M. Komirenko et al. / Physica B 316–317 (2002) 356–358
358
Si/Ge/Si QW
1
T=50 K T=100 K T=200 K T=300 K
d = 2.5 nm _1
α0 = 3470 cm
0.8
α/α 0
ω0 /2π = 228 GHz 0.6
0.4
0.2
0
0
1
2
3
4
5
6
7
8
ω/ω 0 Fig. 2. Amplification coefficient vs. phonon frequency at different temperatures, T:
illustrated in Fig. 2, where we depict the amplification of two lowest symmetric SVW of a Si=Ge=Si QW as a function of phonon frequency, the interplay of the previously mentioned factors leads to the appearance of relatively narrow, spectrally separated amplification bands. Confinement of acoustic waves in a QW provides another possibility of phonon amplification. In cubic bulk crystals all directions of wave propagation in the ð0 0 1Þ-plane are not piezoactive. Confinement of the SHW gives rise to a dependence of the displacement on the z-coordinate and induces their piezoactivity. We have found that symmetric SHW do not interact with carriers
populating the lowest electron subband. Only antisymmetric SHW interact with carriers. They are always represented by the excited branches of the dispersion relations and, hence, are characterized by the onsets of the wavevectors and frequencies. Applying our results to a AlGaAs QW, we have found that although the frequencies of amplified SHW, 180–200 GHz; are close to those of SVW, the piezoelectric interaction leads to lower magnitudes of amplification: 77–110 cm1 : Our estimates, however, suggest that for cubic AlGaN-based heterostructures the amplification can exceed 200 cm1 at frequencies of about 500 GHz: Our findings suggest that the Cerenkov generator=amplifier of confined acoustic phonons via the drift of 2D electrons is an efficient DC-current induced source of coherent high-frequency phonons. The proper modeling and engineering of the high-intensity beams cannot be done without taking into account the features related to the elastic anisotropy of the media. This work was supported by AFOSR and US ARO. V.A.K. would like to acknowledge the support from ERO of US Army.
References [1] S.M. Komirenko, et al., Phys. Rev. B 62 (2000) 7459. [2] L. Wendler, V.G. Grigoryan, Surf. Sci. 206 (1988) 203.