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IR light emission from charge oscillations in semiconductor double wells
Superlattices and Microstructures, Vol. 20, No. 3, 1996
IR light emission from charge oscillations in semiconductor double wells ¨ tz† W. Po University of Illinois at Chicago, 845 W. Taylor Str., Chicago, Illinois, U.S.A.
(Received 20 May 1996) A microscopic approach to the carrier dynamics in semiconductor heterostructures is used to study structural dephasing and phase breaking due to carrier–carrier interactions in GaAlAs– GaAs double wells. Analysis of interband polarizations allows a quantitative account of the destruction of phase coherence and its consequences on the frequency spectrum of the THz dipole radiation emitted by the carriers. c 1996 Academic Press Limited
1. Introduction Already in 1970, superlattices were proposed as candidates for infrared (IR) light sources [1]. In the last few years, IR light emission by optically excited semiconductor heterostructures, such as superlattices and double wells (DWs), has been used as evidence for the presence of coherent charge oscillations [2]. More recently, the potential of DWs as nanoscale IR light pulse sources has been investigated [3]. Recently, we developed a consistent microscopic approach to the carrier dynamics in semiconductor heterostructures based on a generalization of Boltzmann–Bloch equations [4–6] X γ (kk ) + β (kk ) 1 ≷ ≷ − (γβ (kk )+Re 6αγ (kk) ), k f γβ (T, k ) Re 6αγ T, (i h¯ ∂T − α (kk ) + β (kk )) f αβ (T, k ) = 2 h 2 ¯ γ α (kk ) + γ (kk ) 1 ≷ − f αγ + (αγ (kk ) + Re 6γβ (kk) ), k (T, k ) Re 6γβ T, 2h¯ 2 Z +∞ i h¯ ≷ + dω 6αβ (T, ω, k ) 2π −∞ " # 1 1 × P −P ≷ ≷ h¯ ω + 6αβ (kk) /2 − β (kk ) h¯ ω − 6αβ (kk) /2 − α (kk ) γ (kk ) + β (kk ) 0αγ (kk) + γβ (kk ) 1X > < + (T, k ) − , k f γβ 6αγ T, 2 γ 2h¯ 2 γ (kk ) + β (kk ) 0αγ (kk) + γβ (kk ) < > − , k f γβ (T, k ) −6αγ T, 2h¯ 2 † Research for this article funded by U.S. Army Research Office.
0749–6036/96/070273 + 05 $18.00/0
c 1996 Academic Press Limited
274
Superlattices and Microstructures, Vol. 20, No. 3, 1996
Carrier concentration (1010 cm–2)
0.8
0.6
0.4
0.2
0.0 –0.5
0.5
1.5 Time (ps)
2.5
3.5
Fig. 1. Carrier density versus time. Solid line: electrons in the left well, dot–dashed line: electrons in the right well, dotted line: holes. The dashed line gives the carrier density of electrons in the left well when Coulomb scattering is neglected.
α (kk ) + γ (kk ) 0γβ (kk) + αγ (kk ) + ,k + T, 2h¯ 2 α (kk ) + γ (kk ) 0γβ (k ) + αγ (kk ) > < + ,k . − f αγ (T, k )6γβ T, 2h¯ 2 < > f αγ (T, k )6γβ
(1)
α (kk ) are single-particle energies, α denotes the band index, k is the transverse wave vector, T is time, 6 i is component i of the self-energy, and X is the fast frequency component of term X from the free-particle Hamiltonian, in particular, αβ = (β − α )/h¯ . > (T, k ) ≡ (δαβ − f βα (T, k )), f αβ < (T, k ) ≡ − f βα (T, k ), f αβ
where f βα (T ) are time-dependent carrier distribution functions (α = β) and interband polarizations (α 6= β). This approach allows an account of electron phase coherence in the system, as well as its destruction by interaction processes. These equations are derived from Dyson’s equation for non-equilibrium Green’s functions within the Keldysh approach. Accounting for both distribution functions and all interband polarizations, these equations allow quantitative monitoring of the destruction of electron phase coherence on a sub-picosecond time scale. In addition to the laser field and the free-carrier dynamics in the DW, the carrier–carrier Coulomb interaction is included within the screened Hartree–Fock approximation to the self-energy. Thus excitonic effects, as well as scattering terms which provide phase destruction are included in consistent fashion.
LR-polarization (1010 cm–2)
Superlattices and Microstructures, Vol. 20, No. 3, 1996
275
0.2
0.1
0.0 –0.5
0.5
1.5
2.5
3.5
Time (ps) Fig. 2.
Left–Right electron subband polarization versus time with (solid line) and without (dashed line) Coulomb scattering.
2. Results In the present case of asymmetric GaAs–Alx Ga1−x As DWs, there are two sources for coherence, coherence between laser-generated electron-hole pairs (excitons) and, initially, phase coherence between electrons in ˚ ˚ ˚ Al0.2 Ga0.8 As–GaAs– the left and right well. Specifically, we consider an asymmetric 145 A–25 A–100 A Al0.2 Ga0.8 As DW whose lowest two electronic subbands are biased near resonance by means of an external static electric field (E ext ). For the present calculation, we use a three band model including the two lowest electron subbands, L and R, and the top heavy-hole subband (H). Structurally perfect DWs are assumed. Selective generation of direct excitons in the wide well by a 200 fs laser pulse (h¯ ω ≈ 1.54 eV) leads to damped charge oscillations (quantum beating between direct and indirect excitons) provided that carrier concentrations are kept below about 5 × 1010 cm−2 . The damping of charge oscillations, illustrated in Fig. 1 for an electric field of 10.5 kV cm−1 and a final carrier density of about 0.6 × 1010 cm−2 is caused by two effects, phase breaking due to the carrier–carrier Coulomb interaction and, to a lesser extent, by structural dephasing due to the intrinsic electronic structure of the DW. The latter is a consequence of the effective mass differences in GaAs and AlGaAs, as well as non-parabolicity effects. These were evaluated within a multi-band k · p model [7]. They lead to an effective overlap between left- and right-well wave functions (Rabi frequency) which depends on k|| , the in-plane k-vector. In Fig. 1, the dashed lines give the charge oscillations in the absence of Coulomb scattering. It is seen that many-body effects provide the predominant cause for the decay of exciton-based charge oscillations. Figure 2 gives the magnitude of the L–R well interband polarization versus time. For comparison, we show the time-evolution of the L–R polarization without Coulomb scattering (dashed line). Phase breaking
Superlattices and Microstructures, Vol. 20, No. 3, 1996
Radiation field (a. u.)
276
–0.5
Fig. 3.
0.0
0.5
1.0 1.5 Time (ps)
2.0
2.5
3.0
Radiation field versus time for E ext = 6.7 kV cm−1 (dashed line), 10.05 kV cm−1 (solid line), 13.2 kV cm−1 (dot-dashed line).
is evidenced by the rapid decay of the interband polarization within a few ps. Similar behavior is found for the L–H and R–H polarizations which are originally built up by the laser. Light emission by the oscillating charges may be treated classically, with the DW treated as a point source. The resulting light field is then proportional to the second time-derivative of the number of electrons in the right well. Figure 3 gives the electromagnetic field in the far-field region for E ext = 6.7 kV cm−1 , 10.5 kV cm−1 (near resonance) and 13.2 kV cm−1 . We find that the strongest output signal is obtained for an external field which is somewhat larger than the field 9.7 kV cm−1 , which provides minimal splitting between the two lowest electron subbands.
3. Summary and conclusions We have provided a non-phenomenological model for the damping of charge oscillations in semiconductor double wells (DWs) which allows a quantitative analysis of phase breaking in this system. It is found that the carrier–carrier Coulomb interaction plays a crucial role in the dynamics, providing both coherent effects, such as the formation of excitons, as well as phase breaking. Structural dephasing due to details in the electronic structure of GaAlAs–GaAs DWs is found to play a subordinate role when charge oscillations are due to excitons. We find that it plays a more important role when predominantly free carriers are generated. The predicted temporal evolution of the radiation field is in qualitative agreement with experiment. Substantial broadening of the frequency spectrum arises from the scattering contributions of the carrier–carrier Coulomb interaction [3].
Superlattices and Microstructures, Vol. 20, No. 3, 1996
277
References [1] [2] [3] [4] [5] [6] [7]
L. Esaki and R. Tsu, IBM J. Res. Dev. 14, 61 (1970). H. G. Roskos et al., Phys. Rev. Lett. 68, 2216 (1992); C. Waschke, et al., Phys. Rev. Lett. 70, 3319 (1993). M. S. C. Luo et al., Phys. Rev. B 48, 11043 (1993); I. Brener et al., J. Opt. Soc. Am. B 11, 2457 (1994). W. P¨otz, Appl. Phys. Lett. 68, 2553 (1996). ˇ W. P¨otz, M. Ziger and P. Kocevar, Phys. Rev. B 52, 1959 (1995). W. P¨otz, Phys. Rev. B, July (1996). W. P¨otz, W. Porod, and D. K. Ferry, Phys. Rev. B 32, 3868–3875 (1985).
IR light emission from charge oscillations in semiconductor double wells ¨ tz† W. Po University of Illinois at Chicago, 845 W. Taylor Str., Chicago, Illinois, U.S.A.
(Received 20 May 1996) A microscopic approach to the carrier dynamics in semiconductor heterostructures is used to study structural dephasing and phase breaking due to carrier–carrier interactions in GaAlAs– GaAs double wells. Analysis of interband polarizations allows a quantitative account of the destruction of phase coherence and its consequences on the frequency spectrum of the THz dipole radiation emitted by the carriers. c 1996 Academic Press Limited
1. Introduction Already in 1970, superlattices were proposed as candidates for infrared (IR) light sources [1]. In the last few years, IR light emission by optically excited semiconductor heterostructures, such as superlattices and double wells (DWs), has been used as evidence for the presence of coherent charge oscillations [2]. More recently, the potential of DWs as nanoscale IR light pulse sources has been investigated [3]. Recently, we developed a consistent microscopic approach to the carrier dynamics in semiconductor heterostructures based on a generalization of Boltzmann–Bloch equations [4–6] X γ (kk ) + β (kk ) 1 ≷ ≷ − (γβ (kk )+Re 6αγ (kk) ), k f γβ (T, k ) Re 6αγ T, (i h¯ ∂T − α (kk ) + β (kk )) f αβ (T, k ) = 2 h 2 ¯ γ α (kk ) + γ (kk ) 1 ≷ − f αγ + (αγ (kk ) + Re 6γβ (kk) ), k (T, k ) Re 6γβ T, 2h¯ 2 Z +∞ i h¯ ≷ + dω 6αβ (T, ω, k ) 2π −∞ " # 1 1 × P −P ≷ ≷ h¯ ω + 6αβ (kk) /2 − β (kk ) h¯ ω − 6αβ (kk) /2 − α (kk ) γ (kk ) + β (kk ) 0αγ (kk) + γβ (kk ) 1X > < + (T, k ) − , k f γβ 6αγ T, 2 γ 2h¯ 2 γ (kk ) + β (kk ) 0αγ (kk) + γβ (kk ) < > − , k f γβ (T, k ) −6αγ T, 2h¯ 2 † Research for this article funded by U.S. Army Research Office.
0749–6036/96/070273 + 05 $18.00/0
c 1996 Academic Press Limited
274
Superlattices and Microstructures, Vol. 20, No. 3, 1996
Carrier concentration (1010 cm–2)
0.8
0.6
0.4
0.2
0.0 –0.5
0.5
1.5 Time (ps)
2.5
3.5
Fig. 1. Carrier density versus time. Solid line: electrons in the left well, dot–dashed line: electrons in the right well, dotted line: holes. The dashed line gives the carrier density of electrons in the left well when Coulomb scattering is neglected.
α (kk ) + γ (kk ) 0γβ (kk) + αγ (kk ) + ,k + T, 2h¯ 2 α (kk ) + γ (kk ) 0γβ (k ) + αγ (kk ) > < + ,k . − f αγ (T, k )6γβ T, 2h¯ 2 < > f αγ (T, k )6γβ
(1)
α (kk ) are single-particle energies, α denotes the band index, k is the transverse wave vector, T is time, 6 i is component i of the self-energy, and X is the fast frequency component of term X from the free-particle Hamiltonian, in particular, αβ = (β − α )/h¯ . > (T, k ) ≡ (δαβ − f βα (T, k )), f αβ < (T, k ) ≡ − f βα (T, k ), f αβ
where f βα (T ) are time-dependent carrier distribution functions (α = β) and interband polarizations (α 6= β). This approach allows an account of electron phase coherence in the system, as well as its destruction by interaction processes. These equations are derived from Dyson’s equation for non-equilibrium Green’s functions within the Keldysh approach. Accounting for both distribution functions and all interband polarizations, these equations allow quantitative monitoring of the destruction of electron phase coherence on a sub-picosecond time scale. In addition to the laser field and the free-carrier dynamics in the DW, the carrier–carrier Coulomb interaction is included within the screened Hartree–Fock approximation to the self-energy. Thus excitonic effects, as well as scattering terms which provide phase destruction are included in consistent fashion.
LR-polarization (1010 cm–2)
Superlattices and Microstructures, Vol. 20, No. 3, 1996
275
0.2
0.1
0.0 –0.5
0.5
1.5
2.5
3.5
Time (ps) Fig. 2.
Left–Right electron subband polarization versus time with (solid line) and without (dashed line) Coulomb scattering.
2. Results In the present case of asymmetric GaAs–Alx Ga1−x As DWs, there are two sources for coherence, coherence between laser-generated electron-hole pairs (excitons) and, initially, phase coherence between electrons in ˚ ˚ ˚ Al0.2 Ga0.8 As–GaAs– the left and right well. Specifically, we consider an asymmetric 145 A–25 A–100 A Al0.2 Ga0.8 As DW whose lowest two electronic subbands are biased near resonance by means of an external static electric field (E ext ). For the present calculation, we use a three band model including the two lowest electron subbands, L and R, and the top heavy-hole subband (H). Structurally perfect DWs are assumed. Selective generation of direct excitons in the wide well by a 200 fs laser pulse (h¯ ω ≈ 1.54 eV) leads to damped charge oscillations (quantum beating between direct and indirect excitons) provided that carrier concentrations are kept below about 5 × 1010 cm−2 . The damping of charge oscillations, illustrated in Fig. 1 for an electric field of 10.5 kV cm−1 and a final carrier density of about 0.6 × 1010 cm−2 is caused by two effects, phase breaking due to the carrier–carrier Coulomb interaction and, to a lesser extent, by structural dephasing due to the intrinsic electronic structure of the DW. The latter is a consequence of the effective mass differences in GaAs and AlGaAs, as well as non-parabolicity effects. These were evaluated within a multi-band k · p model [7]. They lead to an effective overlap between left- and right-well wave functions (Rabi frequency) which depends on k|| , the in-plane k-vector. In Fig. 1, the dashed lines give the charge oscillations in the absence of Coulomb scattering. It is seen that many-body effects provide the predominant cause for the decay of exciton-based charge oscillations. Figure 2 gives the magnitude of the L–R well interband polarization versus time. For comparison, we show the time-evolution of the L–R polarization without Coulomb scattering (dashed line). Phase breaking
Superlattices and Microstructures, Vol. 20, No. 3, 1996
Radiation field (a. u.)
276
–0.5
Fig. 3.
0.0
0.5
1.0 1.5 Time (ps)
2.0
2.5
3.0
Radiation field versus time for E ext = 6.7 kV cm−1 (dashed line), 10.05 kV cm−1 (solid line), 13.2 kV cm−1 (dot-dashed line).
is evidenced by the rapid decay of the interband polarization within a few ps. Similar behavior is found for the L–H and R–H polarizations which are originally built up by the laser. Light emission by the oscillating charges may be treated classically, with the DW treated as a point source. The resulting light field is then proportional to the second time-derivative of the number of electrons in the right well. Figure 3 gives the electromagnetic field in the far-field region for E ext = 6.7 kV cm−1 , 10.5 kV cm−1 (near resonance) and 13.2 kV cm−1 . We find that the strongest output signal is obtained for an external field which is somewhat larger than the field 9.7 kV cm−1 , which provides minimal splitting between the two lowest electron subbands.
3. Summary and conclusions We have provided a non-phenomenological model for the damping of charge oscillations in semiconductor double wells (DWs) which allows a quantitative analysis of phase breaking in this system. It is found that the carrier–carrier Coulomb interaction plays a crucial role in the dynamics, providing both coherent effects, such as the formation of excitons, as well as phase breaking. Structural dephasing due to details in the electronic structure of GaAlAs–GaAs DWs is found to play a subordinate role when charge oscillations are due to excitons. We find that it plays a more important role when predominantly free carriers are generated. The predicted temporal evolution of the radiation field is in qualitative agreement with experiment. Substantial broadening of the frequency spectrum arises from the scattering contributions of the carrier–carrier Coulomb interaction [3].
Superlattices and Microstructures, Vol. 20, No. 3, 1996
277
References [1] [2] [3] [4] [5] [6] [7]
L. Esaki and R. Tsu, IBM J. Res. Dev. 14, 61 (1970). H. G. Roskos et al., Phys. Rev. Lett. 68, 2216 (1992); C. Waschke, et al., Phys. Rev. Lett. 70, 3319 (1993). M. S. C. Luo et al., Phys. Rev. B 48, 11043 (1993); I. Brener et al., J. Opt. Soc. Am. B 11, 2457 (1994). W. P¨otz, Appl. Phys. Lett. 68, 2553 (1996). ˇ W. P¨otz, M. Ziger and P. Kocevar, Phys. Rev. B 52, 1959 (1995). W. P¨otz, Phys. Rev. B, July (1996). W. P¨otz, W. Porod, and D. K. Ferry, Phys. Rev. B 32, 3868–3875 (1985).