A. J. Taylor, D. J. Erskine, and C. L. Tang, "Ultrafast relaxation dynamics of photoexcited carriers in GaAs and related compounds," J. Opt. Soc. Am. B 2, 663-673 (1985)
The femtosecond intraband relaxation of hot carriers in GaAs, Al0.32Ga0.68As, and the multiple-quantum-well structure is studied using the equal-pulse optical-correlation technique. An overview of the experimental application of this technique to semiconductors is presented. A detailed theoretical analysis of the coherent-artifact contribution to the transmission-correlation peak in the geometry of parallel copropagating beams and a calculation of the saturable-absorption symmetry coefficients for GaAs are given. The relaxation time of carriers from their initially excited states was measured to be in the range 50–100 fsec for the materials studied. The interpretation of the measured relaxation time in terms of electron and hole response functions is discussed. The relevant scattering processes and rates and the corresponding relaxation times calculated from these rates are given.
Dzmitry A. Yarotski, Richard D. Averitt, Nicolas Negre, Scott A. Crooker, Antoinette J. Taylor, Giovanni P. Donati, Andreas Stintz, Luke F. Lester, and Kevin J. Malloy J. Opt. Soc. Am. B 19(6) 1480-1484 (2002)
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See Ref. 11. mΓh,Γs,Γc,L,X are the masses in units of the free-electron mass of the carriers in the heavy and split-off valence bands, in the central-valley conduction band, and in the L and X valleys of the conduction band. Eg is the band gap. Δij are the energy differences between the minima of the various valleys. αi is the nonparabolicity of the various valleys defined by Eq. (28).
“Same” means that a value appropriate to Al0.32Ga0.68As could not be found and that the GaAs value was assumed.
Table 2
Relevant Parameters for the 2.02-eV Transitions in GaAs and Al0.32Ga0.68As
Transitions
Parameters
k (cm−1)
Ec (eV)
Eυ (eV)
Δ (eV)
Eg (eV)
P (eV/cm)
GaAs
0.34
1.42
9.58 × 10−8
Heavy hole
1.06 × 107
1.93
−0.086
Light hole
0.97 × 107
1.87
−0.152
Split off
0.52 × 107
1.57
−0.45
AlGaAs
0.31
1.82
9.00 × 10−8
Heavy hole
0.66 × 107
1.99
−0.028
Light hole
0.51 × 107
1.92
−0.080
Table 3
Wave-Function Coefficients for the Possible 2.02-eV Transitions in GaAs and Al0.32Ga0.68As
Transitions
Bands
Coefficients
ai
bi
ci
GaAs
Heavy hole
Conduction
0.91
0.032
0.42
Light hole
Valence
0.13
−0.96
−0.23
Light hole
Conduction
0.91
0.032
0.40
Split off
Valence
−0.22
−0.55
0.80
Split off
Conduction
0.97
0.024
0.27
AlGaAs
Heavy hole
Conduction
0.97
0.017
0.25
Light hole
Valence
0.15
−0.78
−0.61
Light hole
Conduction
0.97
0.016
0.24
Table 4
Transition-Matrix Element rcvγδ for the Various Valence- to Conduction-Band Transitions for the Direct-Gap Semiconductors Described by the Wave Functions of Eqs. (17) and (18)a
Matrix Elements
Transitions
γ = 1/2
γ = −1/2
rchγδ′
δ = 3/2
0
δ = −3/2
0
rclγδ′
δ = 1/2
−iρrẑ/31/2
δ = −1/2
−iρrẑ/31/2
rcsγδ′ = rclγδ′
r is the reduced matrix element for the transitions. ρ and η are derived from the coefficients ai, bi, and ci by ρ = accυ − aυcc and η = acbυ − aυbc. l refers to the light-hole transition, s to the split-off transition, and h to the heavy-hole transition. k is assumed to be in the z direction.
Table 5
Y Parameters for Direct Band-Gap Semiconductors Described by the Wave Functions of Eqs. (17) and (18)a
Y Parameter
Transitions
Heavy Hole
Light Hole or Split Off
Yzz
ac2|r|2/3
(ρ2 + η2)|r|2/3
Yzzzz
2ac4|r|4/15
(3ρ4 + 2η4 + 2ρ2η2)|r|4/15
Yxxzz
ac4|r|4/10
(2ρ4 + 3η4 + 8ρ2η2)|r|4/30
Yxzzx
ac4|r|4/10
(2ρ4 + 3η4−2ρ2η2)|r|4/30
r is the reduced matrix element for the transitions. ρ and η are derived from the coefficients ai, bi, and ci by ρ = accυ − aυcc and η = acbυ − aυbc, where υ = 1 for the light-hole transition and υ = s for the split-off transition.
Table 6
Y Parametersa for the Possible 2.02-eV Transitions in GaAs and Al0.32Ga0.68As
Transitions
Y Parameters
Yzz
Yzzzz
Yxxzz
Yxzzx
GaAs
Heavy hole
0.28
0.091
0.069
0.069
Light hole
0.28
0.087
0.074
0.056
Split off
0.33
0.135
0.093
0.027
Total
0.29
0.092
0.073
0.060
AlGaAs
Heavy hole
0.31
0.118
0.089
0.089
Light hole
0.32
0.105
0.104
0.029
Total
0.31
0.112
0.096
0.059
The total Y parameter for the 2.02-eV transition can be found by averaging each Y parameter over the possible transitions, weighting each one by the density of states squared. Yzz is given in units of |r|2, and Yzzzz, Yxxzz, and Yxzzx are given in units of |r|4, where r is the reduced matrix element for the transitions.
Table 7
Parametersa Used in the Calculation of Scattering Rates
Calculated Scattering Ratesa from the Heavy and Light OCR
Type of Scattering
Scattering Rates
Electron OCR
Hole OCR
Heavy
Light
Heavy
Light
GaAs
RPOP+
1.7
1.7
5.5
5.5
RPOP−
6.4
6.4
18.5
18.5
RACS
0.6
0.6
4.7
6.2
RALLOY
0
0
0
0
RIV
25(12)
19(0)
0
0
AlGaAs
RPOP+
2.2
2.2
6.8
6.4
RPOP−
7.9
7.9
0
19.8
RACS
0.5
0.4
3.5
6.3
RALLOY
1.0
1.0
0.4
0.4
RIV
15(9)
9(0)f
0
0
Rates are in units of 1012 sec−1. RPOP+,− is polar optical scattering for absorption (+) and emission (−). RACS is acoustic and includes emission and absorption of a phonon, RALLOY is random potential alloy scattering. Note that the latter two rates do not affect the carriers energy but only its momentum; thus they affect only To rather than Tr. RIV is T → L IV deformation-potential scattering by emission and absorption of a phonon using a deformation potential of 1 × 109 eV/cm. The values in parentheses are the Γ → X rates.
See Ref. 11. mΓh,Γs,Γc,L,X are the masses in units of the free-electron mass of the carriers in the heavy and split-off valence bands, in the central-valley conduction band, and in the L and X valleys of the conduction band. Eg is the band gap. Δij are the energy differences between the minima of the various valleys. αi is the nonparabolicity of the various valleys defined by Eq. (28).
“Same” means that a value appropriate to Al0.32Ga0.68As could not be found and that the GaAs value was assumed.
Table 2
Relevant Parameters for the 2.02-eV Transitions in GaAs and Al0.32Ga0.68As
Transitions
Parameters
k (cm−1)
Ec (eV)
Eυ (eV)
Δ (eV)
Eg (eV)
P (eV/cm)
GaAs
0.34
1.42
9.58 × 10−8
Heavy hole
1.06 × 107
1.93
−0.086
Light hole
0.97 × 107
1.87
−0.152
Split off
0.52 × 107
1.57
−0.45
AlGaAs
0.31
1.82
9.00 × 10−8
Heavy hole
0.66 × 107
1.99
−0.028
Light hole
0.51 × 107
1.92
−0.080
Table 3
Wave-Function Coefficients for the Possible 2.02-eV Transitions in GaAs and Al0.32Ga0.68As
Transitions
Bands
Coefficients
ai
bi
ci
GaAs
Heavy hole
Conduction
0.91
0.032
0.42
Light hole
Valence
0.13
−0.96
−0.23
Light hole
Conduction
0.91
0.032
0.40
Split off
Valence
−0.22
−0.55
0.80
Split off
Conduction
0.97
0.024
0.27
AlGaAs
Heavy hole
Conduction
0.97
0.017
0.25
Light hole
Valence
0.15
−0.78
−0.61
Light hole
Conduction
0.97
0.016
0.24
Table 4
Transition-Matrix Element rcvγδ for the Various Valence- to Conduction-Band Transitions for the Direct-Gap Semiconductors Described by the Wave Functions of Eqs. (17) and (18)a
Matrix Elements
Transitions
γ = 1/2
γ = −1/2
rchγδ′
δ = 3/2
0
δ = −3/2
0
rclγδ′
δ = 1/2
−iρrẑ/31/2
δ = −1/2
−iρrẑ/31/2
rcsγδ′ = rclγδ′
r is the reduced matrix element for the transitions. ρ and η are derived from the coefficients ai, bi, and ci by ρ = accυ − aυcc and η = acbυ − aυbc. l refers to the light-hole transition, s to the split-off transition, and h to the heavy-hole transition. k is assumed to be in the z direction.
Table 5
Y Parameters for Direct Band-Gap Semiconductors Described by the Wave Functions of Eqs. (17) and (18)a
Y Parameter
Transitions
Heavy Hole
Light Hole or Split Off
Yzz
ac2|r|2/3
(ρ2 + η2)|r|2/3
Yzzzz
2ac4|r|4/15
(3ρ4 + 2η4 + 2ρ2η2)|r|4/15
Yxxzz
ac4|r|4/10
(2ρ4 + 3η4 + 8ρ2η2)|r|4/30
Yxzzx
ac4|r|4/10
(2ρ4 + 3η4−2ρ2η2)|r|4/30
r is the reduced matrix element for the transitions. ρ and η are derived from the coefficients ai, bi, and ci by ρ = accυ − aυcc and η = acbυ − aυbc, where υ = 1 for the light-hole transition and υ = s for the split-off transition.
Table 6
Y Parametersa for the Possible 2.02-eV Transitions in GaAs and Al0.32Ga0.68As
Transitions
Y Parameters
Yzz
Yzzzz
Yxxzz
Yxzzx
GaAs
Heavy hole
0.28
0.091
0.069
0.069
Light hole
0.28
0.087
0.074
0.056
Split off
0.33
0.135
0.093
0.027
Total
0.29
0.092
0.073
0.060
AlGaAs
Heavy hole
0.31
0.118
0.089
0.089
Light hole
0.32
0.105
0.104
0.029
Total
0.31
0.112
0.096
0.059
The total Y parameter for the 2.02-eV transition can be found by averaging each Y parameter over the possible transitions, weighting each one by the density of states squared. Yzz is given in units of |r|2, and Yzzzz, Yxxzz, and Yxzzx are given in units of |r|4, where r is the reduced matrix element for the transitions.
Table 7
Parametersa Used in the Calculation of Scattering Rates
Calculated Scattering Ratesa from the Heavy and Light OCR
Type of Scattering
Scattering Rates
Electron OCR
Hole OCR
Heavy
Light
Heavy
Light
GaAs
RPOP+
1.7
1.7
5.5
5.5
RPOP−
6.4
6.4
18.5
18.5
RACS
0.6
0.6
4.7
6.2
RALLOY
0
0
0
0
RIV
25(12)
19(0)
0
0
AlGaAs
RPOP+
2.2
2.2
6.8
6.4
RPOP−
7.9
7.9
0
19.8
RACS
0.5
0.4
3.5
6.3
RALLOY
1.0
1.0
0.4
0.4
RIV
15(9)
9(0)f
0
0
Rates are in units of 1012 sec−1. RPOP+,− is polar optical scattering for absorption (+) and emission (−). RACS is acoustic and includes emission and absorption of a phonon, RALLOY is random potential alloy scattering. Note that the latter two rates do not affect the carriers energy but only its momentum; thus they affect only To rather than Tr. RIV is T → L IV deformation-potential scattering by emission and absorption of a phonon using a deformation potential of 1 × 109 eV/cm. The values in parentheses are the Γ → X rates.