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, Al_{0.32}Ga_{0.68}As, 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.

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See Ref. 11. m_{Γ}_{h}_{,Γ}_{s}_{,Γ}_{c,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. E_{g} is the band gap. Δ_{ij} are the energy differences between the minima of the various valleys. α_{i} is the nonparabolicity of the various valleys defined by Eq. (28).
“Same” means that a value appropriate to Al_{0.32}Ga_{0.68}As could not be found and that the GaAs value was assumed.

Table 2

Relevant Parameters for the 2.02-eV Transitions in GaAs and Al_{0.32}Ga_{0.68}As

Transitions

Parameters

k (cm^{−1})

E_{c} (eV)

E_{υ} (eV)

Δ (eV)

E_{g} (eV)

P (eV/cm)

GaAs

0.34

1.42

9.58 × 10^{−8}

Heavy hole

1.06 × 10^{7}

1.93

−0.086

Light hole

0.97 × 10^{7}

1.87

−0.152

Split off

0.52 × 10^{7}

1.57

−0.45

AlGaAs

0.31

1.82

9.00 × 10^{−8}

Heavy hole

0.66 × 10^{7}

1.99

−0.028

Light hole

0.51 × 10^{7}

1.92

−0.080

Table 3

Wave-Function Coefficients for the Possible 2.02-eV Transitions in GaAs and Al_{0.32}Ga_{0.68}As

Transitions

Bands

Coefficients

a_{i}

b_{i}

c_{i}

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 r_{cv}^{γδ} 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

r_{ch}^{γδ}′

δ = 3/2

$$i{a}_{c}r(\stackrel{-}{x}+i\u0177)/{6}^{1/2}$$

0

δ = −3/2

0

$$i{a}_{c}r(\stackrel{-}{x}-i\u0177)/{6}^{1/2}$$

r_{cl}^{γδ}′

δ = 1/2

−iρrẑ/3^{1/2}

$$-i\eta r(\stackrel{-}{x}+i\u0177)/{6}^{1/2}$$

δ = −1/2

$$i\eta r(\stackrel{-}{x}-i\u0177)/{6}^{1/2}$$

−iρrẑ/3^{1/2}

r_{cs}^{γδ}′ = r_{cl}^{γδ}′

r is the reduced matrix element for the transitions. ρ and η are derived from the coefficients a_{i}, b_{i}, and c_{i} by ρ = a_{c}c_{υ} − a_{υ}c_{c} and η = a_{c}b_{υ} − a_{υ}b_{c}. 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

Y_{zz}

a_{c}^{2}|r|^{2}/3

(ρ^{2} + η^{2})|r|^{2}/3

Y_{zzzz}

2a_{c}^{4}|r|^{4}/15

(3ρ^{4} + 2η^{4} + 2ρ^{2}η^{2})|r|^{4}/15

Y_{xxzz}

a_{c}^{4}|r|^{4}/10

(2ρ^{4} + 3η^{4} + 8ρ^{2}η^{2})|r|^{4}/30

Y_{xzzx}

a_{c}^{4}|r|^{4}/10

(2ρ^{4} + 3η^{4}−2ρ^{2}η^{2})|r|^{4}/30

r is the reduced matrix element for the transitions. ρ and η are derived from the coefficients a_{i}, b_{i}, and c_{i} by ρ = a_{c}c_{υ} − a_{υ}c_{c} and η = a_{c}b_{υ} − a_{υ}b_{c}, where υ = 1 for the light-hole transition and υ = s for the split-off transition.

Table 6

Y Parameters^{a} for the Possible 2.02-eV Transitions in GaAs and Al_{0.32}Ga_{0.68}As

Transitions

Y Parameters

Y_{zz}

Y_{zzzz}

Y_{xxzz}

Y_{xzzx}

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. Y_{zz} is given in units of |r|^{2}, and Y_{zzzz}, Y_{xxzz}, and Y_{xzzx} are given in units of |r|^{4}, where r is the reduced matrix element for the transitions.

Table 7

Parameters^{a} Used in the Calculation of Scattering Rates

Calculated Scattering Rates^{a} from the Heavy and Light OCR

Type of Scattering

Scattering Rates

Electron OCR

Hole OCR

Heavy

Light

Heavy

Light

GaAs

R_{POP}^{+}

1.7

1.7

5.5

5.5

R_{POP}^{−}

6.4

6.4

18.5

18.5

R_{ACS}

0.6

0.6

4.7

6.2

R_{ALLOY}

0

0

0

0

R_{IV}

25(12)

19(0)

0

0

AlGaAs

R_{POP}^{+}

2.2

2.2

6.8

6.4

R_{POP}^{−}

7.9

7.9

0

19.8

R_{ACS}

0.5

0.4

3.5

6.3

R_{ALLOY}

1.0

1.0

0.4

0.4

R_{IV}

15(9)

9(0)^{f}

0

0

Rates are in units of 10^{12} sec^{−1}. R_{POP}^{+,−} is polar optical scattering for absorption (+) and emission (−). R_{ACS} is acoustic and includes emission and absorption of a phonon, R_{ALLOY} 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 T_{o} rather than T_{r}. R_{IV} is T → L IV deformation-potential scattering by emission and absorption of a phonon using a deformation potential of 1 × 10^{9} eV/cm. The values in parentheses are the Γ → X rates.

See Ref. 11. m_{Γ}_{h}_{,Γ}_{s}_{,Γ}_{c,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. E_{g} is the band gap. Δ_{ij} are the energy differences between the minima of the various valleys. α_{i} is the nonparabolicity of the various valleys defined by Eq. (28).
“Same” means that a value appropriate to Al_{0.32}Ga_{0.68}As could not be found and that the GaAs value was assumed.

Table 2

Relevant Parameters for the 2.02-eV Transitions in GaAs and Al_{0.32}Ga_{0.68}As

Transitions

Parameters

k (cm^{−1})

E_{c} (eV)

E_{υ} (eV)

Δ (eV)

E_{g} (eV)

P (eV/cm)

GaAs

0.34

1.42

9.58 × 10^{−8}

Heavy hole

1.06 × 10^{7}

1.93

−0.086

Light hole

0.97 × 10^{7}

1.87

−0.152

Split off

0.52 × 10^{7}

1.57

−0.45

AlGaAs

0.31

1.82

9.00 × 10^{−8}

Heavy hole

0.66 × 10^{7}

1.99

−0.028

Light hole

0.51 × 10^{7}

1.92

−0.080

Table 3

Wave-Function Coefficients for the Possible 2.02-eV Transitions in GaAs and Al_{0.32}Ga_{0.68}As

Transitions

Bands

Coefficients

a_{i}

b_{i}

c_{i}

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 r_{cv}^{γδ} 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

r_{ch}^{γδ}′

δ = 3/2

$$i{a}_{c}r(\stackrel{-}{x}+i\u0177)/{6}^{1/2}$$

0

δ = −3/2

0

$$i{a}_{c}r(\stackrel{-}{x}-i\u0177)/{6}^{1/2}$$

r_{cl}^{γδ}′

δ = 1/2

−iρrẑ/3^{1/2}

$$-i\eta r(\stackrel{-}{x}+i\u0177)/{6}^{1/2}$$

δ = −1/2

$$i\eta r(\stackrel{-}{x}-i\u0177)/{6}^{1/2}$$

−iρrẑ/3^{1/2}

r_{cs}^{γδ}′ = r_{cl}^{γδ}′

r is the reduced matrix element for the transitions. ρ and η are derived from the coefficients a_{i}, b_{i}, and c_{i} by ρ = a_{c}c_{υ} − a_{υ}c_{c} and η = a_{c}b_{υ} − a_{υ}b_{c}. 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

Y_{zz}

a_{c}^{2}|r|^{2}/3

(ρ^{2} + η^{2})|r|^{2}/3

Y_{zzzz}

2a_{c}^{4}|r|^{4}/15

(3ρ^{4} + 2η^{4} + 2ρ^{2}η^{2})|r|^{4}/15

Y_{xxzz}

a_{c}^{4}|r|^{4}/10

(2ρ^{4} + 3η^{4} + 8ρ^{2}η^{2})|r|^{4}/30

Y_{xzzx}

a_{c}^{4}|r|^{4}/10

(2ρ^{4} + 3η^{4}−2ρ^{2}η^{2})|r|^{4}/30

r is the reduced matrix element for the transitions. ρ and η are derived from the coefficients a_{i}, b_{i}, and c_{i} by ρ = a_{c}c_{υ} − a_{υ}c_{c} and η = a_{c}b_{υ} − a_{υ}b_{c}, where υ = 1 for the light-hole transition and υ = s for the split-off transition.

Table 6

Y Parameters^{a} for the Possible 2.02-eV Transitions in GaAs and Al_{0.32}Ga_{0.68}As

Transitions

Y Parameters

Y_{zz}

Y_{zzzz}

Y_{xxzz}

Y_{xzzx}

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. Y_{zz} is given in units of |r|^{2}, and Y_{zzzz}, Y_{xxzz}, and Y_{xzzx} are given in units of |r|^{4}, where r is the reduced matrix element for the transitions.

Table 7

Parameters^{a} Used in the Calculation of Scattering Rates

Calculated Scattering Rates^{a} from the Heavy and Light OCR

Type of Scattering

Scattering Rates

Electron OCR

Hole OCR

Heavy

Light

Heavy

Light

GaAs

R_{POP}^{+}

1.7

1.7

5.5

5.5

R_{POP}^{−}

6.4

6.4

18.5

18.5

R_{ACS}

0.6

0.6

4.7

6.2

R_{ALLOY}

0

0

0

0

R_{IV}

25(12)

19(0)

0

0

AlGaAs

R_{POP}^{+}

2.2

2.2

6.8

6.4

R_{POP}^{−}

7.9

7.9

0

19.8

R_{ACS}

0.5

0.4

3.5

6.3

R_{ALLOY}

1.0

1.0

0.4

0.4

R_{IV}

15(9)

9(0)^{f}

0

0

Rates are in units of 10^{12} sec^{−1}. R_{POP}^{+,−} is polar optical scattering for absorption (+) and emission (−). R_{ACS} is acoustic and includes emission and absorption of a phonon, R_{ALLOY} 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 T_{o} rather than T_{r}. R_{IV} is T → L IV deformation-potential scattering by emission and absorption of a phonon using a deformation potential of 1 × 10^{9} eV/cm. The values in parentheses are the Γ → X rates.