Abstract

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.

© 1985 Optical Society of America

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  1. J. L. Oudar, A. Migus, D. Hulin, G. Grillon, J. Etchepare, A. Antonetti, Phys. Rev. Lett. 53, 384 (1984).
    [CrossRef]
  2. C. Shank, R. Fork, R. Yen, J. Shah, B. Greene, A. Gossard, C. Weisbuch, Solid State Commun. 47, 981 (1983).
    [CrossRef]
  3. R. Seymour, M. Junnarkar, R. Alfano, Solid State Commun. 41, 657 (1982).
    [CrossRef]
  4. D. von der Linde, R. Lambrich, Phys. Rev. Lett. 42, 1090 (1979).
    [CrossRef]
  5. A. Smirl, J. Lindle, S. Moss, Phys. Rev. B 18, 5489 (1978).
    [CrossRef]
  6. C. L. Tang, D. J. Erskine, Phys. Rev. Lett. 51, 840 (1983).
    [CrossRef]
  7. D. J. Erskine, A. J. Taylor, C. L. Tang, Appl. Phys. Lett. 45, 54 (1984).
    [CrossRef]
  8. A. J. Taylor, D. J. Erskine, C. L. Tang, in Proceedings of the IVth Workshop on Ultrafast Phenomena (Springer-Verlag, Berlin, 1984).
  9. A. J. Taylor, D. J. Erskine, C. L. Tang, Appl. Phys. Lett. 43, 989 (1983).
    [CrossRef]
  10. E. P. Ippen, C. V. Shank, “Techniques for measurement,” in Ultrashort Light Pulses, S. Shapiro, ed. (Springer-Verlag, New York, 1977).
    [CrossRef]
  11. J. Blakemore, J. Appl. Phys. 53, R123 (1982).
    [CrossRef]
  12. B. S. Wherrett, A. L. Smirl, T. F. Bogess, IEEE J. Quantum Electron. QE-19, 680 (1983).
    [CrossRef]
  13. N. Bloembergen, Nonlinear Optics (Benjamin, Reading, Mass., 1965).
  14. T. F. Heinz, S. L. Palfrey, K. B. Eisenthal, Opt. Lett. 9, 359 (1984).
    [CrossRef] [PubMed]
  15. E. O. Kane, J. Phys. Chem. Solids 1, 249 (1957).
    [CrossRef]
  16. A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton U. Press, Princeton, N.J., 1957).
  17. W. Fawcett, A. Boardman, S. Swain, J. Phys. Chem. Solids 31, 1963 (1970).
    [CrossRef]
  18. J. Hauser, M. Littlejohn, T. Glisson, Appl. Phys. Lett. 28, 459 (1976).
    [CrossRef]
  19. M. Littlejohn, J. Hauser, T. Glisson, D. Ferry, J. Harrison, Solid State Electron. 21, 107 (1978).
    [CrossRef]
  20. M. Littlejohn, J. Hauser, T. Glisson, J. Appl. Phys. 48, 4587 (1977).
    [CrossRef]
  21. D. Pines, D. Bohm, Phys. Rev. 85, 338 (1952).
    [CrossRef]
  22. B. Ridley, Quantum Processes in Semiconductors (Clarendon, Oxford, 1979).

1984 (3)

J. L. Oudar, A. Migus, D. Hulin, G. Grillon, J. Etchepare, A. Antonetti, Phys. Rev. Lett. 53, 384 (1984).
[CrossRef]

D. J. Erskine, A. J. Taylor, C. L. Tang, Appl. Phys. Lett. 45, 54 (1984).
[CrossRef]

T. F. Heinz, S. L. Palfrey, K. B. Eisenthal, Opt. Lett. 9, 359 (1984).
[CrossRef] [PubMed]

1983 (4)

A. J. Taylor, D. J. Erskine, C. L. Tang, Appl. Phys. Lett. 43, 989 (1983).
[CrossRef]

C. L. Tang, D. J. Erskine, Phys. Rev. Lett. 51, 840 (1983).
[CrossRef]

B. S. Wherrett, A. L. Smirl, T. F. Bogess, IEEE J. Quantum Electron. QE-19, 680 (1983).
[CrossRef]

C. Shank, R. Fork, R. Yen, J. Shah, B. Greene, A. Gossard, C. Weisbuch, Solid State Commun. 47, 981 (1983).
[CrossRef]

1982 (2)

R. Seymour, M. Junnarkar, R. Alfano, Solid State Commun. 41, 657 (1982).
[CrossRef]

J. Blakemore, J. Appl. Phys. 53, R123 (1982).
[CrossRef]

1979 (1)

D. von der Linde, R. Lambrich, Phys. Rev. Lett. 42, 1090 (1979).
[CrossRef]

1978 (2)

A. Smirl, J. Lindle, S. Moss, Phys. Rev. B 18, 5489 (1978).
[CrossRef]

M. Littlejohn, J. Hauser, T. Glisson, D. Ferry, J. Harrison, Solid State Electron. 21, 107 (1978).
[CrossRef]

1977 (1)

M. Littlejohn, J. Hauser, T. Glisson, J. Appl. Phys. 48, 4587 (1977).
[CrossRef]

1976 (1)

J. Hauser, M. Littlejohn, T. Glisson, Appl. Phys. Lett. 28, 459 (1976).
[CrossRef]

1970 (1)

W. Fawcett, A. Boardman, S. Swain, J. Phys. Chem. Solids 31, 1963 (1970).
[CrossRef]

1957 (1)

E. O. Kane, J. Phys. Chem. Solids 1, 249 (1957).
[CrossRef]

1952 (1)

D. Pines, D. Bohm, Phys. Rev. 85, 338 (1952).
[CrossRef]

Alfano, R.

R. Seymour, M. Junnarkar, R. Alfano, Solid State Commun. 41, 657 (1982).
[CrossRef]

Antonetti, A.

J. L. Oudar, A. Migus, D. Hulin, G. Grillon, J. Etchepare, A. Antonetti, Phys. Rev. Lett. 53, 384 (1984).
[CrossRef]

Blakemore, J.

J. Blakemore, J. Appl. Phys. 53, R123 (1982).
[CrossRef]

Bloembergen, N.

N. Bloembergen, Nonlinear Optics (Benjamin, Reading, Mass., 1965).

Boardman, A.

W. Fawcett, A. Boardman, S. Swain, J. Phys. Chem. Solids 31, 1963 (1970).
[CrossRef]

Bogess, T. F.

B. S. Wherrett, A. L. Smirl, T. F. Bogess, IEEE J. Quantum Electron. QE-19, 680 (1983).
[CrossRef]

Bohm, D.

D. Pines, D. Bohm, Phys. Rev. 85, 338 (1952).
[CrossRef]

Edmonds, A. R.

A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton U. Press, Princeton, N.J., 1957).

Eisenthal, K. B.

Erskine, D. J.

D. J. Erskine, A. J. Taylor, C. L. Tang, Appl. Phys. Lett. 45, 54 (1984).
[CrossRef]

C. L. Tang, D. J. Erskine, Phys. Rev. Lett. 51, 840 (1983).
[CrossRef]

A. J. Taylor, D. J. Erskine, C. L. Tang, Appl. Phys. Lett. 43, 989 (1983).
[CrossRef]

A. J. Taylor, D. J. Erskine, C. L. Tang, in Proceedings of the IVth Workshop on Ultrafast Phenomena (Springer-Verlag, Berlin, 1984).

Etchepare, J.

J. L. Oudar, A. Migus, D. Hulin, G. Grillon, J. Etchepare, A. Antonetti, Phys. Rev. Lett. 53, 384 (1984).
[CrossRef]

Fawcett, W.

W. Fawcett, A. Boardman, S. Swain, J. Phys. Chem. Solids 31, 1963 (1970).
[CrossRef]

Ferry, D.

M. Littlejohn, J. Hauser, T. Glisson, D. Ferry, J. Harrison, Solid State Electron. 21, 107 (1978).
[CrossRef]

Fork, R.

C. Shank, R. Fork, R. Yen, J. Shah, B. Greene, A. Gossard, C. Weisbuch, Solid State Commun. 47, 981 (1983).
[CrossRef]

Glisson, T.

M. Littlejohn, J. Hauser, T. Glisson, D. Ferry, J. Harrison, Solid State Electron. 21, 107 (1978).
[CrossRef]

M. Littlejohn, J. Hauser, T. Glisson, J. Appl. Phys. 48, 4587 (1977).
[CrossRef]

J. Hauser, M. Littlejohn, T. Glisson, Appl. Phys. Lett. 28, 459 (1976).
[CrossRef]

Gossard, A.

C. Shank, R. Fork, R. Yen, J. Shah, B. Greene, A. Gossard, C. Weisbuch, Solid State Commun. 47, 981 (1983).
[CrossRef]

Greene, B.

C. Shank, R. Fork, R. Yen, J. Shah, B. Greene, A. Gossard, C. Weisbuch, Solid State Commun. 47, 981 (1983).
[CrossRef]

Grillon, G.

J. L. Oudar, A. Migus, D. Hulin, G. Grillon, J. Etchepare, A. Antonetti, Phys. Rev. Lett. 53, 384 (1984).
[CrossRef]

Harrison, J.

M. Littlejohn, J. Hauser, T. Glisson, D. Ferry, J. Harrison, Solid State Electron. 21, 107 (1978).
[CrossRef]

Hauser, J.

M. Littlejohn, J. Hauser, T. Glisson, D. Ferry, J. Harrison, Solid State Electron. 21, 107 (1978).
[CrossRef]

M. Littlejohn, J. Hauser, T. Glisson, J. Appl. Phys. 48, 4587 (1977).
[CrossRef]

J. Hauser, M. Littlejohn, T. Glisson, Appl. Phys. Lett. 28, 459 (1976).
[CrossRef]

Heinz, T. F.

Hulin, D.

J. L. Oudar, A. Migus, D. Hulin, G. Grillon, J. Etchepare, A. Antonetti, Phys. Rev. Lett. 53, 384 (1984).
[CrossRef]

Ippen, E. P.

E. P. Ippen, C. V. Shank, “Techniques for measurement,” in Ultrashort Light Pulses, S. Shapiro, ed. (Springer-Verlag, New York, 1977).
[CrossRef]

Junnarkar, M.

R. Seymour, M. Junnarkar, R. Alfano, Solid State Commun. 41, 657 (1982).
[CrossRef]

Kane, E. O.

E. O. Kane, J. Phys. Chem. Solids 1, 249 (1957).
[CrossRef]

Lambrich, R.

D. von der Linde, R. Lambrich, Phys. Rev. Lett. 42, 1090 (1979).
[CrossRef]

Lindle, J.

A. Smirl, J. Lindle, S. Moss, Phys. Rev. B 18, 5489 (1978).
[CrossRef]

Littlejohn, M.

M. Littlejohn, J. Hauser, T. Glisson, D. Ferry, J. Harrison, Solid State Electron. 21, 107 (1978).
[CrossRef]

M. Littlejohn, J. Hauser, T. Glisson, J. Appl. Phys. 48, 4587 (1977).
[CrossRef]

J. Hauser, M. Littlejohn, T. Glisson, Appl. Phys. Lett. 28, 459 (1976).
[CrossRef]

Migus, A.

J. L. Oudar, A. Migus, D. Hulin, G. Grillon, J. Etchepare, A. Antonetti, Phys. Rev. Lett. 53, 384 (1984).
[CrossRef]

Moss, S.

A. Smirl, J. Lindle, S. Moss, Phys. Rev. B 18, 5489 (1978).
[CrossRef]

Oudar, J. L.

J. L. Oudar, A. Migus, D. Hulin, G. Grillon, J. Etchepare, A. Antonetti, Phys. Rev. Lett. 53, 384 (1984).
[CrossRef]

Palfrey, S. L.

Pines, D.

D. Pines, D. Bohm, Phys. Rev. 85, 338 (1952).
[CrossRef]

Ridley, B.

B. Ridley, Quantum Processes in Semiconductors (Clarendon, Oxford, 1979).

Seymour, R.

R. Seymour, M. Junnarkar, R. Alfano, Solid State Commun. 41, 657 (1982).
[CrossRef]

Shah, J.

C. Shank, R. Fork, R. Yen, J. Shah, B. Greene, A. Gossard, C. Weisbuch, Solid State Commun. 47, 981 (1983).
[CrossRef]

Shank, C.

C. Shank, R. Fork, R. Yen, J. Shah, B. Greene, A. Gossard, C. Weisbuch, Solid State Commun. 47, 981 (1983).
[CrossRef]

Shank, C. V.

E. P. Ippen, C. V. Shank, “Techniques for measurement,” in Ultrashort Light Pulses, S. Shapiro, ed. (Springer-Verlag, New York, 1977).
[CrossRef]

Smirl, A.

A. Smirl, J. Lindle, S. Moss, Phys. Rev. B 18, 5489 (1978).
[CrossRef]

Smirl, A. L.

B. S. Wherrett, A. L. Smirl, T. F. Bogess, IEEE J. Quantum Electron. QE-19, 680 (1983).
[CrossRef]

Swain, S.

W. Fawcett, A. Boardman, S. Swain, J. Phys. Chem. Solids 31, 1963 (1970).
[CrossRef]

Tang, C. L.

D. J. Erskine, A. J. Taylor, C. L. Tang, Appl. Phys. Lett. 45, 54 (1984).
[CrossRef]

A. J. Taylor, D. J. Erskine, C. L. Tang, Appl. Phys. Lett. 43, 989 (1983).
[CrossRef]

C. L. Tang, D. J. Erskine, Phys. Rev. Lett. 51, 840 (1983).
[CrossRef]

A. J. Taylor, D. J. Erskine, C. L. Tang, in Proceedings of the IVth Workshop on Ultrafast Phenomena (Springer-Verlag, Berlin, 1984).

Taylor, A. J.

D. J. Erskine, A. J. Taylor, C. L. Tang, Appl. Phys. Lett. 45, 54 (1984).
[CrossRef]

A. J. Taylor, D. J. Erskine, C. L. Tang, Appl. Phys. Lett. 43, 989 (1983).
[CrossRef]

A. J. Taylor, D. J. Erskine, C. L. Tang, in Proceedings of the IVth Workshop on Ultrafast Phenomena (Springer-Verlag, Berlin, 1984).

von der Linde, D.

D. von der Linde, R. Lambrich, Phys. Rev. Lett. 42, 1090 (1979).
[CrossRef]

Weisbuch, C.

C. Shank, R. Fork, R. Yen, J. Shah, B. Greene, A. Gossard, C. Weisbuch, Solid State Commun. 47, 981 (1983).
[CrossRef]

Wherrett, B. S.

B. S. Wherrett, A. L. Smirl, T. F. Bogess, IEEE J. Quantum Electron. QE-19, 680 (1983).
[CrossRef]

Yen, R.

C. Shank, R. Fork, R. Yen, J. Shah, B. Greene, A. Gossard, C. Weisbuch, Solid State Commun. 47, 981 (1983).
[CrossRef]

Appl. Phys. Lett. (3)

D. J. Erskine, A. J. Taylor, C. L. Tang, Appl. Phys. Lett. 45, 54 (1984).
[CrossRef]

A. J. Taylor, D. J. Erskine, C. L. Tang, Appl. Phys. Lett. 43, 989 (1983).
[CrossRef]

J. Hauser, M. Littlejohn, T. Glisson, Appl. Phys. Lett. 28, 459 (1976).
[CrossRef]

IEEE J. Quantum Electron. (1)

B. S. Wherrett, A. L. Smirl, T. F. Bogess, IEEE J. Quantum Electron. QE-19, 680 (1983).
[CrossRef]

J. Appl. Phys. (2)

J. Blakemore, J. Appl. Phys. 53, R123 (1982).
[CrossRef]

M. Littlejohn, J. Hauser, T. Glisson, J. Appl. Phys. 48, 4587 (1977).
[CrossRef]

J. Phys. Chem. Solids (2)

E. O. Kane, J. Phys. Chem. Solids 1, 249 (1957).
[CrossRef]

W. Fawcett, A. Boardman, S. Swain, J. Phys. Chem. Solids 31, 1963 (1970).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. (1)

D. Pines, D. Bohm, Phys. Rev. 85, 338 (1952).
[CrossRef]

Phys. Rev. B (1)

A. Smirl, J. Lindle, S. Moss, Phys. Rev. B 18, 5489 (1978).
[CrossRef]

Phys. Rev. Lett. (3)

C. L. Tang, D. J. Erskine, Phys. Rev. Lett. 51, 840 (1983).
[CrossRef]

J. L. Oudar, A. Migus, D. Hulin, G. Grillon, J. Etchepare, A. Antonetti, Phys. Rev. Lett. 53, 384 (1984).
[CrossRef]

D. von der Linde, R. Lambrich, Phys. Rev. Lett. 42, 1090 (1979).
[CrossRef]

Solid State Commun. (2)

C. Shank, R. Fork, R. Yen, J. Shah, B. Greene, A. Gossard, C. Weisbuch, Solid State Commun. 47, 981 (1983).
[CrossRef]

R. Seymour, M. Junnarkar, R. Alfano, Solid State Commun. 41, 657 (1982).
[CrossRef]

Solid State Electron. (1)

M. Littlejohn, J. Hauser, T. Glisson, D. Ferry, J. Harrison, Solid State Electron. 21, 107 (1978).
[CrossRef]

Other (5)

B. Ridley, Quantum Processes in Semiconductors (Clarendon, Oxford, 1979).

E. P. Ippen, C. V. Shank, “Techniques for measurement,” in Ultrashort Light Pulses, S. Shapiro, ed. (Springer-Verlag, New York, 1977).
[CrossRef]

A. J. Taylor, D. J. Erskine, C. L. Tang, in Proceedings of the IVth Workshop on Ultrafast Phenomena (Springer-Verlag, Berlin, 1984).

N. Bloembergen, Nonlinear Optics (Benjamin, Reading, Mass., 1965).

A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton U. Press, Princeton, N.J., 1957).

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Figures (7)

Fig. 1
Fig. 1

The solid curve is a TCP for a sample of AlGaAs, and the dashed curve is an AC of the laser pulse.

Fig. 2
Fig. 2

Optical setup for the equal-pulse correlation technique. PBS, BS, 50% polarizing and 4% ordinary beam splitters, respectively; λ/2, half-wave plate; CC, retroreflecting corner cubes; SPKR, speaker; STPMOT, stepping-motor-driven linear actuator; PD1, PD2, P-I-N photodiodes. L1, L2, 40× and 20× microscope objectives, respectively; S, sample; CA, current amplifier; ÷, dividing circuit; LIA, lock-in amplifier; Comp., computer.

Fig. 3
Fig. 3

Band structures of a) GaAs and b) Al0.32Ga0.68As, showing the three valence bands and the central and L valleys of the conduction band. h, l, and s mark the levels optically coupled by the allowed transitions for a 2.02-eV photon from the heavy-hole, light-hole, and split-off valence bands, respectively, to the conduction band.

Fig. 4
Fig. 4

Relaxation time Tr versus photogenerated carrier density for a) GaAs and AlGaAs and b) MQW.

Fig. 5
Fig. 5

Scattering rates for emission and absorption of POP’s as a function of energy for the conduction band of GaAs. This includes the effects of nonparabolic valleys but neglects screening.

Fig. 6
Fig. 6

Scattering rates for emission and absorption of IV phonons, Γ → L, as a function of initial electron energy above the central valley.

Fig. 7
Fig. 7

Calculated free-carrier-carrier rate out of the OCR versus carrier density for GaAs.

Tables (8)

Tables Icon

Table 1 Band-Structure Parametersa

Tables Icon

Table 2 Relevant Parameters for the 2.02-eV Transitions in GaAs and Al0.32Ga0.68As

Tables Icon

Table 3 Wave-Function Coefficients for the Possible 2.02-eV Transitions in GaAs and Al0.32Ga0.68As

Tables Icon

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

Tables Icon

Table 5 Y Parameters for Direct Band-Gap Semiconductors Described by the Wave Functions of Eqs. (17) and (18)a

Tables Icon

Table 6 Y Parametersa for the Possible 2.02-eV Transitions in GaAs and Al0.32Ga0.68As

Tables Icon

Table 7 Parametersa Used in the Calculation of Scattering Rates

Tables Icon

Table 8 Calculated Scattering Ratesa from the Heavy and Light OCR

Equations (51)

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E ( t , r ) = exp ( i k y ) j E j ( t ) ê j
P ( 3 ) ( t , r ) exp ( i k y ) i j k l ê i E j ( t ) × 0 d w E k * ( t w ) E l ( t w ) A i j k l ( t w ) .
A i j k l ( t ) = { Y i j Y k l [ 1 exp ( t / T 0 ) ] + Y i j k l exp ( t / T 0 ) } G ( t ) .
r c υ γ δ = c γ | r | υ δ .
Y i j Y k l = 9 / ( 32 π 2 N 1 ) γ = 1 N 2 β = 1 N 2 δ = 1 N 1 Σ α = 1 N 1 × ( d Ω r c υ γ δ * · ê i * r c υ γ δ · ê j ) × ( d Ω [ r c υ γ α * · ê k * r c υ γ α · ê l + r c υ β δ * · ê k * r c υ β δ · ê l ] ) ,
Y i j k l = 9 / ( 8 π 2 N 1 ) γ = 1 N 2 β = 1 N 2 δ = 1 N 1 Σ α = 1 N 1 × d Ω [ ( r c υ γ δ * · ê i * r c υ γ δ · ê j ) × ( r c υ γ α * · ê k * r c υ γ α · ê l + r c υ β δ * · ê k * r c υ β δ · ê l ) ] .
Δ I = Im [ E * ( t ) · P ( 3 ) ( t ) ] .
TCP ( τ ) i j k l 0 d w A i j k l ( w ) × E i * ( t ) E j ( t ) E k * ( t w ) E l ( t w ) d t .
TCP o ( τ ) 0 d w { A x x z z ( w ) × d t [ E * ( t ) E ( t ) E * ( t w + τ ) E ( t w + τ ) ] + A z z x x ( w ) d t [ E * ( t + τ ) E ( t + τ ) × E * ( t w ) E ( t w ) ] + A x z z x ( w ) × d t [ E * ( t ) E ( t + τ ) E * ( t w + τ ) E ( t w ) ] + A z x x z ( w ) d t [ E * ( t + τ ) E ( t ) E * ( t w ) × E ( t w + τ ) ] + exp ( 2 i ω τ ) A x z x z ( w ) × d t [ E * ( t ) E ( t + τ ) E * ( t w ) E ( t w + τ ) ] + exp ( 2 i ω t ) A z x z x ( w ) d t [ E * ( t + τ ) E ( t ) + E * ( t w + τ ) E ( t w ) ] } .
TCP o ( τ ) 0 d w A x x z z ( w ) [ AC ( τ w ) + AC ( τ + w ) ] + 0 d w A x z z x ( w ) [ ζ ( w , τ ) + ζ ( w , τ ) ] ,
ζ ( w , τ ) = d t E * ( t ) E ( t + τ ) E * ( t + τ w ) E ( t w ) .
TCP p ( τ ) 0 d w A z z z z ( w ) [ AC ( τ w ) + AC ( τ + w ) ] + 0 d w A z z z z ( w ) [ ζ ( w , τ ) + ζ ( w , τ ) ] .
FCA p , o = CA ( 0 ) p , o / TCP ( 0 ) p , o .
FCA o = ( T o / T r ) Y x z z x / [ ( T o / T r ) ( Y x z z x + Y x x z z ) + Y z z 2 ] .
R = TCP ( 0 ) p / TCP ( 0 ) o .
R = 2 [ Y z z 2 + Y z z z z ( T o / T r ) ] / × [ Y z z 2 + ( Y x z z x + Y x x z z ) ( T o / T r ) ] ,
T o / T r = Y z z 2 ( 2 R ) / [ R ( Y x z z x + Y x x z z ) 2 Y z z z z ] .
ϕ h 3 / 2 = [ ( X + i Y ) ] / 2 1 / 2 ,
ϕ h 3 / 2 = [ ( X i Y ) ] / 2 1 / 2 ,
ϕ i 1 / 2 = a i [ i S ] + b i [ ( X + i Y ) ] / 2 1 / 2 + c i [ Z ] ,
ϕ i 1 / 2 = a i [ i S ] + b i [ ( X i Y ) ] / 2 1 / 2 + c i [ Z ] .
a i = k P ( E i + 2 / 3 Δ ) / N ,
b i = 2 1 / 2 Δ ( E i E G ) / 3 N ,
c i = ( E i E G ) ( E i 2 Δ / 3 ) / N ,
r c υ γ δ = M r c υ γ δ ,
M = ( cos β cos α sin α sin β cos α cos β sin α cos α sin β sin α sin β 0 cos β ) .
G ( t ) = a G e ( t ) + ( 1 a ) G h ( t ) , 0 a 1 .
G ( t ) M υ G e ( t ) + M c G h ( t ) ,
R POP = e ζ ( 2 m γ ) 1 / 2 ( 1 + 2 α E ) F o ( E , E ) × { N ( ω o p ) absorption N ( ω o p ) + 1 emission ,
F o ( E , E ) = ( 1 / C ) [ A ln | γ 1 / 2 ( E ) + γ 1 / 2 ( E ) | | γ 1 / 2 ( E ) γ 1 / 2 ( E ) | + B ] ,
A = { 2 ( 1 + α E ) ( 1 + α E ) + α [ γ ( E ) + γ ( E ) ] } 2 ,
B = 2 α γ 1 / 2 ( E ) γ ( E ) } 4 ( 1 + α E ) ( 1 + α E ) + α [ γ ( E ) + γ ( E ) ] } ,
C = 4 ( 1 + α E ) ( 1 + α E ) ( 1 + 2 α E ) ( 1 + 2 α E ) ,
γ ( E ) = 2 k 2 / 2 m = E ( 1 + α E ) ,
E = { E + ω op for absorption E ω op for emission ,
e ζ = m e 2 ω op ( 1 o 1 ) / 2 ,
N ( ω op ) = [ exp ( ω op / k B T ) 1 ] 1 .
R ACS = 2 1 / 2 D ACS 2 k B T E 1 / 2 m 3 / 2 / ( π 4 ρ υ L 2 ) ,
R ALLOY = ( 3 π m 3 / 2 / 8 4 2 1 / 2 ) x ( 1 x ) × γ ( E ) ( 1 + 2 α E ) Ω | Δ U | 2 S ,
R IV = Z j m j 3 / 2 D i j 2 1 / 2 π ρ ω i j 2 E j 1 / 2 G ( E i , E j ) × { N ( ω op ) absorption N ( ω op ) + 1 emission ,
E j = E i Δ i j ± ω i j ,
G ( E i , E j ) = ( 1 + α i E i ) ( 1 + α i E i ) / [ ( 1 + 2 α i E i ) ( 1 + 2 α j E j ) ] .
q o 2 = 4 π e 2 n / k B T c ,
R POP + , = e ζ ( 2 m E ) 1 / 2 { N ( ω op ) N ( ω op ) + 1 × { ½ ln [ ( q max 2 + q o 2 ) / ( q min 2 + q o 2 ) ] + ½ q o 2 [ ( q max 2 + q o 2 ) 1 + ( q min 2 + q o 2 ) 1 } ,
q max , min = k [ ( 1 + ω op / E ) 1 / 2 ± 1 ] ,
q max , min = k [ 1 ± ( 1 ω op / E ) 1 / 2 ] .
σ ( α , k c m ) = U { k c m 2 ( 1 + cos α ) + q o 2 / 4 ] 2 + [ k c m 2 ( 1 cos α ) + q 0 2 / 4 ] 2 + [ k c m 2 ( 1 + cos α ) + q o 2 / 4 ] 1 × [ k c m 2 ( 1 cos α ) + q o 2 / 4 ] 1 } ,
i a c r ( x + i ŷ ) / 6 1 / 2
i a c r ( x i ŷ ) / 6 1 / 2
i η r ( x + i ŷ ) / 6 1 / 2
i η r ( x i ŷ ) / 6 1 / 2

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