Abstract

The results of measurements of nonlinear refraction at the absorption edge in InAs between 68 and 90 K taken with an HF laser are compared with those of a band-gap resonant model in which the contribution of the light-hole band is included and found to account for more than 40% of the observed nonlinear refraction. A generalized expression for the nonlinear index is derived by using the complete Fermi–Dirac distribution function. Good agreement between theory and experiment is obtained, with no free parameters.

© 1984 Optical Society of America

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  1. T. S. Moss, Phys. Status Solidi B 101, 555 (1980);A. Elçi, D. Rogovin, Phys. Rev. B 24, 5796 (1981).
    [CrossRef]
  2. S. W. Koch, S. Schmitt-Rink, H. Haug, Phys. Status Solidi B 106, 135 (1981).
    [CrossRef]
  3. P. K. Sen, Solid State Commun. 43, 141 (1982).
    [CrossRef]
  4. D. A. B. Miller, S. D. Smith, B. S. Wherrett, Opt. Commun. 35, 221 (1980);B. S. Wherrett, N. A. Higgins, Proc. R. Soc. London Ser. A 379, 67 (1982).
    [CrossRef]
  5. D. A. B. Miller, C. T. Seaton, M. E. Prise, S. D. Smith, Phys. Rev. Lett. 47, 197 (1981).
    [CrossRef]
  6. J. R. Hill, G. Parry, A. Miller, Opt. Commun. 43, 151 (1982).
    [CrossRef]
  7. R. K. Jain, D. G. Steel, Opt. Commun. 43, 72 (1982).
    [CrossRef]
  8. D. A. B. Miller, IEEE J. Quantum Electron. QE-17, 306 (1981).
    [CrossRef]
  9. P. Lavallard, R. Bichard, C. Benoit a la Guillaume, Phys. Rev. B 16, 2804 (1976).
    [CrossRef]
  10. A. F. Gibson, C. A. Rosito, C. A. Raffo, M. F. Kimmitt, Appl. Phys. Lett. 21, 356 (1972).
    [CrossRef]
  11. J. R. Dixon, J. M. Ellis, Phys. Rev. 123, 1560 (1961).
    [CrossRef]
  12. E. O. Kane, J. Phys. Chem. Solids 1, 249 (1957).
    [CrossRef]
  13. M. I. Iglitsin, E. V. Soloveva, Sov. Phys. Solid State 7, 2770 (1966).
  14. E. Adachi, J. Phys. Soc. Jpn. 24, 1178 (1968).
    [CrossRef]
  15. M. Neuberger, Handbook of Electronic Materials (Plenum, New York, 1971), Vol. 2.
    [CrossRef]
  16. C. D. Poole, E. Garmire, Appl. Phys. Lett. 44, 363 (1984).
    [CrossRef]

1984 (1)

C. D. Poole, E. Garmire, Appl. Phys. Lett. 44, 363 (1984).
[CrossRef]

1982 (3)

P. K. Sen, Solid State Commun. 43, 141 (1982).
[CrossRef]

J. R. Hill, G. Parry, A. Miller, Opt. Commun. 43, 151 (1982).
[CrossRef]

R. K. Jain, D. G. Steel, Opt. Commun. 43, 72 (1982).
[CrossRef]

1981 (3)

D. A. B. Miller, IEEE J. Quantum Electron. QE-17, 306 (1981).
[CrossRef]

S. W. Koch, S. Schmitt-Rink, H. Haug, Phys. Status Solidi B 106, 135 (1981).
[CrossRef]

D. A. B. Miller, C. T. Seaton, M. E. Prise, S. D. Smith, Phys. Rev. Lett. 47, 197 (1981).
[CrossRef]

1980 (2)

T. S. Moss, Phys. Status Solidi B 101, 555 (1980);A. Elçi, D. Rogovin, Phys. Rev. B 24, 5796 (1981).
[CrossRef]

D. A. B. Miller, S. D. Smith, B. S. Wherrett, Opt. Commun. 35, 221 (1980);B. S. Wherrett, N. A. Higgins, Proc. R. Soc. London Ser. A 379, 67 (1982).
[CrossRef]

1976 (1)

P. Lavallard, R. Bichard, C. Benoit a la Guillaume, Phys. Rev. B 16, 2804 (1976).
[CrossRef]

1972 (1)

A. F. Gibson, C. A. Rosito, C. A. Raffo, M. F. Kimmitt, Appl. Phys. Lett. 21, 356 (1972).
[CrossRef]

1968 (1)

E. Adachi, J. Phys. Soc. Jpn. 24, 1178 (1968).
[CrossRef]

1966 (1)

M. I. Iglitsin, E. V. Soloveva, Sov. Phys. Solid State 7, 2770 (1966).

1961 (1)

J. R. Dixon, J. M. Ellis, Phys. Rev. 123, 1560 (1961).
[CrossRef]

1957 (1)

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

Adachi, E.

E. Adachi, J. Phys. Soc. Jpn. 24, 1178 (1968).
[CrossRef]

Benoit a la Guillaume, C.

P. Lavallard, R. Bichard, C. Benoit a la Guillaume, Phys. Rev. B 16, 2804 (1976).
[CrossRef]

Bichard, R.

P. Lavallard, R. Bichard, C. Benoit a la Guillaume, Phys. Rev. B 16, 2804 (1976).
[CrossRef]

Dixon, J. R.

J. R. Dixon, J. M. Ellis, Phys. Rev. 123, 1560 (1961).
[CrossRef]

Ellis, J. M.

J. R. Dixon, J. M. Ellis, Phys. Rev. 123, 1560 (1961).
[CrossRef]

Garmire, E.

C. D. Poole, E. Garmire, Appl. Phys. Lett. 44, 363 (1984).
[CrossRef]

Gibson, A. F.

A. F. Gibson, C. A. Rosito, C. A. Raffo, M. F. Kimmitt, Appl. Phys. Lett. 21, 356 (1972).
[CrossRef]

Haug, H.

S. W. Koch, S. Schmitt-Rink, H. Haug, Phys. Status Solidi B 106, 135 (1981).
[CrossRef]

Hill, J. R.

J. R. Hill, G. Parry, A. Miller, Opt. Commun. 43, 151 (1982).
[CrossRef]

Iglitsin, M. I.

M. I. Iglitsin, E. V. Soloveva, Sov. Phys. Solid State 7, 2770 (1966).

Jain, R. K.

R. K. Jain, D. G. Steel, Opt. Commun. 43, 72 (1982).
[CrossRef]

Kane, E. O.

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

Kimmitt, M. F.

A. F. Gibson, C. A. Rosito, C. A. Raffo, M. F. Kimmitt, Appl. Phys. Lett. 21, 356 (1972).
[CrossRef]

Koch, S. W.

S. W. Koch, S. Schmitt-Rink, H. Haug, Phys. Status Solidi B 106, 135 (1981).
[CrossRef]

Lavallard, P.

P. Lavallard, R. Bichard, C. Benoit a la Guillaume, Phys. Rev. B 16, 2804 (1976).
[CrossRef]

Miller, A.

J. R. Hill, G. Parry, A. Miller, Opt. Commun. 43, 151 (1982).
[CrossRef]

Miller, D. A. B.

D. A. B. Miller, IEEE J. Quantum Electron. QE-17, 306 (1981).
[CrossRef]

D. A. B. Miller, C. T. Seaton, M. E. Prise, S. D. Smith, Phys. Rev. Lett. 47, 197 (1981).
[CrossRef]

D. A. B. Miller, S. D. Smith, B. S. Wherrett, Opt. Commun. 35, 221 (1980);B. S. Wherrett, N. A. Higgins, Proc. R. Soc. London Ser. A 379, 67 (1982).
[CrossRef]

Moss, T. S.

T. S. Moss, Phys. Status Solidi B 101, 555 (1980);A. Elçi, D. Rogovin, Phys. Rev. B 24, 5796 (1981).
[CrossRef]

Neuberger, M.

M. Neuberger, Handbook of Electronic Materials (Plenum, New York, 1971), Vol. 2.
[CrossRef]

Parry, G.

J. R. Hill, G. Parry, A. Miller, Opt. Commun. 43, 151 (1982).
[CrossRef]

Poole, C. D.

C. D. Poole, E. Garmire, Appl. Phys. Lett. 44, 363 (1984).
[CrossRef]

Prise, M. E.

D. A. B. Miller, C. T. Seaton, M. E. Prise, S. D. Smith, Phys. Rev. Lett. 47, 197 (1981).
[CrossRef]

Raffo, C. A.

A. F. Gibson, C. A. Rosito, C. A. Raffo, M. F. Kimmitt, Appl. Phys. Lett. 21, 356 (1972).
[CrossRef]

Rosito, C. A.

A. F. Gibson, C. A. Rosito, C. A. Raffo, M. F. Kimmitt, Appl. Phys. Lett. 21, 356 (1972).
[CrossRef]

Schmitt-Rink, S.

S. W. Koch, S. Schmitt-Rink, H. Haug, Phys. Status Solidi B 106, 135 (1981).
[CrossRef]

Seaton, C. T.

D. A. B. Miller, C. T. Seaton, M. E. Prise, S. D. Smith, Phys. Rev. Lett. 47, 197 (1981).
[CrossRef]

Sen, P. K.

P. K. Sen, Solid State Commun. 43, 141 (1982).
[CrossRef]

Smith, S. D.

D. A. B. Miller, C. T. Seaton, M. E. Prise, S. D. Smith, Phys. Rev. Lett. 47, 197 (1981).
[CrossRef]

D. A. B. Miller, S. D. Smith, B. S. Wherrett, Opt. Commun. 35, 221 (1980);B. S. Wherrett, N. A. Higgins, Proc. R. Soc. London Ser. A 379, 67 (1982).
[CrossRef]

Soloveva, E. V.

M. I. Iglitsin, E. V. Soloveva, Sov. Phys. Solid State 7, 2770 (1966).

Steel, D. G.

R. K. Jain, D. G. Steel, Opt. Commun. 43, 72 (1982).
[CrossRef]

Wherrett, B. S.

D. A. B. Miller, S. D. Smith, B. S. Wherrett, Opt. Commun. 35, 221 (1980);B. S. Wherrett, N. A. Higgins, Proc. R. Soc. London Ser. A 379, 67 (1982).
[CrossRef]

Appl. Phys. Lett. (2)

A. F. Gibson, C. A. Rosito, C. A. Raffo, M. F. Kimmitt, Appl. Phys. Lett. 21, 356 (1972).
[CrossRef]

C. D. Poole, E. Garmire, Appl. Phys. Lett. 44, 363 (1984).
[CrossRef]

IEEE J. Quantum Electron. (1)

D. A. B. Miller, IEEE J. Quantum Electron. QE-17, 306 (1981).
[CrossRef]

J. Phys. Chem. Solids (1)

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

J. Phys. Soc. Jpn. (1)

E. Adachi, J. Phys. Soc. Jpn. 24, 1178 (1968).
[CrossRef]

Opt. Commun. (3)

J. R. Hill, G. Parry, A. Miller, Opt. Commun. 43, 151 (1982).
[CrossRef]

R. K. Jain, D. G. Steel, Opt. Commun. 43, 72 (1982).
[CrossRef]

D. A. B. Miller, S. D. Smith, B. S. Wherrett, Opt. Commun. 35, 221 (1980);B. S. Wherrett, N. A. Higgins, Proc. R. Soc. London Ser. A 379, 67 (1982).
[CrossRef]

Phys. Rev. (1)

J. R. Dixon, J. M. Ellis, Phys. Rev. 123, 1560 (1961).
[CrossRef]

Phys. Rev. B (1)

P. Lavallard, R. Bichard, C. Benoit a la Guillaume, Phys. Rev. B 16, 2804 (1976).
[CrossRef]

Phys. Rev. Lett. (1)

D. A. B. Miller, C. T. Seaton, M. E. Prise, S. D. Smith, Phys. Rev. Lett. 47, 197 (1981).
[CrossRef]

Phys. Status Solidi B (2)

T. S. Moss, Phys. Status Solidi B 101, 555 (1980);A. Elçi, D. Rogovin, Phys. Rev. B 24, 5796 (1981).
[CrossRef]

S. W. Koch, S. Schmitt-Rink, H. Haug, Phys. Status Solidi B 106, 135 (1981).
[CrossRef]

Solid State Commun. (1)

P. K. Sen, Solid State Commun. 43, 141 (1982).
[CrossRef]

Sov. Phys. Solid State (1)

M. I. Iglitsin, E. V. Soloveva, Sov. Phys. Solid State 7, 2770 (1966).

Other (1)

M. Neuberger, Handbook of Electronic Materials (Plenum, New York, 1971), Vol. 2.
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Nonlinear transmission of the 3.096-μm HF laser line through InAs Fabry–Perot étalon at 73 K. Solid curves are data taken at various locations on the étalon. (thickness 140 μm, wedge angle ≤ 0.5 mrad). Dashed curves represent theoretical fit to the data using the measured absorption coefficient (α = 14 cm−1) and n2 = −1.8 × 10−5 cm2/W. Low-intensity round-trip detuning angles are indicated.

Fig. 2
Fig. 2

Nonlinear index of refraction n2 and absorption coefficient α versus temperature in InAs for 3.096-μm radiation. Solid curve is obtained by using Eq. (8) of text with no free parameters. Position of the photon energy relative to the band-gap energy is indicated on top scale [ = (ħωEG)/kT].

Fig. 3
Fig. 3

K(∊, ξ0) (solid curves, left-hand scale) and relative contributions to K from the light-hole and heavy-hole bands, K1h/Khh (dashed curves, right-hand scale) versus in InAs for several values of equilibrium quasi-Fermi level. Shown in parentheses are values of normalized carrier concentration N′, which are related to the corresponding quasi-Fermi levels through the Fermi–Dirac integral: N = ( N 0 / 2 ) ( 2 π 2 / m e k T ) 3 / 2 = 2 π 1 / 2 0 [ 1 + exp ( ρ ξ 0 ) ] 1 ρ 1 / 2 d ρ .

Equations (13)

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n 2 ( ω ) = d n d I | I = 0 = c π 0 [ d α ( ω ) d I ] I = 0 × [ ( ω ) 2 ( ω ) 2 ] 1 d ( ω ) ,
d α ( ω ) d I | I = 0 = α 0 ( ω ) 1 + ( μ l h / μ h h ) 3 / 2 { d d ξ [ f e ( μ h h m e ) ] + ( μ l h μ h h ) 3 / 2 d d ξ [ f e ( μ l h m e ) ] } ξ = ξ 0 ( d ξ d N ) ξ = ξ 0 ( d N d I ) I = 0 .
f e ( x ) = [ exp ( x ξ ) + 1 ] 1 .
N = 2 π 2 ( m e k T 2 ) 3 / 2 0 ρ 1 / 2 f e ( ρ ) d ρ .
α I / ω = N ( N N 0 ) / N 0 τ ,
α 0 ( ω ) = A ( ω E G ) 1 / 2 .
A = 2 5 / 2 e 2 3 c n 0 m 2 E G ( m P 2 2 ) ( μ h h 3 / 2 + μ l h 3 / 2 ) ,
n 2 α τ = 2 π e 2 2 3 n 0 m k T E G 3 ( μ h h m e ) 3 / 2 ( m P 2 2 ) K ( , ξ 0 ) ,
K ( , ξ 0 ) = [ 0 ρ 1 / 2 F 0 ( ρ ) d ρ ] 1 0 [ F 0 ( μ h h m e ) + ( μ 1 h μ h h ) 3 / 2 F 0 ( μ 1 h m e ) ] 1 / 2 d ,
F 0 ( x ) = d d ξ f e ( x ) | ξ = ξ 0 = [ f e ( x ) ( 1 f e ( x ) ] ξ = ξ 0 .
| d E G / d N | N = N 0 K T d ξ / d N | N = N 0 .
| d E G / d N | N = N 0 5 × 10 17 meV cm 3 ,
K T d ξ / d N | N = N 0 3 × 10 16 meV cm 3

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