Electric-Field Effects on Optical Absorption near Thresholds in Solids

David E. Aspnes

doi:10.1103/PhysRev.147.554

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Electric-Field Effects on Optical Absorption near Thresholds in Solids

David E. Aspnes

Materials Research Laboratory and Department of Electrical Engineering, University of Illinois, Urbana, Illinois

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Phys. Rev. 147, 554 – Published 15 July, 1966

DOI: https://doi.org/10.1103/PhysRev.147.554

Abstract

The weak-field effective-mass-approximation calculations of the absorption coefficient in the presence of an electric field for direct and indirect transitions at a normal ( $M_{0}$ ) threshold are extended to an arbitrary orientation of the electric field in an anisotropic solid. To do this, a systematic method of evaluating density-of-states integrals arising in the electroabsorption or Franz-Keldysh effect is presented. Certain integrals obtained in prior calculations at the $M_{0}$ threshold which have not been previously evaluated are given in closed form. This method is also used to derive the change in absorption coefficient, $Δ α (E)$ , occuring at the saddle-point edges $M_{1}$ and $M_{2}$ , as well as the edge $M_{3}$ , for an arbitrary field direction in an anisotropic solid. It is shown that there is a direct correlation between $Δ α (E)$ for the $M_{0}$ and $M_{3}$ edges, and also for the $M_{1}$ and $M_{2}$ thresholds. Reduced masses of opposite signs, as in the $M_{1}$ and $M_{2}$ edges, give rise to two branches in $Δ α (E)$ .

References (27)

W. Franz, Z. Naturforsch. 13, 484 (1958)
L. V. Keldysh, Zh. Eksperim. i Teor. Fiz. 34, 1138 (1958) [English transl.: Soviet Phys.—JETP 7, 788 (1958)]
K. Tharmalingam, Phys. Rev. 130, 2204 (1963)
J. Callaway, Phys. Rev. 130, 549 (1963) ibid.134, A998 (1964)
C. M. Penchina, Phys. Rev. 138, A924 (1965)
M. Chester and L. Fritsche, Phys. Rev. 139, A518 (1965)
Y. Yacoby, Phys. Rev. 140, A263 (1965)
A. Frova, P. Handler, F. A. Germano, and D. E. Aspnes, Phys. Rev. 145, 575 (1966)
A. Frova and P. Handler, Phys. Rev. 137, A1857 (1965)
K. L. Shaklee, F. H. Pollak, and M. Cardona, Phys. Rev. Letters 15, 883 (1965)
B. O. Seraphin, Phys. Rev. 140, A1716 (1965)
B. O. Seraphin and N. Bottka, Phys. Rev. 139, A560 (1965)
F. Stern, in Solid State Physics, edited by F. Seitz and D. Turnbull (Academic Press Inc., New York, 1963), Vol. 15, p. 299 Phys. Rev. 133, A1653 (1964)
J. C. Phillips, in Solid State Physics, edited by F. Seitz and D. Turnbull (Academic Press Inc., New York, 1966), Vol. 18, p. 55
L. D. Landau and E. M. Lifschitz, Quantum Mechanics (Pergamon Press, Inc., New York, 1959), pp. 70-71, 491
R. J. Elliott, Phys. Rev. 108, 1384 (1957)
L. Van Hove, Phys. Rev. 89, 1184 (1953)
D. Brust, Phys. Rev. 134, A1337 (1964)
J. Bardeen, F. J. Blatt, and L. H. Hall, in Photoconductivity Conference (John Wiley & Sons, Inc., New York, 1956), p. 146
H. A. Antosiewicz, in Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun (U. S. Department of Commerce, National Bureau of Standards, Washington, D. C., 1964), Appl. Math. Ser. 55, p. 446
[20], p. 447
Landau and Lifschitz ([15]) [3,5]
[20], p. 448
[20], pp. 448-449
B. O. Seraphin and N. Bottka, Phys. Rev. 145, 628 (1966)
J. C. Phillips and B. O. Seraphin, Phys. Rev. Letters 15, 107 (1965) J. C. Phillips, in Proceedings of the International School of Physics "Enrico Fermi," 1966 [Nuovo Cimento Suppl. (to be published)]
C. B. Duke, Phys. Rev. Letters 15, 625 (1965) C. B. Duke and M. E. Alferieff, Phys. Rev. 145, 583 (1966)

[c1] W. Franz, Z. Naturforsch. 13, 484 (1958)

[c2] L. V. Keldysh, Zh. Eksperim. i Teor. Fiz. 34, 1138 (1958) [English transl.: Soviet Phys.—JETP 7, 788 (1958)]

[c3] K. Tharmalingam, Phys. Rev. 130, 2204 (1963)

[c4] J. Callaway, Phys. Rev. 130, 549 (1963) ibid.134, A998 (1964)

[c5] C. M. Penchina, Phys. Rev. 138, A924 (1965)

[c6] M. Chester and L. Fritsche, Phys. Rev. 139, A518 (1965)

[c7] Y. Yacoby, Phys. Rev. 140, A263 (1965)

[c8] A. Frova, P. Handler, F. A. Germano, and D. E. Aspnes, Phys. Rev. 145, 575 (1966)

[c9] A. Frova and P. Handler, Phys. Rev. 137, A1857 (1965)

[c10] K. L. Shaklee, F. H. Pollak, and M. Cardona, Phys. Rev. Letters 15, 883 (1965)

[c11] B. O. Seraphin, Phys. Rev. 140, A1716 (1965)

[c12] B. O. Seraphin and N. Bottka, Phys. Rev. 139, A560 (1965)

[c13] F. Stern, in Solid State Physics, edited by F. Seitz and D. Turnbull (Academic Press Inc., New York, 1963), Vol. 15, p. 299 Phys. Rev. 133, A1653 (1964)

[c14] J. C. Phillips, in Solid State Physics, edited by F. Seitz and D. Turnbull (Academic Press Inc., New York, 1966), Vol. 18, p. 55

[c15] L. D. Landau and E. M. Lifschitz, Quantum Mechanics (Pergamon Press, Inc., New York, 1959), pp. 70-71, 491

[c16] R. J. Elliott, Phys. Rev. 108, 1384 (1957)

[c17] L. Van Hove, Phys. Rev. 89, 1184 (1953)

[c18] D. Brust, Phys. Rev. 134, A1337 (1964)

[c19] J. Bardeen, F. J. Blatt, and L. H. Hall, in Photoconductivity Conference (John Wiley & Sons, Inc., New York, 1956), p. 146

[c20] H. A. Antosiewicz, in Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun (U. S. Department of Commerce, National Bureau of Standards, Washington, D. C., 1964), Appl. Math. Ser. 55, p. 446

[c21] [20], p. 447

[c22] Landau and Lifschitz ([15]) [3,5]

[c23] [20], p. 448

[c24] [20], pp. 448-449

[c25] B. O. Seraphin and N. Bottka, Phys. Rev. 145, 628 (1966)

[c26] J. C. Phillips and B. O. Seraphin, Phys. Rev. Letters 15, 107 (1965) J. C. Phillips, in Proceedings of the International School of Physics "Enrico Fermi," 1966 [Nuovo Cimento Suppl. (to be published)]

[c27] C. B. Duke, Phys. Rev. Letters 15, 625 (1965) C. B. Duke and M. E. Alferieff, Phys. Rev. 145, 583 (1966)

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Electric-Field Effects on Optical Absorption near Thresholds in Solids

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