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The Electron-Hole Interaction and Linear Optical Constants

  • L. X. Benedict (a1)

Abstract

We present an ab initio computational scheme to calculate ε(ω) including the screened electron-hole interaction. Results are presented for GaP and optically-pumped GaAs. We also discuss a time-dependent formulation which has some conceptual advantages.

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5. More precisely, k is the wave vector of the hole. The wave vector of the electron is k + q, where hq is the photon momentum. It is assumed that q = 0.
6. Levine, Z.H. and Louie, S.G., Phys. Rev. B 25, 6310 (1982).
7. Hybertsen, M.S. and Louie, S.G., Phys. Rev. B 37, 2733 (1988).
8. Haydock, R., Comp. Phys. Com. 20, 11 (1980).
9. Numerical Data and Functional Relationships in Science and Technology, edited by Hellwege, K.H., Landolt-Bornstein Tables (Springer, New York, 1982).
10. Shirley, E.L., Phys. Rev. B 54, 16464 (1996).
11. For these calculations we use a k-point shift, so that the F-point (k= 0) is shifted to . Such symmetry breaking is not problematic, because it becomes irrelevant when results are converged.
12. The band stretching is taken from the calculation appearing in Zhu, X. and Louie, S.G., Phys. Rev. B 43, 14142 (1991), while the band gap is from the experimental result listed in Ref. 9.
13. Kittel, C., Introduction to Solid State Physics, (John Wiley and Sons Inc., New York, 1986).
14. The data appearing here are the T = 15 K data in Zollner, S., Garriga, M., Kirchner, J., Humlicek, J., Cardona, M., and Neuhold, G., Phys. Rev. B 48, 7915 (1993).
15. Meskini, N., Mattausch, H.J., and Hanke, W., Sol. Stat. Com. 48, 807 (1983).
16. Hughes, J.L.P. and Sipe, J.E., Phys. Rev. B 53, 10751 (1996).
17. Strictly speaking, ø(τ)) describes a fictitious dynamics of an electron-hole pair, since |ø(τ = 0)) ξ|P) = H−λ J|0) involves the Hamiltonian. Viewed differently, |P) can be thought of as frequency-dependent (H−1 in the expression for |P) arises from the I/ω2 prefactor of Eq. 5).
18. The author has not found a single, simple reason for this trend. It may be a combi- nation of variations in electron density, band width (and therefore electron and hole hopping parameters), bond length, etc.
19. The quantity fo dωωτ2 (ω) is equal for non-interacting theories and conserving in- teracting theories, due to the cancellation of repulsive and attractive terms of the interaction. It should be mentioned, however, that the theory we use (characterized by Eqs. 1, 2 and known as the Tamm-Dancoff approximation) is not a conserving approximation. Whether conserving or non-conserving, an attractive term, Hattr, can be defined for which (P|Hattr|P) is non-zero, measuring the downward shift of oscillator strength.
20. Chemla, D.S., Semiconductors and Semimetals, 58, 175 (1999), and references therein.
21. Huang, L., Callan, J.P., Glezer, E.N., and Mazur, E., Phys. Rev. Lett. 80, 185 (1998).
22. LX. Benedict, unpublished.
23. This approximation neglects contributions from virtual transitions between excited electron and hole states. Such contributions can be thought of as giving rise to a widened gap, which would alter tBG(q), particularly near q = 0. Since the free carrier terms dominate at small q, we neglect this correction. We hasten to add that the static screening approximation is not as justified here as it is in the n = 0 case, due to the presence of excited charge-carrier plasmons. Unfortunately, the difficulty of solving the Bethe-Salpeter equation with a dynamically screened interaction for an infinite system precludes the possibility of taking this important effect into account in this work.

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