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Band-to-Band Transitions

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Semiconductor Physics
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Abstract

Optically induced band-to-band transitions are resonance transitions and related to the band structure by the momentum matrix-element and the joint density of states. For transitions near the band edge, the theory of optical transitions between the valence and conduction bands can be simplified with an effective-mass approximation, assuming parabolic band shapes and arriving at quantitative expressions for the absorption as a function of the photon energy. Depending on the conduction-band behavior, strong direct or weak indirect transitions occur at the band edge. In addition, a contribution of forbidden transitions modifies the absorption further away from the band edge. Deviations from the ideal, periodic crystal lattice provide tailing states extending beyond the band edge, usually as an Urbach tail which decreases exponentially with distance from the band edge. In quantum wells the two-dimensional joint density of states leads to a steplike increase of the absorption for increasing photon energy.

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Notes

  1. 1.

    The rewriting of Eq. 5 can be understood from the behavior of the Dirac delta function

    $$ \int g(x)\ \ \ \delta \left[f(x)\right]\; dx=g\left({x}_0\right)\ \ \ {\left|\frac{d\ \ \ f(x)}{dx}\right|}_{x={x}_0}^{-1}. $$
    (6)
  2. 2.

    0, 1, 2, or 3 negative factors a i in Eq. 10 yield for M 0 to M 3 a maximum, saddle point, saddle point, or minimum, respectively, in three dimensions; 0, 1, or 2 negative factors for P 0 to P 2 a maximum, saddle point, or minimum, respectively, in two dimensions; and 0 or 1 negative factors for Q 0 or Q 1 a maximum or minimum, respectively, in one dimension.

  3. 3.

    The momentum matrix-element with dimension W2s4cm−2 should not be confused with the often used oscillator strength

    $$ {f}_{\mathrm{cv}}=\frac{2}{3{m}_0\; h\nu}{\left|\mathbf{e}\cdot {\mathbf{M}}_{\mathrm{cv}}\right|}^2, $$
    (20)

    which is dimensionless and of the order of one, while the matrix element is not. The factor 1/3 in Eq. 19 is due to averaging, with \( {\left|{M}_x\right|}^2={\left|{M}_y\right|}^2={\left|{M}_z\right|}^2=1/3{\left|\mathbf{M}\right|}^2 \); factor 2 accounts for the spin.

  4. 4.

    For semiconductors without inversion symmetry (e.g., GaAs), the valence-band extrema do not occur exactly at k = 0, and the extrapolations of the measured lines in Fig. 13b do not precisely meet in one point at B = 0; a detailed analysis is given by Zwerdling et al. (1957).

  5. 5.

    Such structures are multiple quantum wells or arrays of quantum wires with a sufficient thickness of the separating barrier material or ensembles of quantum dots.

  6. 6.

    For a study of quantum wires, see, e.g., Ihara et al. (2007).

  7. 7.

    An exponential Urbach tail (with Urbach energy in the meV range) was also observed in absorption spectra of high-quality GaAs/AlGaAs quantum wells (Bhattacharya et al. 2015). The broadening is assigned to disorder originating from the electric field of zero-point oscillations of LO phonons in polar semiconductors, yielding a fundamental limit to the Urbach slope.

  8. 8.

    1 dB (decibel) is equal to the logarithm (base ten) of the ratio of two power levels having the value 0.1, that is, log10(p l/p 0) = 0.1 or p l/p 0 = 1.259. For the example given above of 0.16 dB/km, one measures a reduction of the light intensity by 3.8% per km. The intensity is reduced to 33% after 30 km.

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Böer, K.W., Pohl, U.W. (2018). Band-to-Band Transitions. In: Semiconductor Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-69150-3_13

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