Interband Optical Absorption in Quantum Well Solar Cells

Abstract

In quantum well solar cells, the eigenstates are bound in one direction and extended in the other two directions. The Empirical k·p Hamiltonian method can also be applied to this case. The envelopes of the four Γ-point Bloch functions contain a one-dimensional bound function in the direction of the growth of the quantum well layer multiplied by a two-dimensional plane wave in the plane perpendicular to the growth (horizontal). The envelope depends on the wave vector of the extended functions and is different for each Γ-point Bloch function. The dipole matrix of the optical transitions differs from the one used in preceding chapters in this book; the optical dipole operator previously used would be non-Hermitical when the initial and the final eigenfunctions are extended in some dimension. This is a very important aspect studied with detail in this chapter. The transitions caused by vertical photons conserve the horizontal wavevector; otherwise the matrix element is zero. An experimental quantum well solar cell has been modeled and its quantum efficiency has been simulated in reasonably good agreement with the measured curve. A clear description of the transitions produced is provided.

Appendix

In our calculations, the practical execution of the integral of Eq. (5.19) is made as follows:

The function inside the integral depends on the array of horizontal wavevectors (k _x , k _y ). We build an array containing in each term $\left\{ {E_{hor,} \frac{{\left| {\left\langle {{{}^{qn}\zeta_{cb,S} }} \mathrel{\left | {\vphantom {{{}^{qn}\zeta_{cb,S} } {{}^{qn^{\prime}}\zeta_{hh,X} }}} \right. \kern-0pt} {{{}^{qn^{\prime}}\zeta_{hh,X} }} \right\rangle } \right|^{2} }}{{E_{tr} }}} \right\}$, both elements being functions of the horizontal wavevector. The integral, restricted to a domain D, must be a function of the energy so that we must sum only those terms of $\frac{{\left| {\left\langle {{{}^{qn}\zeta_{cb,S} }} \mathrel{\left | {\vphantom {{{}^{qn}\zeta_{cb,S} } {{}^{qn^{\prime}}\zeta_{hh,X} }}} \right. \kern-0pt} {{{}^{qn^{\prime}}\zeta_{hh,X} }} \right\rangle } \right|^{2} }}{{E_{tr} }}$ such that their corresponding E _hor is smaller than a given value of E _hor , building in this way the function of E _hor . This function, which rather smooth, is interpolated by a polynomial and its derivative is then taken for use in Eq. (5.18).