Analysis of semiconductor coupled waveguides with interband absorption

Abstract

The main goal of this paper is the discussion of the energy transfer mechanisms between two concentric waveguides. An analytical model is developed for the electromagnetic propagation in two coupled concentric semiconductor waveguides. This model includes the calculation of the semiconductor refractive index taking into account the carrier densities, temperature and frequency. The model allows the analysis of the mechanisms of energy transfer between the two waveguides. As expected two types of energy loss are present, one related to the curvature of the waveguides and the other related to the absorption of the radiation in the semiconductor. Finally, the analyzed structure was optimized with regards to the above parameters.

Appendix

The dielectric constant of the waveguide is a function of the carrier densities in the semiconductors. As a consequence, the propagation constant of the waveguide also depends on the carrier densities.

In a more generic way the Eq. (3) can be written as:

$$\varepsilon^{\prime\prime} = 8\left( {\pi r_{cv} } \right)^{2} \tanh \left[ {\frac{{\hbar \omega - \mathop \sum \nolimits_{\alpha } u_{\alpha } }}{2kT}} \right]\mathop \sum \limits_{\lambda } \left| {\emptyset_{\lambda } \left( {r = 0} \right)} \right|^{2} \delta_{T} \left( {\hbar \omega - W_{\lambda } } \right)$$

The summation has two contributions, one associated with bound states and another with continuous states, and can be written as:

$$\varepsilon^{\prime\prime} = 8\left( {\pi r_{cv} } \right)^{2} \tanh \left[ {\frac{{\hbar \omega - \sum\nolimits_{\alpha } {u_{\alpha } } }}{2kT}} \right]\left\{ { - \frac{1}{{\pi a_{0}^{3} }}\sum\limits_{\lambda = 1}^{{g^{{\frac{1}{2}}} }} {\left[ {\frac{{(\lambda^{2} - g)(2\lambda^{2} - g)}}{{2\lambda^{3} g^{2} }}\prod\limits_{\begin{subarray}{l} n = 1 \\ n \ne \lambda \end{subarray} }^{\infty } {\frac{{n^{2} [(n\lambda )^{2} - (g - \lambda^{2} )^{2} ]}}{{((n\lambda )^{2} - g^{2} )(n^{2} - \lambda^{2} )}}} } \right]} } {\delta_{T} \left( {\hbar \omega - W_{R} \left( {\frac{{g - \lambda^{2} }}{g\lambda }} \right)^{2} } \right) + \mathop{\int}\limits_{0}^{\infty } {dW_{k} W_{k}^{1/2} \delta_{T} (\hbar \omega - W_{k} )} \prod\limits_{n = 1}^{\infty } {\left[ {1 + \frac{{2n^{2} - g^{2} }}{{(n^{2} - g)^{2} + (gn)^{2} x}}} \right]} } \right\}$$

where $g = \frac{12}{{\pi^{2} a_{0} k_{s} }}$, $\delta_{T} \left( x \right) = \frac{1}{{\pi T{ \cosh }\left[ {\frac{x}{T}} \right]}}$, $x = \frac{{W_{k} }}{kT}$, $a_{0} = \frac{{\hbar^{2} {\mathcal{E}}_{0} }}{{mq^{2} }}$ is the Bohr radius, u _α is Fermi level for electrons (α = e) and holes (α = h), m _α is the electron effective mass if α = e and for holes if α = h, $m^{ - 1} = \sum\nolimits_{\alpha } {m_{\alpha }^{ - 1} }$ and r _cv is the element of the transition matrix in the direction of the electric field.

The screening length is determined by:

$$\left( {a_{0} k_{s} } \right)^{2} = \frac{4}{\pi }\left( {\frac{kT}{{W_{R} }}} \right)^{1/2} \mathop \int \limits_{0}^{\infty } dxx^{1/2} \mathop \sum \limits_{\alpha } m_{\alpha }^{3/2} f\left( {x - \frac{{u_{\alpha } - {\raise0.7ex\hbox{${W_{G} }$} \!\mathord{\left/ {\vphantom {{W_{G} } 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}}{kT}} \right)\left[ {1 - f_{\alpha } \left( {x - u_{\alpha } } \right)} \right]$$

where ${\text{f}}_{{{\upalpha }}}$ is the usual Fermi function for free electrons and holes.

The Fermi level for electron and holes is given by:

$$N_{\alpha } = \frac{1}{{2\uppi^{2} a_{0}^{3} }}\left( {\frac{{{\text{T}}m_{\alpha } }}{{W_{R} m}}} \right)^{3/2} \mathop \int \limits_{0}^{\infty } {\text{d}}x x^{1/2} {\text{f}}\left( {x - \frac{{u_{\alpha } - {\raise0.7ex\hbox{${{\text{W}}_{\text{G}} }$} \!\mathord{\left/ {\vphantom {{{\text{W}}_{\text{G}} } 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}}{\text{kT}}} \right)$$

where N _α is the carrier density.

The results of the paper where obtained with the parameters:

m _α  = 0.0665 m ₀, m _h  = 0.52 m ₀, W _R  = 0.0042 eV, m ₀ = 9.109 × 10⁻³¹kg, W _G  = 1.424 eV, q = 1.602 × 10⁻¹⁹ C

The real part of the dielectric constant, $\varepsilon^{\prime}$, is determined from the Kramers–Kronig relation Eq. (5).

The connection between the carrier densities and the characteristics of the structure are implicit in the characteristic equation. This equation results from the application of the boundary conditions, and is used to find the waveguide propagation constant. This equation is a function of the refractive index, the width of the waveguide and the curvature radius. So when the dielectric constant change with the carrier densities, the refractive index will change and the propagation constant will change too.