2.4 Generation profile

The effects of the device structure and layer properties on the generation rate of electron-hole pairs can be taken into account by calculating the optical absorption profile, or generation profile, using the optical transfer matrix model as presented in Refs. [ 8 ] and [ 9 ] . In this model a multilayer device is considered with \(N\) layers. The main properties of interest for a layer with index \(j\) are its thickness \(d_{j}\) and the complex index of refraction \(\tilde{n}_{j} = n_{j} + ik_{j}\) of the layer material. Assumed is that all layers are homogeneous, isotropic and interfaces between layers are parallel. It is also assumed that every photon generates an electron-hole pair.

The generation rate \(G_{j}\) in layer \(j\) as a function of the position \(x\) in the device and wavelength \(\lambda \) is given by

\begin{equation} \label{eq_G_single} G_{j}\left(x, \lambda \right) = \frac{\lambda }{hc}Q_{j}\left(x,\lambda \right) \end{equation}

where \(h\) is Planck’s constant and \(c\) the speed of light. \(Q_{j}\left(x,\lambda \right)\) is the time average of the energy dissipated per second in layer \(j\) for wavelength \(\lambda \) and is given by

\begin{equation} \label{eq_Q_Ediss} Q_{j}\left(x,\lambda \right) = \frac{1}{2}c\epsilon _{0}\alpha _{j}n_{j}\left|E_{j}\left(x\right)\right|^2 \end{equation}

where \(\epsilon _{0}\) is the vacuum permittivity, \(\alpha _{j}\) is the absorption coefficient and \(E_{j}\left(x\right)\) the optical electric field at a position \(x\) in layer \(j\).

In general, the optical electric field at ambient side and substrate side of the device are related via a scattering matrix (transfer matrix) as

\begin{equation} \label{eq_opt_Efield} E^{+,-}_{0} = SE^{+,-}_{m} \end{equation}

where \(E^{+,-}\) represents either the forward and the backward component of the optical electric field. The scattering matrix \(S\) consists of the product of interface and layer matrices.

An interface matrix describes the behaviour of a wave at an interface between two layers in the device. Using the Fresnel complex reflection \(r_{jk}\) and transmission \(t_{jk}\) coefficients, it is given by

\begin{equation} \label{eq_Imatrix} I_{jk}=\frac{1}{t_{jk}} \begin{bmatrix} 1 & r_{jk} \\ r_{jk} & 1 \\ \end{bmatrix} = \begin{bmatrix} \frac{\tilde{n}_{j}+\tilde{n}_{k}}{2\tilde{n}_{j}} & \frac{\tilde{n}_{j}-\tilde{n}_{k}}{2\tilde{n}_{j}} \\ \frac{\tilde{n}_{j}-\tilde{n}_{k}}{2\tilde{n}_{j}} & \frac{\tilde{n}_{j}+\tilde{n}_{k}}{2\tilde{n}_{j}} \\ \end{bmatrix} \end{equation}

A layer matrix describes the propagation of a wave through a layer and is given by

\begin{equation} \label{eq_Lmatrix} L_{j}=\begin{bmatrix} \exp \left(-i\xi _{j}d_{j}\right) & 0 \\ 0 & \exp \left(i\xi _{j}d_{j}\right) \\ \end{bmatrix} \end{equation}

where \(\xi _{j} = \frac{2\pi \tilde{n}_{j}}{\lambda }\). Multiplied, \(\xi _{j}d_{j}\) represents the phase change a wave experiences when travelling trough the layer.

To determine the optical electric field inside a layer \(j\) for position \(x\), \(E_{j}\left(x\right)\), the scattering matrix \(S\) is split into two subsets, separated by layer \(j\)

\begin{equation} S=S'L_{j}S'' \end{equation}

where \(S'\) represents the components of the scattering matrix before \(L_{j}\) and \(S''\) the components of the scattering matrix after \(L_{j}\). They are defined as

\begin{equation} \label{eq_Sprime} S'_{j} = \begin{bmatrix} S’_{j,11} & S’_{j,12} \\ S’_{j,21} & S’_{j,22} \\ \end{bmatrix} = \left(\prod _{v=1}^{j-1}I_{\left(v-1\right)v}L_{v}\right)I_{\left(j-1\right)j} \end{equation}

\begin{equation} \label{eq_Sdprime} S''_{j} = \begin{bmatrix} S”_{j,11} & S”_{j,12} \\ S”_{j,21} & S”_{j,22} \\ \end{bmatrix} = \left(\prod _{v=j+1}^{N}I_{\left(v-1\right)v}L_{v}\right)I_{N\left(N+1\right)} \end{equation}

The total optical electric field in layer \(j\) at position \(x\) is given by

\begin{equation} \label{eq_Efield_layer_gen} E_{j}\left(x\right) = E_{j}^{+}\left(x\right) + E_{j}^{-}\left(x\right) = \left[t_{j}^{+}\exp \left(i\xi _{j}x\right)+t_{j}^{-}\exp \left(-i\xi _{j}x\right)\right]E_{0}^{+} \end{equation}

where \(t_{j}^{+}\) and \(t_{j}^{-}\) are the internal transfer coefficients in the positive and negative direction [ 10 ] . This expression can be rewritten using the scattering matrices

\begin{equation} \label{eq_Efield_layer_detail} E_{j}\left(x\right) = \frac{S''_{j,11}\exp \left(-i\xi _{j}\left(d_{j}-x\right)\right)+S''_{j,21}\exp \left(i\xi _{j}\left(d_{j}-x\right)\right)}{S'_{j,11}S''_{j,11}\exp \left(-i\xi _{j}d_{j}\right)+S'_{j,12}S''_{j,21}\exp \left(i\xi _{j}d_{j}\right)}E_{0}^{+} \end{equation}

where \(E_{0}^{+}\) is the optical electric field at the first interface, which is related to the irradiance \(I_{0}\).

Finally, to take into account the full solar spectrum, we need to integrate the generation rate \(G\left(x,\lambda \right)\) over \(\lambda \)

\begin{equation} \label{eq_G_full} G\left(x\right) = \int _{\lambda _{min}}^{\lambda _{max}}G\left(x,\lambda \right)d\lambda \end{equation}

where \(\lambda _min\) and \(\lambda _{max}\) are the lower and upper bound of the spectrum in terms of wavelength.