Optoelectronical sensors

Cooperation with: CiS Institut für Mikrosensorik gGmbH (Erfurt), Silicon Sensor International AG (Berlin), MPI-Halbleiterlabor (München)

Description: The project is concerned with the mathematical modeling and numerical simulation of optoelectronical semiconductor sensors for Microsystem Technology. Optical sensors and radiation detectors play a key role in robot engineering, materials science, X-ray microscopy, and many other areas of modern technology. In a variety of technical processes there is a need for sensors that are able to register external signals and process them for the control unit. Semiconductor structures are ideally suited for converting incoming information such as light, X-rays or particle radiation into an electrical signal. Their electrical properties change under the influence of radiation due to the generation of pairs of electrons and holes which are separated in the electrical field and are registered at the contacts as currents of electrons and holes, respectively. Semiconductor detectors have great advantages in terms of small dimensions, quantum efficiency, high signal amplification, and chip integrability.

A simultaneous mathematical modeling of light diffusion and absorption as well as charge generation and transport is needed for both the physical understanding and the optimization of optical semiconductor detectors. An adequate model are Maxwell equations for the optical processes and drift-diffusion and grid temperature equations for the electronical processes. These equations are coupled via optical, Avalanche, and thermal source terms.

The aim of the project is to extend and further develop the simulation program WIAS-TeSCA (Two- and three-dimensional Semi-Conductor Analysis package) with regard to new requirements in the development of these devices. This includes the analytical foundation, implementation, and testing of efficient solution algorithms for the coupled system of equations. The results will be evaluated with the help of real-life optical sensors, namely position sensors and Avalanche photodiodes, taken from the manufacturing line of our project partners.

In a first step the project was concerned with the integration of a module to calculate light diffusion in structured optical films. The interaction between light rays and charge transport is modeled with the generation rates in the corresponding partial differential equations. This optical charge generation is described with the ansatz

$\begin{displaymath} G_{opt}(x,y)= \frac {\lambda}{h c} \cdot \eta \cdot \alpha \cdot I(x,y)\,,\end{displaymath}$

where $\lambda$ represents the wave length of the incoming light, $\eta$ is the quantum efficiency (number of generated electron-hole pairs per absorbed photon), $\alpha$ is the absorption constant, h is the Planck constant, c is the speed of light in a vacuum and I is the intensity of the electromagnetic field. On assuming a perpendicular rays direction, the light intensity can be approximated by $I(x,y)= \exp(-\alpha y) \cdot I(x,0)$ ,where y is the coordinate in the direction of the incoming light. In general, however, the light intensity also depends on the optical and geometric characteristics of the structure and on properties of the incoming light, such as spectral distribution, polarization, coherency, and contact angle. Since the wave length is comparable with the dimensions of relevant sensors, bending effects come into play. Hence geometrical optics approaches are no longer sufficient, and the light intensity has to be obtained as a solution of the time-harmonic Maxwell equations.

Under the invariance assumptions valid here, depending on the light polarisation (TE or TM, resp.), these can be reduced to two-dimensional Helmholtz equations of the form

$\begin{displaymath} \begin{gathered} \begin{aligned} \Delta u + k_0^2\varepsilo... ... \hspace{0.6cm} \text{respectively,}\end{aligned}\end{gathered}\end{displaymath}$

where $\varepsilon_{opt}$ denotes the complex-valued refractive index. The function u denotes the component of the electric (TE) or magnetic field (TM), respectively, which is perpendicular to the cross-section of the device. Additionally, the solution has to satisfy radiation conditions at infinity.

The optical equations have to be solved in an expanded two-dimensional domain. Due to the periodicity, the equations can be solved in a bounded periodic cell, capturing the details of the sensor structure, such as optical grids and micro lenses. The radiation conditions as well as the incoming light are modeled with the help of nonlocal boundary conditions.

To solve the Helmholtz equations we make use of the program DiPoG which was developed at WIAS to simulate and optimize periodic diffraction gratings. The program is based on the Finite-Element Method and the forward solver calculates the efficiency of gratings under conical incidence of plane electromagnetic waves. In order to integrate this module into the program WIAS-TeSCA, we implemented in particular an efficient method to model thick layers within the optical structure, different routines to postprocess the calculated field distributions and an effective exchange of the field and intensity distributions within the semiconductor structure.

We successfully performed test computations to simulate a position sensor for the project partner CiS Institut für Mikrosensorik gGmbH. In this case there are several optical layers on the CiS photodiode, two of which have a chrome strip and can be moved against each other. As the structure dimensions are within a few $\mu$ m, the light diffusion in the optical grid is determined by diffraction effects. The computed entities are the characteristics of the photodiode, the photocurrent with fixed voltage depending on the lateral translations of the structured layers with respect to each other.

**Fig. 1:** Intensity distribution of the electromagnetic field and Poynting vector in the air film between the grid structures
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**Fig. 2:** Intensity distribution of the electromagnetic field within the photodiode for different relative positions of the strip structures
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**Fig. 3:** Current-displacement characteristics of the position sensor for different strip and gap width
$\makeatletter \@ZweiProjektbilderNocap[h]{0.46 \textwidth}{cis-curve1.eps}{cis-curve2.eps} \makeatother$