It is possible to calculate the expected photoelectric current, or the electron-hole generation rate from the results of the FDTD simulation. This information can then be used to perform electrical modeling of the device.

## Optical power absorption

It is possible to calculate the power flowing into the active region as described at Absorption per unit volume. This can be accomplished most easily by inserting an advanced Power Absorbed object from the Object Library, as shown below.

This monitor will give the normalized power absorbed per unit volume. For example, two cross sections in the x-z plane at different values of y are shown below.

## Photoelectric conversion

Once the optical power absorbed in the active region is known, it is possible to use the following photoelectric conversion formula to calculate the optical generation rate, \(G\), which is the number of electrons excited per unit volume per unit time, by

$$ {G(\vec{r}, \xi)=\frac{P_{abs}(\vec{r}, \omega)}{\hbar \cdot \omega}=\frac{P_{\text {source}}(\omega)}{\hbar \cdot \omega} \frac{P_{\text {abs}}^{\text {FDTD }}(\vec{r}, \omega)}{P_{\text {source }}^{\text {FDTD }}(\omega)}} $$

where \(\omega \) is the angular frequency and \(\hbar\) is the reduced Planck's constant. Here we have assumed that each photon is absorbed by exciting an electron-hole, which is an excellent assumption at optical wavelengths in silicon. In situations where this is not a good assumption, a quantum efficiency factor can be included in the above formula.

## Source power normalization

To calculate the generation rate for an real experimental setup, the normalized absorbed power,

$$ {\frac{P_{\text {abs}}^{\text {FDTD }}(\vec{r}, \omega)}{P_{\text {source }}^{\text {FDTD }}(\omega)}} $$

which is calculated from the FDTD simulation, should be multiplied by the experimental source power, \(P_{source}\) in \(Watts\).

## \(cos(\theta)\) correction for source illumination

If you are illuminating with a plane wave at an angle, then

$$ {P_{\text {source }}=I \mathrm{A} \cos (\theta)} $$

where \(I\) is the intensity of your source in \(Watts/m^2\), \(A\) is the pixel area and \(\theta\) is the angle of incidence.

If you are illuminating the device with a broadband source then multiply by

$$ {P_{\text {source }}(\omega)=E_{\text {ev }}(\omega) \mathrm{A} \cos (\theta)} $$

where \(E_{ev}\) is the spectral irradiance of the source in \(Watts/m^2/Hz\), \(A\) is the pixel area and \(\theta\) is the angle of incidence. To get the total generation rate, we then need to integrate \(G(\omega)\) over the frequency range of the source.

We may also integrate over the angle \(\theta\) to calculate the generation rate for uniform illumination.

For details on how this calculation can be used to couple with electrical modeling, please see the reference below.

### Related publications

A. Crocherie et al., “From photons to electrons: a complete 3D simulation flow for CMOS image sensor”, IEEE 2009 International Image Sensor Workshop (IISW).