This section describes how to calculate the internal quantum efficiency of a CMOS image sensor pixel using Green's function method. This method gives a more complete picture of image sensor efficiency than optical simulations by themselves by incorporating the electrostatic profile of the pixel when determining how much photo-generated charge is collected.

### Background

The internal quantum efficiency (IQE) is the ratio of net electrical carriers collected to the number of incident photons absorbed. The IQE depends strongly on the electrical design of the image sensor, and the doping profile in particular. In the reference paper of Gazeley [1], the authors establish a Green's function approach to calculating the IQE for an arbitrary optical generation rate by determining a spatially-varying weighting function that represents the collection probability for a photo-generated electron-hole pair.

The Green's function approach in Gazeley's paper requires a linear system of equations, which places two constraints on the Poisson and drift-diffusion equations used for the simulation of charge transport.

- The electric field used in the calculation is independent of the carrier distribution
- No carrier density dependent (non-linear) recombination terms are included in the continuity equations

While these conditions are clearly at odds with the physical description of the drift-diffusion/Poisson equations, it is still possible to construct a useful simulation where these approximations are reasonable.

The first condition is only required during the calculation of the weighting function, and doesn't restrict how the (independent) electrostatic potential is calculated.

The second condition can be satisfied by assuming that the contribution of the thermal recombination processes in the system are negligible. This assumption is reasonable at moderate illumination strengths in silicon (where SRH recombination would be dominant, but would remain a small contribution to the carrier density distribution). It is also important to note that radiative recombination (the dual of the photon absorption/ehp generation process) is neglected: although unphysical, silicon is an indirect band gap semiconductor, and the radiative recombination rate is typically negligible.

### The Weighting Function

The weighting function \(W(\mathbf{x}) \)[1] describes the probability that a photo-generated carrier at position \(\mathbf{x} \) is collected at the contact of the kth PD,

$$W(\mathbf{x})=\int_{P D_{k}} \Phi_{x} G(\mathbf{x}, \mathbf{y}) \cdot \mathbf{n} d \mathbf{y}$$

where G is the Green's function.

The weighting function is found by integrating the flux of the Green's function (i.e. the current density in response to a delta function generation rate stimulus) over the surface of the kth PD. In practice, the PDk surface is defined as any surface where 100% of the carriers within the surface are collected; a contact interface is one such surface, or one can be constructed using current flux monitors.

The calculation of W(**x**) requires G(**x**,**y**) for all **x** and **y**, which is equivalent to the knowing the carrier density u(**y**) (u=n,p) in response to a delta generation rate point source located at **x**. The carrier densities are calculated in the course of a CHARGE simulation.

### Calculating the IQE

The weighting function only needs to be calculated once, and can be used for arbitrary optical inputs. For example, the IQE can be calculated over a range of wavelengths, source input angles, microlens shift, etc. by running multiple optical simulations, extracting the local optical generation rate (OGR). By integrating the product of that OGR with the weighting function, we will know the charge per second collected by a contact. IQE can then be obtained by normalizing OGR*W with the total generation rate.

$$Internal\ quantum\ efficiency\ (IQE) = \frac{Charges\ collected\ by\ the\ green\ pixel }{Total\ charges\ generated\ by\ absorption}$$

### Photo Current

By integrating the product of that OGR with the weighting function, we will know the charge per second collected by a contact, which is the photo current.

### Dark Current

Separate self-consistent CHARGE simulations are required to obtain the dark current.

### Related publications

W. Gazeley and D. McGrath, "Quantum Efficiency Simulation Using Transport Equations," International Image Sensor Workshop, R06 (2011)