Time-dependent internal Quantum Efficiency and diffusion Modulation Transfer Function of N/P photodiodes

Abstract

The minority carrier continuity equation has been solved with the Green’s function approach in a N/P photodiode under the low-level injection assumption. The analytical solution obtained with this approach depends on the three spatial coordinates and on time. The diffusion and the collection of the excess minority carriers have been studied during the transitional period corresponding to very short integration times. The internal Quantum Efficiency and the diffusion Modulation Transfer Function have been calculated according to time. The simulations showed that they evolve with time until their steady-state values. When the integration time is very short, this evolution has to be taken into account for the estimation of the sensitivity of a photodiode and the contrast on an image of a sensor based on several adjacent N/P-type photodiodes.

Appendix

The transformed continuity equation for the excess minority electrons in the region III is

$$\begin{aligned} \frac{\partial ^2 \psi _n}{\partial x^2} +\frac{\partial ^2 \psi _n}{\partial y^2} +\frac{\partial ^2 \psi _n}{\partial z^2} +\frac{g}{D_n}\exp \left( t/\tau _n\right) = \frac{1}{D_n}\frac{\partial \psi _n}{\partial t}, \end{aligned}$$

(23)

with the boundary conditions

$$\begin{aligned} \left\{ \begin{array}{ll} \psi _n(x,y,z,t)=0, \ &{} {\text{at all boundaries,}} \ t>0 \\ \psi _n(x,y,z,0)=F(x,y,z), \ &{} {\text{in the region, at }}P \ t=0 \\ \end{array} \right. \end{aligned}$$

(24)

To determine the appropriate Green’s function, we consider the homogeneous version of this problem as

$$\begin{aligned} \frac{\partial ^2 \psi _n}{\partial x^2} +\frac{\partial ^2 \psi _n}{\partial y^2} +\frac{\partial ^2 \psi _n}{\partial z^2} =\frac{1}{D_n}\frac{\partial \psi _n}{\partial t} \end{aligned}$$

(25)

We apply the technique of separation of variables and we assume a separation in the form

$$\begin{aligned} \psi _n(x,y,z,t)=\varGamma (t)X(x)Y(y)Z(z), \end{aligned}$$

(26)

which leads to the differential equations: $\frac{X''}{X} = -\beta ^2, \frac{Y''}{Y} = -\gamma ^2, \frac{Z''}{Z} = -\eta ^2$ and $-(\beta ^2+\gamma ^2+\eta ^2) \varGamma = \frac{1}{D_n}\varGamma '$, where $\beta , \gamma$ and $\eta$ are constants. The solutions to these differential equations are the eigenfunctions of the Sturm-Liouville problem and are given by

$$\begin{aligned} X(\beta _a,x)= & {} \sin \left( \beta _a x\right) \text { with } \beta _a=\frac{a\pi }{l_1}, a\in {\mathbb {N}^{*}} \end{aligned}$$

(27a)

$$\begin{aligned} Y(\gamma _b,y)= & {} \sin \left( \gamma _b y\right) \text { with } \gamma _b=\frac{b\pi }{l_2}, b\in {\mathbb {N}^{*}} \end{aligned}$$

(27b)

$$\begin{aligned} Z(\eta _c,z)= & {} \sin \left( \eta _c(z-z_P)\right) \text { with } \eta _c=\frac{c\pi }{l_3-z_P}, c\in {\mathbb {N}}^* \end{aligned}$$

(27c)

$$\begin{aligned} \varGamma (t)= & {} \exp \left[ -D_n (\beta _a^2+\gamma _b^2+\eta _c^2)t\right] \end{aligned}$$

(27d)

For other boundary conditions than Eq. (24), one can easily calculate the eigenfunctions or use the Table 2-2 of Ozisik (1993). The complete solution is written as

$$\begin{aligned} \psi _n(x,y,z,t)= {} \sum \limits _{a=1}^\infty \sum \limits _{b=1}^\infty \sum \limits _{c=1}^\infty C_{abc} \exp \left[ -\alpha (\beta _a^2+\gamma _b^2+\eta _c^2)t\right] X(\beta _a,x)Y(\gamma _b,y)Z(\eta _c,z) \end{aligned}$$

(28)

The coefficients $C_{abc}$ are (Ozisik 1993)

$$\begin{aligned} C_{abc}=\int \limits _{x'=0}^{l_1} \int \limits _{y'=0}^{l_2} \int \limits _{z'=z_P}^{l_3} \frac{X(\beta _a,x')}{N(\beta _a)} \frac{Y(\gamma _b,y')}{N(\gamma _b)} \frac{Z(\eta _c,z')}{N(\eta _c)} F(x',y',z')dx'dy'dz' \end{aligned}$$

(29)

where $N(\beta _a), N(\gamma _b)$ and $N(\eta _c)$ are the normalizing factors of eigenfunctions. The substitution of Eq. (29) into Eq. (28) gives the solution as

$$\begin{aligned} \psi _n(x,y,z,t)= & {} \sum \limits _{a=1}^\infty \sum \limits _{b=1}^\infty \sum \limits _{c=1}^\infty \frac{\exp \left[ -D_n(\beta _a^2+\gamma _b^2+\eta _c^2)t\right] }{N(\beta _a) N(\gamma _b) N(\eta _c)} X(\beta _a,x)Y(\gamma _b,y)Z(\eta _c,z)\nonumber \\&\times \int \limits _{x'=0}^{l_1} \int \limits _{y'=0}^{l_2} \int \limits _{z'=z_P}^{l_3} X(\beta _a,x')Y(\gamma _b,y')Z(\eta _c,z')F(x',y',z')dx'dy'dz' \end{aligned}$$

(30)

We deduce from Eq. (30) the Green’s function

$$\begin{aligned} G(x,y,z,t|x',y',z',\tau )= & {} \sum \limits _{a=1}^\infty \sum \limits _{b=1}^\infty \sum \limits _{c=1}^\infty \frac{\exp \left[ -D_n(\beta _a^2+\gamma _b^2+\eta _c^2)(t-\tau )\right] }{N(\beta _a) N(\gamma _b) N(\eta _c)}\nonumber \\&\times X(\beta _a,x)Y(\gamma _b,y)Z(\eta _c,z)X(\beta _a,x')Y(\gamma _b,y')Z(\eta _c,z') \end{aligned}$$

(31)

In order to calculate the Green’s function of the problem, we have considered an arbitrary initial condition F(x, y, z). After obtaining the Green’s function, the solution to (23) is given for any generation function, and any boundary condition by integration (Ozturk et al. 1999). In this article all the boundary conditions are of the first kind, except for the region I where there is a boundary condition of the third kind which is homogeneous. Moreover we consider that at $t=0$ there is no light and no excess minority carriers. Hence the solution to (23) is

$$\begin{aligned} \psi _n(x,y,z,t)= & {} \int _{\tau =0}^{t} \int _{x'=0}^{l_1} \int _{y'=0}^{l_2} \int _{z'=z_P}^{l_3} G(x,y,z,t|x',y',z',\tau )\nonumber &\times \exp \left( \tau /\tau _n\right) g(x',y',z',\tau ) d\tau dx'dy'dz' \end{aligned}$$

(32)

The generation function can be written in the form $g(x,y,z)= l(x,y) \alpha \exp (-\alpha z)$ for any incoming light. The expression of the l function is $l(x,y)=M$ for a spatially uniform photon flux and $l(x,y)=M'\delta (x-l_1/2,y-l_2/2)$ for a point source of light. The expression (32) becomes

$$\begin{aligned} \psi _n(x,y,z,t)= & {} \sum \limits _{a=1}^\infty \sum \limits _{b=1}^\infty \sum \limits _{c=1}^\infty \frac{\exp \left[ -D_n(\beta _a^2+\gamma _b^2+\eta _c^2)t\right] }{N(\beta _a) N(\gamma _b) N(\eta _c)} X(\beta _a,x)Y(\gamma _b,y)Z(\eta _c,z) \nonumber &\times \int \limits _{\tau =0}^{t}\exp \left[ \left( D_n(\beta _a^2+\gamma _b^2+\eta _c^2) +\frac{1}{\tau _n}\right) \tau \right] d\tau \nonumber &\int \limits _{z'=z_P}^{l_3} Z(\eta _c,z')\alpha \exp (-\alpha z')dz'\nonumber &\times \int \limits _{x'=0}^{l_1} \int \limits _{y'=0}^{l_2} X(\beta _a,x')Y(\gamma _b,y')l(x',y')dx'dy' \end{aligned}$$

(33)

The function f is the same in the regions I and III, and is defined as

$$\begin{aligned} f(a,b) \equiv \int \limits _{x'=0}^{l_1} \int \limits _{y'=0}^{l_2} X(\beta _a,x')Y(\gamma _b,y')l(x',y')dx'dy' \end{aligned}$$

(34)