An Unique SPICE Model of Photodiode with Slowly Changeable Carriers’ Velocities

Abstract

This paper deals with a relatively new SPICE model of a P-i-N photodiode. The model includes a change of velocities of electrons and holes, due to the voltage drop on the edges of the photodiode, which depends on the time form of input excitation. We have derived the model of the P-i-N photodiode for digital input excitation i.e. for Heaviside’s square wave time excitation. The model is incorporated in SPICE program and simulated with it. The model and the limitations of the model itself are observed. The output results are compared with the similar ones. It is suggested when it is practical to use the model, and determined the photodiode working domain regime when the model gives accurate results.

Appendix

The diffusion current is obtained using the continuity equation for holes in N region in the case of Dirac’s time input excitation:

$$ \frac{{\partial {p_n}}}{{\partial t}} - {D_p}\frac{{{\partial^2}{p_n}}}{{\partial {x^2}}} + \frac{{{p_n} - {p_{{n0}}}}}{{{\tau_p}}} = \alpha I\exp \left( { - \alpha x} \right)\delta \left( {t - {t_0}} \right). $$

(A.1)

In above equation p _n is the holes’ concentration in N region, D _p is diffusion constant for holes, τ _p is the holes’ life time in N region, p _n0 is the thermal-equilibrium hole’s concentration in N region. Here we proposed border conditions in steady state as ${p_n}\left( {x = {W_n} \sim d} \right) = 0$ and ${p_n}\left( {x = \propto } \right) = {p_{{no}}}$. This approximation is involved with the aim to simplify solving the case. Solution of the Eq. A.1 is

$$ {p_n}\left( {x,t} \right) = {p_{{n0}}} - {p_{{n0}}}\exp \left( {\frac{{d - x}}{{{L_p}}}} \right) + \alpha I\exp \left( { - \alpha x} \right)\exp \left( { - \gamma \left( {t - {t_o}} \right)} \right)h\left( {t - {t_0}} \right), $$

(A.2)

where γ=(1-α ² L _p ² )/τ _p , and x>d. The holes’ diffusion current in N region can be obtained as

$$ {I_{{pdif}}} = - qS{D_p}{\left. {\frac{{\partial {p_n}}}{{\partial x}}} \right|_{{x = d}}}. $$

(A.3)

Considering that the last member in Eq. A.2 has the biggest value, we exclude the rest members and obtain

$$ {I_{{pdif}}} = qS{D_p}{\alpha^2}I\exp \left( { - \alpha d} \right)\exp \left( { - \gamma \left( {t - {t_0}} \right)} \right). $$

(A.4)

Using the Eq. A.4 and parameter τ instead of (t-t ₀ ), we obtain I _pdif for the case of Heaviside’s time input excitation as

$$ {I_{{pdif}}} = qS{D_p}{\alpha^2}\left( {{\rm int} 01} \right)\exp \left( { - \alpha d} \right)\int\limits_{{ - \infty }}^{\infty } {h(t - {t_1}} - \tau )\exp \left( { - \gamma \tau } \right)d\tau, $$

(A.5)

where int 01 is the intensity of Heaviside’s function. Solving the above integral we get Eq. 12.