Semiconductors — Question 7

Question 7

Diffusion and recombination set a carrier penetration length

For x greater than zero, steady excess holes satisfy $D_p\,d^2\Delta p/dx^2-\Delta p/\tau=0$, with $\Delta p(0)=p_s$ and $\Delta p(\infty)=0$. Use $D_p=.00100\,\mathrm{m^2/s}$ and $\tau=4.00\,\mu\mathrm s$.

1. Solve the boundary-value problem and evaluate the diffusion length.

2. Find the injected hole diffusion current density at the boundary and the distance where the excess density falls to one tenth.

Question 7 - Solution

Diffusion and recombination set a carrier penetration length

(a) Method and calculation

$$\Delta p=Ae^{x/L_p}+Be^{-x/L_p},\qquad L_p=\sqrt{D_p\tau}.$$$$\boxed{\Delta p=p_se^{-x/L_p},\quad L_p=63.25\,\mu\mathrm m}.$$

(b) Interpretation and further calculation

$$\boxed{J_p(0)=-eD_p\left.\frac{d\Delta p}{dx}\right|_0=eD_pp_s/L_p}.$$$$\boxed{x_{10}=L_p\ln10=145.6\,\mu\mathrm m}.$$

Physical check

The growing exponential is excluded by the far boundary condition. Integrating the recombination rate over depth reproduces the boundary particle flux.