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I have a system of coupled non-linear ODEs.

$$J = \mu*e*n(x)*E(x) + \mu*K*T*\frac{dn(x)}{dx}$$ $$\frac{dE(x)}{dx} = \frac{4*\pi*e}{\epsilon}[N_D(x) -n(x)]$$

The first equation is drift-diffusion, the second is Gauss' Law. I am interested in solving this system self-consistently for the carrier profile, $n(x)$, and the electric field, $E(x)$, using finite differences. The current, $J$, the mobility, $\mu$, elementary charge, $e$, are known. The doping profile $N_D(x)$ is known, and I have boundary conditions for the carrier concentration $n(0) = N_D(0)$ and $n(L) = N_D(L)$. I am unsure how to proceed with solving these equations.

I am following the analysis of Baranger and Wilkins in the attached review. Appendix B.

Thanks in advance!

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  • $\begingroup$ Hi Kyle. $\mu$. $e$, $K$, $T$, and $\epsilon$ are indeed constants. I want to approximate the solution of $n(x)$ and $E(x)$ on a spatial domain, say x = [0,L], where the doping $N_D(x)$ varies over the domain. $J$ is the current, but it is a function of drift and diffusion, not just drift, which is the expression you provided. $\endgroup$ – user543898 Mar 20 '18 at 22:56
  • $\begingroup$ J is a constant which is known; the only formula for J is as given. Thanks for your help! $\endgroup$ – user543898 Mar 22 '18 at 1:36
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You can decouple your system by injecting your equation (2) into your equation (1). Doing so, you will get a nonlinear second order differential equation in terms of the electric field $E$. The second equation reads as,

$$n = N - \frac{\epsilon}{4\pi e} \dot{E}$$

from which you derive,

$$\dot{n} = \dot{N} - \frac{\epsilon}{4\pi e} \ddot{E}$$

Letting,

\begin{eqnarray*} a(x) &=& \frac{4\pi e}{\epsilon}\dot N(x) -\frac{4\pi}{\epsilon \mu kT}J\\ b(x) &=& \frac{4\pi e^2}{\epsilon kT}N(x)\\ c &=& \frac{e}{kT} \end{eqnarray*}

The second-order DE can be rewritten, $$\ddot{E}=a(x) + b(x) E + c E\dot{E}$$

This equation can be casted into the form of Abel's equation of the second kind (of the first derivative order) considering as new dependent variable $E$ and new dependent field $u(E)=\dot E$ so that $\ddot{E}=u\dot u$. It can also be casted into the form of Abel's equation of the first kind considering the further change of dependent field $v(E)=\frac{1}{u(E)}$.

Unfortunately I was not able to integrate any of them in closed form.

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  • $\begingroup$ Thanks so much for your comment. I'm having a little trouble understanding you you went from the differential equation: $$\ddot{E}(x) = a(x) + b(x)*E(x) - \frac{e}{K*T}*E(x)*\dot{E}(x)$$ To the functional relation you stated for the field. I understand that the boundary conditions we wish to apply are vanishing field gradient at x = 0, L. Also, is anything in the calculation contingent upon the smoothness of N(x)? For the application of interest, N(x) is piece-wise continuous. $\endgroup$ – user543898 Mar 31 '18 at 21:45
  • $\begingroup$ The ODE is known as as Abel's equation of the second kind. I consulted a reference on Abel's equation of the second and first kind. Suitable variable changes and dependent function substitution allow to lower the ODE order from second to first; I was able to integrate the first order ODE side by side to get the functional relation stated here. I did not make any further asumptions about the smoothness of the doping profile $N(x)$. A good check would be to try to derive the second order equation starting from the functional relation. $\endgroup$ – Pete Apr 2 '18 at 20:52
  • $\begingroup$ My derivation of the functional relation on E was too fast. I made an error. I correct my post indicating a possible way to follow in order to deal with an Abel's equation of the first or second kind (of first derivative order). $\endgroup$ – Pete Apr 3 '18 at 23:35

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