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I am trying to understand an extra terms that appears when I derive the drift-diffusion equations for semiconductors. The extra term (see below) comes from applying the chain rule to the advection term. The extra term behaves as a source term, however, I can't justify its physical meaning.

I think the semiconductor transport equations are analogous to solving the an advection-diffusion problem: the semiconductor equations possess a diffusion term and drift (advection) to describe current flow. These equations are combined with the continuity equation so that particle number is conserved. The reaction-advection-diffusion equation has the same properties,

$$\frac{\partial n}{\partial t} = \nabla\cdot\left(\boldsymbol{v}n + D\nabla n \right) + S$$

where $n$ is the electron density, $\boldsymbol{v}$ is the drift velocity, $D$ is the diffusion coefficient and $S$ is the source function (generation and recombination of electrons).

In semiconductors the drift velocity is a function of electric field, $\boldsymbol{v}=\mu \boldsymbol{E}$ (where $\mu$ is a material constant - the mobility). When there is no divergence in electric field the drift velocity is zero and the transport is dominated by diffusion. In turn the electric field is found my solving Poisson equations.

Note that the differental operator acts on $\boldsymbol{v}$ and $n$ which are both function of $x$ therefore we must apply the chain rule to the adevection term,

$$\nabla\cdot(\boldsymbol{v}n) = \boldsymbol{v}\frac{\partial n}{\partial x} + n \frac{\partial\boldsymbol{v}}{\partial x}$$

Substituting in the expression for drift velocity gives,

$$\nabla\cdot(\boldsymbol{v_n}n) = \mu\boldsymbol{E}\frac{\partial n}{\partial x} + \mu n \frac{\partial\boldsymbol{E}}{\partial x}$$

Therefore the final expression is,

$$\frac{\partial n}{\partial t} = \underbrace{D\frac{\partial^2 n}{\partial x^2}}_{\textrm{Diffusion}} + \underbrace{\overbrace{\mu\boldsymbol{E}}^{\boldsymbol{v}}\frac{\partial n}{\partial x}}_{\textrm{Advection}} + \underbrace{\mu n \frac{\partial \boldsymbol{E}}{\partial x}}_{\textrm{?}} + \underbrace{S}_{\textrm{Source}}$$

The term marked with the question mark acts like an additional source term which generates/removes (depending on the sign) electrons when the field divergence is non-zero. If you take the example of the pn-junction, there is always a build-in electric field so how can this ever reach steady-state?

Simulation

I have simulated the formation of the pn-junction (neutral p-type on the left, joined to neutral n-type on the right) and I do indeed get generation of carriers in the region where there is no divergence of the electric field (i.e. the depletion region). To solve the equations I am doing a Crank-Nicolson time sweep. At every step I solve the Poisson equation to give the electric field profile, which is used in the subsequent time step to calculate electron (and hole) density.

Does anybody have an explanation for additional term? Or maybe there is something fundamentally wrong with my approach to solving the equations i.e. is re-calculating the Poisson equation at each time step valid and substituting the electric field into the advection-diffusion equations a valid approach?

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The answer is yes. Drift diffusion = advection-diffusion. You will find this rather frequently: different disciplines, trying to describe the phenomena in their field, assign names to the equations they derive that have meaning for this discipline. Other disciplines come up with the same equations, in different contexts, and name them differently. It's still the same equation.

(Another example: Neutron diffusion approximation == Poisson equation.)

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  • $\begingroup$ Thanks for your comments Wolfgang, but I answered my own question (before you posted your answer); I had some useful discussion offline. $\endgroup$ – boyfarrell Mar 20 '13 at 14:00
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(I realised the issue, after having a discussion offline).

The answer is yes, and the equations are correct.

The error is in assuming that the electron and hole densities should remain constant when both halves of the pn-junction are brought together. Obviously when the free carrier flow into each other under diffusion they "annihilate" causing a change in carrier number. This will continue until the divergence of the electric field becomes zero at equilibrium. Provided the electric field is calculated self-consistently with the Poisson equation, the resulting transient is physically correct.

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