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Discretizing the electrical equations

Although we assume potential varies linearly between mesh points, the same assumption can not be made for the charge density. Therefore to ensure numerical stability, the Scharfetter-Gummel approach is used so that the derivative to equation 11 and 12 are not evaluated between the left and right mesh point, but at the mid points. To do this we firstly cast the drift diffusion equations as,

$\displaystyle J_{nx} \rvert_{i+1/2}=n \mu \rvert_{i+1/2} \bigg[ \frac{dE_c}{dx}...
...ac{kT}{N_c}\frac{dN_{c}}{dx} \bigg]_{1+1/2}+\mu \rvert_{1+1/2} kT \frac{dn}{dx}$ (21)

this is in the form,

$\displaystyle \frac{dn}{dx}+An=C$ (22)

we can identify the constants as

$\displaystyle A=\frac{1}{kT}\bigg[ \frac{dE_c}{dx}-\frac{kT}{N_c}\frac{dN_{c}}{dx} \bigg]_{1+1/2}$ (23)

and,

$\displaystyle C=\frac{1}{kT}\frac{J_{nx} \rvert_{i+1/2}}{\mu \rvert_{i+1/2}} .$ (24)

Equation 22 can be written as

$\displaystyle \frac{d}{dx} \bigg( e^{Ax} n \bigg)=Ce^{Ax}$ (25)

which can be integrated between the mesh point $ i$ and $ i+1$ to give

$\displaystyle e^{Ax_{i+1}}n_{i+1}-e^{Ax}n_{i}=\frac{C}{A} \bigg( e^{Ax_{i+1}}-e^{Ax} \bigg)$ (26)

rearranging gives,

$\displaystyle =An_{1}\frac{e^{Ax_{i+1}}}{e^{Ax_{i+1}}-e^{Ax_{i}}}-An_{1}\frac{e^{Ax_{i+1}}}{e^{Ax_{i+1}}-e^{Ax_{i}}}$ (27)

$\displaystyle C=\frac{n_{1}}{h_{i+1/2}}\frac{-A h_{i+1/2}}{e^{-Ah_{i+1/2}}-1}-\frac{n_{i+1}}{h_{i+1/2}}\frac{Ah_{i+1/2}}{e^{Ah_{i+1/2}}-1}$ (28)

where the definition $ h_{i+1/2}=x_{i+1}-x_{i}$ has been used, then using the Bernoulli relation

$\displaystyle B(x)=\frac{x}{e^{-x}-1}$ (29)

one can reformulate the equation as

$\displaystyle J_{nx} \rvert_{i+1/2} = q \frac{D_{n}\rvert_{i+1/2}}{h_{i+1/2}} \bigg[ B(-\zeta \rvert_{i+1/2}) n_{i+1}- B(\zeta \rvert_{i+1/2}) n_{i} \bigg]$ (30)

Care has to be taken to evaluating the Bernoulli expression about zero, so the Taylor expansion is used. Note, picking the right number of terms has a significant impact on numerical noise in the solver.

$\displaystyle B(x)=-\frac{1.0}{2.0}+\frac{x}{6.0}-\frac{x^3}{180}+\frac{x^{5}}{5040}$ (31)

The argument to the Bernoulli expression is $ \zeta \rvert_{i+1/2}$ is given as

$\displaystyle \zeta \rvert_{i+1/2}=\frac{h_{i+1/2}}{kT} \bigg( \frac{dE_{c}}{dx}-\frac{kT}{N_c}\frac{dN_{c}}{dx} \bigg) \rvert_{i+1/2}$ (32)

A thermal driving term can also be added to the expression, and $ N_{c}$ should be reformulated as an effective mass when a temperature gradient is present in the device, due to it's thermal dependence.

$\displaystyle \zeta \rvert_{i+1/2}=\frac{1}{kT} \Big( E_{c}\rvert_{i+1}-E_{c}\r...
...2 \frac{N_{c}\rvert_{i+1}-N_{c}\rvert_{i} }{N_{c}\rvert_{i+1}+N_{c}\rvert_{i} }$ (33)

An analogous procedure can be used to derive the the hole current flux which will give

$\displaystyle J_{p}=q \frac{D_{p}\rvert_{i+1/2}}{h_{i+1/2}} \bigg[ B(-\zeta \rvert_{i+1/2}) p_{i+1}- B(\zeta \rvert_{i+1/2}) p_{i} \bigg]$ (34)

$\displaystyle \zeta_{p} \rvert_{i+1/2}=\frac{1}{kT} \Big( E_{v}\rvert_{i+1}-E_{...
...)+\frac{N_{v}\rvert_{i+1}-N_{v}\rvert_{i} }{N_{v}\rvert_{i+1}+N_{v}\rvert_{i} }$ (35)

The standard finite difference scheme can then be used to rewrite the above equations ready for a computer to solve.


next up previous
Next: Solving the electrical equations Up: The physical model Previous: Free-to-free carrier recombination
rod 2017-12-08