5.6 Contact Voltage Variable

Having a separate solution variable for the contact voltage avoids numerical problems with large arguments of the Bernoulli function B. Using a Scharfetter-Gummel discretization scheme the expression for the current between two grid points i and j reads

$$\Delta \Delta$$ I_ij = (5.25)

$$\Delta \psi_{j}^{} \psi_{i}^{}$$ = (5.26)

with C_i being material parameters. Applying the contact voltage directly to the boundary grid point could cause large arguments of B and hence numerical problems. This is avoided by having a separate variable for the contact voltage. At the beginning of the iteration procedure the constitutive relation for $\psi_{C}^{}$ is violated and will only successively be adapted which guarantees numerical stability (see Fig. 5.2).

Figure 5.2: Effect of a separate potential variable on the initial-guess of the potential: a) with a separate potential variable the potential stays smooth inside the semiconductor region. b) directly applying the contact potential gives a large discontinuity of the potential.