2.7 Boundary Conditions

The boundary conditions that are found in [11,19] also apply for magnetic effects. However, for galvanomagnetic simulations, a particular problem arises. It is known that when a magnetic field perpendicular to a current distribution is applied, a Hall voltage appears across the device to compensate the deflection of the current. As a-priori rule it should be known where the discretization must be efficiently fine to resolve this Hall voltage. If this is not possible, dense grids must be used in large structures, which results in expensive simulations with a fairly large demand on computational resources.

The Hall field can be computed as

$$\mathbf{E_H}=\frac{\mathbf{J_n}\times \mathbf{B}}{qn} = -R_n(\mathbf{J_n}\times \mathbf{B})$$ (2.20)

This voltage counteracts the Lorentz force and affects the electric potential and the boundary conditions for the Poisson equation.

For the electric potential the Gauß law states that the potential has to be a continuous function when crossing a surface

$$\epsilon_{I} \frac{\partial \psi}{\partial N}\vert _{I} = \epsilon_{II} \frac{\partial \psi}{\partial N}\vert _{II} + \rho_{\mathrm{surf}}$$ (2.21)

where $\epsilon_{I}$ and $\epsilon_{II}$ are the respective permittivity constants of the materials adjacent to the interface, and $\rho_{\mathrm{surf}}$ is the surface charge density. The non-zero magnetic field differs in the following way.

Computing a fictitious induced surface charge from the Hall field, so that the boundary condition for the electric potential reads:

$$\mathbf{E}\cdot \mathbf{N} = \frac{\rho_{\mathrm{surf}}+\rho_{\mathrm{ind}}}{\epsilon}$$ (2.22)

with

$$\rho_{\mathrm{ind}}=\epsilon \mathbf{E_H}\cdot \mathbf{N}$$ (2.23)

For general bipolar operating conditions, the isothermal case of the boundary condition for the electric potential is:

$$\frac{\partial \psi}{\partial N}\vert _{sc} = -\frac{\rho_{\mathr... ...ac{\sigma_p^2 R_p}{\sigma}(\mathbf{B}\times\nabla(\phi_p-\psi)) \cdot\mathbf{N}$$ (2.24)

where $\sigma = \sigma_n+\sigma_p$ is the transversal ambipolar electric conductivity, and

$$R_H=\frac{\sigma_n^2 R_n + \sigma_p^2 R_p}{\sigma^2}$$ (2.25)

is the ambipolar Hall coefficient.

The boundary conditions along internal interfaces or contacted outer surface portions are not affected by the magnetic field.