3.5 The New Method

The new method for specifying interface conditions is capable of handling all types of boundary conditions used in device simulation, including the extreme cases of Dirichlet (3.5) and Neumann (3.6) boundary conditions.

u = a (3.5)

$$= b$$ n ^. gradu (3.6)

In (3.5) the value for the scalar quantity u is defined whereas in (3.6) the normal component of the gradient of u is specified.

The method consists of a transformation of the equations describing the flux across the interface and the elimination of large common factors. Thereby the spectral condition number $\ensuremath{\kappa_{\mathrm{s}}}$ of the system matrix [24][25] is reduced.

$$\kappa_{\mathrm{s}}^{} {\frac{\vert\lambda\vert _{\rm max}}{\vert\lambda\vert _{\rm min}}}$$ (3.7)

|$\lambda$|_max and |$\lambda$|_min are the eigenvalues with maximum and minimum magnitude of the system matrix. For iterative linear equation solvers the spectral condition number is a measure for the rate of convergence. The larger the value of $\kappa_{\mathrm{s}}^{}$ the poorer is the condition of the system matrix. The accuracy of the solution of the linearized equation system influences the convergence of the nonlinear solving scheme. Insufficient accuracy in the solution of the linear equation system will increase the number of iteration steps needed to achieve convergence for the nonlinear problem or even prevent convergence for equation systems with small convergence radius.

The unified treatment of interface conditions is demonstrated for modeling the current flow across a heterojunction interface. When a finite box discretization scheme (see Fig. 3.6) is used, one obtains for the discretization point i on the interface

$$\sum_{j_{1}}^{} \perp$$ box i₁ :          (3.8)

$$\sum_{j_{2}}^{} \perp$$ box i₂ :          (3.9)

Figure 3.6: Finite box discretization.

The subscripts 1 and 2 denote quantities associated to regions 1 and 2, respectively. For J_ij the standard model for the bulk current density is used:

$${\frac{\partial n}{\partial x}}$$ (3.10)

which is discretized to

$${\frac{n_{i} - n_{j}}{d_{ij}}}$$ (3.11)

The current flow across the heterojunction interface J_$\perp$ is modeled by [26]:

$$\perp \alpha \left(\vphantom{e_{\mathrm{b}}}\right. \left.\vphantom{e_{\mathrm{b}}}\right) \left(\vphantom{v_{2}\cdot n_{2} -\frac{m_{2}}{m_{1}}\cdot v_{1}\cdot n_{1}\cdot \exp\left(-e_{\mathrm{b}}\right)}\right. {\frac{m_{2}}{m_{1}}} \left(\vphantom{-e_{\mathrm{b}}}\right. \left.\vphantom{-e_{\mathrm{b}}}\right) \left.\vphantom{v_{2}\cdot n_{2} -\frac{m_{2}}{m_{1}}\cdot v_{1}\cdot n_{1}\cdot \exp\left(-e_{\mathrm{b}}\right)}\right)$$ (3.12)

$$\sqrt{\frac{2\cdot \ensuremath{\mathrm{k_{B}}}\cdot T_{i}}{\pi\cdot m_{i}}}$$ (3.13)

$${\frac{\Delta E_{C}\left(E_{\perp}\right)}{\ensuremath{\mathrm{k_{B}}}\cdot T_{1}}}$$ (3.14)

This model has been chosen because it is easy to implement and gives good results for a wide class of devices with moderately changing band edges and devices containing a delta doping. It is not suitable for spike shaped barriers with very short tunneling lengths. For such devices other models which consider tunneling as a nonlocal generation-recombination mechanism have to be used to appropriately describe the interface [27].

The value of the factor $\alpha$ depends on the shape of the energy barrier and the physical effects taken into account. For a simple thermionic emission model without tunneling (see Fig. 3.7a) $\alpha$ is equal to 1. v_i is the thermionic emission velocity and e_b the effective barrier height of the heterojunction. $\Delta$E_C$\left(\vphantom{E_{\perp}}\right.$E_$\perp$$\left.\vphantom{E_{\perp}}\right)$ is the difference of the conduction band edge energies which depends on the normal component of the electric field at the interface. T_i is the carrier temperature and m_i the relative carrier mass.

If tunneling is negligible ( e_b $\gg$ 1 and $\alpha$ $\approx$ 1), the current flow is suitably approximated by the thermionic emission model (3.12). The boundary condition defined by (3.12) is of Neumann type and does not cause convergence problems. However, very often tunneling (see Fig. 3.7b) must be taken into account, e.g., by a field-dependent barrier height lowering as it is proposed in [28]. Tunneling as well as carrier heating can reduce the effective barrier height significantly, and in its limit it approaches zero. Thus, large values of $\alpha$ can occur as $\lim_{e_{\mathrm{b}}\rightarrow0}^{}$$\alpha$$\left(\vphantom{e_{\mathrm{b}}}\right.$e_b$\left.\vphantom{e_{\mathrm{b}}}\right)$ = $\infty$. Since J_$\perp$ must remain finite, this limit simply implies that the boundary condition changes to Dirichlet type:

$$\lim_{e_{\mathrm{b}}\rightarrow0}^{} {\frac{1}{\alpha}} \perp$$ = 0 = f (n₁, n₂), (3.15)

$$\left(\vphantom{v_{2}\cdot n_{2} - \frac{m_{2}}{m_{1}}{\cdot}v_{1}\cdot n_{1}}\right. {\frac{m_{2}}{m_{1}}} \left.\vphantom{v_{2}\cdot n_{2} - \frac{m_{2}}{m_{1}}{\cdot}v_{1}\cdot n_{1}}\right)$$ f (n₁, n₂) (3.16)

Figure 3.7: The conduction band edge at the interface for (a) thermionic emission and (b) thermionic field emission.

Hence, a low perpendicular component of the electric field on the interface and the absence of carrier heating result in a Neumann type condition for current flow across the interface, whereas for increasing electric field or carrier temperature the interface model (3.12) determines the carrier concentration itself rather than the current flow across the interface. Therefore the type of interface condition can change when varying the bias point during curve tracing or even during a single simulation for example along the channel of a heterojunction FET (HFET).

Large values of $\alpha$ which occur for effective barrier heights near zero increase the spectral condition number of the system matrix. This will be demonstrated by a one-dimensional example.