3.7 Device Simulation

Finally, the data structure is ready for processing with device simulation. The main inputs for device simulation are:

1. Doping concentration of different doping species (e.g. Arsen, Phosphorus, Antimony, Boron etc.) on a mesh.
2. Structural information about the shape of the region which has to be evaluated with device simulation. This information includes material types of layers, topological variation of layers and the detailed surface and interface shapes of the materials present in the structure.
3. Named contacts as source for adjustable boundary conditions of the device simulation.

An example for a typical input structure for device simulation is shown in Figure 3.12. The three main input classes (doping, structural and contact information) can be clearly seen.

Figure 3.12: Example for a typical input structure for device simulation comprising of doping concentration (a) including the different doping species e.g. Boron (b) on a mesh and the topological structure and contact definition (d).

(a)

(b)

(c)

(d)

The semiconductor device simulators are fairly similar in their solution approach. They all solve a system of partial differential equations describing the potential distribution and carrier transport in a doped semiconducting material. The standard semi-classical transport theory is based on the BOLTZMANN equation [131],[132]

$$\left(\frac{\partial}{\partial t} + \vec{F}\nabla_{\vec{p}}+\frac... ...}}\right)f(\vec{r},\vec{p},t)=\left(\frac{\partial f}{\partial t}\right)_{coll}$$ (3.1)

where $\vec{r}$ is the position, $\vec{p}$ is the impulse, $\vec{F}$ is the electric field vector and $f(\vec{r},\vec{p},t)$ is the distribution function. In the simplest approach for solving this equation the collision term on the right hand side of (3.1) is substituted with a phenomenological term

$$\left(\frac{\partial f}{\partial t}\right)_{coll} = \frac{f_{eq}-f(\vec{r},\vec{p},t)}{\tau}$$ (3.2)

where $f_{eq}$ indicates the (local) equilibrium distribution function, and $\tau$ is a microscopic relaxation time. It is very useful to express the distribution function in terms of velocity, rather than impulse, since it will be easier to calculate electrical currents. In equilibrium one may use the MAXWELL-BOLTZMANN distribution function

$$f_{eq}(\vec{r},\vec{v}) = n(\vec{r}) \left(\frac{2 \pi k_B T_0}{m^*}\right)^{-\frac{3}{2}} e^{-\frac{m^*\vert\vec{v}\vert^2}{2 k_B T_0}}$$ (3.3)

where $n(\vec{r})$ is the carrier density, $T_0$ is the lattice temperature and $m^*$ is the effective mass. The use of (3.3) for semiconductors is justified in equilibrium as long as degeneracy is not present. the carrier density $n(\vec{r})$ is directly related to the distribution function according to

$$n(\vec{r}) = \int d\vec{v} f(\vec{r},\vec{v})$$ (3.4)

which is of general applicability. The significance of the momentum relaxation time can be understood if the electric field is switched off instantaneously and a space-independent distribution is considered. The resulting BOLTZMANN equation is then

$$\frac{\partial f}{\partial t} = \frac{f_{eq}-f}{\tau}$$ (3.5)

which shows that the ralaxation time is a characteristic decay constant for the return to the equilibrium state.
The often used drift-diffusion current equations

$$J_n = q n(x) \mu_n F(x) + q D_n \frac{dn}{dx}$$

$$J_p = q p(x) \mu_p F(x) - q D_p \frac{dp}{dx}$$ (3.6)

can be easily derived directly from the BOLTZMANN equation as outlined in Appendix D. All device simulators use the drift-diffusion approach as the simplest model to cover the transport effects inside the semiconductor material.