Device Simulation Overview
History and Purpose
This is a first draft
- explain purpose of TCAD
Technology Computer Aided Design (TCAD), has a long successful history of modeling charge transport and electrostatic potential in semiconductor devices, especially Si. Robust finite-volume (FV) methods have been developed to solve Poisson's equation, coupled with the electron and hole continuity equations, in 1 to 3D semiconductor solids. The main limitation on a TCAD tool is mesh size, or how many points are required to ensure that the finite-volume discretization scheme is a stable and accurate approximation to a continuous system.
- explain where TCAD tools fit into the bigger picture
On the average computer, only a few semiconductor devices can be simulated at a time. In more complicated device structures, sometimes only a small important region of a device can be simulated. Therefore several other tools and modeling techniques are needed in addition to TCAD for a full picture of a semiconductor system, and quite powerful ones for a whole IC. For example, density-functional theory (DFT) or molecular dynamics (MD) calculations are needed to model electronic and material properties before simulating at the device level. At the device level, TCAD tools are best at modeling the electronic structure of semiconductor systems, handling larger structures and accounting for boundary conditions more realistically. Finally, the results of TCAD simulations are used to create a simplified picture of devices, which is used by compact-model circuit simulators (such as SPICE) to connect lumped-element devices together in a large network.
Basic TCAD: A Poisson and Drift/Diffusion Solver
TCAD solvers solve 3 things on a grid: Electrostatic potential, via Poisson's equation, and electron and hole concentration, via the electron and hole continuity equations. On a physical contact, electrostatic potential translates to voltage ($V$), and net charge flux (holes - electrons) translates to electric current ($I$). This results in a basic 3 equations, 3 unknowns problem.
Poisson's equation is as follows:
<math>\epsilon\nabla^{2}\Psi = -Q</math>
where <math>\epsilon</math> is the material-dependent permittivity, <math>\Psi</math> is electrostatic potential, %dielectric constant is Er, relative permittivity and <math>Q</math> is charge density. In the most common TCAD case, <math>Q</math> is accounted for by adding up the concentrations of electrons (n), holes (p), and n- and p-type ionized impurities, such as arsenic or boron, called donors and acceptors (<math>N_d^-</math> and <math>N_a^+</math>). Finally, multiplying by <math>q</math>, the charge on an electron, we have: <math>Q = q(n - p + N_d^{-} - N_a^{+}).</math> The continuity equations for electrons and holes are as follows:
<math>\frac{dn}{dt}=\frac{1}{q}\nabla\cdot(q\mu_n n {E} + q D_n \frac{dn}{dx})</math>
<math>\frac{dp}{dt}=-\frac{1}{q}\nabla\cdot(q\mu_p p {E} - q D_p \frac{dp}{dx})</math>
where $\mu_{n,p}$ are the electron and hole mobilities, $\mathscr{E}$ is electric field, and $D_{n,p}$ are the electron and hole diffusion coefficients. Equations \eqref{NLR:eq:continuity} state that there are two main contributions to the transport, $\frac{d}{dt}$, or motion over time of our two free charged carriers, electrons and holes. The first term, called the ``drift" term, arises from any electric field gradient ($\mathscr{E}$) in the device. It is the first term in the parenthetical sums in \eqref{NLR:eq:continuity}. Its experimentally-determined proportionality constant, $\mu$, is called mobility. Electric field gradients arise from the distribution of the mobile charges in the device, as well as any bias voltage conditions on the contacts, and are calculated from the electrostatic potential, $\Psi$, via \begin{equation}\label{NLR:eq:Efield} \mathscr{E}=-\nabla\Psi. \end{equation}
The second contribution to carrier transport is from any concentration gradient of $n$ or $p$, the $\frac{dn}{dx}$ term in \eqref{NLR:eq:continuity}. This is a ``diffusion" term, and its experimentally-determined proportionality constant is $D$, diffusivity.
Equations \eqref{NLR:eq:Poisson} and \eqref{NLR:eq:continuity} present three coupled partial differential equations with three variables, $\Psi$, $n$ and $p$. The effects of drift and diffusion coupled with Poisson's equation do not by any means give an exhaustive account of all the physics involved in the operation of a semiconductor device. For example, these equations neglect collision or scattering terms \cite{NLR:Smith}. Furthermore, the continuity equations as expressed in \eqref{NLR:eq:continuity} above are derived from Boltzmann statistics, and are an approximation to the more accurate way of expressing transport as $n\nabla E_{Fn}$ and $p\nabla E_{Fp}$, where $E_F$ is the Fermi level. Equation \eqref{NLR:eq:continuity} assumes Maxwell-Boltzmann (MB) statistics in that it uses the Einstein relation \begin{equation}\label{NLR:eq:einstein} \frac{kT}{q}=\frac{D}{\mu}. \end{equation} where $k$ is Boltzmann's constant, and T is temperature. This Maxwell-Boltzmann simplification does not apply under all conditions. More advanced models that build on \eqref{NLR:eq:continuity}, as well as a discussion of when these models are needed, are presented in later sections.