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 <math>\mu_{n,p}</math> are the electron and hole mobilities, <math>{E}</math> is electric field, and <math>D_{n,p}</math> are the electron and hole diffusion coefficients. The continuity equations state that there are two main contributions to the transport, <math>\frac{d}{dt}</math>, 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 (<math>{E}</math>) in the device. It is the first term in the parenthetical sums in the continuity equations. Its experimentally-determined proportionality constant, <math>\mu</math>, 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, <math>\Psi</math>, via
<math>{E}=-\nabla\Psi.</math>
The second contribution to carrier transport is from any concentration gradient of n or p, the <math>\frac{dn}{dx}</math> term in the continuity equations. This is a "diffusion" term, and its experimentally-determined proportionality constant is D, diffusivity.
The Poisson and continuity equations present three coupled partial differential equations with three variables, <math>\Psi</math>, 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. Furthermore, the continuity equations as expressed above are derived from Boltzmann statistics, and are an approximation to the more accurate way of expressing transport as <math>n\nabla E_{Fn}</math> and <math>p\nabla E_{Fp}</math>, where <math>E_F</math> is the Fermi level. Our equations assume Maxwell-Boltzmann (MB) statistics in that they uses the Einstein relation
<math>\frac{kT}{q}=\frac{D}{\mu}.</math>
where k is Boltzmann's constant, and T is temperature. This Maxwell-Boltzmann simplification does not apply under all conditions. More advanced models build on these equations.
Boundary Conditions / Contact Equations
In order to solve the 3 equations and 3 unknowns described above above, we need boundary conditions. Here is a cartoon of a typical MOSFET structure
- put in a figure of a typical MOSFET and it's energy band diagram
There are two different types of boundary conditions relevant to this kind of structure. The first is a simple metal-contact work-function difference. Above you can see a representative energy band diagram for a MOS device with p-type bulk and a metal contact. The work function, <math>\Phi_m</math>, which we chose here to measure in volts, is the minimum energy required to move an electron from the metal (residing in the Fermi level of the metal, <math>E_{Fm}</math>) to the vacuum level <math>E_l</math>. This is similar to the electron affinity, <math>\chi_S</math> in Si, the minimum energy required to move an electron from the conduction band, <math>E_C</math>, to the vacuum. If we define electrostatic potential as the band-bending in the vacuum level,
<math>\Psi=-E_l,</math>
with both <math>\Psi</math> and <math>E_l</math> measured in volts; and if we define the zero reference point as the Fermi level in the metal (<math>E_{Fm}</math>) at zero applied bias,
<math>E_{Fm}=0 \mbox{ at } V_{applied}=0,</math>
then we can express our work-function-difference boundary condition as
<math>-\Psi-\Phi_m=E_F,</math>
on all the nodes that touch both metal and oxide, where in general <math>\Psi</math> is constant within a metal and <math>E_F</math> is continuous between materials.
An ohmic contact is more complicated. Physically, it is a contact to the bottom of the Si wafer that is required to keep both the actual and simulated system from floating, or having undefined voltage values. We model this as a boundary condition "somewhere in the bulk," which in practice means "on the first Si nodes that are far enough away from the wafer surface to be considered bulk." The bulk is in equilibrium, which means that, first and second, the Fermi levels are equal to each other, and equal to the voltage applied at the ohmic contact:
<math>E_{Fn}=E_{Fp}=V_{applied};</math>
and third, local conservation of charge is maintained:
<math>Q = q(n - p + N_d^{-} - N_a^{+})=0.</math>
These equations provide three boundary conditions for our three variables. Other boundary conditions necessary for simulation include Dirichlet and Von Neumann boundary conditions on non-contacted edges, and the condition that <math>\Psi</math> is continuous everywhere.