Bologna / 02_TCAD_laboratory_A_simulation_primer_GBB_20140326H1020

Generalities

•The power of an iterative method lies in its ability to achieve convergence efficiently, i.e. as fast as possible. In general, convergence and speed are two conflicting issues, since more robust convergence means a higher number of iterations and heavier mesh (i.e. an higher number of points in which the solution must be computed).

•For semiconductor devices, two different approaches are usually employed to solve the coupled set of equations that comprises the Drift-Diffusion model, depending upon the problem (it is also possible to combine the two methods for the same problem when needed):

1.The Newton iteration (or fully-coupled method) and

2.The Gummel iteration (or de-coupled method)

•For both of them, once the non-linear partial-differential-equations of the Drift-Diffusion model are discretized in space, the Newton’s method for the solution of non-linear systems is applied.

•For both of them, in general, the solution converges non-linearly, meaning that the error, defined as the difference in the unknown between two subsequent iteration is a non-linear function of the iteration counter.

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Newton method

In both Newton and Gummel iteration schemes, the Newton iterative method is applied to solve the non-linear system. Let’s briefly review the concept of the Newton method by considering a single-variable equation. The first step of the method consists in manipulating the equation in the residual form = 0, being a function of some real unknown and let its derivative being ’. Thus, to solve the equation one needs to find out the zeroes of the function. To get successively better approximation of the zeroes we start with a first guess 0. Then, provided that the function is reasonably well-behaved, a better approximation of 0, 1, can be easily calculated. In fact:

Being 0 a first guess as a zero of .

Suppose 1 is a zero of then

	′ 0	=	0 − (0)		1	= 0	−	(0)
			1	− 0				′(0)

Geometrically (1,0) is the intersection with the

-axis of a line tangent

to at 0, 0 . The process is repeated as

		(		)
	= −
+1		′(			)

until a sufficiently accurate value is reached.

( )

1 0

two iterations enough to get good accuracy

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Newton iteration

•In the fully-coupled Newton iteration (also called Bank-Rose scheme in semiconductor device simulators) the total system of unknowns is solved together, meaning that there is only one system to be solved. The systems derives from the discretization of (i.e. includes) all equations to be solved: Poisson equations, Continuity equations and Transport equations.

Initial guess of the solution

	Solve a whole system including
		Poisson, Continuity and	Newton
		Transport equations
			iteration
	n
		converged?
	?

y

•The keyword for the Newton iteration in Sentaurus Device is Coupled.

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Gummel iteration

•Each iteration of the Gummel method treats one equation at a time, solving for the given equation (i.e. Poisson equation or Continuity and Transport equations) with respect to its primary unknown (i.e., for Poisson equation the electric field, for the Continuity and Transport equations the carrier densities) updating at each step only the values of the primary unknown. An iteration is completed when the procedure has been performed on each independent variable.

Initial guess of the solution

Solve Poisson equation

		n		converged?
			?		Gummel
				y
					iteration
		Solve Continuity and Transport
				equations
		n		converged?
			?
				y

• The keyword for the Gummel iteration in Sentaurus Device is Plugin.	G. Betti Beneventi 24

Combined Newton-Gummel iteration

•A combined scheme is often used where heating effects (accounted for with the Fourier heating postulate) come into play. First of all, the Drift-Diffusion model is computed until convergence is achieved with a Newton iteration, then electric field and current density solutions are plugged into the Fourier equation, which is computed until convergence is achieved in a Gummel scheme.

Initial guess of the solution

Solve a whole system including

Poisson, Continuity and

Transport equations

n

converged?

?

y

Solve Heat equation

n

converged?

?

y

Newton iteration

Gummel iteration

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Comparison between Newton and Gummel iterations

•Newton’s iteration converge with a lower number of iteration compared to the Gummel iteration (quadratic rate of convergence vs. linear rate of convergence). However the single Newton iteration takes more time than a Gummel iteration.

•Gummel iteration converges relatively slowly compared to Newton iteration but the method will often tolerate poor initial guess.

•In certain problems where it is difficult to choose good initial guess, starting with Gummel to refine an initial guess, then switching to Newton after some iterations to achieve quicker convergence can be useful.

•In general use Gummel only when the transport problem can be decoupled from the electrostatic problem (i.e. at low fields where diffusion dominates, no band-to-band-tunneling, no avalanche, no field-dependent mobility). In those cases, Gummel is quicker than Newton, since Newton keeps updating quantities that are essentially constant or weakly changing.

•As initial guess, one often relies on the equilibrium solution.

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Error handling/accuracy of simulations

During a Solve statement, Sentaurus Device tries to determine the value of an equation variable , such that the computed updated ∆ (after the n-th iteration) is small enough. That is, it iterates until

		∆
				< 1	where	= 10-Digits and = Error
				+	and is a scaling constant equal to 0.1

What does it mean? Let’s simplify the inequality in two limiting cases:

∆

For ∞ < Relative Error criterion

For 0	|∆| <		Absolute Error criterion

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Outline

•Introduction

•Definition of equilibrium and out-of-equilibrium

•Static, Transient and AC simulations

•Simplify the simulation domain

•Numerical methods from a TCAD user perspective

•Meshing

•Numerical methods

 Synopsys Sentaurus TCAD Solvers

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Choosing the appropriate solver

•At each step of the simulation a linear system is solved

•Three linear solvers are available in Sentaurus Sdevice, which basic features are listed in the following table

solver	type	memory	good for
		requirements
SUPER	direct	high	1D
	(systematic
	triangularization
	of the matrixes)
PARDISO	direct parallel	high	2D
			modification of
			default parameters
			not recommended
ILS (based on	iterative parallel	low	3D
the GMRES)	(first guess than		requires
	refinement)		modification of
			default parameters

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