Technology Computer Aided Design (TCAD) · PDF fileTechnology Computer Aided Design (TCAD)...

1 1 G. Betti Beneventi

Technology Computer Aided

Design (TCAD) Laboratory

Lecture 2,

A simulation primer

Giovanni Betti Beneventi E-mail: [email protected] ; [email protected]

Office: Engineering faculty, ARCES lab. (Ex. 3.2 room), viale del Risorgimento 2, Bologna

Phone: +39-051-209-3773

Advanced Research Center on Electronic Systems (ARCES)

University of Bologna, Italy

[Source: Synopsys]


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


Outline

Introduction





perspective

• Meshing




Introduction

• In this lecture we will talk about different topics, all of

them to be discussed before starting the laboratory

activity.

• Some content of this class is related to the review of

basic concepts, like the choice of a simulation domain,

the equilibrium and out-of-equilibrium definitions, the

definitions of static, transient, and AC simulations.

• In addition, a TCAD user perspective background on the

numerical methods and on the solvers used in Synopsys

Sentaurus TCAD is provided.


Outline

• Introduction

Definition of equilibrium and out-of-

equilibrium




perspective

• Meshing




Equilibrium and out-of-equilibrium

• In a system under equilibrium condition, each process and its inverse must self-

balance independently from any other process that may occur in the system. In

other words, equilibrium is the special case where each fundamental process

and its inverse self-balance, it is also known as detailed balance principle.

• If a net current flows, or if light is shined upon the semiconductor creating free

carriers, we are out-of-equilibrium.

• A system under steady-state, transient, or ac signal is always out-of-equilibrium.

The only exception is the presence of a capacitor, than can be biased but in

which the net current flow is zero due to the presence of the oxide

V electrode electrode

e

I semiconductor

e

Applied Voltage Shined light

E= hn


Outline

• Introduction


Static, Transient and AC simulations



perspective

• Meshing




Static, transient, and AC simulations (1)

• Static condition or steady-state (DC bias point). In steady-state conditions,

for each property of the systems, its partial derivative with respect to time is

zero, i.e. , nothing is changing with time.

• To perform steady-state simulations, the Sentaurus TCAD keyword is Quasistationary. The Quasistationary command is used to ramp a

device from a solution to another through the modification of the boundary conditions (that can be Voltage, Current, or Temperature).

• For example, ramping a drain voltage of a device from 0 to 5V is performed

by

Quasistationary( Goal {Voltage=5 Name=Drain } ){

Coupled { Poisson Electron Hole } }

Step Boundary Re-solve the Device



• Transient response. A transient response or natural response is the time-

varying response of a system to a change from equilibrium.

• In Sentaurus, the keyword that must be used to perform transient simulation is Transient. The command must start with a device that has already been

solved under stationary conditions. The simulation then proceeds by iterating

between incrementing time and re-solving the device.

• An example of performing a transient simulation is:

Transient( InitialTime = 0.0 FinalTime=1.0e-5 ){

Coupled { Poisson Electron Hole }

}

Increment time Re-solve the Device



• Small signal AC analysis. Performing a small signal or AC analysis means

simulate the behavior of system when a relatively small harmonic signal is

superimposed to a steady-state condition or DC bias point.

• Small-signal modeling is a common analysis technique in electrical

engineering which is used to approximate the behavior of nonlinear devices

with linear equations. This linearization is formed about the DC bias point of

the device (that is, the voltage/current levels present when no AC signal is

applied), and can be accurate for small excursions about this point. • The keyword for AC analysis in Sentaurus Sdevice is ACCoupled.

Graphical, qualitative, illustration of steady-state, transient, and AC signals.


Example M. A. Alam (2008),

"ECE 606: Principles of Semiconductor Devices,“

http://nanohub.org/resources/5749.

EQUILIBRIUM

10 people going out and 10 people

coming in Italy (population of Italy is

“stationary”).

In addition, immigration and emigration

counterbalance country-wise (each

process counterbalance its opposite in

each country): we are in equilibrium

condition.

Italy, France, Germany and UK form our ‘system’. Italian population is the ‘quantity of interest’.

Immigration/emigration in/from the other countries are our ‘processes’.

TRANSIENT

12 people going out and 8

people coming in Italy. Italian

population is not conserved,

i.e. Italian population is not

stationary. In addition,

immigration and emigration

do not counterbalance

country-wise: we are out-of-

equilibrium.

STEADY STATE

10 people going out and 10

people coming in Italy

(population of Italy is

stationary)

But immigration and

emigration process do not

counterbalance country-

wise: we are out-of-

equilibrium.

Italy

France

Germany

UK

4

4

4 4

2 2

Italy

France

Germany

UK

5 3

3 4

3 2

Italy

France

Germany

UK

2

6

1 4

2 5


Outline

• Introduction



Simplify the simulation domain


perspective

• Meshing




Simulation on 1D, 2D and 3D domains (1)

• Reality is always 3D. However, some problems can be thought (approximated) as they

would occur in less than 3D. This can be done when there is invariance of the problem

on some directions. In other words, a dimension can be suppressed from the simulation

if the physical internal properties of the simulated device would not change if the

dimension itself would be made infinite. To make such approximations (not always easy

to see!), an a-priori knowledge of the problem is needed.

Possible to think about a 2D equivalent (on

the yz plane) of what happens in the

transistor channel. That is the device features

can be thought to be invariant in the x

direction (within the channel). The larger the

L the better the 2D approximation.

Standard planar MOSFET FinFET

L very thin and raised

source and drain.

Intrinsically a 3D device,

very difficult to simplify

into a 2D equivalent!

2D

L L x

y

z

Example


Simulation on 1D, 2D and 3D domains (2)

• The simplification of a 3D problem into 2D or even 1D, if possible, is strongly encouraged .

Indeed, the simplification of the simulation domain, means a lower number of nodes in which the

numerical simulation must be computed. Reducing the number of the mesh nodes decreases the

computational burden and therefore the time needed for the simulation to be performed.

e.g.: same device, for 2D simulation ~ 103 nodes, for a 3D simulations ~105 nodes ~ 2 orders of

magnitude of difference in the node numbers!

• One of the most frequently employed simplification is simulating a 3D device having cylindrical

symmetry using a 2D domain and solving the model equations in cylindrical coordinates.

• To simulate a 2D domain in cylindrical coordinates, the partial-differential equations must be

expressed in the cylindrical coordinate system. In Sentaurus Synopsys this is accomplished by using the Cylindrical keyword in the Math section of the Sdevice input file (see later)

• Because of the simple geometry, all the devices simulated in the course need only a 2D planar (*)

simulation domain.

equivalent 2D domain to be simulated

in cylindrical coordinates

(*) planar = Cartesian coordinates vs. cylindrical coordinates

Example Nanowire (NW) MOSFET

source

drain

gate


Outline

• Introduction




Numerical methods from a TCAD user

perspective

• Meshing




Scope

• In this tutorial we discuss the basic features of the numerical method

used in TCAD Sentaurus Devices from a TCAD user perspective.

The purpose of this tutorial is not describing into details the

numerical methods implemented in the software, but discussing their

general features, in order for the user to understand which numerical

methods is most useful to be employed for a given problem and to

cope with possible convergence issues.

• In general, in an iterative solution method, we start with a guess for

the solution (often the zero vector) and then we successively renew

this guess, getting closer to the solution at each stage. The

iterations continue until the solution converges to a desired accuracy

xi –xi-1 < e, where x is the solution vector, that is the vector of the

values of the problem unknowns, i is the index counting the iteration

number, and e is the vector which defines the needed accuracy.


Outline

• Introduction





perspective

Meshing




Considerations on numerical mesh (1)

• Define and effectively optimize numerical meshes in which the problem can be solved

assuring convergence and, at the same time, reasonable simulation times, is a

challenge. However, some general rules can be applied:

• The grid spacing must be sufficiently dense so that all the relevant features of the

geometry (e.g. doping) are accurately represented.

• Points must be allocated to accurately approximate the physical quantities of

interest (e.g. potentials, fields, carrier concentrations, currents).

• This means that high grid densities must be allocated in regions where the

geometrical and physical quantities of interest undergo rapid changes (e.g. junctions).

• Conversely, the spacing between points could be relaxed in the areas where values

are expected to stay relatively constant without adding any significant contribution to

the overall error (e.g. quasi-neutral regions, deep regions inside the device). In fact,

because the overall computation time depends on the total number of grid points, grid

point number must be minimized for computational efficiency.

• In advanced finite-element software, as well as in Synopsys Sentaurus there are tools

for automated grid generation and automatic adaptation of grids.


Considerations on numerical mesh (2)

• Once that convergence is achieved within reasonable simulation time, an operation

that must always been performed prior to elaborate the results of a simulation is to

check the invariance of the solution with respect to variations of the numerical mesh.

That is, once a mesh has been created to assure convergence in reasonable time, try

to further reduce the mesh spacing and see if the solutions does not change. If the

solution does not change mesh is good !

Example. Numerical grid in a pn diode

junction

p region n region


Outline

• Introduction





perspective

• Meshing

Numerical methods



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.


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 𝑓 𝑥 = 0, being 𝑓 a function of some real

unknown 𝑥 and let its derivative being 𝑓’. 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 𝑥0. Then, provided that the function is reasonably well-behaved, a

better approximation of 𝑥0, 𝑥1, can be easily calculated. In fact:

Being 𝑥0 a first guess as a zero of 𝑓 𝑥 .

Suppose 𝑥1 is a zero of 𝑓 𝑥 then

𝑓′ 𝑥0 =0 − 𝑓(𝑥0)

𝑥1 − 𝑥0

𝑥1 = 𝑥0 −𝑓(𝑥0)

𝑓′(𝑥0)

Geometrically (𝑥1,0) is the intersection with the

𝑥-axis of a line tangent

to 𝑓 at 𝑥0, 𝑓 𝑥0 . The process is repeated as

𝑥𝑛+1 = 𝑥𝑛 −𝑓(𝑥𝑛)

𝑓′(𝑥𝑛)

until a sufficiently accurate value is reached.

𝑥0

𝑥

𝑓(𝑥)

𝑥1

two iterations enough to

get good accuracy


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.

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

Initial guess of the solution

Solve a whole system including

Poisson, Continuity and

Transport equations

converged?

?

n

y

Newton

iteration


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.

• The keyword for the Gummel iteration in Sentaurus Device is Plugin.


Solve Poisson equation

converged?

?

n

y Gummel

iteration Solve Continuity and Transport

equations

converged?

?

y

n


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.


Solve a whole system including

Poisson, Continuity and

Transport equations

converged?

?

n

y

Newton

iteration

Solve Heat equation

converged?

? y

n

Gummel

iteration


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.


Error handling/accuracy of simulations

During a Solve statement, Sentaurus Device tries to determine the value of an

equation variable 𝑥, such that the computed updated ∆𝑥 (after the n-th iteration)

is small enough. That is, it iterates until

∆𝑥𝑥∗

𝜀𝑅𝑥𝑥∗ + 𝜀𝐴

< 1 where 𝜀𝑅 = 10-Digits and 𝜀𝐴 = Error

and 𝑥∗ is a scaling constant equal to 0.1

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

For 𝑥 ⟶ ∞ ∆𝑥

𝑥< 𝜀𝑅

For 𝑥 ⟶ 0 |∆𝑥| < 𝜀𝐴 𝑥∗

Relative Error criterion

Absolute Error criterion


Outline

• Introduction





perspective

• Meshing


Synopsys Sentaurus TCAD Solvers


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

requirements

good for

SUPER direct

(systematic

triangularization

of the matrixes)

high 1D

PARDISO direct parallel high 2D

modification of

default parameters

not recommended

ILS (based on

the GMRES)

iterative parallel

(first guess than

refinement)

low 3D

requires

modification of

default parameters

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