Download Presentation

Simulations for the design of
Semiconductor devices
Harshit Arora
3rd year undergraduate,
Department of electrical engineering,
IIT Kanpur.
Mentor:
Dr. Heiner Ryssel
Contents of the talk
†
†
†
†
†
†
Role of computer simulation.
Steps involved in simulating a physical device.
Simulation of semiconductor device.
Basic criterion to be satisfied by a semiconductor simulator.
Stages in physical simulation of a semiconductor device.
Steady state simulation of bipolar semiconductor devices.
Role of Computer Simulations
† Computer Simulations provide a vital link between the
Experimental and the theoretical world.
† It compliments the experiment and theory when the
presence of certain constraints makes the exact physical or
the mathematical analysis impossible.
What does simulation do?
Steps involved in simulating a physical
phenomenon
† The first and the foremost step in simulation is mathematical
modeling of the phenomenon.
“A model of a physical device is a mathematical entity with
precise laws relating its variables.”
† A mathematical model is always distinct from the physical
device, though its behavior ordinarily approximates that of
the physical device represented. Thus a model is never strictly
equivalent to the device it represents.
† Modeling Æ Approximations (different levels)
† Simulation Æ Trade-off (different considerations)
Simulation of semiconductor devices
† What is the need?
“A simulator is basically required to bring down the number
of iterative steps involved in the fabrication of a
semiconductor device with desired properties”.
† What place does a simulator occupy in the entire device development
process?
O/P of a semiconductor device simulator
A simulator used for a semiconductor device must give the
following as output.
† Terminal characteristics: for any general semiconductor
device these include the following:
* I-V characteristics.
* Capacitance Vs voltage.
* Conductance Vs Voltage.
† 2D Plots: These include the plots of potential, field, fermilevel, doping, carrier concentration, generation and
recombination rates.
O/P of a semiconductor device simulator
† 3D Plots: These include the plots of quantities mentioned in
2D but in a 3D plane. Thus these plots give a more realistic
picture of the actual distribution.
† Vectors: These include the plots of fields and current density
as a function of voltage and position vector.
Ways of obtaining semiconductor device
characteristics
† There are primarily three ways for obtaining characteristics
for a semiconductor device:
Stages of physical simulation of a
semiconductor device
† Process Simulation: involves process sensitivity
investigation, alternative process investigation and process
yield improvement.
† Device Simulation: involves understanding physical
effects, electrical characteristics prediction and device
reliability study.
† Technology Characterization: involves device
parameter extraction and their optimization for circuit design,
full cell extraction for technology development and
characterization.
Device simulation
It involves the following:
† Developing physical models: mathematical equations
governing injection, transport, recombination, band gap and
mobility are framed.
† Boundary Conditions: These are the special conditions
assigned to the boundaries and interfaces occurring in the
semiconductor device. These serve as the boundary
conditions for the mathematical equations devised in step 1,
thus reducing them to boundary value problems.
Device simulation
Boundaries and interfaces can be categorized into one of the
following:
Ohmic/Schottky
Insulator/Neumann
Interface charges/traps
Lumped/distributed RC
† Numerical Methods: After a mathematical model has been
devised, a numerical method is employed to solve the
generated set of equations.
Device simulation
Depending upon the complexity and stiffness of the equations one
of the following solving techniques is used:
- Decoupled or Gummel method.
- Coupled or Newton method.
- Linear matrix.
- Jacobian matrix.
† Analysis: The analysis of the device involved can be:
- Steady state.
- Transient.
- Small signal AC.
Device simulation( Boundary conditions)
† Schottky vs Ohmic Contacts :
In case of ohmic contact we consider no barrier at the junction
while in the schottky contact electrons and holes experience a
barrier at the junction.
Numerical methods
† Decoupled or Gummel method:
It is a technique used to solve a set of linear equations. We
seek solution to a set of equation represented in matrix
form as
Formula used:
Device simulation
Steady state solution of bipolar
semiconductor equations
† The general semiconductor
equations may be written as
† Choice of variables
- Natural variable formulation (V, n, p)
- Quasi-Fermi level formulation ( V,
)
Steady state solution of bipolar
semiconductor equations
- Slotboom variables (V, ηn, ηp)
† Here we choose to work with the slotboom variables. We
should express the current and the charge density in terms of
quasi-Fermi level φn, φp:
Steady state solution of bipolar
semiconductor equations
† We define new variables
Steady state solution of bipolar
semiconductor equations
† Modified set of equation becomes the following:
† An important observation which can be made is that all the
above three equations are of the same type. Thus from here
on we concentrate solving on just one.
Steady state solution of bipolar
semiconductor equations
† Grid definition
† Discretization of derivatives
Steady state solution of bipolar
semiconductor equations
† Boundary conditions: A unique solution to the poisson’s
equation equation is found only upon application of boundary
conditions.
Steady state solution of bipolar
semiconductor equations
† Boundary Conditions for Poisson’s equation:
-Dirichlet Conditions :
where s indicates the value specified along the
boundary surface.
• Dirichlet BCs generally apply to metal/semiconductor
interfaces where metal is treated as an ideal conductor (i.e. an
equipotential surface), the value of which is given by the
applied bias.
• Schottky contacts must include the built-in potential, φB.
Steady state solution of bipolar
semiconductor equations
-
Neumann conditions
where n indicates the normal with respect to the
boundary surface, s, and En is the normal electric field.
•Neumann BCs are generally applied on free surfaces where
the normal field is set to zero (zero current flow), or to a
value determined by the surface charge associated with
surface states.
Steady state solution of bipolar
semiconductor equations
† Discretization scheme for the continuity
equations:
The discretization of the continuity equation in conservative form
requires the knowledge of the current densities
on the mid-points of the mesh lines connecting neighboring
grid nodes. Since solutions are available only on the grid
nodes, interpolation schemes are needed to determine the
solutions.
Steady state solution of bipolar
semiconductor equations
† Continuity Equation:
Steady state solution of bipolar
semiconductor equations
There are two schemes that one can use:
- linearized scheme: V, n, p, μ and D vary linearly between
neighboring mesh points
- Scharfetter-Gummel scheme: electron and hole densities
follow exponential variation between mesh points
Steady state solution of bipolar
semiconductor equations
† Linearized scheme: Within the linearized scheme, one has
from
Steady state solution of bipolar
semiconductor equations
† Scharfetter Gummel scheme
One solves the electron current density equation:
for n(V), subject to the boundary conditions: n(Vi ) = ni and
n(Vi +1) = ni +1
Steady state solution of bipolar
semiconductor equations
The solution of this first-order differential equation leads to:
Where,
is the Bernaulli function
Steady state solution of bipolar
semiconductor equations
† Flowchart of a typical Drift-Diffusion program
Steady state solution of bipolar
semiconductor equations
† Newton’s method
The three equations that constitute the DD model, written in
residual form are: Fv (v,n,p) = 0 Fn(v,n,p) = 0 Fp(v,n,p) = 0
Starting from an initial guess, the corrections are calculated by
solving:
Steady state solution of bipolar
semiconductor equations
† Gummel Vs Newton method
In general Gummels method is preferred at low bias because of
its faster convergence and low cost per iteration At medium
and high bias the Newtons method becomes more convenient
since the convergence rate of Gummels method becomes
worse as the coupling between equations becomes stronger at
hogher bias But since Gummels method has a fast initial error
reduction it is often convenient to couple the two procedures
using Newtons method after several Gummels iterations.
Conclusion
† Computer simulation bridges the theoretical and experimental
worlds.
† Often a suitable choice of variable brings down the
complexity of the problem drastically.
† A bipolar semiconductor device simulation problem basically
involves solution of three equations simultaneouslly at each
mesh point.
† At times interpolation of the node quantities might be
required for a more perfect analysis and hence interpolation
techniques might be needed for the same.
Thank you….