Thesis

# MODELLING OF SEMICONDUCTOR DEVICES FOR ICS AND VLSI

Thesis for: M.Sc, Advisor: M. Elbanna, T. Namoor

**ABSTRACT**

This thesis is about Simulation of semiconductor devices. Simulation of

semiconductor devices aims at solving the semiconductor partial differential equations

(PDE) for different device geometry and parameters. Device simulators help the device

engineer to create, test and verify novel structures. This increases our physical knowledge

without even going to the laboratory. In the era of the ever-decreasing device structures

simulation methods must be validated and the device engineer must rely on it, since the

cost of manufacturing a device to test is high.

This thesis discusses the requirements needed to implement such a simulator. Several

computational challenges and how they can be solved are presented. It was the aim of this

thesis to implement a device simulator using an object-oriented programming (OOP)

methodology. OOP is a modern programming approach which makes it easy to increase

the program capabilities while keeping the program neat and tight.

The thesis was divided into two main components: geometry and computation. The

geometry component deals with the geometrical aspects ( geometry intersection, point

inclusion, mesh generation and neighborhood adjacency relationships ). We used

Semiconductor Wafer Representation (SWR)model, the well known standard by

Stanford, for implementing the geometry component. SWR does not provide details about

the implementation, it mainly concerned about giving the interface. We tried to keep

close to the standard as we could. Since our simulator is designed to be used for different

device structures, It is important for the mesh generation procedures to deal with

irregularities in the device. Triangles are chosen as the mesh elements. Since we use

OOP, a triangle is inherited from a more abstract polygon class so that different polygonal

elements can be used as well. The simulator is implemented to deal with polygonal

devices but, curved boundaries can be added by inheriting a curve from the basic Edge

class as well.

The geometry algorithms: point inclusion, region intersections, storage of device, etc..

are chosen to be efficient and general and that is the real challenge since, efficiency

doesn't always go well with generality. Therefore, we are biased to generality when

generality is a must, otherwise efficiency has the higher priority. Since the variables

(electric field, electric current, potentials,..etc..) may change with different orders of

magnitude in small regions, adaptive mesh is required. A quad-tree algorithm is used for

generating the mesh. The algorithm goes by spatially decomposing the device domain

into squares when the level of a nodal point is higher than the level of the square. This

will generate an adaptive spatial decomposition. Triangulation in the interior of the device is

done using certain patterns. The boundary zone (known as delta zone) is isolated from the

device region and triangulated using Constrained Delaunay Triangulation ( CDT )

algorithm. CDT is a complex and expensive (from the computational point of view )

algorithm. Fortunately, it is used only in the delta-zone.

The computational component is a real challenge as well. For different device

structures as well as in different modes of operation certain parameters and functions may

dominate while others may be negligible. The simulator has to deal with this. Different

mobility, recombination and generation models can be incorporated to the simulator. The

simulator user has to be free to use the model he wants. Not only that, but the user can

even implement his own models and test them. Again, OOP plays the important role here.

Models are inherited from a general, abstract function class. The user can do what ever he

wants inside the function, but since all functions have the same interface, the simulator

will not distinguish between them.

There are different device models, Hydrodynamic, Energy Transport, Drift-

Diffusion,..etc.. Even the Drift-Diffusion model, which is the simplest model, is complex

in its own. The Drift-Diffusion model consists of three coupled nonlinear partial

differential equations. There are different discretization schemes for discretizing the

system of equations. We used a hybrid mixed finite element scheme. This scheme gives

us some sort of continuity for the electric flux ( required when using different materials )

and current density. It is common for a practical device consisting of different sub-regions

to be discretized using thousands of elements. This will lead to thousands of variables and

millions of matrices elements. Fortunately, the finite element scheme is sparse in nature.

Using different sparse techniques depending on the nature of the matrix is another use of

OOP. Nonlinear Poisson's equation is linearized. The linearized system was solved using

a global Newton method algorithm. Global Newton method converges to the exact

solution whatever the initial guess is.

Different results, a PN junction and a MOS device are shown. Different channel

lengths are entered to the simulator and the I-V characteristics are drawn. The results

shown are highly compatible with the analytical formulae.

MATLAB was originally used for displaying the geometry, mesh as well as plotting

the I-V characteristics. Another version is implemented which uses the windows

application interfaces for entering the device, setting the different parameters as well as

generating the mesh.

SEMILAB is the result of this thesis. It is a mini simulator, which shows how

sophisticated a device simulator is.

semiconductor devices aims at solving the semiconductor partial differential equations

(PDE) for different device geometry and parameters. Device simulators help the device

engineer to create, test and verify novel structures. This increases our physical knowledge

without even going to the laboratory. In the era of the ever-decreasing device structures

simulation methods must be validated and the device engineer must rely on it, since the

cost of manufacturing a device to test is high.

This thesis discusses the requirements needed to implement such a simulator. Several

computational challenges and how they can be solved are presented. It was the aim of this

thesis to implement a device simulator using an object-oriented programming (OOP)

methodology. OOP is a modern programming approach which makes it easy to increase

the program capabilities while keeping the program neat and tight.

The thesis was divided into two main components: geometry and computation. The

geometry component deals with the geometrical aspects ( geometry intersection, point

inclusion, mesh generation and neighborhood adjacency relationships ). We used

Semiconductor Wafer Representation (SWR)model, the well known standard by

Stanford, for implementing the geometry component. SWR does not provide details about

the implementation, it mainly concerned about giving the interface. We tried to keep

close to the standard as we could. Since our simulator is designed to be used for different

device structures, It is important for the mesh generation procedures to deal with

irregularities in the device. Triangles are chosen as the mesh elements. Since we use

OOP, a triangle is inherited from a more abstract polygon class so that different polygonal

elements can be used as well. The simulator is implemented to deal with polygonal

devices but, curved boundaries can be added by inheriting a curve from the basic Edge

class as well.

The geometry algorithms: point inclusion, region intersections, storage of device, etc..

are chosen to be efficient and general and that is the real challenge since, efficiency

doesn't always go well with generality. Therefore, we are biased to generality when

generality is a must, otherwise efficiency has the higher priority. Since the variables

(electric field, electric current, potentials,..etc..) may change with different orders of

magnitude in small regions, adaptive mesh is required. A quad-tree algorithm is used for

generating the mesh. The algorithm goes by spatially decomposing the device domain

into squares when the level of a nodal point is higher than the level of the square. This

will generate an adaptive spatial decomposition. Triangulation in the interior of the device is

done using certain patterns. The boundary zone (known as delta zone) is isolated from the

device region and triangulated using Constrained Delaunay Triangulation ( CDT )

algorithm. CDT is a complex and expensive (from the computational point of view )

algorithm. Fortunately, it is used only in the delta-zone.

The computational component is a real challenge as well. For different device

structures as well as in different modes of operation certain parameters and functions may

dominate while others may be negligible. The simulator has to deal with this. Different

mobility, recombination and generation models can be incorporated to the simulator. The

simulator user has to be free to use the model he wants. Not only that, but the user can

even implement his own models and test them. Again, OOP plays the important role here.

Models are inherited from a general, abstract function class. The user can do what ever he

wants inside the function, but since all functions have the same interface, the simulator

will not distinguish between them.

There are different device models, Hydrodynamic, Energy Transport, Drift-

Diffusion,..etc.. Even the Drift-Diffusion model, which is the simplest model, is complex

in its own. The Drift-Diffusion model consists of three coupled nonlinear partial

differential equations. There are different discretization schemes for discretizing the

system of equations. We used a hybrid mixed finite element scheme. This scheme gives

us some sort of continuity for the electric flux ( required when using different materials )

and current density. It is common for a practical device consisting of different sub-regions

to be discretized using thousands of elements. This will lead to thousands of variables and

millions of matrices elements. Fortunately, the finite element scheme is sparse in nature.

Using different sparse techniques depending on the nature of the matrix is another use of

OOP. Nonlinear Poisson's equation is linearized. The linearized system was solved using

a global Newton method algorithm. Global Newton method converges to the exact

solution whatever the initial guess is.

Different results, a PN junction and a MOS device are shown. Different channel

lengths are entered to the simulator and the I-V characteristics are drawn. The results

shown are highly compatible with the analytical formulae.

MATLAB was originally used for displaying the geometry, mesh as well as plotting

the I-V characteristics. Another version is implemented which uses the windows

application interfaces for entering the device, setting the different parameters as well as

generating the mesh.

SEMILAB is the result of this thesis. It is a mini simulator, which shows how

sophisticated a device simulator is.

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