fdtd student report

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ED12S003 1

Index Termselectric field, Finite difference method,Laplace,metal contact, silicon substrate, voltage potential

AbstractThe Electric field and the voltage distributions

inside a simple silicon substrate is calculated by solving the

Laplace equation of the voltage potential using Finite Difference

method. The field distributions on various alignments of the

metal contact are simulated and studied

I. INTRODUCTIONHE objective is to study the voltage and electric field

distribution inside a silicon substrate when the it is

operating. Usually metal contacts are provided at the top of the

silicon substrate. The arrangement is analogous to a parallel

plate capacitor arrangement where the voltage is applied on a

metal plate and a dielectric substrate is filled between the

metal and the ground. As such, the voltage distribution inside

the silicon can be modeled as a parallel plate capacitor

arrangement.

II. THEORYThe governing equation for the voltage distribution is the

Laplace equation. In our case the arrangement is considered as

a two dimensional one with x and y co-ordinates. In order to

numerically solve the Laplace equation, the equation is

converted into a Finite difference form. The Finite difference

form calculates the value of the voltage from the knowledge of

the four neighbouring points around it. Numerically, it can be

solved as follows. A two dimensional matrix is taken and the

voltage values are initialized to zero. The voltage at point(i,j)

is calculated using the formula given below. The process

however is repeated until the values converge to a desired

degree of tolerance. The electric field is then calculated as agradient of the voltage distribution obtained.

V_curr(i,j) = ((1/4)*(V_prev(i-1,j) + V_prev(i+1,j) +

V_prev(i,j-1) + V_prev(i,j+1))) ;

Different Alignments of the metal contact

A. The field inside the silicon substrateThe simplest case is to consider the field inside the substrate

alone unmindful of the environment around it. The case is

modeled using a simple parallel plate capacitor arrangement.

The voltage values at all points inside the grid are calcualted

using the finite difference equation. For the voltage values, atthe end of the matrix, suitable boundary conditions need to be

applied. The two most commonly used boundary conditions

are the Dirichlet and the Neumann conditions.

The Dirichlet conditions assumes a constant value at the

boundary, usually 0 volts. It is used in situations when the

boundary extends over a large area and we assume that the

voltage at the end tends to zero volts since the distance is

large. The Neumann condition however assumes that the

change along the boundary is zero. For our case, the Neumann

condition would give better result since the area of the

capacitor is confined to a small region.

A region of 20*20 was taken. Neumann conditions were

applied and the voltage was calculated until a tolerance of 0.01was achieved. The figure of the voltage distribution and the

electric field distribution are shown as follows.

Figure 1: Voltage distribution

Elementary analysis of potentials and fields

between contacts on a

Simple Silicon substrate (August 2012)

Ragothaman R, ED12S003, Dept of Engineering Design, IIT Madras

T


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ED12S003 2

Figure 2 : Electric field distribution

The voltage distribution is seen as a linear plot along the

direction of the plate thickness. Since the potential distribution

is linear, the electric field distribution must be a constant. Theelectric field values tend to converge to a single value as the

iteration is increased.

B. Distribution in the air surrounding the substrateThe next case is to calculate the voltage distribution in the

air that is surrounding the silicon substrate. The fundamental

Laplace equation still holds inside the capacitor. However at

the boundary, the equations are slightly modified to take into

account the effect of change in permittivity along the air-

silicon interface. The voltage at the interface on all four sides

is calculated based on the formula given below.

V_curr(i,j) = (2/(e_a + e_s))*((1/4)*(e_s*V_prev(i,j-1) +

e_a*V_prev(i,j+1) + ((e_a+e_s)/2)*V_prev(i-1,j) +

((e_a+e_s)/2)*V_prev(i+1,j))) ;

The outermost boundary calculation once again requires the

use of boundary conditions. In this case, since the air extends

to a large distance around the silicon, the Dirichlet conditons

are applied setting the outermost grid points to zero volts. The

voltage and the electric field distributions are shown as

follows.

Figure 3 : Voltage distribution

Figure 4 : Electric field distribution (A close up view)

It is seen that the voltage distribution extends on the sides of

the capacitor. The fields spread out and culminate at the lower

plate of the capacitor. This field is referred to as the fringing

fields. Also, it is to be noted that the field also extends on thetop into the air. The distribution however is different from that

inside a capacitor since the permittivities of air and silicon are

different. Field extends more into the silicon.

C. Distribution for a smaller localised metal contactThe metal contact on top cannot be placed on the entire

stretch of the substrate. It is usually placed in a much smaller

region. The scenario is similar to the previous case except thatthe value of the voltage at the interface must be calculated for

two layers namely the metal layer and the layer immediately

above it. The voltage and the field distributions are plotted as

given below.



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Figure 6 : Electric field (A close up view)

It is seen that the metal contact on the silicon exhibits a

pattern much like that of the field given by a point source

since the metal contact is very small when compared to that of

the silicon substrate. Also fringing fields are seen in this casealso. However if the position of the metal contact is moved to

the centre of the silicon, the strength of the fringing field canbe reduced greatly.

D. Distribution of a metal contact and a ground contactarrangement

The ground contact is also made on the top of the silicon

substrate. The voltage distribution in this case can be analyzed

using the previous case with the difference that the ground

contact is taken into account for calculating the field. The

voltage and the electric field distribution are as follows.

Figure 7: Voltage distribution

Figure 8 : Electric field distribution (A close up view)

It is clear from the Figure 5 and the Figure 7 that the

inclusions of the ground contact on the top of the wafer

confines the voltage distribution till the ground plate. The

voltage field does not extend beyond the ground plate in theright-side direction. Also the field inside the capacitor also

dies down quickly because of the influence of the ground

contact. The electric field ofFigure 8 shows the picture more

clearly.

E. Insulated GroundIn this case, the ground is kept at zero volts but rather it is

insulated. Let us assume that the insulator used for packaging

has the same permittivity as that of the silicon. This case can

be modelled by formulating the boundary conditions at the

bottom of the silicon. Like the previous examples, the dirichlet

condition is applied for the air boundary that is extending to

several metres.



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Figure 10 : Electric field Distribution

The voltage distribution clearly shows that when compared

to a metal backing, an insulated backing allows fields to

extend outside the silicon substrate considerably. Although theground contact nearer to the metal contact restricts field to

cross the boundary in the top right region, a considerable

amount of field is seen in the left side and the bottom side.

III. CONCLUSIONFrom the simulations, the following points can be inferred.

The field of the substrate extends into the surrounding air as a

fringing field. The strength of the field is reduced if a smaller

metal contact is positioned at the centre of the wafer. A

ground contact nearer to the metal contact helps to confine the

field(majority of the field) to the region between them. Aninsulated field allows the field to be extended into the insultor

as compared to a metal contact which does not allow electric

field to be excited within it.

ACKNOWLEDGMENT

The author likes to thank Prof. Ananth.K for providing the

assignment problem and kindling interest in the author to

solve them. The author also likes to thank Mr. Pankaj for his

advice on the expected outputs and the MATLAB commands.

REFERENCES

[1] Momoh, O.D., M.N.O. Sadiku, and C.M. Akujuobi. PotentialComputation in a Conducting Prolate Spheroidal Shell Using Exodus

Method. In 2010 14th Biennial IEEEConference on ElectromagneticField Computation (CEFC), 1, 2010.

[2] Ying Zhang, Wang, J.M., Liang Xiao, Huizliong Wu, "Adaptivedifference method and singular treatment approach for fast parameters

extraction of interconnects in MEI system", Communications, Circuits

and Systems, 2005. Proceedings. 2005 International Conference on,

Volume: 2, 27-30 May 2005

[3] Hammel, J., and J. Verboncoeur. Freespace Boundary Conditions forPoissons Equation in 2D. In Vacuum Electronics Conference, 2004.

IVEC 2004. Fifth IEEE International, 136 137, 2004.

[4] da Silva, F.C., A.M. Soares, and L.R.A.X. de Menezes. New AbsorbingBoundary Conditions for the Finite Difference Method Based onDiscrete Solutions of Laplace Equation. In Microwave and

Optoelectronics Conference, 2003. IMOC 2003. Proceedings of the 2003

SBMO/IEEE MTT-S International, 2:1037 1041 vol.2, 2003.

Ragothaman. R (M2010) received his under-graduate in the field of

Electronics and Communications from the Pondicherry University. The

Author has two years industrial expertise in writing codes for a RTOS

processor. Presently, the author is pursuing his MS degree in the Departmentof Engineering Design, under the guidance of Dr. Kavitha Arunachalam

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