1. Independent Study Report
Finite element simulation for nano devices
Submitted To: Dr. Moh’d Rezeq
Student: Isra Lababidi
ID: 100020272
08-Jan-13
2. 2
Table of Contents
Table of Figures.............................................................................................................................................3
Abstract.........................................................................................................................................................4
Introduction ..................................................................................................................................................5
Schottky diode: Theoretical Analysis ............................................................................................................6
Thermionic Emission vs . Tunneling Effect................................................................................................8
Metal-Semiconductor Contact: Model Analysis ...........................................................................................9
Simulation Analysis and Results..................................................................................................................11
Conclusion and Future Work ......................................................................................................................21
References ..................................................................................................................................................22
Appendix A: Conventional Schottky contact equations..............................................................................23
Appendix B: Nano Schottky contact equations ..........................................................................................24
3. 3
Table of Figures
Figure 1: MS contact an instant after contact formation ..............................................................................................6
Figure 2: under equilibrium conditions .........................................................................................................................6
Figure 3: Energy band diagram of an MS contact..........................................................................................................7
Figure 4: Representative Nano Metal-Semiconductor Model.......................................................................................9
Figure 5: Depletion width vs. metal paricle radius for different doping concentrations ............................................11
Figure 6: Maximum electric field plot for different doping concentrations................................................................11
Figure 7: Built in potential at the interface for different doping concentrations........................................................12
Figure 8: COMSOL model of an MS contact with a nano metal particle of radius R = 3nm ........................................14
Figure 9: Electric potential surface distribution of an MS contact with a nano metal particle of radius R = 3nm......14
Figure 10: Electric field surface distribution of an MS contact with a nano metal particle of radius R = 3nm ...........15
Figure 11: Electric potential 2D line graph of an MS contact with a nano metal particle of radius R = 3nm ..............15
Figure 12: Electric field 2D line graph of an MS contact with a nano metal particle of radius R = 3nm.....................16
Figure 13: COMSOL model of an MS contact with a nano metal particle of radius R = 20nm ....................................16
Figure 14: Electric potential surface distribution of an MS contact with a nano metal particle of radius R = 20nm ..17
Figure 15: Electric field surface distribution of an MS contact with a nano metal particle of radius R = 20nm .........17
Figure 16: Electric potential 2D line graph of an MS contact with a nano metal particle of radius R = 20nm ............18
Figure 17: Electric potential 2D line graph of an MS contact with a nano metal particle of radius R = 20nm ............18
Figure 18: Electric potential profiles in semiconductor region different metal particle radii .....................................19
Figure 19: Electric field profiles in semiconductor region different metal particle radii.............................................19
Figure 20: FWHM plot .................................................................................................................................................20
4. 4
Abstract
Reducing the size of metal particle in metal-semiconductor contacts has revealed
significant improved characteristics over classical planar MS Schottky contacts. The theory
behind this report states that for a metal radius much smaller than the conventional barrier
thickness for planar contacts (nanometers sizes), the new depletion width and thus the potential
profile in the semiconductor depend directly on the radius of the nano metal size. Such an effect
would only be valid for moderately doped semiconductor substrate while it diminishes for highly
doped ones. The report include theoretical and analytical model for nano Schottky contacts
along with simulation using COMSOL software to show the enhanced built in potential, electric
field at the interface. Comparisons between conventional and nano scaled technologies are
discussed throughout the report along with supporting equations along with numerical and
graphical results.
5. 5
Introduction
Research and industry are now more concerned about nano technologies than ever, using
nano scaled devices has revealed more advantages than normal sized ones. Using nano devices
allow higher efficiency and less energy consumption. Metal-semiconductor contacts are one of
the most important devices in solid-state devices. When metal and semiconductor materials are
in contact two possible behaviors could occur; it can be either Ohmic (non-rectifying) or
rectifying. The type of behavior mainly depends on the materials used. Ohmic contacts mainly
exist for heavily doped semiconductors, where high doping would narrow the depletion region
allowing current to pass in either direction. Such contacts are called Ohmic as they obey Ohm’s
law, meaning that the I-V characteristics curve is linear and symmetric. The main concern is the
rectifying behavior of MS contact which is to be discussed next. [1]
6. 6
Schottky diode: Theoretical Analysis
Schottky diode or barrier refers to the rectifying MS contact, and examples of such a
behavior would exist for an n-type semiconductor with work function Φs less than the metal’s
work function ΦM. The semiconductor’s work function can be described by the following
equation
Φs = X + (Ec + EF) Equation 1: Semiconductor work function
It must be noted that both X and ΦM are material constants so they’re unaffected by the
contacting process.
When a metal and semiconductor materials are in contact, electrons flow from the
semiconductor to the metal (higher to lower potential energy) as shown in Figure 1. The net loss
of electrons creates a surface depletion region and a growing barrier to electron transfer. This
process would continue until the transfer rate is the same in both directions and the Fermi level
EF is the same throughout the structure as illustrated in Figure 2. There are three main parameters
in MS contact; barrier height, built in potential, and depletion width. [7]
Figure 1: MS contact an instant after contact formation
Figure 2: under equilibrium conditions
7. 7
ΦB is the surface potential-energy barrier height, which is the potential difference
between the old Ec and the new Fermi level given by Equation (2)
ΦB = ΦM - X Equation 2: MS barrier height
The final energy band diagram is shown in Figure 3, Φi is the built in potential, which is
the difference between the old Ec level and the new Ec level, which depends on the Fermi level of
the semiconductor; xd is the depletion width, the area where potential energy changes.[7]
Figure 3: Energy band diagram of an MS contact
The nature of charge depletion over the surface on the metal side automatically follows
that inside the metal, the electric field E = 0 and electric potential V = constant. Consequently,
the concern would be for the semiconductor side of the MS contact. The electric field and charge
density are related through the Poisson’s equation (Appendix A: Equation 7) which is solved to
find V, E and W. All equations deduced for electric field, potential, and depletion width of a
conventional Schottky diode are included in Appendix A.
Although the I-V characteristics of an MS Schottky diode are very similar to those of pn
junction diode, the detailed current flow is different. The Schottky barrier diode has certain
characteristics that give it a higher advantage compared to the normal pn diode. Normally, the
minority carriers are responsible for the current in pn diodes, where the carriers are injected from
the high to the low doped area while the injection from the lightly doped is negligible. In MS
diode however, the relatively low potential barrier seen by the electrons of the semiconductor
(Figure 2), electrons injection injected into the metal dominates in current flow hence it is said to
be a majority carrier device. [7] The following equation represented for MS diode:
8. 8
` Equation 3: MS diode equation
where B is the Schottky barrier height, VA is applied voltage, A is area, and A*
is
Richardson’s constant.
Schottky diodes however, have lower drop voltage compared to a normal pn diode,
hence it has a faster switching ability, higher efficiency, and lower power consumption. At the
same time, the reverse bias current is generally much larger than in pn diode.
The nano technology effect of shrinking the size of the metal particle is plotted in Figure
3 and labeled (E nano); this shows how moving to nano sizes leads to a narrower depletion
region and an increase in the charge density as the electrons would not distribute over a large
area. The electric field can then be described by the following equation:
𝑬 = 𝑲𝒆
𝑸
𝒓 𝟐
Equation 4: Electric field of a metal particle
So as the radius of the metal particle decreases, electric field increases.
Thermionic Emission vs . Tunneling Effect
One of the most important affects of employing such a technology over conventional
method is the tunneling effect. Thermionic emission is the normal process of moving electrons
(or holes) between the metal and semiconductor regions. However, when the depletion region
decreases, in other words, electric field increases as discussed previously, the potential curve
would decrease sharply leading to electrons being able to jump quickly through the barrier with a
much lower loss of energy. The process described is called the tunneling effect. This effect is
considered significant to nano devices technology as it leads fast switching with low power
consumption. [2][3]
Since theory is not enough to stand alone, following are the new model analysis and its
simulation. The main purpose behind these sections is to prove how for small sizes of metal
particle the potential difference in MS contact depends on the amount of electrons available on
the nano particle, unlike planar MS contacts where the potential depends on the work function.
kTkT
qV
TAIII
BA
ewhere1e 2*
ss
A
9. 9
Metal-Semiconductor Contact: Model Analysis
For feasible analysis of theory and simulation, the model in Figure 4 was created and
employed. Figure 4 visualizes a nano metal particle with a spherical shape, embedded in an n-
type semiconductor substrate with a uniform doping concentration Nd. The depletion region
around the particle would form a spherical shell with a positive charge density ρ = q*Nd.
According to metal-semiconductor contact, the electric field inside the depletion layer is a
superposition of the field from the metal surface and the screening field from the charge density
in the depletion region. The effective potential should be a function of charges being transferred
and the radius of the nano particle; for that reason, the built in potential at the nano particle
should increase within the surface charge density. The change in the electric field is related to the
change in the radius size of the nano particle.[8]
Analyzing the model with the same process described in the theoretical analysis section,
new equations describing the nano Schottky model were deduced. All equations are included in
Appendix B. It can be concluded that unlike previous models, the potential energy at the
interface is not fixed at the planar interface value but it rather depends on the charge density on
the nano metal side.
R
r
w R: nano-metal particle radius
r: semiconductor bulk
w: depletion region
Figure 4: Representative Nano Metal-Semiconductor Model
10. 10
In order to model the interface shown in Figure 4, it’s important to find questions to be
used for simulation but at the same time does not lead to the same equations in Appendix B, or
else the results of simulation would be a representation of the equations not the model itself. For
this cause, it was focused on the charge density. The general definition for the charge density is
the total amount of charge over certain geometry. Therefore, the surface and volume charge
density, ρs and ρv respectively, can be described by the following equations
𝝆 𝒔 =
𝑸
𝟒𝝅𝑹 𝟐
Equation 5: Surface Charge Density
𝝆 𝒗 =
𝑸
𝟒𝝅
𝟑
(𝒓 𝟑− 𝑹 𝟑)
Equation 6: Volume Charge Density
11. 11
Simulation Analysis and Results
Before moving to the main simulation, the equations obtained for the new model
(Appendix B) were used in order to obtain important figures that illustrate how the new theory
would be valid for moderately doped semiconductor region while it’s not applied for high doped
one. Looking at Figures 5 to 7, it can be clearly observed that for moderately doped material the
new equations deduced are applicable while it’s not the case for high doped. Confirming such a
characteristic, the main simulation could be made for moderately doped semiconductor.
Figure 5: Depletion width vs. metal paricle radius for different doping concentrations
Figure 6: Maximum electric field plot for different doping concentrations
10
-9
10
-8
10
-7
10
-6
10
-5
10
-4
0
0.5
1
1.5
2
2.5
x 10
-7
Metal Particle Radius(nm)
DepletionWidth(m)
Nd
= 1*1016
cm-3
Nd
= 1*1018
cm-3
10
-9
10
-8
10
-7
10
-6
10
-5
10
-4
10
6
10
7
10
8
10
9
10
10
Metal Particle Radius(nm)
MaximumInterfaceElectricField(V/m)
Nd
= 1*1016
cm-3
Nd
= 1*1018
cm-3
12. 12
Figure 7: Built in potential at the interface for different doping concentrations
The main simulation tool used for implementing the metal-semiconductor model shown
above was COMSOL software, which is powerful software for studying different physical
models. It has many predefined physics interfaces for different applications. The physics used for
simulating and viewing the properties of an M-S contact was the Electrostatics physics has the
equations, boundary conditions (charge conservation and initial electric potential), and space
charges (surface and volume) for modeling electrostatic fields, solving for the electric potential
and electric field as well. The equations used by the Electrostatics depend on Gauss’s law of
electric displacement field.
COMSOL offers variety of methods for model building, for simplicity, the method
chosen for this model was the axisymmetric one which requires only one half of the model and a
revolving point which leads to a 3D model. In this case, half circles were built and the x origin
was set as the revolving axis, which is considered by the software as metal particle embedded
inside a semiconductor sphere (or bulk).
After building the geometrical model, materials for the two domains (metal and
semiconductor) was defined. The metal domain material was set to copper while the
semiconductor was silicon. After that, the parameters and variables of the model were defined.
Table 1 lists the parameters used for the model:
10
-9
10
-8
10
-7
10
-6
10
-5
10
-4
0
0.5
1
1.5
2
2.5
3
Metal Particle Radius(nm)
Interfacebuilt-inPotential(V)
Nd
= 1*1016
cm-3
Nd
= 1*1018
cm-3
13. 13
Table 1:Variables used for MS COMSOL model
Parameter Description Value
q Electron charge 1.6 × 10-19
C
T Temperature 300 K
Nd
Doping concentration of
semiconductor
1 × 1016
cm-3
ro_v Volume charge density q× Nd = 1600 C.m-3
phi_s
Initial electric potential at
semiconductor boundary
0.2054 V
R Metal particle radius Varies from 3 to 500000 (nm)
w Depletion width Varies according to R (nm)
r Semiconductor section radius w + R (nm)
On the other hand, the variables of the model were the total charge Q and the surface
charge density ro_s. The total charge Q was deduced from Equation (6) in the model analysis
section, while Equation (5) was used for the surface charge density.
Before moving to simulating COMSOL model, the depletion width w was obtained for
the different radii of nano metal particle. Appendix B includes the reference equations used for
calculating w (Equations 11 - 14) and results table (Table 2).
The next step was to simulate the COMSOL model by varying the nano metal particle
radius (R) along with its corresponding depletion width (w) from Table 2.
The following figures represent samples of the simulation model and results for two nano
particle sizes, R = 3nm and R = 20 nm. The figures include the MS interface, surface
distribution, and 2D line graph of both electric potential and electric field.
14. 14
Figure 8: COMSOL model of an MS contact with a nano metal particle of radius R = 3nm
Figure 9: Electric potential surface distribution of an MS contact with a nano metal particle of radius R = 3nm
15. 15
Figure 10: Electric field surface distribution of an MS contact with a nano metal particle of radius R = 3nm
Figure 11: Electric potential 2D line graph of an MS contact with a nano metal particle of radius R = 3nm
Half maximum
voltage
Depletion width at half maximum
16. 16
Figure 12: Electric field 2D line graph of an MS contact with a nano metal particle of radius R = 3nm
Figure 13: COMSOL model of an MS contact with a nano metal particle of radius R = 20nm
17. 17
Figure 14: Electric potential surface distribution of an MS contact with a nano metal particle of radius R = 20nm
Figure 15: Electric field surface distribution of an MS contact with a nano metal particle of radius R = 20nm
18. 18
Figure 16: Electric potential 2D line graph of an MS contact with a nano metal particle of radius R = 20nm
Figure 17: Electric potential 2D line graph of an MS contact with a nano metal particle of radius R = 20nm
Examining the figures above, it can be clearly deduced that the higher the size of the
nano particle, the lower the maximum value of electric field and potential, which corresponds to
the new theory proposed. At the same time, it was observed that the electric field and potential
decrease exponentially. The larger the radius of the metal particle, the lower the decay is.
Figures 18 and 19 represent some electric potential (V) and electric field (E) profiles
inside the semiconductor region for R = 3 to 20 nm. In the metal particle region, E is equal to
zero since the charged particle would exist only at the surface, and hence V would be constant
inside it.
19. 19
Figure 18: Electric potential profiles in semiconductor region different metal particle radii
Figure 19: Electric field profiles in semiconductor region different metal particle radii
0 20 40 60 80 100 120
0
0.5
1
1.5
2
2.5
3
3.5
Electric Potential profiles of an MS interface with a Metal Particle of Radius R
Distance from the surface of the metal particle (nm)
ElectricPotential(V)
R = 3 nm
R = 5 nm
R = 10 nm
R = 20 nm
0 20 40 60 80 100 120
0
2
4
6
8
10
12
x 10
8
Electric Field profiles of an MS interface with a Metal Particle of Radius R
Distance from the surface of the metal particle (nm)
ElectricField(V/m)
R = 3 nm
R = 5 nm
R = 10 nm
R = 20 nm
20. 20
Figure 20 shows the FWHM plot which is a significant way that’s used to describe the
resolution of the images based on Gaussian function; it uses a logarithmic scale for x axis and a
linear y axis. In this context FWHM is important to show the variation of the depletion width for
different radii of nano particle, the size of the particle was varied from 3 nm to 500000 nm.
Figures 11 17 illustrates how the data for FWHM are obtained. Analyzing the FWHM plot, it can
be observed that up to R = 5000 nm, the plot is approximately linear which indicates that
depletion width is a function of the nano particle size. However, the value of FWHM saturates
above R = 5000 nm, indicating for large nano particle sizes the depletion width is independent of
R which reflects to the classical theory.
Figure 20: FWHM plot
10
0
10
1
10
2
10
3
10
4
10
5
10
6
0
20
40
60
80
100
120
Full Width at Half Maximum of Voltage vs. Metal Particle Radius
Metal Particle Radius (nm)
FullWidthatHalfMaximum(nm)
21. 21
Conclusion and Future Work
The research work done for this report has revealed the improved characteristics of
Schottky contacts when moving to nano sizes of metal particles. Theoretical analysis for nano
MS model diverted from the classical theory in using the size of the nano metal particle as a
dependency variable. The narrower depletion width of new Schottky MS contact showed
enhanced built in potential, electric field, and hence easier control over flow of current and lower
power consumption. Results from COMSOL simulation were consistent with theoretical model.
These results can lead to breaking improvements in devices that use conventional MS contacts,
pn junctions, and CMOS devices. Future work would include implementing the model in a better
simulation model for different devices.
22. 22
References
[1] "Metal-semiconductor junction "Semiconductor Devices: Modelling and Technology", Nandita
Dasgupta, Amitava Dasgupta.(2004).
[2] G. D. J. Smit, S. Rogge and T. M. Klapwijk, "Enhanced Tunneling across nanometer-scale metal-
semiconductor interference," Applied physics Letters, vol. 80, no. 14, pp. 2568-2570, 2002.
[3] L.L. Goldin, "Tunneling Effect, The introduction in quantum physics". Nauka NT-MDT, Novikova,
1988.
[4] G. D. J. Smit, S. Rogge and T. M. Klapwijk, "Scaling of nano-Schottky-diodes," Applied Physics
Letters, vol. 81, no. 20, pp. 3852-3854, 2002.
[5] M. A. Laughton, (2003) "Schottky diode," "17. Power Semiconductor Devices" pp. 25–27.
[6] S.M.Sze, "Metal-Semiconductor Contacts," in Semiconductor Devices, Hsinchu, Taiwan, Argosy
Publishing Services, 2001, pp. 225-236.
[7] R. F. Pierret, "MS Contacts and Schottky Diodes," in Semiconductor Device Fundamentals, Wesly,
Accison-Wesly, 1996, pp. 477-500.
[8] M. Rezeq and M. Ismail, "The significant effect of the size of a nano-metal particle on the
interface with a semiconductor substrate," IEEE, 2011.
[9] Razavy, Mohsen (2003), "Quantum tunneling,". Quantum Theory of Tunneling. World Scientific.
23. 23
Appendix A: Conventional Schottky contact equations
𝒅𝑬
𝒅𝒙
=
𝝆
𝑲𝒔 𝜺𝒐
=
𝒒𝑵𝒅
𝑲𝒔 𝜺𝒐
……. 0≤x≤xd Equation 7: Poisson's equation
𝑬(𝒙) =
− 𝒒𝑵𝒅
𝑲𝒔 𝜺𝒐
(𝑾 − 𝒙) ……. 0≤x≤xd Equation 8: Electric field of conventional MS contact
𝑽(𝒙) =
− 𝒒𝑵𝒅
𝑲𝒔 𝜺𝒐
(𝑾 − 𝒙) 𝟐
……. 0≤x≤xd Equation 9: Electric potential of conventional MS contact
𝑾 = (
𝟐 𝑲𝒔 𝜺𝒐
𝒒 𝑵𝒅
(𝑽𝒃𝒊 − 𝑽𝑨))
𝟏
𝟐
Equation 10: Depletion width of a conventional MS
contact
Where Vbi is the built in potential and VA is the applied voltage
24. 24
Appendix B: Nano Schottky contact equations
Reference equations 11 through 14 for nano MS contacts were implemented in Matlab to get the
depletion width w for different metal particle sizes. The resulting values were tabulated in Table
2; values of w were used in COMSOL simulation which was discussed in Simulation Analysis
and Results section.
s
s
xR
xR
wR
wRxV
2
)(
)(
)(
)(
2
3
3
)(
23
2
Equation 11: Electric potential of nano MS contact
33
11
3R4
Q
RwR
R
cVcVV
s
bp
s
bpbn
Equation 12: Built in Potential of nano MS contact
2
s
20000
R4
Q
cEEEE penpn Equation 13: Electric field at the interface of nano MS contact
0
2
)(
)(
2
3
33
1 23
2
2
0
0
R
R
wR
wR
R
wRw
EV
s
pbp
Equation 14: Solving for enhanced w
