Design of Nanoscale
Electromechanical Switches
Design Group 24
Brianna Cleary, ME
Lian Xin Huang, ME
Faculty Advisor: Changhong Ke
May 08, 2009
Submitted in partial fulfillment of the requirements of ME 494/EECE 488 in the Spring Semester of 2009.
Thomas J. Watson School of Engineering and Applied Science
State University of New York at Binghamton
Abstract
This project is the design of a nanoscale electromechanical switch which is similar to a
nano electromechanical system (NEMS). In order to understand the behavior of NEMS, the
switch is modeled in a multi-physics package as an external electrical signal is applied to the
switch. A multi-physics package combines the structural and electro mechanics of a system and
delivers the response of the system. The multi-physics package used for the finite element
analysis (FEA) of NEMS is ANSYS. The system is configured to either be a doubly clamped or
singly clamped beam positioned over an infinite conductive substrate. It is to represent a nano-
scaled electromechanical switch that converts an electrical signal into a mechanical result. To
fully understand the response of the system, the switch is modeled with the parameters set in
each case. The parameters that make up the beam properties are the radius, length of beam,
distance between substrate and beam, and Young s Modulus. Once the model was established in
ANSYS, a range of external electrical signals were applied to the beam to find at what voltage
the system is in equilibrium or instable.
Design of Nanoscale Electromechanical Switches:
Table of Contents
Table of Contents
I. Introduction.............................................................................................................................. 1
A. Nanotechnology ................................................................................................................... 1
B. Nano Electromechanical Systems (NEMS) ......................................................................... 2
II. Objective Identification ........................................................................................................... 5
III. NEMS Design .......................................................................................................................... 6
A. Beam Configuration ............................................................................................................. 6
B. Beam Material...................................................................................................................... 6
C. Beam Parameters ................................................................................................................. 7
D. Predicted Behavior ............................................................................................................... 8
IV. Solutions .................................................................................................................................. 9
A. Reduced Order Modeling..................................................................................................... 9
B. ROM144 ............................................................................................................................ 12
C. Multi-field Solver............................................................................................................... 13
V. Design Verification................................................................................................................ 15
A. Reduced Order Model ........................................................................................................ 15
1. Build Process ................................................................................................................. 15
2. Test Procedure ............................................................................................................... 16
3. Test Results .................................................................................................................... 18
B. ROM144 Model ................................................................................................................. 24
1. Build Process ................................................................................................................. 24
2. Test Procedure ............................................................................................................... 25
3. Test Results .................................................................................................................... 25
C. Multi-field Solve ................................................................................................................ 26
1. Build Process ................................................................................................................. 26
2. Test Procedure ............................................................................................................... 26
D. Requirements Matrix ......................................................................................................... 27
E. Current Status..................................................................................................................... 27
VI. Case Study (Final Design) ..................................................................................................... 28
VII. Documentation....................................................................................................................... 30
A. Requirement Matrix ........................................................................................................... 30
B. Schedule ............................................................................................................................. 30
C. Budget ................................................................................................................................ 30
D. Expenses ............................................................................................................................ 30
E. Research Data .................................................................................................................... 30
i
Design of Nanoscale Electromechanical Switches:
Table of Contents
F. Calculation ......................................................................................................................... 30
G. Reduced Order Model Code .............................................................................................. 31
H. ROM144 Code ................................................................................................................... 31
I. Multi-field Solver Code ..................................................................................................... 31
J. Test Plan ............................................................................................................................ 31
VIII. Conclusion ............................................................................................................................. 32
IX. Work Cited............................................................................................................................. 33
X. Acknowledgements................................................................................................................ 34
XI. APPENDIX............................................................................................................................ 35
APPENDIX A- Requirement Matrix ......................................................................................... 36
APPENDIX B- Spring 2009 Schedule ...................................................................................... 37
APPENDIX C- Spring 2008 Budget.......................................................................................... 39
1. Estimated Budget ........................................................................................................... 39
2. Estimated Labor ............................................................................................................. 39
3. Actual Budget ................................................................................................................ 40
APPENDIX D- Expenses .......................................................................................................... 41
APPENDIX E  Research Data ................................................................................................. 42
1. Static Stability Characteristics ....................................................................................... 42
2. Pull-in Voltages for Various Cases ................................................................................ 42
APPENDIX F- Calculation ........................................................................................................ 43
1. Excel File ....................................................................................................................... 43
2. MATLAB Code ............................................................................................................. 44
APPENDIX G- Reduced Order Model Code ............................................................................ 45
1. ANSYS Code-Linear Analysis for Double Clamped Beam .......................................... 45
2. ANSYS Code-Nonlinear Linear Analysis for Double Clamped Beam ......................... 47
APPENDIX H- ROM 144 ......................................................................................................... 50
1. Model Parameters .......................................................................................................... 50
2. Generation Pass.............................................................................................................. 52
3. Use Pass ......................................................................................................................... 54
APPEMDIX I- Multi-field Solver ............................................................................................. 55
APPENDIX J- Test Plan ............................................................................................................ 58
ii
Design of Nanoscale Electromechanical Switches:
List of Figures
List of Figures
Figure 1 Illustrates the nano scale in comparison to common units of measurement..................... 1
Figure 2 Schematic of the Electrostatic Actuator ............................................................................ 2
Figure 3 Static Stability of Force vs. Deflection (V1Figure 4 a&b Schematic of nanotube based NEMS Device [10]..................................................... 6
Figure 5 Strip of a Graphene Sheet Rolled into a Tube [12, pg 4-2] .............................................. 7
Figure 6 Schematic of the finite configuration of a cantilever nanotube device subjected to
electrostatic forces and van der Waals forces [10]......................................................... 8
Figure 7 Schematic of the finite configuration of a fixed-fixed nanotube device subjected to
electrostatic forces and van der Waals forces [10]......................................................... 8
Figure 8 Reduced Order Model and Elements ............................................................................... 10
Figure 9 Block Diagram (Level 0) ................................................................................................. 11
Figure 10 Mechanical Block Diagram ........................................................................................... 11
Figure 11 Electrical Block Diagram .............................................................................................. 11
Figure 12. Cross Section used for Second Moment of Inertia ....................................................... 12
Figure 13 Atmosphere (Electrostatic Element) around the Beam (Structure Element) ................. 13
Figure 14 Schematic of the Force & Displacement Transfer between the Beam and Atmosphere
[13] ............................................................................................................................... 14
Figure 15 Plot of Capacitance vs Gap Distance............................................................................. 16
Figure 16 Beam and Capacitances ................................................................................................ 17
Figure 17 Deformed fixed-fixed beam due to applied voltage less than pull-in voltage .............. 17
Figure 18 Deformed fixed-fixed beam due to applied voltage greater than pull-in voltage ........ 17
Figure 19 Effect of Changing Length : H=100nm, R=10nm, E=1TPa, DC, Linear ..................... 20
Figure 20 Effect of Changing Radius : H=100nm, L=3000nm, E=1TPa, DC, Linear .................. 20
Figure 21 Effect of Changing Gap Distance: R=10nm, L=3000nm, E=1TPa, DC, Linear ........... 21
Figure 22 Nanotube vs. Silicon: H=100nm, R=10nm, L=3000nm, DC, Linear ............................ 22
Figure 23 Single Clamped vs. Double Clamped: H=100nm, R=10nm, L=500nm, E=1TPa, Linear
..................................................................................................................................... 22
Figure 24 Double Clamped: Linear vs. Non-Linear: R=10nm, H=100nm, E=1TPa, L=4000nm . 23
Figure 25 Single Clamped: Linear vs. Nonlinear: H=100nm, R=10nm, L=500nm, E=1TPa ....... 24
Figure 26. ROM144 model of the beam and atmosphere .............................................................. 25
Figure 27 Multi-field model of Beam and Atmopshere................................................................. 26
Figure 28 Nano-Switch Configuration........................................................................................... 28
Figure 29 Requirement Matrix....................................................................................................... 36
Figure 30 Static Stability Characteristics according to ANSYS [6] ............................................. 42
Figure 31 Numerical and Theoretical Pull-In Voltages [10] ......................................................... 42
iii
Design of Nanoscale Electromechanical Switches:
List of Tables
List of Tables
Table 1 Cases are used to verify the ANSYS model developed [10] ............................................ 7
Table 2. Volume Load Transfer between Beam and Air[13]......................................................... 14
Table 3 Cumulative Pull-in Voltage Results.................................................................................. 19
Table 4. ROM144 and Theory Pull-in voltage .............................................................................. 26
Table 5 Set of Parameters for Case Study...................................................................................... 29
Table 6 Von Mises Stress and Von Mises Strain ........................................................................... 29
Table 7 Estimated Budget .............................................................................................................. 39
Table 8 Estimated Labor ................................................................................................................ 39
Table 9 Actual Budget ................................................................................................................... 40
Table 10 Plot Capacitance Per Unit Length vs. Gap Distance....................................................... 43
iv
Design of Nanoscale Electromechanical Switches:
Introduction
I. Introduction
A. Nanotechnology
 Coals and diamonds, sand and computer chips, cancer and healthy tissue: throughout
history, variations in the arrangement of atoms have distinguished the cheap from the cherished,
the diseased from the healthy. [1, pg. 2] Eric Drexler introduced the science of nanotechnology
in his 1986 book Engines of Creations. Drexler explains the history, significance, predicts certain
outcome, and cautions about the dangers of nanotechnology. [1]
Nanotechnology is the field of building functional materials, devices, and systems that
have the ability to control atoms and molecules. Since the arrangement of atoms of an element
gives that particular element its specific characteristics, nanotechnology is a whole new frontier
into the world of science. The ability to alter and arrange atoms will lead scientist and engineers
to develop products that are much faster, smarter, and smaller. This breakthrough will have an
unsurpassed technological effects for the medical, electronic, and energy industries. [2]
Figure 1 Illustrates the nano scale in comparison to common units of measurement
Nanotechnology works within the range of 1 nm to about 100nm. To help visualize the
scale in which nanotechnology uses, Figure 1 illustrates the units of measurement involved. [3]
1
Design of Nanoscale Electromechanical Switches:
Introduction
Observing nano scale materials and devices are important because materials reduced to a
smaller scale can begin to show different attributes compared to their larger scale properties.
Determining material properties like strength, conduction, magnetic tendencies and light
reflection at the nano scale can help develop more favorable devices and systems. [4] For
example, an element considered to be an insulator in the meter scale can actually have
semiconductor tendencies in the nano scale. Due to property changes at the nano level, it is
important to realize that nano science requires one to forget what is known and to start learning
all over again. [2]
B. Nano Electromechanical Systems (NEMS)
Electromechanical devices consist of a transducer that converts electrical signal into
mechanical energy and vice versa. To understand the mechanical element of the applied voltage,
Figure 2 illustrates the mechanical schematic of an electrostatic actuator. The mechanical
element of this system deflects as an external electrical signal is applied to the switch.
Ael
Figure 2 Schematic of the Electrostatic Actuator
The electrostatic force and spring force acting on the movable plate is obtained by
Equation (1) where V is the applied voltage, C is the capacitance Equation (2) of the plate area,
Ael, k is the stiffness of the spring, and 5Øß is the permittivity of the vacuum. The deflection is
denoted as the distance between the movable plate and capacitor plate in Equation (3), where d is
the original gap and x is the moved distance of the plate.
2
Design of Nanoscale Electromechanical Switches:
Introduction
Equation (1)
1 5Ø6Ü
5Ø9Ü = 5ØIÜ2 - 5ØXÜ5ØeÜ
2 5ØQÜ - 5ØeÜ
5Ø4Ü5ØRÜ5ØYÜ
Equation (2)
5Ø6Ü = 5Øß
5ØQÜ - 5ØeÜ
Equation (3)
5ØÿÞ = 5ØQÜ - 5ØeÜ
The electrostatic forces with various voltages and mechanical forces are plotted with
respect to the deflection of the movable plate in Figure 2.
F
Unstable
V2
V3
V4
V1
Pull-In
Voltage
Stable
´
Figure 3 Static Stability of Force vs. Deflection (V1Since the beam is modeled to be a switch, the system is considered to be in the  on
position when the beam collapses onto the substrate. The pull-in voltage is the peak voltage
applied to the beam that causes it to collapse onto the substrate. The only point that the Pull-in
voltage can be obtain is when the electrostatic force is tangent to the spring force of the beam.
The Pull-in voltage labeled in Figure 3 is obtained when the slope of the electrostatic force and
spring force equal each other shown in Equation (4). [5]
5ØQÜ 5ØQÜ 1 5Øß5Ø4Ü5ØRÜ5ØYÜ
5ØXÜ5ØeÜ = 5ØIÜ2
Equation (4)
2
5ØQÜ5ØÿÞ 5ØQÜ5ØÿÞ 2 5ØQÜ - 5ØeÜ
3
Design of Nanoscale Electromechanical Switches:
Introduction
The important points of the plot are when the spring force and electrostatic force are
equal to each other. As illustrated in Figure 3, the system is considered unstable when the
distance of the movable plate and capacitor plate is approaching zero. The system is considered
stable when the deflection is approaching the original gap distance, d. Figure 3 is a similar to
ANSYS s Static Stability Characteristics found in APPENDIX E  Research Data.[6]
In order to make processes faster and smaller, electromechancial systems are being
analyzed in a nano scale. The demand to used NEMS for new applications is growing because of
their superior speed, smaller size, and mass. An example of NEMS is a switch that can be used
as a memory device for computers or other electronic devices. The switch is the mechanical
element and when voltage is applied the beam is to be displaced. [7]
4
Design of Nanoscale Electromechanical Switches:
Objective Identification
II. Objective Identification
It cannot be assumed that the behaviors of nanoscaled devices are similar to their larger
scale components. The purpose of this project is to design of a nanoscaled electromechanical
switch. This is accomplished by the investigation of electrical and mechanical characteristics of
nanotube based electromechanical devices.
5
Design of Nanoscale Electromechanical Switches:
Design
III. NEMS Design
A. Beam Configuration
The two different types of mechanical elements in which the switch was modeled after is
a cantilever beam and fixed-fixed beam. The cantilever beam has no translation in the x and y
direction at one given end, where as the fixed-fixed beam has no translation in the x and y axis at
both ends of the beam. The schematic of a cantilever beam and fixed-fixed beam is illustrated in
Figure 4.
Even though both beam configurations have tendencies to displace as force is applied,
they behave a little differently depending on their lengths. Since both ends are constrained, fixed-
fixed beams are stiffer than cantilever and need a longer length in order to deflect.
Figure 4 a&b Schematic of nanotube based NEMS Device [10]
B. Beam Material
Silicon is commonly used in Micro electromechanical systems (MEMS) however if used
in NEMS, the silicon wire starts
. to leak electricity. [8] Instead of modeling the beams as silicon, carbon nanotubes are
used to model the switches. Carbon nanotubes have a small size, great strength, favorable
electronic properties, low density, high stiffness, and flexibility. To complete the design, the
carbon nanotubes are suspending over an infinite conductive substrate, creating a functional
NEMS device. [9]
The switches have a radius because nanotubes and nanowires are rolled sheets of carbon
atoms in the shape of a tube. Different patterns emerge when the sheet is rolled in a specific
6
Design of Nanoscale Electromechanical Switches:
Design
direction as shown in Figure 5. Depending on the arrangement of carbon atoms, the carbon
nanotube can be metallic or semiconductive. [2]
Figure 5 Strip of a Graphene Sheet Rolled into a Tube [12, pg 4-2]
C. Beam Parameters
The parameters of the beam are known to have an effect on the behavior of the beam
once an external electrical signal is applied to it. The design parameters include the radius of the
beam(R), the external electrical signal (V), gap between the substrate and beam (H), length of the
beam (L), and Elastic Modulus (E).
E H L
Case Beam (TPa) (nm) (nm) R (nm)
1
1 Fixed-Fixed 100 4000 10
1
2 Fixed-Fixed 100 3000 10
1
3 Fixed-Fixed 100 2000 10
1
4 Fixed-Fixed 150 3000 10
1
5 Fixed-Fixed 200 3000 10
1
6 Fixed-Fixed 100 3000 20
1
7 Fixed-Fixed 100 3000 30
1
8 Cantilever 100 50 10
1
9 Cantilever 100 50 10
Table 1 Cases are used to verify the ANSYS model developed [10]
Table 1 shows the cases analyzed in Professor Ke s  Numerical Analysis of Nanotube
Based NEMS Devices  Part II: Role of Finite Kinematics, Stretching and Charge
Concentrations. These cases are used to verify the ANSYS model developed.
7
Design of Nanoscale Electromechanical Switches:
Design
D. Predicted Behavior
Figure 6 and Figure 7 show the predicted behavior of the cantilever beam and fixed-fixed
beam as voltage is applied. Since the beams are electromechanical devices, the electrical
element, voltage, is having a mechanical effect on the system, deformation. Deformation is due
to the electrostatic forces and van der Waals forces subjected to the beam. These forces push and
pull the beam towards the conductive substrate. The electrostatic force per unit length of the
nanotube is a function capacitance shown in Figure 6 and Figure 7. The electrostatic Force
between the substrate and beam can be found with use of Equation (5).
1 5ØQÜ5Ø6Ü
Equation (5)
5Ø^Ü5ØRÜ5ØYÜ5ØRÜ5ØPÜ = 5ØIÜ2
2 5ØQÜ5Ø_Ü
5ØQÜ5Ø6Ü
where V is the voltage and is capacitance per unit length.
5ØQÜ5Ø_Ü
Figure 6 Schematic of the finite configuration of a Figure 7 Schematic of the finite configuration of a
cantilever nanotube device subjected to fixed-fixed nanotube device subjected to electrostatic
electrostatic forces and van der Waals forces [10] forces and van der Waals forces [10]
8
Design of Nanoscale Electromechanical Switches:
Solutions
IV. Solutions
In the electrostatic field analysis of NEMS, it is important to determine both
capacitance and electrostatic forces which are typically used to actuate devices such as
nano switches. Coupling the electrostatics and structural physics allows the actual
electrostatic actuation of a NEMS device to be simulated, capturing the relationship
between these two environments. As a structure is deformed, the electrostatic field
distribution and the force generated will change. ANSYS provides six different methods
of analyzing electrostatic-structural systems. The three methods that were used in this
project is Reduced Order Modeling (Trans126), ROM144, and Multi-field solver.
A. Reduced Order Modeling
Since it is very difficult and expensive to physically test nano switches, a multi-
physics package is need to properly analyze nano electromechanical system. ANSYS is a
reliable package that is used for finite element analysis and can solve a wide variety of
mechanical problems. [12] Two models that will be used to analyze the characteristics of
this device are Reduced Order Modeling and Full Field Modeling. The Reduce Order
Model and Full Field Model are used to find the pull-in voltage specific and verify each
case stated in Table 1. APPENDIX E  Research Data includes the published pull-in
voltage data in  Numerical analysis of nanotube based NEMS devices - Part II
The model of the device should include a charged fixed-fixed or cantilever beam,
ground plate and surrounding air. In order to simplify the case, surrounding air is
excluded in the first modeling method, Reduced-Order Model, as shown in Figure 8.
According to Coupled-Field Analysis Guide, Reduced-order means  the electrostatic
characteristics of an electromechanical device are captured in terms of the device s
capacitance over a range of displacement and formulated in a simple coupled beam-like
element. [13] In other words, the device s capacitance is used to visualize the
electrostatic characteristics. In ANSYS, TRANS126, as shown in Figure 8, is an
electromechanical transducer that is used to simulate the capacitance. The more
capacitances created, the more accurate the analysis would be. The capacitance along the
side surface per unit length for an infinitely long tube can be expressed as [14]
9
Design of Nanoscale Electromechanical Switches:
Solutions
Equation (6)
Where r is the gap distance between the beam and the ground plate, R is the external
radius of the beam, and 5Ø:ß5ØÎß is the permittivity of vacuum .
Since ANSYS can only take polynomial form of equation, so Equation (6) needs to be
transformed into polynomial, such as
Equation (7)
For calculation purpose, where H is the initial gap distance between the
beam and the ground plate. For different combination of initial gap distance H and
external radius R, there will be a different polynomial equation for capacitance C.
Figure 8 Reduced Order Model and Elements
There are two types of analyses concerned in the Reduced Order Model: linear
analysis and nonlinear analysis. In linear analysis, the beam deformation is only due to
bending. In nonlinear analysis, beam deformation is not only due to bending but also
stretching. The nonlinear effect is only significant with double clamped beam. In other
words, the pull-in voltage of single clamped beam with nonlinear analysis is not much
different than that of linear analysis.
Figure 9 shows the level 0 block diagram of the ANSYS Simulation of Nano-
switch. The whole idea of the ANSYS simulation is applying voltage to the beam and
10
Design of Nanoscale Electromechanical Switches:
Solutions
getting the beam deformation as output. Pull-in voltage can be determined based on the
deformation. Pull-in voltage is the maximum voltage that can be applied to the beam
before the beam collapses to the substrate. Figure 10 is the mechanical block diagram
which shows that the produced electrostatic force is applied on the beam (double clamped
or single clamped), and then the beam is deformed. In order to under how to get the
electrostatic force, the electrical block diagram as shown in Figure 11 needs to be
introduced. The applied voltage is applied on the beam, and the substrate is grounded by
applying zero voltage. Due to the voltage potential difference, the beam and the substrate
can form capacitances between them. As a result, the capacitances produce the
electrostatic forces between the beam and substrate. As the beam deflects downward, the
capacitance changes resulting change in electrostatic force.
Figure 9 Block Diagram (Level 0)
Figure 10 Mechanical Block Diagram
Figure 11 Electrical Block Diagram
11
Design of Nanoscale Electromechanical Switches:
Solutions
B. ROM144
ROM144 is a dynamic reduced order macromodel of a complex 3-D electrostatic-
structural system. [15] It creates a mathematical representation of the coupled system
derived from the full Finite Element Analysis to find the pull-in voltage. There are four
key stages to creating this analysis model. [13]
1) Model Preparation: Creates the finite element model to be used.
The cases used in this analysis have very large length compared to the radius
of the beam. This caused issues with running the analysis of the system. The
beams cross section was made to be a square and in order to find the height
and width was found by equaling the second moment of inertia of a square
cross section to a beam with a circular cross section. As shown in Figure 12,
the square and cicular cross section is to find the side length of square beam to
be used in ANSYS.
a
r
a
Figure 12. Cross Section used for Second Moment of Inertia
5Ø<Ü5Ø`Ü = 5Ø<Ü5ØPÜ
Equation (8)
5ØNÜ4 5Øß5Ø_Ü4
=
Equation (9)
12 4
2) Generation Pass: At this point, the program executes the static analysis used to
determined the deformation of the structure under different operating
conditions. With this information, the generation pass creates mathematical
relationships to produce the reduced order model that will be used to find the
12
Design of Nanoscale Electromechanical Switches:
Solutions
behavior of the system. The model is stored into a ROM database and
polynomial coefficient file.
3) Used Pass: The ROM database and polynomial coefficient file is imported and
the analysis of the system is then performed to find the Pull-in voltage.
4) Expansion Pass: The review of the stresses and displacements of the FEA
model can be shown. Animations may also be created to show the response of
the system due to a certain load. [16]
C. Multi-field Solver
In order to begin the multi-field solver, the code creates fields defined by different
element types to identify the beam and surrounding air. The air and beam were treated as
separate entities as independent model with different mesh. Volume load transfer is
enabled by indicated the couple field load region. The solver than iterates between the
two fields until load is fully transferred.
As shown in Figure 13, the atmosphere around the beam is modeled as the
electrostatic element of the system. The two entities are meshed after being modeled and
the load transfer interface is then identified. Load transfer is the process of one meshed
field transmits quantities to another meshed field. As shown in Figure 14, once voltage is
applied forces are transmitted from the electrostatic field to the structural field and the
displacements are transmitted from the structural domain to electrostatic field. Table 2
shows the load transfer between the structural element and electrostatic element in a
couple physics analysis.
Figure 13 Atmosphere (Electrostatic Element) around the Beam (Structure Element)
13
Design of Nanoscale Electromechanical Switches:
Solutions
Volume
Structural Electrostatic
Load
(Beam) (Atmosphere)
Transfer
Send Displacement Forces
Receive Forces Displacements
Figure 14 Schematic of the Force &
Table 2. Volume Load Transfer between Beam
Displacement Transfer between the Beam and
and Air[13]
Atmosphere [13]
Once the constraints and load transfer surfaces or volumes of the model are
addressed, the ANSYS multi-field solver solution procedure is ready to be completed.
The final step is to define the fields and capture the field solutions of the model in
ANSYS. [13]
14
Design of Nanoscale Electromechanical Switches:
Design Verification
V. Design Verification
A. Reduced Order Model
1. Build Process
First of all, the polynomial coefficients (5Ø6Ü0, 5Ø6Ü1, 5Ø6Ü2, 5Ø6Ü3, 5ØNÜ5Ø[Ü5ØQÜ 5Ø6Ü4) of capacitance per
unit length, Equation (10) need to be determined. After obtaining the coefficients, the
total capacitance for each beam element is calculated by Equation (10), where L is the
total length of the beam, and N is the number of nodes on the beam.
5Ø?Ü
Equation (10)
5Ø6Ü = 5Ø6Ü5ØQÜ
5ØAÜ
Figure 15 is obtained for the case that H=100nm, R=10nm using Excel data
shown in Appendix F section a. It shows the distributed capacitance has non-linear
relationship to the gap distance. The distributed capacitance decreases with increase of
gap distance. The MATLAB code in Appendix F section b is used to verify the
capacitance coefficients.
Using H=100nm, R=10nm, and the number of points n=200 as input for the
MATLAB function file in Appendix F, the coefficient output is
p =[ -14.5344e-3, 4.1875e-3, -463.5314e-6, 36.9939e-6]
This means
5Ø6Ü5ØQÜ = -1.4534E - 2 " r3 + 4.1875E - 3 " r2 - 4.63531E - 4 " r + 3.69939E - 5
From
Figure 15, the distributed capacitance as a function of gap distance is
5Ø6Ü5ØQÜ = -1.312045Ø8Ü - 02 " 5Ø_Ü3 + 3.908755Ø8Ü - 03 " 5Ø_Ü2 - 4.459475Ø8Ü - 04 " 5Ø_Ü + 3.663995Ø8Ü - 05
15
Design of Nanoscale Electromechanical Switches:
Design Verification
Since the distributed capacitance expressions obtained using both methods are
very close, it is reasonable to use either expression. For the rest of analysis of
capacitance, excel is used as main method, and the MATLAB is used as verification.
Capacitance Per Unit Length vs. Gap Distance
2.50E-05
2.40E-05
Theoretical Values
2.30E-05
2.20E-05
Poly. (Theoretical Values)
2.10E-05
2.00E-05
1.90E-05
1.80E-05
y = -1.31204E-02x3 + 3.90875E-03x2 - 4.45947E-04x + 3.66399E-05
1.70E-05
R² = 9.99926E-01
1.60E-05
0.03 0.05 0.07 0.09 0.11
Gap Distance r [micro-meter]
Figure 15 Plot of Capacitance vs Gap Distance
2. Test Procedure
The following are the steps to create and solve the Reduced-Order Model on
ANSYS and the ANSYS code is attached in the Appendix F for detail description:
1. Define parameters: length of beam L, Radius of beam R, gap distance H, Elastic
Modulus of beam E, and applied voltage V.
2. Define material properties of beam and element types for both beam and capacitance.
The capacitance coefficients are input as real constant of TRANS126 element.
3. Create the beam and capacitances with their own element type, as shown in Figure
16. Beam is the horizontal light blue element; capacitances are the vertical Purple
Elements (Figure 8 Shows the Detail View of One Capacitance)
4. Define boundary conditions, either fixed-fixed or cantilever.
16
[pF/micro-meter]
Capacitance Per Unit Length
Design of Nanoscale Electromechanical Switches:
Design Verification
5. Apply the specified voltage on the nodes of the beam, and ground the nodes of the
ground plate by applying zero voltage.
6. Solve the static analysis with linear solution (only consider bending).
Figure 17 shows a deformed fixed-fixed beam due to an applied voltage that is
less than the pull-in voltage. On the other hand, Figure 18shows the deformed fixed-
fixed beam when an applied voltage that is greater than the pull-in voltage. The middle
part of the beam is flat because it snaps to the ground plate.
Figure 16 Beam and Capacitances
Figure 17 Deformed fixed-fixed beam due to applied voltage less than pull-in voltage
Figure 18 Deformed fixed-fixed beam due to applied voltage greater than pull-in voltage
In step 5, the maximum displacement of the beam with certain voltage V can be
obtained. The pull-in voltage is the voltage such that the displacement of the beam
reaches its maximum before it snaps to the ground plate. As the voltage increasing, the
attraction force between the beam and ground plate increases and the gap decreases.
Considering the beam as a spring-mass system and with a gap distance 5Ø_Ü, the spring
restoring force is proportional to 5Ø_Ü according to Hooke s law; and the electrostatic force
is proportional to 1/5Ø_Ü2 due to Coulomb s law. When the gap distance decreases to a
certain value, the electrostatic force is much larger than the spring restoring force. As a
17
Design of Nanoscale Electromechanical Switches:
Design Verification
result, the beam snaps to the ground plate. On the contrary, the beam and the ground plate
snap apart when the voltage decreases to a certain point.
Debugging the ANSYS code was most important step before actual testing. The
next step was to hold all except one parameters constant, and then input different voltage
in the ANSYS code. The maximum deflection at the middle of beam associate with each
input voltage was recorded to excel file. Minimum of 15 data points were needed to show
the effect of changing each parameter.
3. Test Results
Last but not least, a cumulative table pull-in voltage of each case with changing
one parameter at a time was constructed. The overall table of collected pull-in voltage
for each case is shown in Table 3. Most of the cases have less than Ä…5% error. Cases 1-3
show the effect of changing length of beam that the pull-in voltage increases as the length
decreases. Cases 2, 4 and 5 imply that pull-in voltage increases with increase of radius of
beam. Cases 2, 6, and 7 show that increase of initial gap distance resulting increase of
pull-in voltage. Cases 8 and 9 are comparison between single and double clamped
boundary conditions. Pull-in voltage for double clamped beam is much higher than single
clamped beam. Pull-in voltage of Case 9 is almost 6.5 times that of case 8. Case 10 is for
silicon wire. Compared to Case 2, the pull-in voltage for Case 10 is about 62% less. A
mentioned before, nonlinear analysis is performed as well. Columns 7 and 10 are
comparisons between linear and nonlinear analysis. For double clamped beam, the pull-in
voltage significantly increases with nonlinear analysis. However, there is not much
difference between linear and nonlinear analysis for single clamped beam. Therefore,
large deformation in single clamped beam is negligible.
18
Design of Nanoscale Electromechanical Switches:
Design Verification
1 D 1 100 4000 10 3.15 3.2 -1.56% 9.01 9.06 -0.55%
2 D 1 100 3000 10 5.6 5.69 -1.58% 15.88 16.14 -1.61%
3 D 1 100 2000 10 12.6 12.81 -1.64% 35.34 36.31 -2.67%
4 D 1 100 3000 20 18.55 19.21 -3.44% 32.58 31.57 3.20%
5 D 1 100 3000 30 37.32 38.57 -3.24% 52.25 51.96 0.56%
6 D 1 150 3000 10 9.35 9.45 -1.06%
7 D 1 200 3000 10 13.37 13.53 -1.18%
8 S 1 100 500 10 31.2 27.28 14.37% 32.64 31.66 3.10%
9 D 1 100 500 10 201.47
10 D 0.15 100 3000 10 2.17 6.31
Table 3 Cumulative Pull-in Voltage Results
From Figure 19, the initial gap distance is 100nm, the radius of beam is 10nm,
elastic modulus of beam is 1 trillion Pascal, and the beam is double clamped under linear
analysis. Therefore, only length of the beam is being varied. Pull-in Voltage increases as
beam length decreases. In other words, it requires higher input voltage for shorter beam
to obtain the same deflection as the longer beam.
19
E
L
R
H
Case
[nm]
[nm]
[nm]
V_IP
V_IP
V_IP
V_IP
[Volts]
[TPa]
[Volts]
[Volts]
[Volts]
(Theo-
%Error
%Error
(Linear)
Boundary
Conditions
NonLinear)
(NonLinear)
(Theo-Linear)
Design of Nanoscale Electromechanical Switches:
Design Verification
Effect of Changing Length
120
100
80
60
40
20
0
0 5 10 15
Voltage V [volts]
L=4000nm L=3000nm L=2000nm
Figure 19 Effect of Changing Length : H=100nm, R=10nm, E=1TPa, DC, Linear
The effect of only changing radius of beam is shown in Figure 20. Pull-in Voltage
increases as beam radius increases.
Effect of Changing Radius
120
100
80
60
40
20
0
0 10 20 30 40
Voltage V [volts]
R=10nm R=20nm R=30nm
Figure 20 Effect of Changing Radius : H=100nm, L=3000nm, E=1TPa, DC, Linear
20
Gap Distance r(x) [nm]
Gap Distance r(x) [nm]
Design of Nanoscale Electromechanical Switches:
Design Verification
Figure 21 shows the effect of changing gap distance, pull-in Voltage increases as
gap distance increases.
Effect of Changing Gap Distance
250
200
150
100
50
0
0 5 10 15
Voltage V [volts]
H=100nm H=150nm H=200nm
Figure 21 Effect of Changing Gap Distance: R=10nm, L=3000nm, E=1TPa, DC, Linear
Changing material of beam can cause the change in pull-in voltage because
different material has different elastic modulus which can tell how stiff the beam it. If the
elastic modulus is large, then the beam is stiffer, as a result, the switch has higher pull-in
voltage. In short, Figure 22 pull-in voltage increases as elastic modulus of beam
increases.
21
Gap Distance r(x) [nm]
Design of Nanoscale Electromechanical Switches:
Design Verification
Nanotube vs. Silicon
120
100
80
60
40
20
0
0 2 4 6
Voltage V [volts]
Nanotube E=1TPa Silicon E=150GPa
Figure 22 Nanotube vs. Silicon: H=100nm, R=10nm, L=3000nm, DC, Linear
Single Clamped vs. Double Clamped
120
100
80
60
40
20
0
0 50 100 150 200 250
Voltage V [volt]
Single Clamped Double Clamped
Figure 23 Single Clamped vs. Double Clamped: H=100nm, R=10nm, L=500nm, E=1TPa, Linear
Figure 23 shows the effect of changing boundary condition. Double clamped
beam has higher pull-in voltage than single clamped beam.
22
Gap Distance r(x) [nm]
Gap Distance r(x) [nm]
Design of Nanoscale Electromechanical Switches:
Design Verification
Double Clamped: Linear vs. Non-Linear
120
100
80
60
40
20
0
0 5 10
Voltage V [volts]
Linear Analysis NonLinearAnalysis
Figure 24 Double Clamped: Linear vs. Non-Linear: R=10nm, H=100nm, E=1TPa, L=4000nm
All cases have been discussed above are using linear analysis, but in real life, the
beam does not behavior absolute linearly. Therefore, nonlinear analysis is necessary for
large deformation of the beam. Figure 24 shows that Non-linear analysis provides higher
pull-in voltage. This conclusion is only suitable for double clamped beam. Since there is
nothing to constrain the free end of the single clamped beam, so the beam will deform
freely. According to Figure 25, the nonlinear analysis was not significant for single
clamped beam.
23
Gap Distance r(x) [nm]
Design of Nanoscale Electromechanical Switches:
Design Verification
Single Clamped: Linear vs. Nonlinear
120
100
80
60
40
20
0
0 5 10 15 20 25 30 35
Voltage V [volts]
Single Clamped Nonlinear SC
Figure 25 Single Clamped: Linear vs. Nonlinear: H=100nm, R=10nm, L=500nm, E=1TPa
It was hard to get the pull-in voltage for effect of changing initial gap distance.
Once the applied voltage reached certain number which is at least twice less than the
analytical pull-in voltage, the nonlinear analysis diverged. The major problem for
Reduced Order Model analysis method is difficulty of controlling convergence.
Sometimes the analysis converges to an incorrect value might due to convergence setting
in the ANSYS code.
B. ROM144 Model
1.Build Process
The 6.5 Sample Miniature Clamped-Clamped Beam Analysis from Chapter 6.
Reduced Order Modeling of ANSYS Coupled-field Analysis Guide was used to as a
model for the ROM144 code. However, the nanoswitch length is much larger than the
radius, therefore the meshing of both the atmosphere and beam needed to be altered.
Figure 26 shows how the length of the beam (blue piece) is so large compared to the
thickness of the beam. The atmosphere (purple piece) consist of the area of air above,
below, and to the side of the beam.
24
Gap Distance r(x) [nm]
Design of Nanoscale Electromechanical Switches:
Design Verification
Figure 26. ROM144 model of the beam and atmosphere
The relationship between the second moment of inertia of a square and circular
beam was used to find the height and width of the beam. The Generation Pass was altered
to make it easier on the user to obtain results from the program. In the Used Pass section
of the code, a guess for the pull-in voltage is needed. Since this program is dealing with
different cases, the user must refer to the Design Guide to see what needs to be inputted.
2.Test Procedure
Since the pull-in voltage of each cases was specified in Professor Ke s Paper, the
debugging processes was ongoing.
The original example of code was original intended for a Micro Beam. The code
was meshed according to the relationship of the beams length, width, height, and gap
distance. It would not run properly with the specific cases that were to be studied with the
Nano switch. Since the length of the beam was so large compared to the radius of the
beam, there needed to be many divisions in order for the beam to mess properly. This
caused ANSYS to run slowly and not produce results efficiently. This led to the decision
to avoid modeling a circular beam and keep to the square beam. The assumption that the
beam will behave similarly when the square beam s second moment of inertia equals the
circular beam was made for this code.
3.Test Results
A beam with a length of 4000nm, radius of 10 nm, Young s Modulous of 1 TPa,
and gap distance of 100 nm was used in the ROM144 code to find the pull-in voltage.
Table 4 displays the pull in voltage found by the ROM144 code and the theoretical pull in
25
Design of Nanoscale Electromechanical Switches:
Design Verification
voltage, along with the error. A five percent error is an acceptable error since the analysis
is nonlinear and the behavior may be affected due to a 3D analysis.
V (ROM144) V (theory) Percent Error
9.52 9.06 5.07%
Table 4. ROM144 and Theory Pull-in voltage
C. Multi-field Solve
1.Build Process
The multi-field solver code was loosely based around example 3.4 Sample
Electrostatic Actuated Beam Analysis from Chapter 3 The ANSYS Multi-field "! solver
from ANSYS Coupled-Field Analysis Guide. In the beginning of the spring semester,
there were many different approaches to the modeling and meshing the beam and
atmosphere. A majority of the semester consisted of creating the model and Figure 27
shows the final model for the Multi-field Solver.
Figure 27 Multi-field model of Beam and Atmosphere
The load transfer interface was assigned to be between the beam volume and the
atmosphere volume. Voltage less than the pull in voltage was assigned to the beam and
ground voltage was applied to the outter edges of the electrostatic element.
2. Test Procedure
26
Design of Nanoscale Electromechanical Switches:
Design Verification
The analysis began but it was excessively slow for ANSYS. It is believed that
since the beam is so long compared to the radius, the model flags assigned in the code are
not correct. Unfortantly, there was not enough time in the semester to debug the coding to
produce results.
D. Requirements Matrix
Located in Appendix A is the Requirement Matrix with the customer sign-off.
E. Current Status
This project began in the fall 2008 academic semester . At that time, general
knowledge of electromechanical behaviors and ANSYS was developed. At the end of the
semester, the Reduced Order Model (ROM) was complete and data obtain by ANSYS
included a range of zero voltages to the pull-in voltage. The data was compared to the
theoretical data and yielded minimal error. The displacement for each particular oltage
input was graphed to show the various relationships. In the Case Study section displays
the relationship of changing length, radius, gap distance, non-linear vs linear analysis,
double clamped vs single clamped beams, and silicon vs carbon nanotube beam.
The Spring semester included the nonlinear analysis of the beam with the use of
ROM and 2 new approaches. The two new approaches were ROM144 and Multi-field
Solver which both analysis the beam in 3D. At the end of this semester, ROM144 used
nonlinear analysis to obtain the pull-in voltage for the particular case. Multi-field
Analysis was not able to obtain an result due to a lack of expertise and time to specify the
characteristics of analysis required in the code.
In order to continue working on the design of a nanoswitch, more data can be
obtained to generalize the relationship between the parameters of the switch. The linear
and nonlinear analysis using the Reduced Order Model may be used to find data.
One case was verified using the ROM144 code found the pull-in voltage based on
nonlinear analyses with an acceptable error. Finding the pull-in voltages for the other
cases is a task for future work.
Completing the Multi-field Model in ANSYS and verifying it will complete the
3D nonlinear analysis of the system. The website should further be developed to educate
students on nanotechnology specifically Nano electromechnical systems (NEMS).
27
Design of Nanoscale Electromechanical Switches:
Case Study (Final Design)
VI. Case Study (Final Design)
The goal of the design of nano-switch is to analyze the electromechanical
characteristic of the switch by varying parameters of the switch system. The switch is
simplified into a double clamped or single clamped beam above a substrate as shown in
Figure 28. Parameters of the switch system are:
1) Length of the Beam, L
2) Radius of the Beam, R
3) Initial Gap, H
4) Applied Voltage, V
5) Beam Material, E
Fixed-Fixed Beam Cantilever Beam
Figure 28 Nano-Switch Configuration
There are several criteria of designing nano-switch: manufacturability, reliability,
sustainability, economics, and flexibility. First, the length, radius, gap distance should not
be so small that it is hard to manufacture. Second the simulation should be repeatable and
able to get the same deformation of the beam with the same voltage input using the same
ANSYS code. Third, the maximum Von Mises stress on the beam should not exceed the
yield stress when the beam collapses to the substrate. If the beam yields, then the beam
will not be able to recover. In other words, the nano-switch is broken. Finally, being
economical means choosing relatively cheap and acceptable material, but for the nano-
switch, nano carbon tube is the best choice. Therefore, in order to save money, the
dimension of the switch should be as small as possible, but at the same time it needs to be
manufacturable.
28
Design of Nanoscale Electromechanical Switches:
Case Study (Final Design)
There are lots of different set of parameters that can handle 5 volts as input
voltage to the switch. Based on the linear analysis of Reduced Order Model, the
following set of parameters works well with 5-volt input voltage.
Boundary E [TPa] H [nm] L [nm] R [nm] 5Ø}Ü5ØÄ™Ü5Ø
Ü5ØćÜ5ØćÜ-5ØŠÜ5ØŹÜ[volts]
Conditions
Double 1 100 4000 10 3.15
Clamped
Table 5 Set of Parameters for Case Study
Since the applied voltage, 5 volts, is larger than the switch s pull-in voltage, 3.15
volts, the switch will turn on once the 5-volt voltage is applied on it. The following table
shows the Von Mises Stress and Von Mises Strain when the switch is on. The measured
stress and strain are much smaller than the theoretical ones, so the switch is flexible and
reliable. Of course, the dimension is able to be manufactured. In conclusion, the set of
parameters for the switch system works for 5-volt applied voltage.
Von Mises Stress Yield Strength of Von Mises Acceptable Strain
Carbon Nanotube Strain
2.572 GPa 13~53 GPa 0.2572% 10%
Table 6 Von Mises Stress and Von Mises Strain
29
Design of Nanoscale Electromechanical Switches:
Documentation
VII. Documentation
Throughout the Course, it was required to develop documents to enhance the
experience of working on a project. The following are documents required to be
completed during the Spring 2009 academic semester.
Included in the Documents are the calculations to model the nanoswitch in
ANSYS.
A. Requirement Matrix
The Requirement Matrix defines the cutomer s requirements, the proposed
solution to meet requirements, test necessary to satisfies & verifies that requirement,
and assigns who is responsible for verifying each requirement.
The Requirement Matris is crucial in the initial stages, design, prototype
construction, testing, and customer sign-off of the project.
Located in APPENDIX A.
B. Schedule
Located in APPENDIX B
C. Budget
Located in APPENDIX C.
a. Estimated Budget
b. Estimated Labor
c. Actual
D. Expenses
Located in APPENDIX D.
E. Research Data
Located in APPENDIX E.
a. Static Stability Characteristics
b. Theoretical and numerical pull-in voltages for specific beam parameters
F. Calculation
Located in APPENDIX F.
a. Excel File
30
Design of Nanoscale Electromechanical Switches:
Documentation
b. Matlab Code
G. Reduced Order Model Code
Located in APPENDIX G.
H. ROM144 Code
Located in APPENDIX H.
I. Multi-field Solver Code
Located in APPENDIX I.
J. Test Plan
Located in APPENDIX J.
31
Design of Nanoscale Electromechanical Switches:
Conclusion
VIII. Conclusion
Nanotechnology will have a very important role in the future of technology. Since
the behavior of elements at the nano scale can be considerably different, it is crucial to
investigate the subject. Nano electromechanical systems will surpass the world of
MEMS, making processes faster, smaller, and with less waste.
The nano electromechanical switch designed in ANSYS was able to display the
beam deformation as a result of the applied voltage. The Reduced Order Model does
display a general trend of displacement due to an increase of voltage when the applied
voltage is less than the pull-in voltage. The Reduced Order Model takes a very simple
approach in the linear and nonlinear analysis of the nano switch. The pull-in voltage
obtained by both linear and non-linear analysis in Reduced Order Model had less than 4%
of error compared to the theoretical and numerical results. Therefore, the Reduced Order
Model works well to analyze the switch.
The ROM144 creates mathematical representation of a 3-D electrostatic-structural
system to find the pull-in voltage of a system. Since the length of the double clamped
beam is much larger compared to the radius, it was assumed that the second moment of
inertia of a circular and square beam equal. The second moment was used to find the
height and width of square beam to be modeled in ANSYS. The square beam allowed
ANSYS to run much quicker and the pull-in voltage obtained had an error of 5%
compared to the theoretical results.
The creation of the multi-field solver started in Spring 2009. The model was
created and analysis was difficult to run. There are many parameters to be address in
order to perform the analysis. However, results were not obtained due to lack of
knowledge of analysis parameters and a time constraint.
It is important to use multiple methods to verify the behavior of the nano switch.
Therefore, it was important to use the Reduced Order Model, ROM144, and Multi-field
Solver to compare data when drawing any conclusion about the electromechanical
characteristics of the nano switch.
32
Design of Nanoscale Electromechanical Switches:
Work Cited
IX. Work Cited
[1] Drexler, Eric. Engines of Creation. New York: Anchor Books, 1986.
[2] Bonsor, Kevin, and Jonathan Strickland. How Nanotechnology Works. 25 October 2007. 3
December 2008 <>.
[3] "SVTC Nanotech Report: Regulating Emerging Technologies in Silicon Valley and Beyond."
2 April 2008. Silicon Valley Toxics Coalition (SVTC). 2008 December 2008
.
[4] National Nanotechnology Initiative, United States. "Nanotechnology: Big Things from a Tiny
World." National Nanotechnology Initiative (NNI). 2 December 2008
.
[5] Kaajakari, Ville. "MEMS Tutorial: Pull-In Voltage in Electrostatic Microactuators." Ville
Kaajakari's homepage . 8 December 2008
.
[6] Release 11.0 Documentation for ANSYS. Section 2.9 Electromechanical Analysis. 08 12
2008 .
[7] Roukes, Michael. "Nanoelectromechanical systems face the future." Physics World February
2001.
[8] Waters, Darren. "Nano switch hints at future chips." 17 April 2008. BBC News. 02
December 2008 .
[9] Ke, Changhong et al. "Analysis of Double Clambed Nanotube Devices in the Finite
Deformation Regime." Journal of Applied Mechanics, Transactions ASME (2005): 445-
449.
[10] Ke, Changhong et al.. "Numerical analysis of nanotube based NEMS devices - Part II: Role
of finite kinematics, stretching and charge concentrations." Journal of Applied
Mechanics, Transactions ASME (2005): 726-731.
[11] Dorf, Richard C. Sensors, Nanoscience, Biomedical Engineering, and Instruments: Sensors
Nanoscience Biomedical Engineering. CRC Press: Boca Raton, 2006.
[12] University of Alberta - ANSYS Tutorials. 2001. 2 December 2008
.
[13]  Direct Coupled-Field Analysis. ANSYS Coupled- Field Analysis Guide. ANSYS,Inc.
ANSYS Release 10.0 (2005): p7-1 to 7-72
[14] Hayt, W., and Buck, J.,. Engineering Electromagnetics, 6th ed.,. New York: McGraw-Hill,
2001.
[15] ANSYS Microsystem Analysis Key Feature. 2008. 15 April 2009
.
[16] ANSYS Microsystem Analysis Key Feature. 2008. 28 April 2009
.
33
Design of Nanoscale Electromechanical Switches:
Acknowledgements
X. Acknowledgements
We would like to take this time to thank Professor Ke for giving us the
opportunity to work on the design of nanoscale electromechanical switches. The
experience and knowledge gained from this project is far beyond a classroom. His time
and attention towards this project and towards the development of our education is
greatly appreciated.
Without Professor Rogers and Professor Selleck, this great opportunity would
certainly not exist. Their time and efforts in teaching and mentoring go beyond the
classroom and lab. They encouraged the senior mechanical engineering students to
venture beyond the classroom but were readily available for any assistance.
Thank you to Ezgi for creating a wonderful website to help display our hard work.
We hope that this website will create more interest and educate students on
nanotechnology.
After four years, the Binghamton University Mechanical Engineering Department
and Watson Dean s Office has made Senior Project a possibility. Thanks to the NSF
Nanotechnology Undergraduate Education (NSF-NUE) Program for support in our
presentation. Thank you for encouraging and facilitating this program. It has been
beneficial for not only the students involved but for the companies these students will
work for.
34
Design of Nanoscale Electromechanical Switches:
APPENDIX
XI. APPENDIX
35
Design of Nanoscale Electromechanical Switches:
APPENDIX
APPENDIX A- Requirement Matrix
Figure 29 Requirement Matrix
36
Design of Nanoscale Electromechanical Switches:
APPENDIX
APPENDIX B- Spring 2009 Schedule
Date Action
Wednesday, February 4, 2009 FA Meeting
Create Schedule for Spring 2009
Obtained a set of result for Non-linear Analysis using
ROM
Monday, February 9, 2009 Spring 2009 Schedule & Budget Complete
Wednesday, February 11, 2009 FA Meeting
·ð Study full-field model example
·ð Create linear and pre-stress analysis for full-field
model
·ð Obtain Linear Analysis Resuts using ROM
·ð Work on Non-linear Code
Wednesday, February 25, 2009 FA Meeting
·ð Obtained data from the Multi-field Solver of a square
beam.
·ð Modeled a Square & Cylinder Beam and Atmosphere
with the use of the Multi-field Solver.
·ð Obtained the plots for displacement along the beam
due to different input voltages
·ð Obtained Linear Analysis results using ROM for 15
cases
·ð Obtained Non-Linear Analysis results using ROM for
3 cases
Monday, March 2, 2009 Test Procedure (1st draft)
Wednesday, March 4, 2009 FA Meeting
·ð Meshed the cylindrical atmosphere to be used with the
multi-field solver
·ð Obtained the Linear Analysis Data for Animation
(Step 1)
·ð Obtained Three Charts from Linear Analysis results
using ROM (Step 2)
Friday, March 13, 2009 Test Procedure (final draft)
Midterm Review Status
37
Design of Nanoscale Electromechanical Switches:
APPENDIX
Date Action
Wednesday, March 18, 2009 FA Meeting
·ð Update the mesh of the cylindrical atmosphere to be
used with the multi-field solver
·ð Obtained the Linear Analysis Data for Animation
(Step 1)
·ð Obtained Six Charts using ROM (Step 2)
Friday, April 3, 2009 Design Guide (Rough Draft)
Wednesday, April 12, 2009 FA Meeting
·ð Sectioned the report to be included in the Website
·ð Used Multi-field Solver with two different shapes
(square and triangle) of atmosphere
·ð Obtained data from the ROM144 Code method
·ð Fixed two charts using ROM (step2)
·ð Obtain Non-linear Analysis Data for changing radius
·ð Wrote Design Guide Rough Draft
Friday, April 17, 2009 Design Guide (Final draft)
Wednesday, April 22, 2009 Customer Requirements Sign off
FA Meeting
·ð Obtain data from Multi-field Solver of a square beam
·ð Nonlinear Analysis for Radius effect
·ð Work on Case Study
Friday, April 24, 2009 Presentation and Report (Outline)
Wednesday, April 29, 2009 Presentation and Report (rough draft)
Monday, May 4, 2009 FA Meeting
·ð Review of presentation and Report
·ð Dry Run of Presentation
Friday, May 8, 2009 Final Report & Presentation
38
Design of Nanoscale Electromechanical Switches:
APPENDIX
APPENDIX C- Spring 2008 Budget
1. Estimated Budget
Estimated budget is based on the assumption of being the engineering firm working on
this project. The budget approximates how much the customer is charged for facilitating
the work environment.
Item Cost /
# Item Name unit Quantity Total Cost
ANSYS Multiphysics (V11.0)
1 License $53,000 1 $53,000.00
2 External Hard Drive $120.00 1 $120.00
Total Materials $53,120.00
Total Project
Cost $107,120.00
Table 7 Estimated Budget
2. Estimated Labor
Estimated Labor is also based on the assumption of being an actual engineering firm. The
engineers cost of labor is broken down into time spent on each task.
Name Task Name (must line up with your schedule) Estimated Hours
Develop Full Field Model
1 40
Verify Full Field Model
2 30
Test Procedure
3 40
Obtain Table of Relationship Between Parameters
Cleary
(Full Field Model Analysis)
4 15
Design Guide
5 28
Total Hours 153
Total Labor Cost $15,300.00
1 Reduce Order Model (Non-Linear Analysis) 20
Develop Full Field Model
2 20
Verify Full Field Model
3 15
Test Procedure
4 40
Huang Obtain Table of Relationship Between Parameters
(ROM Non Linear & Linear Analysis)
5 30
Design Guide
6 28
Total Hours 153
Total Labor Cost $15,300.00
Total Project Labor Costs $30,600.00
Table 8 Estimated Labor
39
Design of Nanoscale Electromechanical Switches:
APPENDIX
3. Actual Budget
Actual budget is the proposed costs of Fall 2008 and Spring 2009 academic year.
Item # Item Name Cost / unit Quanity Total Cost
1 ANSYS Multiphysics (V11.0) License $53,000 1 $0.00
2 External Hard Drive $120.00 1 $120.00
Total $120.00
Table 9 Actual Budget
40
Design of Nanoscale Electromechanical Switches:
APPENDIX
APPENDIX D- Expenses
There were no expenses for the spring 2009 academic semester.
41
Design of Nanoscale Electromechanical Switches:
APPENDIX
APPENDIX E  Research Data
1. Static Stability Characteristics
The plot features the electrostatic and spring force acting on a specific beam is
relationship to the distance of the beam to substrate. [6]
Figure 30 Static Stability Characteristics according to ANSYS [6]
2. Pull-in Voltages for Various Cases
The table displays the nonlinear and linear pull-in voltages found by Ke s
"Numerical analysis of nanotube based NEMS devices - Part II. Research data is
data provided to make conclusions on the Reduced Order Model.
Figure 31 Numerical and Theoretical Pull-In Voltages [10]
42
Design of Nanoscale Electromechanical Switches:
APPENDIX
APPENDIX F- Calculation
1. Excel File
Used to Plot Capacitance Per Unit Length vs. Gap Distance
Distance Between Distance Between
Capacitance Capacitance
Nanotube and Plate Nanotube and Plate
(pF/micro-m) (pF/micro-m)
(micro-m) (micro-m)
0.04 2.42674E-05 0.073 1.98E-05
0.041 2.40554E-05 0.074 1.97E-05
0.042 2.38512E-05 0.075 1.97E-05
0.043 2.36545E-05 0.076 1.96E-05
0.044 2.34646E-05 0.077 1.95E-05
0.045 2.32813E-05 0.078 1.94E-05
0.046 2.31042E-05 0.079 1.93E-05
0.047 2.29329E-05 0.08 1.93E-05
0.048 2.27672E-05 0.081 1.92E-05
0.049 2.26066E-05 0.082 1.91E-05
0.05 2.24511E-05 0.083 1.91E-05
0.051 2.23003E-05 0.084 1.90E-05
0.052 2.21539E-05 0.085 1.89E-05
0.053 2.20118E-05 0.086 1.88E-05
0.054 2.18738E-05 0.087 1.88E-05
0.055 2.17397E-05 0.088 1.87E-05
0.056 2.16092E-05 0.089 1.86E-05
0.057 2.14823E-05 0.09 1.86E-05
0.058 2.13587E-05 0.091 1.85E-05
0.059 2.12384E-05 0.092 1.85E-05
0.06 2.11211E-05 0.093 1.84E-05
0.061 2.10069E-05 0.094 1.83E-05
0.062 2.08954E-05 0.095 1.83E-05
0.063 2.07867E-05 0.096 1.82E-05
0.064 2.06806E-05 0.097 1.82E-05
0.065 2.05769E-05 0.098 1.81E-05
0.066 2.04757E-05 0.099 1.80633E-05
0.067 2.03768E-05 0.1 1.80097E-05
0.068 2.02802E-05 0.101 1.79568E-05
0.069 2.01857E-05 0.102 1.79048E-05
0.07 2.00932E-05 0.103 1.78535E-05
0.071 2.00028E-05 0.104 1.7803E-05
0.072 1.99E-05 0.105 1.77532E-05
Table 10 Plot Capacitance Per Unit Length vs. Gap Distance
43
Design of Nanoscale Electromechanical Switches:
APPENDIX
APPENDIX F- Calculation
2. MATLAB Code
Used to verify the Capacitance Equation Error! Reference source not found. into the
olynomial Equation (7) developed by Excel File APPENDIX F.a.
function Capacitance2 (H,R,n)
e0=8.854e-6; %pF/micro m
ri=H; %initial gap
rf=(1/3)*H; %final gap
r=linspace(ri,rf,n); %gap distance between the beam and ground plate
C=(2*pi*e0)./(acosh(1+(r/R))); % exact form of capacitance
p=polyfit(r,C,3)% generate the polynomial coefficients
%the polynomial expression is:
%P(1)*r^3 + P(2)*r^2 + P(3)*r + P(4)
f=polyval(p,r);
plot(r,C,'k*',r,f,'k-')
xlabel('Distance Between the Nanotube and the Substrate')
ylabel('Capacitance')
title('Capacitance vs Distance')
end
44
Design of Nanoscale Electromechanical Switches:
APPENDIX
APPENDIX G- Reduced Order Model Code
1. ANSYS Code-Linear Analysis for Double Clamped Beam
fini
/clear, nostart
/plopts,logo,off
/prep7
/title, Static response of a NEMS structure (Fixed- Fixed Beam)
L=4 ! Length of the beam (micrometer)
rad=0.01 ! Radius of the beam (micrometer)
vlt=3.498 ! Voltage (V)
H=0.1 ! Initial Gap Distance (micrometer)
gapi=H ! Gap (micro meter)
E=1e6 ! Young's Modulus (micro N/(micrometer)^2)
I=3.14/4*rad**4 ! Moment of inertia
N=200 ! Number of nodes
b=L/N ! length of each beam element
et,1,3 ! Beam Properties (Beam 3)
r,1,3.14*rad**2,I,2*rad ! Real constant
mp,ex,1,E ! Beam Material Property
et,2,126,,2,,1 ! Transducer element (trans126), UY-VOLT dof,
!symetric
! Real Constants: (1-5)
!C0,C1,C2,C3,C4 are obtained from the capacitance polynomial equation
C0=0
C1=(3.66399E-05)*b
C2=(-4.45947E-04)*b
C3=(3.90875E-03)*b
C4=(-1.31204E-02)*b
r,2,0,0,gapi
rmore, C0,C1, C2, C3, C4 ! With unit of [pF]
!Define nodes
n,1,0,0
n,N,L,0
fill
n,N+1,0,gapi
n,2*N,L,gapi
fill
! Define element types and asign element types to beam and capacitance
type,2
real,2
e,1,N+1
*repeat, N,1,1
45
Design of Nanoscale Electromechanical Switches:
APPENDIX
type,1
real,1
e,N+1,N+2
*repeat, N-1,1,1
! Define boundary conditions
nsel,s,loc,y,0
d,all,ux,0
d,all,uy,0
nsel,s,loc,x,0
nsel,r,loc,y,gapi
d,all,ux,0
d,all,uy,0
d,all,rotz,0
nsel,s,loc,x,L
nsel,r,loc,y,gapi
d,all,ux,0
d,all,uy,0
d,all,rotz,0
! Apply voltage on the nodes of the beam and the nodes of the substrate
nsel,s,loc,y,gapi
d,all,volt,-vlt
nsel,s,loc,y,0
d,all,volt,0
nsel,all
fini
! Static Analysis
/solu
antyp,static
pstres,on
autots,on
!nsubst,50
outress,all,all
cnvtol,U,,0.005
solve
fini
! Plot beam deflection
/post1
pldisp,2
/efacet,1
plnsol,u,y,0,1.0
46
Design of Nanoscale Electromechanical Switches:
APPENDIX
2. ANSYS Code-Nonlinear Linear Analysis for Double Clamped
Beam
fini
/CLEAR,NOSTART
/prep7
/title, Static Response of a NEMS Structure (Nonlinear Analysis for Fixed- Fixed Beam)
L=2 ! Length of the beam (micrometer)
rad=0.01 ! Radius of the beam (micrometer)
vlt=33 ! Voltage (V)
H=0.1 ! (micrometer)
gapi=H ! Gap (micro meter)
E=1e6 ! Young's Modulus (micro N/(micrometer)^2)
I=3.14/4*rad**4 ! Moment of inertia
N=200 ! Number of nodes
b=L/N ! Size of each beam element
et,1,3 ! Beam Properties (Beam 3)
r,1,3.14*rad**2,I,2*rad ! Real constant
mp,ex,1,E ! Beam Material Property
et,2,126,,2,,1 ! Transducer element (trans126), UY-VOLT dof,
!symetric
! Real Constants: (1-5)
C0=0
!------Range for H from 0.05micrometer to 0.105micrometer-------
!C1=(3.49995E-05)*b
!C2=(-3.80470E-04)*b
!C3=(3.06130E-03)*b
!C4=(-9.55321E-03)*b
!------Range for H from 0.04micrometer to 0.105micrometer-------
!C1=(3.66399E-05)*b
!C2=(-4.45947E-04)*b
!C3=(3.90875E-03)*b
!C4=(-1.31204E-02)*b
!----Range for H from 0.035micrometer to 0.105micrometer--------
C1=(3.76657E-05)*b
C2=(-4.89440E-04)*b
C3=(4.49986E-03)*b
C4=(-1.5097E-02)*b
!----Range for H from 0.03micrometer to 0.105micrometer--------
!C1=(3.88919E-05)*b
!C2=(-5.43899E-04)*b
47
Design of Nanoscale Electromechanical Switches:
APPENDIX
!C3=(5.26730E-03)*b
!C4=(-1.91691E-02)*b
r,2,0,0,gapi
rmore, C0,C1, C2, C3, C4 ! With unit of [pF]
!Define nodes
n,1,0,0
n,N,L,0
fill
n,N+1,0,gapi
n,2*N,L,gapi
fill
! Define element types and asign element types to beam and capacitance
type,2
real,2
e,1,N+1
*repeat, N,1,1
type,1
real,1
e,N+1,N+2
*repeat, N-1,1,1
! Define boundary conditions
nsel,s,loc,y,0
d,all,ux,0
d,all,uy,0
nsel,s,loc,x,0
nsel,r,loc,y,gapi
d,all,ux,0
d,all,uy,0
d,all,rotz,0
nsel,s,loc,x,L
nsel,r,loc,y,gapi
d,all,ux,0
d,all,uy,0
d,all,rotz,0
! Apply voltage on the nodes of the beam and the nodes of the substrate
nsel,s,loc,y,gapi
d,all,volt,-vlt
nsel,s,loc,y,0
d,all,volt,0
nsel,all
fini
save
48
Design of Nanoscale Electromechanical Switches:
APPENDIX
! Static Analysis
/solu
nlgeom, on !Large Deformation
antyp,static !Static analysis
TIME,1
KBC,0
nsubst,30
pstres,on
outress,all,all
save
solve
fini
!nsel,s,loc,y,H
!/POST1
!*
!PRNSOL,U,Y
!fini
! Plot beam deflection
/post1
pldisp,2
/efacet,1
plnsol,u,y,0,1.0
fini
49
Design of Nanoscale Electromechanical Switches:
APPENDIX
APPENDIX H- ROM 144
1. Model Parameters
(modal parameters_march29.rtf)
/filnam,cbeam
/PREP7, Clamped-clamped beam with fixed ground electrode
! µMKSV system of units
! Model parameters
B_L=4 ! Beam length
B_W=.0175 ! Beam width
B_T=.0175 ! Beam thickness
F_L=.1 ! Farfield in beam direction
F_Q=.1 ! Farfield in cross direction
F_O=.1 ! Farfield above beam
E_G=.1 ! Electrode gap
sigm_b=-100
/VIEW,1,1,-1,1
/PNUM,TYPE,1
/NUMBER,1
/PBC,ALL,1
/PREP7
ET,1,SOLID45 ! Structural domain
ET,2,SOLID122 ! Electrostatic domain
EMUNIT,EPZRO,8.85e-6 ! Free space permittivity
MP,PERX,2,1 ! Relative permittivity of air
! Half symmetry
BLOCK,0,B_L,0,B_W/2+F_Q,-E_G,B_T+F_O ! Entire domain
BLOCK,0,B_L,0,B_W/2,0,B_T ! Structural domain
BLOCK,0,B_L,0,B_W/2,-E_G,0
VOVLAP,ALL
LSEL,S,LOC,X,B_L/2 ! Mesh density in axial direction
LESIZE,ALL,,,200,,1
LSEL,S,LOC,Y,B_W/4 ! Mesh density in transverse direction
LESIZE,ALL,,,1,,1
LSEL,S,LOC,Z,B_T/2 ! Mesh density in vertical direction
LESIZE,ALL,,,2,,1
LSEL,ALL
VSEL,S,LOC,Z,B_T/2 ! Mesh structural domain (mapped meshing)
VMESH,ALL
VSEL,ALL
SMRTSIZ,3
MSHAPE,1,3D
50
Design of Nanoscale Electromechanical Switches:
APPENDIX
MSHKEY,0
TYPE,2
MAT,2
VMESH,4
LSEL,S,LOC,Y,b_w/2+f_q ! Mesh density at bottom electrode
LSEL,R,LOC,x,b_l/2
LESIZE,ALL,,,200,,1
LSEL,S,LOC,Y,0 ! Mesh density at bottom electrode
LSEL,R,LOC,Z,b_t+f_o
LESIZE,ALL,,,200,,1
LSEL,S,LOC,Y,(b_w+f_q)/2
LESIZE,ALL,,,10,.1,1
LSEL,ALL
VMESH,ALL
VSEL,S,LOC,Z,b_t/2 ! Movable electrode
ASLV,S,1
ASEL,U,LOC,Y,0
ASEL,U,LOC,X,0
ASEL,U,LOC,X,B_L
NSLA,S,1
CM,COND1A,AREA
CM,COND1,NODE ! Conductor 1 node component
ALLSEL
ASEL,S,LOC,Z,-e_g ! Fixed ground electrode
NSLA,S,1
CM,COND2A,AREA
CM,COND2,NODE ! Conductor 2 node component
ALLSEL
VSEL,U,LOC,Z,b_t/2 ! Region for DVMORPH
CM,AIR,VOLU ! Default name 'AIR'
VSEL,ALL
ESEL,S,MAT,,1
NSLE,S,1
NSEL,R,LOC,Z,b_t/2
CM,NEUN,NODE ! Neutral plane node component
ALLSEL
ET,1,0
PHYSICS,WRITE,ELEC ! Write electrostatic physics file
PHYSICS,CLEAR
ET,1,SOLID45
ET,2,0
MP,EX,1,1e6 ! Material properties Si
MP,NUXY,1,0.066 ! <110>
MP,DENS,1,1.3e-15
MP,ALPX,1,1e-6
51
Design of Nanoscale Electromechanical Switches:
APPENDIX
ASEL,S,LOC,Z,b_t/2
ASEL,R,LOC,Y,b_w/4
NSLA,S,1
CM,FIXA,AREA ! Boundary condition must be
DA,ALL,UX ! applied on solid model entities
DA,ALL,UY
DA,ALL,UZ
ASEL,S,LOC,Z,b_t/2
ASEL,R,LOC,Y,0
NSLA,S,1
CM,BCYA,AREA
DA,ALL,UY
ALLSEL
FINI
/SOLU
tref,0
tunif,sigm_b*(1-0.066)/(169e3*1e-6)
FINI
PHYSICS,WRITE,STRU ! Write structural physics file
! ET,2,SOLID122 ! Plot the entire model
EPLOT
FINI
SAVE ! Save model database
2. Generation Pass
(/generation pass_march29.rtf)
/filnam,gener ! Jobname for the Generation Pass
rmanl,cbeam,db,,3,z ! Assign model database, dimensionality, oper.
direction
resu,cbeam,db ! Resume model database
rmcap,cap12,1,2 ! Define capacitance
rmclist ! List capacitances
rmaster,node(b_l/2,0,0) ! Define master nodes
rmaster,node(b_l/4,0,0)
! Apply element loads
physics,clear
physics,read,STRU
/solu
antype,static
52
Design of Nanoscale Electromechanical Switches:
APPENDIX
nlgeom,on
acel,,,9.81e12 ! Acceleration in Z-direction 9.81e6 m/s**2
lswrite,1
acel,0,0,0
esel,s,type,,1
nsle,s,1
nsel,r,loc,z,0
sf,all,pres,0.1 ! 100 kPa
allsel
lswrite,2
lssolve,1,2
fini
/post1 ! Extract neutral plane displacements
set,1 ! due to element loads
rmndisp,'eload','write'
set,2
rmndisp,'eload','append'
fini
physics,clear
physics,read,STRU
! Perform prestressed modal analysis
/solu
nlgeom,off
pstress,on ! Thermal prestress (see cbeam.inp)
solve
fini
/solu
antype,modal
modopt,lanb,9
mxpand,9
pstress,on
solve
fini
/post1 ! Extract modal displacements at neutral
rmnevec ! plane nodes
fini
rmmselect,3,'nmod',.01-gap,gap-.01 ! Automated mode selection
rmmlist ! List selected mode parameters
rmmrange,2,'UNUSED' ! do not use unsymmetric mode for ROM
53
Design of Nanoscale Electromechanical Switches:
APPENDIX
rmsave,cbeam,rom ! Save ROM database
rmsmple,1 ! nlgeom,on
rmporder,6,,2 ! Set polynomial orders for modes 1 and 3
rmroption,sene,lagrange,0 ! Specify response surface parameter
rmro,cap12,lagrange,1
rmrgenerate ! Generate response surface
rmrstatus,sene ! Print status of response surface
rmrstatus,cap12
rmrplot,sene,func ! Plot response surface
rmrplot,cap12,func
rmsave,cbeam,rom ! Save ROM database
rmlvscale,2,0,0 ! Necessary to consider element loads
! in a VHDl-AMS model
rmxport ! Extract model input files for system simulation
3. Use Pass
(/pass_march29.rtf)
! *** Calculation of voltage displacement functions up to pull-in
/clear
/filnam,use1
rmresu,cbeam,rom
/PREP7
ET,1,144
*do,i,1,20
n,i
*enddo
rmuse,on
e,1,2,3,4,5,6,7,8
emore,9,10,11,12,13,14,15,16
emore,17,18,19,20
FINISH
/gst,off
DCVSWP,'pi',1,2,9,1,.01 ! Run voltage sweep up to Pull-in voltage
54
Design of Nanoscale Electromechanical Switches:
APPENDIX
APPEMDIX I- Multi-field Solver
(/april1_B_multi-field.docx)
Fini
/clear,nostart
/title, Electrostatic clamped beam analysis
/com, ANSYS Multi-field solver
/com, globally conservative Load transfer
/com, Structure: SOLID45 brick elements
/com, Electrostatic: SOLID123 tetrahedral elements
/com, uMKSV units
l=4 ! beam length, um
r=.01
gap=.1 ! gap, um
/prep7
! Structural model
et,1,95 ! 8-node bricks
mp,ex,1,1e6 ! kg/(um)(s)^2
mp,nuxy,1,0.066
mp,dens,1,1.3e-15 ! kg/(um)^3
k,1,0, sqrt(2)*r/2, sqrt(2)*r/2
k,2,0, sqrt(2)*r/2,- sqrt(2)*r/2
k,3,0, -sqrt(2)*r/2,- sqrt(2)*r/2
k,4,0,- sqrt(2)*r/2, sqrt(2)*r/2
k,5,l,0,0
k,6,0,0,0
l,1,6,4
*repeat,4,1
Larc,4,1,6,r
lArc,1,2,6,r
*repeat,3,1,1
L,5,6,200
Al,1,2,6
*repeat,3,1,1,1
Al,1,4,5
Vdrag,1,2,3,4,,,9
MSHAPE,0,3D
MSHKEY,1
Vmesh,all
55
Design of Nanoscale Electromechanical Switches:
APPENDIX
alls
asel,s,loc,x,0
asel,a,loc,x,l
nsla,s,1
da,all,ux
da,all,uy
da,all,uz
alls
nsel,s,loc,y,0
nsel,s,loc,z,0
da,all,uz
alls
vsel,s,volume,,all ! Define Surface interface
bfe,all,fvin,,1 ! define volumetric interface
et,2,122
emunit,EPZRO,8.854e-6 ! pF/um
mp,perx,2,1
morph,on
k,12,0,gap,gap
k,13,0,gap,-gap
k,14,0,-gap,-gap
k,15,0,-gap,gap
l,15,12,4
l,12,13,4
*repeat,3,1,1
L,12,1
*repeat,4,1,1
Lsel,s,line,,27,30,1
Lesize,all,,,10,.05
Alls
Al,5,23,30,27
Al,6,24,27,28
*repeat,3,1,1,1,1,
Vdrag,17,18,19,20,,,9
vsel,s,volu,,5,8,1
mshape,1,3D
vatt,2,,2
vsweep,all
aslv,s
asel,r,loc,x,0
da,all,ux,0 ! Apply structural morphing constraints
56
Design of Nanoscale Electromechanical Switches:
APPENDIX
aslv,s
asel,r,loc,x,l
da,all,ux,0
aslv,s
asel,r,loc,z,gap
da,all,uz,0
aslv,s
asel,r,loc,z,-gap
da,all,uz,0
aslv,s
asel,r,loc,y,-gap
da,all,uy,0
alls
esel,s,mat,,2 ! select billet material
bfe,all,fvin,,1 ! define volumetric interface
asel,s,area,,26,34,4
asel,a,area,,21
nsla,s,1
d,all,volt,9 ! Apply voltage
alls
nsel,s,loc,y,-gap
d,all,volt,0 ! Apply ground potential
allsel,all
fini
/solu
mfan,on ! Activate ANSYS Multi-field solver analysis
mfel,1,1 ! structure field
mfel,2,2 ! electrostatic field
mfor,2,1 ! Order for field solution
mfco,all,1.0e-5 ! Convergence settings
antyp,stat
eqslv,iccg
morph,on
mfcm,2, ! Electrostatic field analysis options
antyp,stat
nlgeom,on
deltim,1 ! Field loop time increment within a stagger
morph,off
kbc,0 ! Ramp voltage load
mfcm,1 ! Structural field analysis options
mfti,5 ! End time
mfou,1 ! Write solution every time step
mfdt,5 ! Stagger time increment
mfit,2 ! Max staggers
mfint,cons ! globally conservative load transfer
mfvo,1,2,forc,1 ! Transfer forces to structure field
mfvo,1,1,disp,2 ! Transfer displacements to electrostatic field
solve ! Solve the ANSYS Multi-field solver problem
save
finish
57
Design of Nanoscale Electromechanical Switches:
APPENDIX
APPENDIX J- Test Plan
Test plan is to help address the criteria and execution of any sort of testing for the project.
It is linked to the Requirement Matrix to verify that each of their requirements has been met.
Once this is executed most or all of you test cases, the customer will verify by looking at your
documented test results and sign off in order to complete the project.
Test Plan
1) Check Command Line used in ANSYS
a. Units
b. Input Voltage
c. Boundary definition of constraints
2) Check Modeling Methods
a. Reduced order model
b. Multi-Field Model
3) Verification of Reduced Order Model
a. Divide the length of the beam to create 200 nodes
b. Obtain table for different cases
c. Use data table to produce graphs that depict the 5 relationships
d. Verify obtained cases table by comparing with the published results
4) Verification of Multi-Field Model
a. Obtain pull-in voltage from Multi-Field Model using ANSYS code
b. Compare the ANSYS results for Multi-Field Model to the Reduced Order Model
c. Verify obtained pull-in voltage by comparing with the published results
5) Check Simulation
a. Equilibrium state
b. Instability state
58
Design of Nanoscale Electromechanical Switches:
APPENDIX
Test Procedure
1) Check Command Line used in ANSYS
a. Units
i. Length units reflect in the nano-scale (10^-9)
ii. The free space permittivity is 8.854e-6 pico-Farad/micro-meter (5Ø]Ü5Ø9Ü/5Øß5ØZÜ)
b. Input Voltage
i. In a range of 0 V ~ Pull-In Voltage
c. Boundary definition of constraints
i. Fully constraints for fixed-fixed beam at both ends
1. No translation in X,Y,Z directions
2. No rotation in X,Y,Z directions
ii. Fully constraints for the cantilever beam only at either end
1. No translation in X,Y,Z directions
2. No rotation in X,Y,Z directions
iii. Apply voltage (Choose 1 or 2)
1. Input voltage applied to beam & ground the substrate
2. Input voltage applied to substrate & ground the beam
2) Check Modeling Methods
a. Reduced order model
i. Obtain capacitance equation format from excel file as a 3rd order
polynomial
ii. The capacitance polynomial coefficients are entered into the real
constants of TRANS126 (e.g.  TRANS126 command line is used in the
Reduced order model)
b. Multi-Field Model
i. Include surrounding air as part of the mesh
3) Verification of Reduced Order Model
a. Divide the length of the beam to create 200 nodes
i. Use ANSYS to obtain the vertical deflection of the nodes for ten (10)
different applied voltages in the range of 0 ~ Pull-in Voltage
b. Obtain table for different cases
i. Get the data from ANSYS for each cases
1. Deflection of the beam due to applied voltage
2. Pull-in Voltage for each case
ii. Length, Radius, Young s Modulus, and gap distance between beam & the
plate are tabulated for different cases
c. Use data table to produce graphs that depict the 5 relationship
i. Effect of changing length (linear & non-linear)
ii. Effect of changing radius (linear & non-linear)
iii. Effect of changing gap distance (linear & non-linear)
59
Design of Nanoscale Electromechanical Switches:
APPENDIX
iv. Effect of changing Young s Modulus (linear & non-linear)
v. Effect of Double clamped and single clamped beams (linear & non-linear)
d. Verify obtained cases table by comparing with the published results
4) Verification of Multi-Field Model
a. Obtain pull-in voltage from Multi-Field Model
b. Compare the ANSYS results for Multi-Field Model to the Reduced Order Model
c. Verify obtained pull-in voltage by comparing with the published results
5) Check Simulation
a. Equilibrium state
i. When the beam is not touching the ground plate and staying still.
Touching the ground means the smallest distance between the beam and
the ground plate is greater than zero.
b. Instability state
i. When the beam is continuously vibrating. Extreme case would be the
beam touches the ground plate
60