SIMULATING THE EFFECTS OF MECHANICAL …...12= 12+( 11− 12− 44)( 12 22+ 12 22+ 12 22) (2-4)...

SIMULATING THE EFFECTS OF MECHANICAL STRESS

ON HALL-EFFECT SENSOR DEVICES

By

NATHAN MILLER

A THESIS PRESENTED TO THE COLLEGE OF ENGINEERING

OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OF

BACHELOR OF SCIENCE SUMMA CUM LAUDE

UNIVERSITY OF FLORIDA

2019

To my Mom and Dad, Kirstin, and Logan

3

ACKNOWLEDGMENTS

I would like to thank my advisor, Dr. Mark Law, for taking me on as an undergraduate

researcher nearly three years ago, for his continual support, and for his endless guidance through

the past several years. My sincere thanks also go to Dr. Keith Green and Dr. Aravind

Appaswamy, our closest contacts from Texas Instruments on this project, as well as other

contributors from TI; my colleagues Dr. Madeline Sciullo, Henry Johnson, and Thomas

Weingartner who each contributed immensely to this project; my committee members Dr. Erin

Patrick and Dr. James Keesling; and my lab mates Dr. Shrijit Mukherjee, Bobby, Lars, Nimesh,

Ribhu and Marquita.

I would also like to thank my parents for always supporting me and pushing me to be the

best I can be, and my brother and sister for their love and support. I would like to thank my

friends Zach, Robert, Logan, Kyle, Mitch, and Dane for their brotherhood through the years; my

colleagues and friends from Eta Kappa Nu Epsilon Sigma; my microchurch community for their

prayers and support; and Skyler, Aamir, Carina, Daniel, Nicole, Madison, John, Dan, Cristina,

and the many others who have influenced my journey.

Above all, I would like to thank God for His providence in getting me to where I am

today.

4

TABLE OF CONTENTS

page

ACKNOWLEDGMENTS ...............................................................................................................3

LIST OF TABLES ...........................................................................................................................5

LIST OF FIGURES .........................................................................................................................6

ABSTRACT .....................................................................................................................................7

INTRODUCTION ...........................................................................................................................8

BACKGROUND AND INTRODUCTION TO HALL-EFFECT SENSORS ................................9

SIMULATIONS TOOLS AND MODELS ...................................................................................15

RESULTS ......................................................................................................................................17

Resistance Changes Due to Mechanical Stress ......................................................................17 Offset Voltage Changes Due to Mechanical Stress ................................................................19 Using Electron Mobility and Electrostatic Potential Changes to Understand Trends ............20

CONCLUSIONS AND FUTURE WORK ....................................................................................38

Summary of Findings .............................................................................................................38 Future Work ............................................................................................................................38

LIST OF REFERENCES ...............................................................................................................40

BIOGRAPHICAL SKETCH .........................................................................................................41

5

LIST OF TABLES

Table page

Table 1-1. Piezoresistance components at room temperature in units of 10-12 cm2/dyne .............14

6

LIST OF FIGURES

Figure page

Figure 2-1. The Hall-effect sensor devices studied in this work. .................................................12

Figure 2-2. A graphical depiction of the Hall-effect.....................................................................12

Figure 2-3. A graphical depiction of the values of the piezoresistance coefficients for n-Si .......13

Figure 2-4. A graphical depiction of Euler’s angles .....................................................................14

Figure 3-1. Images displaying the mesh density used for the square and cross devices. .............16

Figure 4-1. Resistance results for the default cross structure with increasing stress. ...................24

Figure 4-2. Resistance results for the default square structure with increasing stress ..................25

Figure 4-3. Resistance results for the rotated cross structure with increasing stress ....................26

Figure 4-4. Resistance results for the rotated square structure with increasing stress ..................27

Figure 4-5. Summary of offset voltage results for the <100> square structure ............................28

Figure 4-6. Offset voltage results for the square device in various orientations. .........................29

Figure 4-7. Offset voltage results for the cross device in various orientations. ............................31

Figure 4-8. Changes in electron mobility and electrostatic potential for the <100> square

structure at various levels of stress ....................................................................................33


structure at 200MPa with various orientations. .................................................................34


structure at 200MPa with various orientations ..................................................................35

Figure 4-11. Changes in electron mobility and electrostatic potential for the <100> cross

structure at 200MPa with various orientations. .................................................................36


structure at 200MPa with various orientations ..................................................................37

7

Abstract of Thesis Presented to the College of Engineering

of the University of Florida in Partial Fulfillment of the

Requirements for the Bachelor of Science Summa Cum Laude

SIMULATING THE EFFECTS OF MECHANICAL STRESS

ON HALL-EFFECT SENSOR DEVICES

By

Nathan Miller

May 2019

Chair: Mark E. Law

Major: Electrical Engineering

This work presents a study on the effects of mechanical stress on n-type Silicon Hall-

effect sensor devices. Device modeling performed in Sentaurus Device of mobility changes

through piezoresistive effects is used to analyze and understand experimentally observed

changes in device resistance and offset voltage in such devices. Relationships between crystal

orientation, stress direction, and current direction are observed and defined, wherein an offset

voltage is created in a Hall-effect device due to mechanical stress only when an asymmetry

exists in the electron mobility of the device across the axis of the driving current direction.

Future goals discovered through this work are outlined, as current Sentaurus Device

models are incapable of coupling mobility changes using the piezoresistive model and the Hall-

effect simultaneously for a study of the piezo-Hall effect. Studying the source of mechanical

stress through packaging and how to adequately model and account for it are crucial next steps

that industry partners need for improved device performance.

8

CHAPTER 1

INTRODUCTION

Mechanical stress affects semiconductor devices of all types, so it is of great interest to

better understand and predict the influence of package-induced stresses. In particular, silicon-

based devices can be significantly affected by stress due to the piezoresistive nature of silicon.

These effects can have negative impact on the accuracy of devices such as Hall-effect sensors.

These sensors measure a potential difference produced across the device in the presence of an

applied magnetic field with a component orthogonal to the flow of bias current. Mechanical

stress is known to influence charge carrier mobility [1]. This therefore can affect the potential

difference produced by a magnetic field, causing the device to produce inaccurate readings of

magnetic field intensity. These inaccurate readings have been measured in industry applications

by Texas Instruments, leading them to a particular interest in how mechanical stress effects on

Hall-effect sensors can be modeled and understood. By better understanding the sources of

mechanical stress and its effects, methods of counteracting or even utilizing its effects can be

developed in order to create more accurate and novel products for the end consumer. With the

widespread applications that the effects of mechanical stress can have on all sorts of

semiconductor devices, there are many possibilities for the use of this work across the

semiconductor industry.

9

CHAPTER 2

BACKGROUND AND INTRODUCTION TO HALL-EFFECT SENSORS

Figure 2-1 shows four configurations of the Hall-effect sensor devices used in this work.

The original devices which are commonly referred to as the “square” and “cross” are shown at

the top of the figure, and identical devices rotated 45° around the Z-axis are shown below them.

Generally, the current flowing through the device is driven by the terminal named “Force1,”

which is abbreviated as “F1” and shown in the figure. Current driven from Force1 flows across

the device to the “Force2” terminal directly opposite from Force1. The resistance of the device is

then calculated by dividing the voltage across the two driving terminals by the current through

them. The remaining two terminals of the device are named the “Hall1” (abbreviated “H1” in the

figure) and “Hall2” terminals, across which the Hall voltage is measured. The direction of the

current flow across the device can then be varied such that any of the four terminals of the device

can drive current, which will flow to the terminal on the opposite corner of the device. For

example, if current were driven from the terminal named “Hall1,” then the drive current would

flow towards “Hall2” and the true Hall-effect voltage would be the voltage measured between

the Force1 and Force2 terminals.

The Hall voltage across the non-driving terminals is produced by the presence of a

magnetic field shown in Equation 2-1 which serves to bend current out of its straight path

(similar to the image shown in Figure 2-2) between Force1 and Force2 to produce a voltage on

the Hall terminals.

𝐹𝑏 = q𝑉𝑑 𝑥 𝐵 (2-1)

Under typical operation, the voltage measured across these terminals can be used to

calculate the intensity of the ambient magnetic field present around the sensor. In the presence of

stress, however, a voltage across these terminals can be created which is referred to as the “offset

10

voltage.” The offset voltage is defined as the voltage across the Hall terminals in the absence of

magnetic field. This offset voltage in Hall-effect sensors leads to the measurement of a false

magnetic field when there is none present.

The principle effect of mechanical stress analyzed in this work is the piezoresistive effect,

which is defined as the change in resistivity in a semiconductor (in this case, silicon) when

mechanical stress is applied. This change in resistivity follows a series of equations outlined by

Yozo Kanda [1] and is highly dependent on the orientation of crystal axes, stress, and current

with respect to each other. Figure 2-2 shows a plot from Kanda’s paper which summarizes the

relationship between stress and crystal orientation on the piezoresistive coefficients for n-type

silicon for devices with <100> and <110> crystal orientations at room temperature. The

mathematics describing this transformation are described in Equations 2-2, 2-3 and 2-4. Note in

Equation 2-2 that, for brevity, cosine is abbreviated as “c” and sine is abbreviated as “s.” These

equations are derived by Kanda to produce the plots shown in Figure 2-3. The definitions of the

ϕ, θ and ψ angles are shown in Figure 2-4.

[

𝑙1 𝑚1 𝑛1

𝑙2 𝑚2 𝑛2

𝑙3 𝑚3 𝑛3

]

= [

c(𝜙) c(𝜃) s(𝜓) − s(𝜙) s(𝜓) s(𝜙) c(𝜃) c(𝜓) + c(𝜙) s(𝜓) − s(𝜃) s(𝜓)

− c(𝜙) c(𝜃) s(𝜓) − s(𝜙) s(𝜓) −s(𝜙) c(𝜃) s(𝜓) + c(𝜙) c(𝜓) s(𝜃) s(𝜓)

c(𝜙) s(𝜃) 𝑠(𝜙)𝑠(𝜃) 𝑐(𝜃)

]

(2-2)

Equation 2-2 displays the transformation matrix used for finding the longitudinal and

transverse piezoresistive coefficients.

𝜋′11 = 𝜋11 − 2(𝜋11 − 𝜋12 − 𝜋44)(𝑙1

2𝑚12 + 𝑚1

2𝑛12 + 𝑛1

2𝑙12)

(2-3)

Equation 2-3 yields the longitudinal piezoresistive coefficient, 𝜋′11, which is the

piezoresistive coefficient used when stress is parallel to the direction of current.

11

𝜋′12 = 𝜋12 + (𝜋11 − 𝜋12 − 𝜋44)(𝑙1

2𝑙22 + 𝑚1

2𝑚22 + 𝑛1

2𝑛22)

(2-4)

Equation 2-4 yields the transverse piezoresistive coefficient, 𝜋′12, which is the

piezoresistive coefficient used when stress is perpendicular to the direction of current.

𝛥𝜌𝜔

𝜌= ∑ 𝜋′

𝜔𝜆𝑋𝜆

6

𝜆=1

=

[ 𝜋11

′ 𝜋12′ 𝜋12

′

𝜋12′ 𝜋11

′ 𝜋12′

𝜋12′ 𝜋12

′ 𝜋11′

0 0 00 0 00 0 0

0 0 0 0 0 0 0 0 0

𝜋44 0 00 𝜋44 00 0 𝜋44]

[ 𝑆𝑥𝑥

𝑆𝑦𝑦

𝑆𝑧𝑧

𝑆𝑥𝑦

𝑆𝑧𝑥

𝑆𝑧𝑦]

=

[ 𝜋11

′ 𝑆𝑥𝑥 + 𝜋12′ 𝑆𝑦𝑦 + 𝜋12

′ 𝑆𝑧𝑧

𝜋12′ 𝑆𝑥𝑥 + 𝜋11

′ 𝑆𝑦𝑦 + 𝜋12′ 𝑆𝑧𝑧

𝜋12′ 𝑆𝑥𝑥 + 𝜋12

′ 𝑆𝑦𝑦 + 𝜋11′ 𝑆𝑧𝑧

𝜋44𝑆𝑥𝑦

𝜋44𝑆𝑧𝑥

𝜋44𝑆𝑧𝑦 ]

(2-5)

The end result of Kanda’s derivation for converting applied stress to a change in

resistivity is shown in Equation 2-5. All of the results analyzed in this study are produced by

applying a compressive stress in the X direction (referred to as Sxx) on the cross and square

devices shown in Figure 2-1. Therefore, current flowing in the X direction will be affected by the

resistivity changes from the longitudinal piezoresistive coefficient, and current flowing in the Y

and Z directions will be affected by the resistivity changes from the transverse piezoresistive

coefficient. Note that 𝜋44 refers to the piezoresistive component induced from shear stress,

which is not tested here due to the difficulty of applying a shear stress to a silicon wafer.

12

Figure 2-1. The Hall-effect sensors used in this work are shown in two configurations. The

square structure (top left), the square structure rotated 45° around the Z-axis (bottom

left), the cross structure (top right), and the cross structure rotated 45° around the Z-

axis (bottom right) are all used for modeling the effects of semiconductor devices on

Hall-effect sensor performance.

Figure 2-2. A graphical depiction of the Hall-effect [5].

13

A

B

Figure 2-3. A graphical depiction of the values of the piezoresistance coefficients for n-Si at

room temperature for various crystal orientations compared to the direction of stress

[1]. Table 1-2 gives a more general summary.

14

Figure 2-4. A graphical depiction of Euler’s angles used for the coordinate system to derive the

crystal axis transformation matrix used for calculating piezoresistance coefficients of

Si in Equation 2-2 [1].

Table 1-1. Piezoresistance components at room temperature in units of 10-12 cm2/dyne [1].

Material n-Si p-Si

ρ (Ω-cm) 11.7 7.8

π11 -102.2 +6.6

π12 +53.4 -1.1

π44 -13.6 +138.1

15

CHAPTER 3

SIMULATIONS TOOLS AND MODELS

The models to be modified and utilized in this study are created in Sentaurus Device, a

finite volume analysis software developed by Synopsys. Sentaurus simulates characteristics of

semiconductor devices using the Scharfetter-Gummel method [4], creating a tetrahedral mesh

and calculating characteristic trends across the entire device using the integral of all the

differential calculations from each mesh component. Mesh density is made to be higher near the

contacts of the devices, as shown in Figure 3-1, in order to maximize the model accuracy near

the most volatile regions while maintaining a reasonable runtime. Sentaurus implements

advanced mobility models using these processes, as well as Monte Carlo models for carrier

transport, multiband modeling, carrier temperature and effective mass effects, and so on. The

main Sentaurus feature used for this study is the First Order Kanda model, which models

piezoresistive effects using the mathematics shown previously in Equations 2-1 through 2-5.

In addition to Sentaurus Device, Florida Object Oriented Device Simulator (FLOODS, a

subset of FLOOXS) is used to verify Sentaurus results with the goal of developing an even better

simulation tool. The Scharfetter-Gummel approach utilized by Sentaurus has several drawbacks,

namely its inability to perform mobility modeling with both stress and magnetic field applied

simultaneously, also known as the piezo-Hall effect. FLOOXS uses a finite element approach in

which these effects can be coupled together, which is a goal that is still in development.

16

A

B

Figure 3-1. Images displaying the mesh density used for the A) square and B) cross devices.

17

CHAPTER 4

RESULTS

Resistance Changes Due to Mechanical Stress

As shown in Kanda’s work [1] and in Equation 2-5, the application of compressive stress

along a given crystal axis compresses the crystal lattice in the direction of stress, thereby

increasing majority carrier mobility and decreasing resistance of the device in that direction.

Following Poisson’s ratio [6], this also results in a stretching of the lattice in the perpendicular

crystal directions, leading to a decrease in majority carrier mobility and an increase in resistance

in those directions. Understanding these piezoresistive effects is the key that leads to the

modeling of resistance and offset voltage changes in Hall-effect sensor devices due to

mechanical stress.

The simplest device with which to view the effect of the relationship between current

direction and stress direction is the original cross structure device (Figure 2-1). With the default

configuration of this device, the direction of current flow between the Force1 and Force2

terminals is directly along the X axis, and the direction of current flow between the Hall1 and

Hall2 terminals is directly along the Y axis. Figure 4-1 shows the change in resistance of devices

with both <100> and <110> crystal orientations. For the <100> device shown in Figure 4-1 A,

the resistance of the device increases by nearly 10% when 200MPa is applied perpendicular to

the current direction, and the resistance of the device decreases by close to 15% when 200MPa is

applied parallel to the current direction. This is consistent with Equation 2-5 relating the

direction of stress and direction of current to a change in resistivity of the device. Figure 4-1 B

shows the results when the same test is run on a <110> device. Since the current and stress are

no longer aligned parallel or perpendicular to the crystal axes, the resistance for both current

directions decrease, but the decrease in the resistance for the parallel case (approximately 6%) is

18

greater than the decrease in the resistance for the perpendicular case (approximately 3.5%). This

is also consistent with the Kanda [1] derivations. Of all of these tests, the resistance change is

maximal when the current and stress are both parallel to one of the crystal axes (the <100> case),

which is supported by the Kanda derivation as well.

Moving to the square structure tested for the data shown in Figure 4-2, any terminal on

which current is driven results in a 45° angle between the stress direction and the current

direction. Therefore, the resistance changes for all four driving terminals are the same. With

200MPa of stress, this corresponds to a 2.5% change in resistance in the <100> device shown in

Figure 4-2 A, and a nearly 5% resistance change in the <110> device shown in Figure 4-2 B.

To verify that these resistance changes are solely dependent on the direction of current

flow compared to the direction of stress, these results can be compared to the results shown in

Figure 4-3 and Figure 4-4, which show the cross and square structures, respectively, after a 45°

rotation around the Z-axis. Figure 4-3 shows the rotated cross structure, where the direction of

current flow from any driving terminal will now be at a 45° angle to the applied stress on the X-

axis. Due to this change in the angle between stress and current, the rotated cross structure now

displays similar resistance change characteristics to the original square structure in Figure 4-2,

wherein all four driving terminals result in a matching decrease in resistance with the application

of mechanical stress. The <100> device shown in Figure 4-3 A displays a resistance decrease of

approximately 3.5% with 200MPa of applied stress, and the <110> device shown in Figure 4-3 B

shows a decrease of nearly 5%. Figure 4-4 shows the rotated square structure where the direction

of current flow will be parallel to the compressive stress on the X axis when current is driven

between the “Force1” and “Force2” terminals, and perpendicular to the compressive stress when

current is driven between the “Hall1” and “Hall2” terminals. Similar to the results previously

19

shown for the original cross structure, current driven parallel to the direction of stress results in a

maximally decreasing resistance of approximately 13% for the <100> device shown in Figure 4-

4 A and a decrease of 5.5% for the <110> device shown in Figure 4-4 B. For the case of driving

current perpendicular to stress, the <100> device shown in Figure 4-4 A displays an increase in

resistance of 7% with 200MPa of applied stress and the <110> device in Figure 4-4 B displays a

decrease in resistance of 4%. All of these results are consistent with the mathematics derived in

Equations 2-1 through 2-5.

Offset Voltage Changes Due to Mechanical Stress

The same principles which result in a net decrease in resistance across the device are also

what lead to the creation of an offset voltage across the non-driving terminals of the device

where the voltage created by the presence of a magnetic field would typically be measured. In

the presence of zero magnetic field, mechanical stress can lead to the production of an offset

voltage across these terminals.

Figure 4-5 shows the magnitude of offset voltage measurements for all configurations of

the <100> device that were tested in this study. The detailed results of each individual

simulation, and those of the similar <110> devices, are described in the upcoming figures. From

Figure 4-5, it is clear to see that the only tests which produced an offset voltage greater than

approximately zero were the original square structure, where stress and current are at a 45° angle

to each other, and the rotated cross structure, where stress and current are also at a 45° angle to

each other.

Figure 4-6 shows offset voltage measurements from four configurations of the square

structure. For the <100> and <110> devices in Figure 4-6 A and B, the X-axis compressive stress

is at a 45° angle to the current direction for all four driving terminals. This results in an offset

voltage at 200MPa applied stress of positive or negative 53mV depending on which terminal is

20

measured for the <100> device, and 4.3mV for the <110> device. For the rotated square devices

in Figure 4-6 C and D, the X-axis compressive stress is now parallel to the direction of current

flow when current is driven from the Force1 or Force2 terminals, and perpendicular to the

direction of current flow when current is driven from Hall1 or Hall2. In these cases, the offset

voltage measurements for both the <100> and <110> devices are all in the microvolt range and

do not change with mechanical stress. The small, constant offset can therefore be attributed to

the presence of numerical noise in the calculation, rather than any effect from mechanical stress.

For the cross structure, Figure 4-7 shows the results of offset voltage measurement for

various configurations. In the <100> and <110> devices in their original position, Figure 4-7 A

and B shows that there is no change in offset voltage with stress, and the constant offset voltage

that is measured is in the microvolt range, similar to the rotated square structure. Again, this can

be attributed to numerical noise in the simulation causing the constant offset voltage, and there is

no visible effect of mechanical stress on the offset voltage in this configuration. For the rotated

cross structures, the <100> device in Figure 4-7 C shows an offset voltage change of positive or

negative 40mV depending on the direction of current flow, and the <110> device in Figure 4-7 D

shows an offset voltage change of positive or negative 4mV. This is similar to the change seen in

the original square devices, as in both cases the current direction is always at a 45° angle to the

direction of stress.

Using Electron Mobility and Electrostatic Potential Changes to Understand Trends

To understand how offset voltage forms, Sentaurus simulations can be used to make a

“difference plot” of the stressed case and the non-stressed case. These difference plots are a

valuable tool for visualizing how specific device characteristics can change with increasing

stress. Figure 4-8 shows difference plots between the stressed tests and the zero stress test for the

<100> square device characteristics of electron mobility and electrostatic potential for increasing

21

levels of stress. Figure 4-8 A displays the electron mobility change as stress increases, showing

an increase in electron mobility between the terminals in the vertical direction, which is the

direction of applied stress, as red regions of increasing size and visibility. Horizontally across the

device, perpendicular to the direction of stress, the device shows blue regions which indicate a

decrease in electron mobility in the presence of stress. This change in mobility can be directly

mapped to the electrostatic potential difference plots which are displayed in Figure 4-8 B. The

offset voltage of this device is measured as the electrostatic potential difference between the top

right contact and the bottom right contact. As stress increases, a red region of increased

electrostatic potential appears at the top right terminal, and a blue region of decreased

electrostatic potential appears at the bottom left contact. This net difference maps exactly to the

increased offset voltage displayed in Figure 4-6 A with increasing stress.

To create a summary of the behavior of each device under various current and stress

directions, Figure 4-9, Figure 4-10, Figure 4-11, and Figure 4-12 show difference plots of

electron mobility and electrostatic potential for the <100> square, the <110> square, the <100>

cross, and the <110> cross devices respectively. Each figure displays the difference plots for

electron mobility and electrostatic potential when the current and stress are at 45° angles, when

they are parallel, and when they are perpendicular to each other, all for the maximum stress case

of 200MPa.

In Figure 4-9, the <100> square device displays increasing electron mobility in the

direction of stress for the case of current and stress being at 45° to each other, and a decrease

perpendicular to the direction of stress. This translates directly to the offset voltage seen in the

device for this configuration, as explained above. For the parallel and perpendicular stress and

current directions case, these graphics are a bit more difficult to understand, particularly the

22

mobility plots. However, the clearest change is that the mobility generally increases for the

parallel case, and generally decreases for the perpendicular case. This explains the increase in

resistance seen for the perpendicular case and the decrease in resistance seen for the parallel case

displayed in Figure 4-4 A. It is also clear from the electrostatic potential plots for the parallel and

perpendicular cases that the electrostatic potential at the contacts where offset voltage is

measured (opposite the terminals where current is flowing) show no change in electrostatic

potential, which matches the previous observation from Figure 4-6 C that there is no offset

voltage change for these cases. The key to understanding the difference between the 45° case

which shows an offset voltage and the parallel and perpendicular cases which do not show the

creation of an offset voltage is the asymmetry seen in the electron mobility plots. Along the axis

of the main flow of current, the 45° case is the only one in which the electron mobility across the

device is asymmetrical. This asymmetry leads to an accumulation of charge in one side of the

device and a depletion in the other side, thus resulting in the differences in electrostatic potential

of the non-current driving terminal regions and the creation of an offset voltage.

Figure 4-10 displays the same plots for the <110> square device, which in general look

very similar to the <100> device. In particular, the increase in offset voltage displayed by the

electrostatic potential difference plot for the case of stress and current at 45° to each other looks

almost identical to the <100> case, although the net change is slightly less. This can be explained

by the much more convoluted electron mobility difference plot, which displays a more

significant gradient in electron mobility change throughout the center of the device than its

<100> counterpart. Similarly to the <100> device, the case of current and stress at 45° to each

other is the only case in which there is an asymmetry in the mobility plots across the axis of the

driving current direction, and it is therefore the only case which shows an offset voltage. In the

23

cases of parallel and perpendicular stress and current directions, the mobility plots are also more

convoluted than their <100> counterparts but show a general trend of increasing electron

mobility throughout the device, with a stronger increase for the parallel case. This is consistent

with the resistance change plots shown in Figure 4-4 B. In a similar fashion to the <100> case,

the electrostatic potential plots for the parallel and perpendicular cases show no change in

electrostatic potential on the contacts which measure offset voltage, matching the lack of offset

voltage change shown in Figure 4-6 D.

Figures 4-11 and 4-12 show very similar results for the cross structure. Both the <100>

and <110> devices display an offset voltage in the electrostatic potential plots when current is

driven at a 45° angle to the direction of stress. These changes in electrostatic potential can be

explained by small regions of increased mobility in the vertical directions near the current

driving terminals where current flows in the vertical direction towards the offset terminals and

small regions of decreased mobility near the current driving terminals where current flows in the

horizontal direction towards the offset terminals, thus creating an asymmetry in the electron

mobility plots across the axis of the driving current direction. For the parallel and perpendicular

cases, a similar trend in the electron mobility changes can be seen which shows a net decrease in

electron mobility in the <100> parallel case, and a net increase in electron mobility in the <100>

perpendicular case and both <110> cases. This corresponds to the resistance plots shown in

Figure 4-1 and Figure 4-3. In the electrostatic potential plots for the parallel and perpendicular

cases for both the <100> and <110> devices, there is a change in electrostatic potential in the

driving current direction corresponding to the resistance change, but no change in the

electrostatic potential at the offset terminals, matching the plots in Figure 4-7 A and Figure 4-7 B

where no offset voltage occurs with the application of stress.

24

A

B

Figure 4-1. Simulations of Hall-effect sensors using the default cross structure show changes in

the resistance of the device with increasing stress. The resistance changes differently

for A) <100> devices and B) <110> devices, and is also affected by the direction of

current flow.

0.8

0.85

0.9

0.95

1

1.05

1.1

0 50 100 150 200

No

rmal

ized

ch

ange

in R

esis

tan

ce (

-)

Applied Stress (MPa)

Normalized Change in Resistance vs Applied Stress for <100> Cross Device

Stress Perpendicular to Current

Stress Parallel to Current

0.925

0.95

0.975

1

1.025

0 50 100 150 200

No

rmal

ized

Ch

ange

in R

esis

tan

ce (

-)


Normalized Change in Resistance vs Applied Stress for <110> Cross Device

Stress Perpendicular to Current

Stress Parallel to Current

25

A

B

Figure 4-2. Simulations of Hall-effect sensors using the default square structure show changes in

the resistance of the device with increasing stress. The resistance changes differently

for A) <100> devices and B) <110> devices, but the direction of current flow has no

effect.

0.925

0.95

0.975

1

1.025

0 50 100 150 200

No

rmal

ized

ch

ange

in R

esis

tan

ce (

-)


Normalized Change in Resistance vs Applied Stress for <100> Square Device

Driving Force 1 and 2

Driving Hall 1 and 2

0.925

0.95

0.975

1

1.025

0 50 100 150 200

No

rmal

ized

Ch

ange

in R

esis

tan

ce (

-)


Normalized Change in Resistance vs Applied Stress for <110> SquareDevice

Driving Force 1 and 2

Driving Hall 1 and 2

26

A

B

Figure 4-3. Simulations of Hall-effect sensors using the cross structure rotated 45 degrees show

changes in the resistance of the device with increasing stress. The resistance changes

differently for A) <100> devices and B) <110> devices, but the direction of current

flow has no effect. Sxx refers to a compressive stress along the X axis.

0.95

0.96

0.97

0.98

0.99

1.00

1.01

0 50 100 150 200

No

rmal

ized

Res

ista

nce

Ch

ange

(-)

Applied Stress Sxx (MPa)

<100> Cross Sxx Normalized Resistance Change with 45 Degree Rotation

Force1

Force2

Hall1

Hall2

0.94

0.95

0.96

0.97

0.98

0.99

1.00

1.01

0 50 100 150 200

No

rmal

ized

Res

ista

nce

Ch

ange

(-)


<110> Cross Sxx Normalized Resistance Change with 45 Degree Rotation

Force1

Force2

Hall1

Hall2

27

A

B

Figure 4-4. Simulations of Hall-effect sensors using the square structure rotated 45 degrees

show changes in the resistance of the device with increasing stress. The resistance

changes differently for A) <100> devices and B) <110> devices, and the direction of

current has a significant effect. “Hall1” and “Hall2,” which overlap, indicate a current

parallel to the direction of stress, and “Force1” and “Force2,” which also overlap,

indicate a current perpendicular to the direction of stress. Sxx refers to a compressive

stress along the X axis.

0.80

0.85

0.90

0.95

1.00

1.05

1.10

0 50 100 150 200

No

rmal

ized

Res

ista

nce

Ch

ange

(-)


<100> Square Sxx Normalized Change in Resistance with 45 Degree Rotation

Force1

Force2

Hall1

Hall2

0.94

0.95

0.96

0.97

0.98

0.99

1.00

1.01

0 50 100 150 200

No

rmal

ized

Res

ista

nce

Ch

ange

(-)


<110> Square Sxx Normalized Change in Resistance with 45 Degree Rotation

Force1

Force2

Hall1

Hall2

28

Figure 4-5. Simulations of various <100> Hall-effect devices produce differing offset voltage.

Various tests were completed such that current and stress would be either parallel to

each other, orthogonal to each other, or at a 45° angle to each other. The only tests

that produce a significant offset voltage change are the cases where stress and current

are at a 45° angle to each other, and all of the other cases produce nearly zero offset

voltage with applied stress.

-1.00000E-02

0.00000E+00

1.00000E-02

2.00000E-02

3.00000E-02

4.00000E-02

5.00000E-02

6.00000E-02

0 50 100 150 200 250

Off

set

Vo

ltag

e (V

)

Applied Mechanical Stress (MPa)

Magnitude of Offset Voltage Changes for <100> Devices

<100> Cross Parallel Case

<100> Square 45deg Case

<100> Rotated Cross 45deg Case

<100> Rotated Square Parallel Case

<100> Cross Perpendicular Case

<100> Rotated SquarePerpendicular Case

29

A

B

Figure 4-6. Simulations of Hall-effect sensors using the square structure can be used to measure

offset voltage of the device with increasing stress. The offset voltage changes are

shown for A) <100> devices, B) <110> devices, C) <100> devices rotated 45

degrees, and D) <110> devices rotated 45 degrees. The direction of current has a

significant effect. Note that the direction of current in the non-rotated devices is

always at a 45 or 135 degree angle to the direction of stress, where “Hall1” and

“Hall,” overlap, and “Force1” and “Force2” also overlap. In the rotated devices, the

current direction is parallel to the direction of stress for “Force1” and “Force2,” and

perpendicular to the direction of stress for “Hall1” and “Hall2.” Sxx refers to a

compressive stress along the X axis.

-6.0E-02

-4.0E-02

-2.0E-02

0.0E+00

2.0E-02

4.0E-02

6.0E-02

0 50 100 150 200

Off

set

Vo

ltag

e (V

)


100 Square Sxx Offset Voltage vs Stress

Force1

Hall1

Force2

Hall2

-5.0E-03

-4.0E-03

-3.0E-03

-2.0E-03

-1.0E-03

0.0E+00

1.0E-03

2.0E-03

3.0E-03

4.0E-03

5.0E-03

0 50 100 150 200

Off

set

Vo

ltag

e (V

)


110 Square Sxx Offset Voltage vs Stress

Force1

Hall1

Force2

Hall2

30

C

D

Figure 4-6. Continued.

-4.00E-05

-3.00E-05

-2.00E-05

-1.00E-05

0.00E+00

1.00E-05

2.00E-05

3.00E-05

0 50 100 150 200

Off

set

Vo

ltag

e (V

)


<100> Square Sxx Offset Voltage with 45 Degree Rotation

Force1

Force2

Hall1

Hall2

-4.00E-05

-3.00E-05

-2.00E-05

-1.00E-05

0.00E+00

1.00E-05

2.00E-05

3.00E-05

0 50 100 150 200

Off

set

Vo

ltag

e (V

)


<110> Square Sxx Offset Voltage with 45 Degree Rotation

Force1

Force2

Hall1

Hall2

31

A

B

Figure 4-7. Simulations of Hall-effect sensors using the cross structure can be used to measure

offset voltage of the device with increasing stress. The offset voltage changes are

shown for A) <100> devices, B) <110> devices, C) <100> devices rotated 45

degrees, and D) <110> devices rotated 45 degrees. The direction of current has a

significant effect. Note that in the non-rotated devices, the current direction is parallel

to the direction of stress for “Force1” and “Force2,” and perpendicular to the

direction of stress for “Hall1” and “Hall2.” In the rotated devices, the direction of

current is always at a 45 or 135 degree angle to the direction of stress, where “Hall1”

and “Hall,” overlap, and “Force1” and “Force2” also overlap. Sxx refers to a

compressive stress along the X axis.

-6.0E-06

-4.0E-06

-2.0E-06

0.0E+00

2.0E-06

4.0E-06

6.0E-06

0 50 100 150 200

Off

set

Vo

ltag

e (V

)


100 Cross Sxx Offset Voltage vs Stress

Force1

Hall1

Force2

Hall2

-6.0E-06

-4.0E-06

-2.0E-06

0.0E+00

2.0E-06

4.0E-06

6.0E-06

0 50 100 150 200

Off

set

Vo

lage

(V

)


110 Cross Sxx Offset Voltage vs Stress

Force1

Hall1

Force2

Hall2

32

C

D

Figure 4-7. Continued.

-6.00E-02

-4.00E-02

-2.00E-02

0.00E+00

2.00E-02

4.00E-02

6.00E-02

0 20 40 60 80 100 120 140 160 180 200

Off

set

Vo

ltag

e (V

)


<100> Cross Sxx Offset Voltage with 45 Degree Rotation

Force1

Force2

Hall1

Hall2

-4.00E-03

-3.00E-03

-2.00E-03

-1.00E-03

0.00E+00

1.00E-03

2.00E-03

3.00E-03

4.00E-03

0 20 40 60 80 100 120 140 160 180 200

Off

set

Vo

ltag

e (V

)


<110> Cross Sxx Offset Voltage with 45 Degree Rotation

Force1

Force2

Hall1

Hall2

33

A

B

Figure 4-8. Changes in A) electron mobility and B) electrostatic potential for the <100> square

structure can be seen by creating a difference plot between the stressed structure and

the non-stressed structure. The gradient of change can be seen at various levels of

applied stress.

34


structure can be seen by creating a difference plot between the stressed structure with

200 MPa of applied stress and the non-stressed structure.

35




36




37




38

CHAPTER 5

CONCLUSIONS AND FUTURE WORK

Summary of Findings

As shown in both the cross and square structures, compressive mechanical stress in

silicon Hall-effect sensor devices produces a change in electron mobility which varies based on

current direction relative to stress direction. This change in electron mobility can lead to either

increased or decreased resistance in the device depending on current direction. The maximal

decrease in resistance occurs when current and stress are parallel, and a perpendicular

relationship can lead to increased resistance. These mobility effects can also lead to the

appearance of an offset voltage in Hall-effect devices, which can lead to inaccurate

measurements of magnetic field during their application. These findings show how mechanical

stress leads to performance changes, particularly with the creation of offset voltages from

random stresses induced from sources such as device packaging, and can modeled to understand

the creation of offset voltage in devices. This knowledge can potentially be used to develop a

means with which to counteract the effects of mechanical stress.

Future Work

This work has many future applications, considering the possibility of mechanical stress

affecting semiconductor devices of all types. Future progress on the Hall-effect sensor project in

particular will be focused on implementing mobility models affected by mechanical stress in

addition to magnetic field effects to result in a full piezo-Hall mobility model to measure

changes in the magnetic sensitivity of these devices under mechanical stress. Another future

direction is to explore how exactly stresses form from device packaging and how these stresses

can be modeled, avoided, and counteracted.

39

Exploring other devices, these models can be applied to any silicon device due to the

piezoresistive nature of silicon. In addition, other materials and devices such as silicon-

germanium bipolar junction transistors can be explored. In some applications, mechanical stress

can also induce bandgap changes and other effects which can drastically affect device

performance. Understanding the effects of mechanical stresses on semiconductor devices has

proved to be critical in multiple applications, and continuing this study on how to both counteract

and eventually utilize the effects of mechanical stress is an exciting opportunity to explore.

40

LIST OF REFERENCES

[1] Y. Kanda, “A Graphical Representation Of The Piezoresistance Coefficients In Silicon,”

IEEE Trans. Electron Devices, vol. 29, no. 1, pp. 64–70, 1982.

[2] S. Popovic, R, “Hall Effect Devices,” Inst. Phys. Publ. Bristol Philadelphia, p. 426, 2004.

[3] Synopsys Inc., “Sentaurus Device User Guide,” Santa Clara, June, 2006.

[4] D. L. Scharfetter and H. K. Gummel, "Large-signal analysis of a silicon Read diode

oscillator," in IEEE Transactions on Electron Devices, vol. 16, no. 1, pp. 64-77, Jan 1969.

[5] “Measuring Magnetic Fields,” Measuring Mag Fields. [Online]. Available: https://www.nde-

ed.org/EducationResources/CommunityCollege/MagParticle/Physics/Measuring.htm.

[Accessed: 04-Apr-2019].

[6] R. Lakes, “Meaning of Poisson's Ratio,” What is Poisson's ratio? [Online]. Available:

http://silver.neep.wisc.edu/~lakes/PoissonIntro.html. [Accessed: 04-Apr-2019].

41

BIOGRAPHICAL SKETCH

Nathan Miller is a graduating Electrical Engineering student at the University of Florida

with a minor in Mathematics. Nathan has performed internships at GE Aviation and Intel

Corporation and will be returning to Intel Corporation’s Nonvolatile Memory Systems Group

this summer before beginning his PhD studies at Georgia Institute of Technology during the fall

of 2019.

Errata

Simulating the Effects of Mechanical Stress on Hall-Effect Sensor Devices

Undergraduate Thesis

Nathan E. Miller

University of Florida 2019

1. On page 11, line eight, replace “compressive” with “tensile”.

2. On page 17, line two, replace “compressive” with “tensile”.

3. On page 18, third and fourth to last lines, replace “compressive” with “tensile”.

4. On page 19, fourth to last line, replace “compressive” with “tensile”.

5. On page 20, line one, replace “compressive” with “tensile”.

6. On page 21, line seven, replace “bottom right” with “bottom left”.

7. On page 26, last line of Figure 4-3 caption, replace “compressive” with “tensile”.

8. On page 27, second to last line of Figure 4-4 caption, replace “compressive” with

“tensile”.


“tensile”.


“tensile”.

11. On page 38, first line, replace “compressive” with “tensile”.

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