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In plane a & cross-plane thermoelectric characterizations of van der Waals heterostructures

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In plane a & cross-plane thermoelectric characterizations of van der Waals heterostructures

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IN PLANE & CROSS-PLANE THERMOELECTRIC
CHARACTERIZATIONS OF VAN DER WAALS
HETEROSTRUCTURES
by
Nirakar Poudel

A Dissertation Presented to
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(ELECTRICAL ENGINEERING)

December 2018

Acknowledgement
Foremost, I am grateful to my advisor Professor Stephen B. Cronin. He has been
quite an inspirational professional and personal role model for me. Every time, I have
derailed, he has redirected me to right path. When I got too arrogant, he made me humble.
When I was too low and lacking confidence, he gave me confidence. Observing him and
working with him in last five years has helped me develop a lot of leaderships and
management skill. His wife Hanna has been really supportive too and I owe her a lot for
giving me practical life advices in life.

I would like to thank my parents for their unconditional love and giving me the
opportunity to pursue my own dreams and goals without any pressure. I am also grateful
to my dad for being such a great mentor and a friend. There have been many times when I
wanted to quit because I felt extremely helpless. However, the support and constant
encouragement from my parents always gave me the power to persevere through hard
times. I am grateful to Poudel family too for being very supportive. I owe my education to
Govinda, Rama, Dhruva, Mira Poudyals and my cousins, Nirmala, Sajani, Suraj, Niraj,
Rajani, Roshani and Sahisnu, Ravi, Nikki and Binni.. They always gave me a lot of
encouragement and unconditional love. My late grand mom Kulkumari Poudyal and my
maternal grand mom Bhanu Nepal have been other sources of inspiration for me. While I
have acquired a lot of management and leadership skills from my paternal grand mom, I
have learnt the value of resilience from my maternal grand mom. She lost two sons, one at
31, another at 45, had paralyzing stroke 5 years ago and lost her home in the earthquake.
However, she always laughs and smiles and manages to overcome obstacles. It was
inspiring when she took a loan at 75 to rebuild her house. She does not give up, laughs at
herself and no matter what happens, keeps optimistic and pushes forward.

Koirala family has been my second family in LA. Suman Koirala brought me to
Los Angeles to USC and his family created a nurturing and loving environment for me to
flourish in LA. I am thankful to the Koirala parents, three brothers and specially to my
really good friend Suman. He has been such a great inspiration since grade 9.

My life in Los Angeles would have been quite miserable without support of Avi
and Sapana Pandey. They provided a great platform and support for me and helped me
solve many of my personal and professional issues. They have been like another second
family to me for my stay here. I am also extremely grateful to my peers Ashrant, Baibhav,
Neha, Achal,who have directly and indirectly taught me a great deal about life. Ashrant,
Baibhav and Achal have helped me come up with so many innovative solutions to so many
personal and professional issues in Los Angeles. I am thankful to Prathamesh, Sakar,
Rajendra and Laxman too.

There have been many other people in my life without whom I could not
have enjoyed my PHD life. I am really thankful to my best friend Utsha Karki. She has
been an amazing source of inspiration and support for last 10 years of my life. She has
supported me emotionally, financially and just been there to listen to me when I needed to

talk to someone. I can call her at 3 am at night and whine about things and she listens. I
also found another good friend in Astha Thapa in grad school. Interacting with her has been
a lot of fun. She keeps me grounded and real. My friend Sriyas Pandey has been quite a
support in grad school too.

I am also thankful to my college friends Chiran Singh Bhandari, Awais Malik,
Rahul Khakurel, Aarya Uprety, Anup Dhamala, all of whom have been always super
supportive. In addition, I am thankful to all the faculty, staffs and administrators at Viterbi
for creating an environment for me to thrive and flourish
Table of Contents

1 Introduction to Thermoelectricity ............................................................................... 1
1.1 Brief History (1800s to 1970) ................................................................. 1
1.2 Significance of Thermoelectricity for Modern Society .......................... 2
1.3 Current State of Research in Thermoelectricity ...................................... 7
1.4 Unique Aspect of Nanoscale thermal Transport ................................... 13
1.5 Theoretical Foundation of Thermoelectricity ....................................... 15
1.5.1 Boltzman-Landauer Formalism 19
2 Cross-plane Thermoelectric and Thermionic Transport across Au/h-BN/Graphene
Heterostructures ................................................................................................................ 30
2.1 Introduction ........................................................................................... 30
2.2 Methods................................................................................................. 33
2.3 Results and Discussion ......................................................................... 36
2.4 Conclusion ............................................................................................ 42
2.5 Supplemental Information .................................................................... 43
3 In-plane thermoelectric transport across Graphene and Graphene-hBN
heterojunction ................................................................................................................... 44
3.1 Introduction ........................................................................................... 44
3.2 Results and Discussion ......................................................................... 46
3.3 Conclusion ............................................................................................ 50
4 Plasmon Resonant Amplification of a Hot Electron-Driven Photodiode ................. 63
4.1 Introduction ........................................................................................... 63
4.2 Methods................................................................................................. 66
4.3 Results and Discussion ......................................................................... 66
4.4 Conclusion ............................................................................................ 69
5 Future Direction ........................................................................................................ 69
5.1 Cross Plane Thermal Conductivity and Seebeck Coefficient of TMDCs
using GERS ............................................................................................... 69
5.1.1 Introduction 69
5.1.2 Preliminary Results and Future Work70
5.2 Enhancing Thermoelectric Properties of MoS2

using Remote Oxygen
Plasma Treatment...................................................................................... 72
5.2.1 Introduction 72
5.2.2 Methods 74
5.2.3 Preliminary Results and Future Work76
6 Appendix A ............................................................................................................... 81
6.1 Fundamental Physics of Plasmonics ..................................................... 81

6.2 The concept of surface plasmon polariton ............................................ 88
References ......................................................................................................... 93

Figure of Contents
Figure 1.1 Carnot Efficiency of a thermoelectric generator as a function of average ZT.
5

Figure 1.2 Efficiency of a thermoelectric generator compared with other energy
technologies.
3

Figure 1.3 Efficiency of Various Energy Generators at different Power Levels.
3

Figure 1.4 The timeline of progress in engineering various materials with given ZT
value.
5

Figure 1.5 Illustration of dependence of Seebeck Coefficient and Electrical Conductivity
as a function of carrier concentration. This also illustrates the optimization issue of ZT.
5

Figure 1.6 (a): L and  valleys of valance band of PbTe. (b) Illustration of how the Na
doped PbTe has a slightly higher value of Seebeck coefficient induced by valley
degeneracy of doping.
5

Figure 1.7 Illustration of Phonon Scattering in a Crystal.
5

Figure 1.8 (a) A harmonic system where longitudinal movement of atoms is in the same
direction at any given time (b) An Harmonicity can originate when atoms vibrate without
synchronizing in any direction (c) Harmonic region and anharmonic region of a
interatomic potential.
5

Figure 1.9: Illustration of a n-type material when an external current Jx is induced on it.
2

Figure 1.10 Peltier Effect in a material.
2

Figure 1.11 Seebeck Effect in a Material.
2

Figure 1.12 Fermi Distributions of two contacts in a band picture.
2

Figure 1.13 Fermi Distribution in a Material.
2

Figure 1.14 Illustration of Peltier effect in the band picture.
2

Figure 1.15 Change in Fermi Distribution with application of small Field.
2

Figure 1.16 Illustration of Seebeck Effect in band picture.
2

Figure 1.17 Change of Fermi distributions with external temperature difference.
2

Figure 2.1 Schematic diagram of the thermionic emission process, illustrated for a
graphene/h-BN/Au heterostructure.
Figure 2.2 Illustration of Fabrication Process.
Figure 2.3 (a) Cross-sectional diagram, (b) optical microscope image, and (c) Raman
spectrum of the graphene/h-BN/Au heterostructure with ITO heater.
Figure 2.4 (a) Calibration data of the graphene 2D-band Raman shift plotted as a function
of temperature obtained in a temperature controlled optical vacuum cryostat. (b)
Graphene 2D-band Raman shift plotted as a function of the heater voltage. (c, d) Cross-
plane temperature difference plotted as a function of heater voltage and heater power.
Figure 2.5 Schematic diagrams of the measurement set up and device geometry for the
(a) ‘non-heating’ and (b) ‘heating’ configurations. (c) Cross-plane AC voltage measured
between the top graphene contact and bottom Au electrode at 2ω for both heating and
non-heating configurations, plotted as a function of the applied AC heater voltage.
Figure 2.6 Thermoelectric voltage plotted as a function of the temperature difference
across the Au/h-BN/graphene heterostructure.
Figure 2.7 Comparison of theoretical model with experimental measurements for thermo-
voltage across Au-hBN-Graphene.
Figure 3.1(a) 3D schematic of graphene-hBN device (b) Raman spectra across various
channels (c) and (d) 20X and 100X microscope image of the active device.
Figure 3.2 (a) 3D schematic of the device (b) Numerical calculations of hot spot
temperature distribution across the hot spot between two graphene channels. (c)

Experimental measurement and fit through simulation of the temperature profile across
the channel.
Figure 3.3 Measured (a) & (b) thermal and electrical profiles (c) EFM image showing a
steep potential drop at the location where the hot spot is shown by the SThM image. (d)
EFM image after the constriction is destroyed by ESD, showing a step change in the
surface potential that spans the entire channel signifying an open circuit. Horizontal scale
bar is 5 μm and the vertical scale bar is 50 nm, 160 K, 4°and 10° phase shift for images a,
b, c, and d, respectively.
Figure 3.4 (a) The calculated maximum temperature rise as a function of the basal-plane
thermal conductivity κ for different interfacial conductance G in increments of
1.5×10
7
W m
−2
K
−1
for a 300-nm-thick dielectric. The inset shows the maximum
temperature rise as a function of increasing G for different κ in the range between
100 W m−1 K−1 (top curve) and 2900 W m
−1
K
−1
in 400 W m
−1
K
−1
increments. (b)
Normalized maximum temperature rise calculated by the analytical model as a function
of the G/Gox ratio (main figure) and the z0 parameter (inset). The solid and dashed lines
are for graphene devices made on a 10- and 300-nm-thick SiO2 dielectric on a high
thermal conductivity substrate, respectively. The graphene basal-plane thermal
conductivity is kept as 600 W m
−1
K
−1
for the main figure, and ranges between 300 and
3000 W m
−1
K
−1
for the inset, where G is taken as 4 × 107 W m
−2
K
−1
.
Figure 3.5 Thermal image obtained using SThM for (a) graphene/SiO2,
graphene/hBN/SiO2, graphene/WillowGlass, graphene/hBN/WillowGlass substrates.
Figure 4.1 (a) Cross-sectional scanning electron microscope (SEM) image and (b)
schematic diagram of the plasmon resonant grating structure. (c) Energy band diagram
illustrating the mechanism of hot electron injection.
Figure 4.2 (a) Photocurrent and (b) photoreflectance plotted as a function incident angle
for light polarized parallel and perpendicular to the grating structure. (c) Schematic
diagram of the experimental measurement configuration.
Figure 4.3(a) Finite difference time domain (FDTD) simulation of the photoreflectance as
a function of the incident angle for s- and p-polarized light. (b,c) Cross-sectional electric
field intensity profiles for illumination at normal and (d,e) 10
o
incidence.
Figure 5.1 (a) Illustration of GERS thermoelectric measurement structure with ZnPC as
top GERS molecule and CuPC as bottom GERS molecule (b) Proposed GERS Geometry
with Crystal Violet and R6G as GERS temperature calibration molecules.
Figure 5.2 (a) CupC in graphene illuminated by 633 nm laser. (b) Temperature
calibration of Raman Peak of CuPC downshift with increasing temperature (c) Raman
spectra of CuPC on graphene (d) Peak downshift with increasing temperature.
Figure 5.3 (a) Optical Image of the final device. The materials in this case is MoSe2 (b)
& (c) The temperature calibrations of the downshift of CuPC (on bottom graphene) and
ZnPC (on top graphene) GERS spectra (d) Raman downshift of MoSe2 downshift.
Figure 5.4. (a) PL enhancement observed in sample after remote oxygen plasma
treatment Optical Image (c) Illustration of the mechanism underlying the enhancement
process.
Figure 5.5 Illustration of effect of α- particles irradiation in (a) ratio of Sulpher to
Molybdenum atom (b) Raman spectra (c) Photoluminescence of MoS2.
4

Figure 5.6 (a), (b) & (c) 3D schematic, side view and top view of the device (d) Optical
image of the device.

Figure 5.7: (a) Ids vs Vg for Vds of 1V (b) Log plot of Ids vs Vg for Vds of 1V (c) Ids vs Vds
at various gate voltages.
Figure 5.8 Configuration of RTD thermometry using a four-probe technique.
Figure 5.9 (a) Seebeck Coefficient as a function of applied back gate voltage for various
Temperature for monolayer MoS2
1
(b) Seebeck Coefficient as a function of difference
between applied back gate voltage and threshold voltage for various layered of MoS2
samples.
4

Figure 5.10 (a) Temperature of RTDs as a function of Heater Voltage (b) Thermal
Voltage as a function of Heater Voltage (c) Thermal Voltage as a function of
Temperature Difference.
Figure 5.11(a) and (b) Threshold voltage shift observed in two different remote oxygen
plasma treated samples.
Figure 5.12 (a) Optical Image of the sample (b) Photoluminescence spectra (c) Raman
spectra before and after remote oxygen plasma treatment.
1

1 Introduction to Thermoelectricity
1.1 Brief History (1800s to 1970)
Thermoelectricity incorporates three different phenomena – the Seebeck
effect, the Peltier effect and the Joule-Thompson effect. Baltic German Physicist Thomas
Johann Seebeck observed direct conversion of heat into electricity in a junction of two
different wires in 1821. Jean Charles Athanase Peltier discovered that there is transfer of
heat in electrical junction of two different materials in 1834. Lord Kelvin observed Joule-
Thomson effect, which describes how heat is transferred through a current carrying
conductor when a temperature gradient is imposed in the wire in 1851. After these early
discoveries in 1800s, many active research projects were carried out in application of
thermoelectricity between 1920 and 1970.
1, 4
The emergence of semiconductors and their
novel alloys led to a major resurgence in studies of thermoelectric devices and materials in
1950. The maximum of 5% efficiency observed in solid state thermoelectric generators
could not compete with the mechanical compression cycles in the 1950s and 1960s.
1, 4
The
excitement in the research in the field dwindled by 1970s and the research in the field came
to a complete stall.
1

One of the major innovations in the field of thermoelectric power generation in
1950 was the Radioisotope Thermoelectric Generators (RITEG), a device that converts
thermal energy produced by decay of a radioactive material (Pu-238) into electricity using
Seebeck effect. NASA has employed these RITEG modules in missions like Apollo,
Pioneer, Viking, Galeleo and Cassini.
4
Although the academic research in field was quite
2

dormant for almost two decades between 1970 and 1990, there was some innovation in
solid state coolers based on Peltier effect using Bi2Te3-Sb2Te3 alloys.
4
These products had
some applications in the niche market of optoelectronics, refrigerators and seat
cooling/heating systems.
4

1.2 Significance of Thermoelectricity for Modern Society
The resurgence of immense interest in 1990s also coincided with the era when
global warming, climate change and sustainability of global energy usage finally became
major policy issues.
1, 6
Global warming and sustainability of global energy usage are
intertwined issues. For example, almost 70 percent of electricity generated in the United
States is generated from coal and natural gases, which generate significant amount of
greenhouse gases like carbon dioxide. To further exacerbate the matter, only around 30
percent of the energy generated by burning coal and natural gas is converted into useful
electricity that powers the economy. This issue of energy mismanagement is a glaring
illustration of the major problem that our society is facing to move towards a more
sustainable and environmentally friendly energy infrastructure. Thermoelectricity research
gained attention again because it provided a potential for fuel efficient technology through
waste heat recovery and thermal energy harvesting
.
3

The most important parameter in thermoelectricity is the concept of Figure of merit
ZT. Abram Fedorovich Ioffe, a pioneer in use of semiconductors in thermoelectric research,
introduced the concept of ‘Figure of merit’ ZT in his textbook, Semiconductor
Thermoelements and Thermoelectric Cooling in 1949.
7
ZT is given by the following
equation
𝑍𝑇 =
𝑆 2
𝜎𝑇
𝜅
where 𝑇 is the temperature, 𝑆 is the Seebeck coefficient of the material, 𝜎 is the
electrical conductivity and 𝜅 is the thermal conductivity. The materialization of
Thermoelectricity’s promise in achieving a cleaner and energy efficient future primarily
depends on ZT. The concept of ZT is analogous to concept of Carnot efficiency 𝜂 for a
conventional heat engine. Carnot efficiency 𝜂 measures the amount of work a heat engine
does per heat consumed. The maximum allowable efficiency 𝜂 𝑚𝑎𝑥
= 1−
𝑇 ℎ𝑜𝑡
𝑇 𝑐𝑜𝑙𝑑 is the
Figure 1.1 Carnot Efficiency of a thermoelectric generator as a function of
average ZT.
5

4

maximum possible efficiency of an ideal heat engine. Figure 1.1 demonstrates a conversion
of average 𝑍𝑇 to Carnot efficiency 𝜂 . A thermoelectric generator working at a temperature
differential of 400 K and ZT of 3.0 will have a generating efficiency of 25%, which is a
comparable value of efficiency for a conventional heat engine.
5

Figure 1.2 shows the comparison of efficiency of current state of art power
generators with the thermoelectric technology. Based on the Figure 1.2, it is unlikely that
TE generators will have any impact on large scale waste heat recovery in near future.
However, one of the main advantage for thermoelectric generators are their scalability as
illustrated by Figure 1.3. While conventional mechanical technologies become extremely
inefficient at low power level, TE generators maintain their efficiency.
Figure 1.2 Efficiency of a thermoelectric generator compared with other energy
technologies.
3

5

The prospect of thermoelectric generators generating or even recovering enough
power to make significant contribution in reducing greenhouse gases in environment is
bleak. However, there are many areas in the modern society where energy inefficiency is
a major issue and thermoelectricity can make significant impact.
The modern economy is transitioning from a paper-based to a digital-based
economy. Financial transactions, medical records, banking, bureaucratic records are all
being digitized. The processing and storage of the digital data require data centers. Data
centers consumed 73 Billion kwh of electricity (1.8%) of the total usage in 2016.
8
Most of
the energy consumed in data centers goes to cooling these setups for optimal performance.
The huge amount of heat generated in these centers originate in the fundamental building
blocks of electronics, i.e. integrated circuits. The continual miniaturizations of integrated
circuits has come at a cost of increasing power consumption and the issue of power
dissipation has become one of the roadblocks to the scaling down trend.
9
The current power
densities in chip level are in between 100 W/cm
2
to 1000W/cm
2
.
9-10
This trend of power
Figure 1.3 Efficiency of Various Energy Generators at different Power Levels.
3

6

consumption can have damaging effect on performance of portable devices by draining the
battery faster as well as exceeding the nominal temperature for proper operation of chips.
Moreover, device engineers these days encounter local thermal hotspots that negatively
affect the performance of the chip.
9
Understanding thermal transport at nanoscale level has
become imperative for thermal management of these electronic devices. Besides,
integrated circuits and efficient energy, thermoelectricity has also been a promising
candidate for energy harvesting in implantable chips. Powering these implantable chips
with current battery technology is cumbersome. Thermoelectric devices provide a way to
use ambient temperature to power the chips. However, the research in this field is still in
its infancy. Fabric based TE devices able to generate 4.3mV at a temperature of 75.2 K has
been demonstrated.
11
This might pave a way for using clothing to power wearable
electronics in the future. Besides, integrated circuits, MOSFET transistors and implantable
devices, the technologies where thermoelectric effect will play important role are Phase
Change Memory devices and photonic crystal and waveguides for optical interconnects.
12

The Physics behind thermoelectric systems will be discussed later. Until 1990s, the
highest reported ZT was around 1 at 300 K for bulk Bi2Te3. In 1993, Millie Dresselhaus
published a work in which she demonstrated that the calculated ZT for a two dimensional
(2D) material was a factor of three higher than its 3D counterpart after a request by US
navy.
13-14
This seminal work triggered research in nanostructured materials for enhancing
ZT.
7

1.3 Current State of Research in Thermoelectricity
Since its resurgence in early 1990s, thermoelectric research has been currently
going through a period of renaissance. The progress in nanotechnology has helped in
pushing the science and engineering of thermoelectricity. Figure 1.4 illustrates the
progress in thermoelectric material research from 1960 to 2016. Between 1960 and 1990,
Bi2Te3 had the highest reported ZT value between 0.5 to 1. Nano-structuring helped
increase ZT between 1990 and 2010. Electronic band structure engineering and phonon
mean free path engineering have helped push the value of ZT to even higher value in recent
years. In 1995, Glen Slack introduced the concept of “phonon glass and an electron single
crystal” for design of optimal thermoelectric material.
15
This philosophy has driven
research in the field to engineer high ZT materials and systems.

Figure 1.4 The timeline of progress in engineering various materials with
given ZT value.
5

8

The engineering of materials with high ZT requires understanding of the
fundamental Landauer-Boltzman equations that govern thermoelectricity. The detailed
Mathematics and Physics of Thermoelectricity can be found in a good Solid State textbook.
We should Revisit ZT again.
𝑍𝑇 =
𝑆 2
𝜎𝑇
𝜅 𝑒 +𝜅 𝑙
𝜅 𝑒 and 𝜅 𝑙 in the above equations represent electronic and lattice contributions to
thermal conductivity. For metals, 𝜅 𝑒 dominates the thermal conduction, while for
semiconductors and insulators 𝜅 𝑙 dominates heat flow. 𝑆 2
𝜎 is also referred to as Power
Factor (PF). Just based on the above equation, ZT can be increased by increasing the PF
while decreasing the thermal conductivity. However, in reality, maximizing ZT is a much
harder problem because of intrinsic trade-offs that arise from the interrelationships between
various parameters hidden underneath the simplicity of the equation.
To increase the 𝑃𝐹 = 𝑆 2
𝜎 , we have to increase both conductivity and
Seebeck Coefficient. Figure 1.5 illustrates the fundamental tradeoff between the Seebeck
coefficient and the electrical conductivity. For a simple analysis, we can take Seebeck
coefficient as
𝑆 𝑛 (𝑇 )= −(
𝑘 𝐵 𝑞 )
(𝐸 𝐽 −𝐸 𝐹 )
𝑘 𝐵 𝑇
Where 𝐸 𝐽 is the energy carried by electron in the conduction band. For Seebeck
coefficient to be large, we want 𝐸 𝐽 −𝐸 𝐹 as large as possible, or 𝐸 𝐹 ≪ 𝐸 𝐶 . This is more
reflective of a non-degenerate semiconductor. For degenerate semiconductor 𝐸 𝐹 is much
9

closer to 𝐸 𝐶 . For the maximum value of electrical conductivity (in this case n type
semiconductor) given by
𝜎 𝑛 = (
2𝑞 2
ℎ
)〈𝑀 〉〈〈𝜆 〉〉 = 𝑛 0
𝑞 
𝑛
We want the Fermi Level to be as close to conduction band as possible 𝐸 𝐹 ≪ 𝐸 𝐶 .
Therefore, there is an optimal distance between the fermi level and the conduction band
that gives a perfect value for Power Factor. As the Figure 1.5 illustrates, when the Fermi
level is increased, the Seebeck Cofficient becomes smaller, while the conductivity starts
increasing. At an optimal value of the carrier concentration (which determines the position
of Fermi Energy), there is a maximum value of Power factor.
Figure 1.5 Illustration of dependence of Seebeck Coefficient and Electrical
Conductivity as a function of carrier concentration. This also illustrates the
optimization issue of ZT.
5

10

The Fermi level of a material can be tuned by changing the doping. One of the
approach to improving the Power factor can be to either find materials with best band
structure or engineer an optimal band structure of materials. Band structure can be modified
by exploiting valley degeneracy, strain engineering and using alloys. Heremans et al have
enhanced thermoelectric efficiency of PbTe by distorting the density of states using
Titanium doping. They were able to report ZT of 1.5 at 773K
16
.
Figures 1.6(a) and (b) illustrates the Seebeck coefficient tuning by exploiting the
valley degeneracy in PbTe. The difference between L-point and -point of valance bands
in PbTe is around 0.15eV.
5
By reducing, the energy offset between these two valleys,
carrier concentration can be redistributed as well as more channels can be created for hole
transport. The ZT of p-type PbTe has been increased through strain engineering via
introduction of impurities like Mg
5, 17
(2 at 873K) and Mn
5, 18
(1.6 at 700K).
Another aspect of improving ZT involves reducing the thermal conductivity (𝜅 𝐿 ).
In fact, most of the significant improvement in ZT of materials have come from reduction
Figure 1.6 (a): L and  valleys of valance band of PbTe. (b) Illustration of how the Na
doped PbTe has a slightly higher value of Seebeck coefficient induced by valley
degeneracy of doping.
5

11

in 𝜅 𝐿 through phonon engineering rather than the improvement in Power Factor. The ratio
𝜎 𝜅 𝐿 gives a good indication of how good of a thermoelectric a material is.
𝜎 𝜅 𝐿 =
〈𝑀 𝑒𝑙
𝐴 ⁄ 〉
〈𝑀 𝑝 ℎ
𝐴 ⁄ 〉
×
〈〈𝜆 𝑒𝑙
〉〉
〈〈𝜆 𝑝 ℎ
〉〉

For, n-Bi2Te3
𝜎 𝜅 𝐿 =
9.2×10
16
𝑚 −2
53×10
16
𝑚 −2
×
21𝑛𝑚
9𝑛𝑚
= 0.41
For Si,
𝜎 𝜅 𝐿 =
14×10
16
𝑚 −2
360×10
16
𝑚 −2
×
10𝑛𝑚
140𝑛 𝑚 = 0.003
For silicon and n-Bi2Te3, while the electrical conductivity are very similar order in
magnitude, the difference in the mean free path of phonons makes n-Bi2Te3 very good
Figure 1.7 Illustration of Phonon Scattering in a Crystal.
5

12

candidate for thermoelectrics while Si is a bad thermoelectric material. Phonon Mean Free
path engineering has been a great strategy to reduce thermal conductivity.
Phonons scatter from grain boundaries, point defects, nano-precipitates and other
phonons as illustrated in Figure 1.7. Therefore, by engineering defects, the scattering rates
of phonons can be significantly increased and hence the thermal conductivity can be
significantly reduced. The record high ZT that has been observed up to date has been 2.6
at a 923 K along the b-direction of SnSe crystals.
19
This came as quite a shock to
Thermoelectric community because a decade before that SnSe was perceived useless.
20

Usually, in a perfect crystal, the bonds between two neighboring atoms are approximately
modeled as simple harmonic oscillator and the spring constant is modelled to be constant.
However, in case of SnSe, the nature of chemical bond means, that the phonon transport
cannot just be modelled based on the parabolic approximation of simple harmonic
oscillator. Therefore, the spring constant is not constant but is a function of displacement
of atomic bond. Each phonon displacement modifies the spring constant for subsequent
phonon and so on. Therefore, there are more scattering and reflection of phonons resulting
in reduced thermal conductivity. Another reason the thermal conductivity starts decreasing
after peaking at some optimal temperature comes from the fact that Umklapp scattering
due to high momentum phonons start dominating the normal scattering process.
13

1.4 Unique Aspect of Nanoscale thermal Transport
Pushing the frontier of any technology also requires advancement in the
fundamental science. Pushing the technologies like phase change memories, nanoparticles
for medical therapies, thermally assisted magnetic recording is possible only through
understanding of heat transport at nanoscale.
12
The term “Nanoscale regime” has been
casually used in the field. However, the science of thermal transport at that nano scale has
come a long way in last two decades. Here, the “nanoscale” regime has been defined as a
regime at which sub-continuum effects become important and the system cannot be
described by bulk properties alone
12
. In this regime, the Fourier heat equation 𝑞⃗ = −𝑘 ∇𝑇
might not predict the flow of thermal energy as it ignores the effect of scattering at the
boundaries as well as the effect of interfaces.
Figure 1.8 (a) A harmonic system where longitudinal movement of atoms is in the same
direction at any given time (b) An Harmonicity can originate when atoms vibrate without
synchronizing in any direction (c) Harmonic region and anharmonic region of a
interatomic potential.
5

14

Therefore, a significant effort in the thermal transport studies of modern era has
been on studying the transport across interfaces. The first theoretical work on interfacial
transport was done by Khalatanikov who developed Acoustic Mismatch Model.
12
Acoustic
Mismatch model doesn’t incorporate various important effects like phonon dispersion, cut
off in the phonon density of states for high frequencies, or weak interfacial bonding.
21

Therefore, Young and Maris derived a more refined model for understanding thermal
contact resistance or Kapitza resistance in which they presented a calculation of phonon
transmission and the Kapitza conductance at the interface between two semi-infinite face
centered cubic lattice with different masses and spring constants.
21
Many theoretical work
based on interfaces do not take into account the effect of surface roughness or any surface
residues that can also affect the thermal transport.
In the modern era, molecular dynamics simulations are performed to characterize
the Kaptiza resistance. In this approach, atoms and molecules follow classical dynamics
based on solutions to Newton’s second law of motion.
12
These molecular dynamics
simulations incorporate all possible phonon modes and polarizations to provide
transmission and reflection coefficients for phonons at the boundaries. However, these
simulations are usually performed for simple systems. Another limitation can also come
from the limitation of computational power. To accurately capture heat flow across
nanowires, sometimes 100 of millions of atoms might need to be incorporated and this is
quite a mammoth task.
12
The two major models that are used to characterize Kapitza
resistance are Acoustic Mismatch Model and Diffusive Mismatch model. Moreover, many
parameters used in these simulations are based on bulk properties and they assume very
strong and perfect bonds at the interface.
22
Another approach to getting insight into the
15

thermal transport in the interfaces is to use the phonon Boltzmann equation. In this
approach, information like phonon dispersion relation, group velocity and the phonon
collision rates are used to evaluate heat current across the two interfaces. In general, the
results of theory and simulations are significantly higher than those reported for
experiments of the effects produced by surface roughness and residues at the interfaces.
12

1.5 Theoretical Foundation of Thermoelectricity
To understand thermoelectric effect, it is quite important to understand the
fundamental science of transport. There are many ways of approaching the problem of
thermoelectric transport. The basic building block of the transport equation is the
Boltzmann transport equation and this can be found in the reference.
23

Another way of looking at thermoelectric transport that I found intuitive and easy
to grasp is the bottom-up approach. The fundamental building block of this approach is
Landauer-Boltzman formalism. In this chapter, I will briefly discuss about the Landauer-
Bolzman or bottom up approach to understanding thermoelectric transport. The
fundamental knowledge covered on this chapter are based on the lectures by Mark
Lundstrom and Supriya Datta in their lectures on nanuHub-U.
2
In my graduate school, their
λ

𝑥
𝑥
−
𝑇 1
𝑇 2
Figure 1.9: Illustration of a n-type material when an external current
Jx is induced on it.
2

16

lectures online were instrumental in helping me understand device physics and
thermoelectric transport.
Let us assume that we have a material as shown in the Figure 1.9 above. If we
assume this is a bulk material, then we are in a regime where the length of the material (L)
is much larger than the mean free path ( λ) of the carrier in that material. For the shake of
simplicity, let us assume that the material of interest is a n-type material, where the majority
charge carriers are electrons. In thermoelectric effect, the basic idea can be thought about
in two ways.
In the first approach, we basically force an external electric field between the two
ends of the material by forcing a current through the material
𝑥 . From Ohm’s law, we have

𝑥 = 𝜎 𝐸 𝑥
That forced current
𝑥 causes an electric field between two ends of that material,
which can also be expressed as
𝐸 𝑥 = −
∆𝑉

Where ∆𝑉 is the potential difference between the two ends while L is the length of
the material. This forced current or electric field or potential gradient, causes charge carrier
to move from one end to another. The charges carry heat with them causing that end to
cool while the other end to heat as shown below.
17

Figure 1.10 Peltier Effect in a material.
2

In Figure 1.10, the flow of electrons from right to left causes the right end to cool
down while left end to heat.. We call this effect thermoelectric effect. In Pelier effect, we
measure the temperature gradient across the two ends while we force the current through
the device. We define a variable Π, called Peltier coefficient, which is the ratio,
𝛱 =

ℎ𝑒𝑎𝑡 𝑐𝑢𝑟𝑟 𝑒𝑛𝑡
𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐𝑎𝑙𝑙 𝑐𝑢𝑟𝑟𝑒𝑛𝑡

Figure 1.11 Seebeck Effect in a Material.
2

λ

𝑥
𝑥
−
𝑇 1
𝑇 2
V
λ

−
𝑇 1
𝑇 2
18

The complimentary effect of Peltier effect is Seebeck effect as shown in Figure
1.11. In this phenomenon, we force a temperature gradient in the material by heating or
cooling one end with respect to another. This temperature gradient can cause the charge to
flow from one end to another, because of which we can induce a voltage difference between
the two ends. We define Seebeck coefficient as 𝑆 = −
𝛥𝑉
𝛥𝑇
.
Based on these interrelationships between heat flow and charge flow, we come up
with following sets of equations that govern thermoelectric phenomenon.
𝑉 𝑥 = −(
𝐽 𝑥 𝜎 +𝑆 𝑑𝑇
𝑑𝑥
)L

𝑞𝑥
= 𝛱
𝑥 −(𝜅 𝑒 +𝜅 𝐿 )
𝑑𝑇 𝑑𝑥
𝛱 = 𝑇𝑆
Where,
𝜅 𝑒 : Electronic contribution
𝜅 𝐿 : Lattice contribution
σ : Electrical Conductivity
𝑇 : Thomson relation
𝛱 : Peltier coefficient
S: Seebeck coefficient

19

Heat is transferred not only by the charge carriers, but can also be transferred by
phonons. In fact, phonons are dominant heat transfer media for non-degenerate
semiconductors and insulators while electrons and holes are dominant for degenerate semi-
conductors and metals. The heat transported by phonons for bulk materials is given by
Fourier heat transport equation.
𝑄 = 𝑘𝐴
𝑑𝑇 𝑑𝑥
Thomson relation provides insight into the inter-relationship between Seebeck and
Peltier coefficient as these two effects are closely coupled with each other.
1.5.1 Boltzman-Landauer Formalism
Let us try to understand the transport mechanism in the bulk n-type material
presented earlier using Boltzman-Landauer formalism.

Figure 1.12 Fermi Distributions of two contacts in a band picture.
2

Let us assume the n-type material is placed between two contacts. The temperature
of each contact and the distribution of the electrons in each contact is given by variable f
and T respectively. In the equilibrium case, the Fermi-level or electrochemical potential of
both ends are the same. The current flow condition is given by the flowing equation.
20

𝐼 =
2𝑞 ℎ
∫𝑇 (𝐸 )𝑀 (𝐸 )(𝑓 1
−𝑓 2
)𝑑𝐸
Where 𝑇 (𝐸 ) is a transmission probability and is given by
𝑇 (𝐸 )=
𝜆 (𝐸 )
𝜆 (𝐸 )+

And 𝑀 (𝐸 ) is the number of channels available for charge carriers to move in the
band. 𝑀 (𝐸 ) is a product of density of the states in the material and the mean velocity of
the carriers in the materials and is given by
. 𝑀 (𝐸 )=
ℎ
4
〈𝑣 〉𝐷 (𝐸 )
The mean velocity 〈𝑣 〉 for 1D, 2D and 3D cases are given by
〈𝑣 〉
1𝐷 = 𝑣 (𝐸 )
〈𝑣 〉
2𝐷 =
2
𝜋 𝑣 (𝐸 )
〈𝑣 〉
2𝐷 =
1
2
𝑣 (𝐸 )
If we stay within the parabolic band approximation, for a carrier with effective mass m*
1
2
𝑚 ∗𝑣 2
= 𝐸 −𝐸 𝑐
From, which we can get an expression
𝑣 =
√
2(𝐸 −𝐸 𝑐 )
𝑚 ∗

The Density of states for 1D, 2D and 3D cases are given by
21

𝐷 1
(𝐸 )=
1
4𝜋 (
2𝑚 ∗
ħ
)
1 2 ⁄
1
√𝐸 −𝐸 𝑐
𝐷 2
(𝐸 )=
𝑚 2𝜋 ħ
2

𝐷 3
(𝐸 )=
1
4𝜋 2
(
2𝑚 ∗
ħ
2
)
1 2 ⁄
√𝐸 −𝐸 𝑐
In the case when no external voltage or temperature gradient is imposed on the
system, the fermi distribution of both ends remain the same as shown in the figure below.

Figure 1.14 Illustration of Peltier effect in the band picture.
2

f(E)
Energy
E
F
1
Figure 1.13 Fermi Distribution in a Material.
2

22

There are two ways to perturb the system shown in Figure 1.13. If we apply a small
voltage V to the right-hand contact, we can perturb the system as shown in Figure 1.14.
This corresponds to change of Fermi distribution in both contacts as shown in the
Figure 1.15.

Figure 1.15 Change in Fermi Distribution with application of small Field.
2

Assuming the applied voltage is quite small and using Taylor expansion we can express
𝑓 1
−𝑓 2
as
𝑓 1
−𝑓 2
≈ −
𝜕 𝑓 0
𝜕𝐸
𝑞𝑉
And for 𝜆 (𝐸 )
𝑇 ≈
𝜆 (𝐸 )

Now after substituting these equations into and we can express it as

E
F
E
F
-Vq
1
f(E)
Energy
23

𝐼 =
2𝑞 2
ℎ
[∫
𝜆 (𝐸 )

𝑀 (𝐸 )(−
𝜕 𝑓 0
𝜕𝐸
)𝑑𝐸 ]𝑉

By equating this equation to 𝐼 = 𝐺𝑉 , where G is the quantum conductance, we can
derive the expression for G as
𝐺 =
2𝑞 2
ℎ
∫
𝜆 (𝐸 )

𝑀 (𝐸 )(−
𝜕 𝑓 0
𝜕𝐸
)𝑑𝐸
And since,
𝐺 = 𝜎 𝐴

We can evaluate the electrical conductivity as
𝜎 (𝐸 )=
2𝑞 2
ℎ
∫𝜆 (𝐸 )
𝑀 (𝐸 )
𝐴 (−
𝜕 𝑓 0
𝜕𝐸
)𝑑𝐸
If we look at the equation

𝑞𝑥
= 𝛱
𝑥 −(𝜅 𝑒 +𝜅 𝐿 )
𝑑𝑇 𝑑𝑥
Whenever we create a perturbation and change in Fermi energy between two
contacts, we can think about this process as a charge q moving to an energy channel with
energy E and hence carrying energy corresponding to 𝐸 −𝐸 𝑓 from one contact to another.
We can define 𝜎 ′(𝐸 ) as differential conductivity as
𝜎 (𝐸 )= ∫𝜎 ′(𝐸 )𝑑𝐸
24

Each differential current ′
𝑥 contains 𝑛 =
𝐽 ′
𝑥 𝑞 number of carriers and if each charge
carries energy 𝐸 −𝐸 𝑓 , the differential heat energy carried is given by following relation

′
𝑞𝑥
= 𝑛 (𝐸 −𝐸 𝑓 )

′
𝑞𝑥
=
− ′
𝑥 𝑞 (𝐸 −𝐸 𝑓 )
The total heat current
𝑞𝑥
is given by

𝑞𝑥
= ∫
′
𝑞𝑥
𝑑𝐸
Where

′
𝑥 = 𝜎 ′(𝐸 )𝜀
And hence,

𝑞𝑥
= ∫−
𝜎 ′(𝐸 )𝜀 𝑞 (𝐸 −𝐸 𝑓 )𝑑𝐸
If we divide both side by
𝑥 = 𝜎 (𝐸 ) 𝜀
We evaluate
𝛱 =

𝑞𝑥

𝑥 =
∫−
𝜎 ′(𝐸 )𝜀 𝑞 (𝐸 −𝐸 𝑓 )𝑑𝐸 𝜎 (𝐸 ) 𝜀

𝛱 =

𝑞𝑥

𝑥 =
∫−
𝜎 ′(𝐸 )
𝑞 (𝐸 −𝐸 𝑓 )𝑑𝐸 ∫𝜎 ′(𝐸 )𝑑𝐸
25

To evaluate electronic contribution to the heat current

𝑞𝑥
= 𝛱
𝑥 −𝜅 𝑒 𝑑𝑇 𝑑𝑥
We first need to look at the coupled equation again
𝑉 𝑥 = −(
𝐽 𝑥 𝜎 +𝑆 𝑑𝑇
𝑑𝑥
)L

𝑞𝑥
= 𝛱
𝑥 −𝜅 𝑒 𝑑𝑇 𝑑 𝑥
In this case, we ignore the lattice contribution to thermal current. There is a trick
that is performed to evaluate 𝜅 𝑒 . In order to do that we exploit the fact that heat is
transferred in the open circuit condition too, which means
𝑥 = 0. In this case the equation
simplifies to
𝑉 𝑥
= 𝜀 = −𝑆 𝑑𝑇 𝑑𝑥
Now, the net flow of charge is due to the difference in the temperature between the
two contacts in the following equation.
𝐼 =
2𝑞 ℎ
∫𝑇 (𝐸 )𝑀 (𝐸 )(𝑓 1
−𝑓 2
)𝑑𝐸
This is also illustrated in Figure 1.16.

26

Figure 1.16 Illustration of Seebeck Effect in band picture.
2

Figure 1.17 Change of Fermi distributions with external temperature difference.
2

In this case, by using Taylor expansion and simplification we can express 𝑓 1
−𝑓 2
as
𝑓 1
−𝑓 2
= −
𝜕 𝑓 0
𝜕𝐸
(𝐸 −𝐸 𝑓 )
𝑇
The heat current carried by a charge q is given by the following equation
𝐼 𝑞 =
2
ℎ
∫(𝐸 −𝐸 𝑓 )𝑇 (𝐸 )𝑀 (𝐸 )(𝑓 1
−𝑓 2
)𝑑𝐸
1

Energy
f(E)
T
1
T
2
>T
1
E
F
27

After substituting the approximation provided above, we can express above
equation as
𝐼 𝑄 = −[
2
ℎ
∫
(𝐸 −𝐸 𝑓 )
2
𝑇 𝑇 (𝐸 )𝑀 (𝐸 )−
𝜕 𝑓 0
𝜕𝐸
𝑑𝐸 ]𝛥𝑇
By equating the above expression to the equation
𝐼 𝑄 = −𝜅 𝑜 𝛥𝑇
And then using the expression
𝐾 𝑜 = 𝜅 0
𝐴

We can express 𝜅 0
as
𝜅 0
=
2
ℎ
∫
(𝐸 −𝐸 𝑓 )
2
𝑇 𝜆 (𝐸 )
𝑀 (𝐸 )
𝐴 −
𝜕 𝑓 0
𝜕𝐸
𝑑𝐸

𝜅 0
= ∫
(𝐸 −𝐸 𝑓 )
2
𝑞 2
𝑇 𝜎 ′(𝐸 )𝑑𝐸
After finding the open circuit contribution to thermal conductivity, we can find
short circuit contribution to thermal conductivity too. In order to do that, we visit the
equation
𝑉 𝑥 = −(
𝐽 𝑥 𝜎 +𝑆 𝑑𝑇
𝑑𝑥
)L
And for short circuit 𝑉 𝑥 = 0
28

𝑥 = −𝜎𝑆
𝑑𝑇 𝑑𝑥
If we substitute the above expression into

𝑞𝑥
= 𝛱
𝑥 −𝜅 𝑒 𝑑𝑇 𝑑𝑥
We get

𝑞𝑥
= −(𝛱𝑆 𝜎 +𝜅 𝑒 )
𝑑𝑇 𝑑𝑥
From which we can get the following relationship
𝜅 𝑒 = 𝜅 𝑜 −𝛱𝑆𝜎
Here we evaluated most of our expressions for electrons. We could also use the
same approach with some careful choice of sign to evaluate these parameters for holes too.
The following relationship summarizes the fundamental thermoelectric transport
equations.
2

𝜎 ′
(𝐸 )=
2𝑞 2
ℎ
𝜆 (𝐸 )
𝑀 (𝐸 )
𝐴 (−
𝜕 𝑓 0
𝜕𝐸
)
𝜎 (𝐸 )= ∫𝜎 ′
(𝐸 )𝑑𝐸
𝑆 = −
1
𝑞𝑇
∫𝜎 ′(𝐸 )(𝐸 −𝐸 𝑓 )𝑑𝐸 ∫𝜎 ′(𝐸 )𝑑𝐸
𝜅 0
=
1
𝑞 2
𝑇 ∫(𝐸 −𝐸 𝑓 )
2
𝜎 ′(𝐸 )𝑑𝐸
𝜅 𝑒 = 𝜅 𝑜 −𝛱𝑆𝜎
29

𝑆 𝑡𝑜𝑡 =
𝑆 𝑛 𝜎 𝑛 +𝑆 𝑝 𝜎 𝑝 𝜎 𝑡𝑜𝑡
Lattice contribution to thermal conductivity is given by the following expression
2

𝜅 𝐿 =
𝜋 2
𝑘 𝐵 2
𝑇 3ℎ
∫𝜆 𝑝 ℎ
(ћω)
𝑀 𝑝 ℎ
(ћω)
𝐴 𝑊 𝑝 ℎ
(ћω)d(ћω)
30

2 Cross-plane Thermoelectric and Thermionic
Transport across Au/h-BN/Graphene
Heterostructures
This chapter is similar to our publication in Scientific Reports 7(1), 14148.
24

2.1 Introduction
Extensive research on two-dimensional (2D) materials has resulted in a wide range
of materials that are available for building van der Waals bonded heterostructures. Vertical
stacking of graphene, transition metal dichalcogenides (TMDCs), boron nitride, and
phosphorene has been used to engineer various types of electronic and optoelectronic
devices including transistors,
25-28
photovoltaics,
29-30
and light emitting diodes,
31
as well as
to explore novel physical and chemical phenomena on a fundamental level. These van der
Waals bonded heterostructures, in principle, provide a system in which it is possible to
control electron and phonon transport independently. 2D materials have recently attracted
a lot of attention as potential new TE materials.
32-33
In a recent report, Liang et al. proposed
a highly efficient solid state thermionic device based on a van der Waals (vDW)
heterojunction of graphene/TMDC (e.g., MoS2, MoSe2, WS2, WSe2) by exploiting the
ultra-low cross-plane conductance of the 2D materials and the thermionic emission over
the Schottky barrier between graphene and the 2D materials.
32
In principle, this Schottky
barrier can be tuned through gating and chemical doping.
32
These calculations predict that
thermionic devices can provide better or comparable power generation and refrigeration
31

efficiency than traditional bulk TE devices.
32
However, these predictions have yet to be
verified experimentally.

The efficiency of solid-state thermionic energy conversion has been predicted to exceed
that of conventional thermoelectric energy conversion based on bulk Peltier and Seebeck
effects, if the thermionic barriers can be properly engineered. Theoretical investigations of
this process date back to 1997.
32, 34-36
However, there have been relatively few experimental
studies on solid-state thermionic energy conversion, mainly because of the difficulty of
fabricating interfaces with the appropriate energy barriers, characterizing thermal transport
across these interfaces, and separating the bulk thermoelectric properties from the
interfacial properties. In thermionic devices, the electrons must flow over the barrier and
propagate in the conduction band without scattering in order to have a high efficiency. This
requires that the mean-free path  of the electrons in the barrier be longer than the width L
of the barrier (i.e.,  > L). This constrains the barrier width L to be rather small. On the
other hand, the barrier should be thick enough to prevent electrons from quantum
mechanically tunneling through the barrier region. The general constraints are  > L > Lmin,
Figure 2.1 Schematic diagram of the thermionic emission process, illustrated for
a graphene/h-BN/Au heterostructure.
32

where Lmin is the minimum thickness to prevent the electron from tunneling through the
barrier, typically 5–10 nm for most semiconductors. In these devices, electrons traverse
relatively thick, low energy barriers (on the order of 2-3kBT). Only electrons in the high
energy tail of the Fermi-Dirac distribution will have enough energy to overcome the
barrier, resulting in a phenomenon similar to evaporative cooling. A generalized schematic
diagram of this process is illustrated in Figure 2.1.
In the work presented here, we study the cross-plane thermoelectric and thermionic
transport across Au/h-BN/graphene heterostructures. This material system is chosen
because BN is known to provide a good barrier for electrons (and holes) in graphene, with
several previous studies on the graphene/BN/graphene heterostructure system.
25, 37
In our
structure, we switched the bottom electrode to Au to provide a better heat sink and to enable
us to perform Raman thermometry on the top electrode.
33

2.2 Methods
In the work presented here, we fabricate graphene Au/h-BN/graphene
heterostructures to explore thermoelectric and thermionic transport across extremely short
length scales. There have been several experimental and theoretical studies of thermal
transport across graphene/SiO2, graphene/Si, and graphene/SiC interfaces.
38-47

Experimental measurements of the thermoelectric transport across a graphene/h-
BN/graphene heterostructure was previously carried out by Chen et al.
37
In this work, the
Seebeck coefficient (S) was reported to be -99.3 V/K and the corresponding power factor
Figure 2.2 Illustration of Fabrication Process.
34

(S
2
σ) was 1.51×10
-15
W/K
2
.
37
In this previous graphene/BN/graphene thermoelectric
measurement, the graphene was used both as a heating element and a temperature monitor.
This limited the maximum voltage that could be applied to the heater to ≤2V, in order to
protect the delicate graphene heater from electrical burnout under high currents.
37

Moreover, using graphene as both the heating element and heat sink on the bottom, made
it difficult to distinguish the Raman spectra from the top and bottom graphene, which
introduces significant inaccuracies in the measurement of the vertical temperature gradient
( T) in the active region of the device. In the work presented here, a new device structure
was employed to separate the heater and temperature monitor, with the Au/graphene/h-BN
stack being isolated from the heater by a 50 nm layer of Al2O3. A 30 nm film of indium tin
oxide (ITO) was used as a transparent electrical heater, enabling Raman spectroscopy (and
thermometry) to be obtained from the graphene top electrode in situ under device operating
conditions. As discussed in the works of Vallabhaneni et al. and Sullivan et al., phonons
can be driven out of local equilibrium in suspended graphene under high laser power
irradiation.
4, 48
However, at the small laser powers used in our study (<0.1mW/µm
2
), the
local thermal equilibrium is maintained. Here, the heating is provided by an external
electrical heater that raises the temperature of different phonons uniformly. As such, we
35

believe that the Raman shift provides a reliable measure of the temperature change in this
system.
In the device fabrication process, Ti/Au (5/30 nm) electrodes are deposited on a
Si/SiO2 substrate using electron beam lithography (EBL) followed by electron beam metal
deposition. This bottom gold electrode provides a good heat sink for this thermoelectric
device, as described below. A viscoelastic dry transfer process was then used to place an
approximately 5 to 10 nm thick h-BN flake on top of the center gold electrode, as shown
ITO
Al
2
O
3
Graphene
h-BN
(a)
(b)
1500 2000 2500
0
10000
20000
30000
40000
50000
hBN ~ 1370 cm
-1
G ~ 1583 cm
-1

Counts (arbitrary units)
Raman Shift (cm
-1
)
2D ~ 2708 cm
-1
(c)
Figure 2.3 (a) Cross-sectional diagram, (b) optical microscope image, and (c)
Raman spectrum of the graphene/h-BN/Au heterostructure with ITO heater.
36

in Figure 2.3b.
49
A wet chemical transfer technique using PMMA as a sacrificial layer was
used to transfer graphene grown by chemical vapor deposition (CVD) on copper foil onto
the Au/h-BN stack. An “I” shaped graphene strip was then patterned, as shown in Figure
2.3b, using electron-beam lithography followed by oxygen plasma etching (RF power of
100 W, base pressure of 200 mTorr) for 50 seconds. An insulating layer of Al2O3 was then
deposited on top of the Au/ graphene/h-BN sandwich structure using atomic layer
deposition. Finally, a 100 m × 50 m, 30 nm-thick film of ITO (indium tin oxide) was
deposited on top of the oxide via RF sputtering after electron-beam lithography.

2.3 Results and Discussion
After fabrication, the device was placed in a high vacuum, temperature controlled
optical cryostat. The Raman spectra of the graphene were acquired using a 532 nm
wavelength laser through the transparent ITO heater at various stage temperatures and
heater voltages. One such spectrum is plotted in Figure 2.3c. Here, distinct peaks can be
seen for the G and 2D band Raman modes of the graphene, as well as the optical phonon
of h-BN at 1370cm
-1
. The downshifts of the 2D Raman modes of graphene with increasing
temperature and heater voltage can be seen in Figures 2.4a and 2.4b, respectively. This data
was then used to determine the temperature of the graphene, and hence the vertical
temperature drop ( T) across the Au/graphene/h-BN heterostructure as a function of the
DC heater voltage, as plotted in Figure 2.4c. As expected, this T follows a parabolic
dependence on the heater voltage and a linear dependence on the heater power (Figure
2.4d. In order to verify that the bottom Au electrode remains close to room temperature,
we fabricated a separate test structure, in which a monolayer of MoS2 was inserted on top
37

of the Au electrode in a Au/MoS2/BN/graphene configuration. This device geometry is
illustrated in Figure S1 of the Supplement section. The corresponding temperatures of the
graphene and MoS2 are plotted as a function of heater voltage in Figure S2, as measured
by Raman spectroscopy. This data shows that for every 1 K increase in temperature of
graphene, there is a corresponding temperature increase of 0.16 K on average in bottom
gold electrode. We use a corrrection factor of 𝛥𝑇 /𝑇 𝐺𝑟𝑎𝑝 ℎ𝑒𝑛𝑒
= 0.84 to account for the slight
change in temperature of the bottom gold electrode as a function heating power.

280 320 360 400 440
2703
2704
2705
2706
2707
2708
2709

Raman Shift (cm
-1
)
Temperature (K)
Slope: -0.033 cm
-1
K
-1
0 2 4 6 8 10 12
2706.5
2707.0
2707.5
2708.0
2708.5

Raman Shift (cm
-1
)
Heater Voltage (V)
-5 0 5 10 15 20 25 30 35 40
0
4
8
12

T(K)
Heater Power (mW)
(c) (d)
(a)
(b)
0 2 4 6 8 10 12
0
20
40
60

T(K)
Heater Voltage (V)
Figure 2.4 (a) Calibration data of the graphene 2D-band Raman shift
plotted as a function of temperature obtained in a temperature controlled
optical vacuum cryostat. (b) Graphene 2D-band Raman shift plotted as a
function of the heater voltage. (c, d) Cross-plane temperature difference
plotted as a function of heater voltage and heater power.
38

Following this calibration procedure, thermoelectric measurements were
performed using an AC lock-in technique. Here, an AC voltage V( ) was applied to the
ITO heater at a frequency of 100 Hz, while the thermoelectric voltage was measured across
the Au/h-BN/graphene heterostructure at 2  (i.e., 200Hz) using a lock-in amplifier. In our
previous work, we found that thermovoltage was frequency independent in this range
indicating that the thermal time constant of the system is much faster than the AC
modulation .
37
The raw data from this measurement is shown in Figure 2.5c, which plots
the AC thermoelectric voltage measured at 2  as a function of the AC heater voltage
applied. Here, the sample was measured in two different configurations, ‘heating’ and
‘non-heating’, as illustrated in Figures 2.5a and 2.5b, respectively. In the ‘heating’
configuration, one side of the ITO heater is grounded, enabling heating to occur in the
heater when an AC voltage is applied. In the ‘non-heating’ configuration, the AC heater
voltage is applied to both sides of the heater, while the underlying Au electrode is
grounded. In this configuration, the voltage is applied across the device stack, but does not
produce any heating. This non-heating configuration is used to evaluate the effect of second
harmonic generation produced across the stack due to non-linearities in the I-V
characteristics of the Au/h-BN/graphene device. Whenever an AC voltage is applied to an
electronic component with a non-linear I-V response, a second harmonic can be generated,
which is not thermal in nature. This is particularly important in the case of our 2D sample
geometry, since the heater is capacitively coupled to the thermoelectric device. The results
shown in Figure 2.5c indicate that this second harmonic generation is negligible, and the
difference between these two measurements is, in fact, a reliable measure of the
thermoelectric voltage generated in these devices. Based on the calibration data shown in
Figure 2.4c, we can convert the heater voltage in Figure 2.5c to temperature difference
39

( T) based on the DC Raman measurements, as plotted in Figure 2.6. Here, the
thermoelectric voltage exhibits a linear dependence on the temperature gradient, with a
Seebeck coefficient of -215 V/K. The thickness of h-BN is estimated to be around 5 to 10
( 3 to 5nm) layers thick based on the optical contrast of the flake on the 300 nm SiO2
substrate. It should be noted that this thermovoltage is created at the graphene/h-BN
0 1 2 3 4 5
0.0
0.4
0.8
1.2
1.6
2.0
0 2 4
0.0
0.8
1.6

V(2 )/ V
V( )/V

Non-heating
Heating
Difference
V
AC
(2 ) (mV)
AC Applied Voltage (V)
(a)
(b)
(c)
Figure 2.5 Schematic diagrams of the measurement set up and device geometry
for the (a) ‘non-heating’ and (b) ‘heating’ configurations. (c) Cross-plane AC
voltage measured between the top graphene contact and bottom Au electrode at
2ω for both heating and non-heating configurations, plotted as a function of the
applied AC heater voltage.
40

interface rather than in the constituent materials, and originates from thermionic emission
mechanism.
When the temperature difference is applied across the Au/graphene/h-BN
heterostructure, the hot electrons in the graphene are ballistically transported to Au by a
thermionic emission process, as illustrated in Figure 2.1. We assume that the injection of
electrons from the graphene into the Au will not change the Fermi level position, which is
justified by the very large density of states in the Au. The thermionic emission from
graphene is different from Au and can be described by equation
27

𝐺 = 𝐴 𝐺 𝑇 ℎ
3
exp[−
𝑒 Φ
𝐵𝑛
𝑘 𝐵 𝑇 ℎ
],
where 𝐴 𝐺 is revised Richardson constant for graphene, 𝑇 ℎ
is the temperature of graphene,
𝑘 𝐵 is Boltzmann constant, is elementary charge and Φ
𝐵𝑛
is the Schottky barrier height at
the graphene/h-BN interface. The thermionic emission from Au is determined by the
equation
𝐴𝑢
= 𝐴 𝐴𝑢
𝑇 𝑐 2
exp[−
𝑒 Φ
𝐵𝑛
−𝑉 𝑜𝑐
𝑘 𝐵 𝑇 𝑐 ], where 𝐴 𝐴𝑢
is Richardson constant for Au, 𝑇 𝑐 is
the temperature of Au, and 𝑉 𝑜𝑐
is the open voltage generated by the temperature difference
S = -215 µV/K
-2 0 2 4 6 8 10
0.0
0.4
0.8
1.2
1.6
2.0
Thermovoltage (mV)

T/K
Figure 2.6 Thermoelectric voltage plotted as a function of the temperature
difference across the Au/h-BN/graphene heterostructure.
41

across heterostructure. The net electric current across the graphene/h-BN/Au
heterostructure can be determined by the difference of
𝐺 𝑎𝑛𝑑
𝐴𝑢
. When operating under
open circuit conditions, the net electric current is zero, and 𝑉 𝑜𝑐
= Φ
𝐵𝑛
(1−
1
1+
∆𝑇 𝑇 𝑐 )+
𝑘 𝐵 𝑇 𝑐 𝑒 ln (
𝐴 𝐺 𝐴 𝐴𝑢
(∆𝑇 +𝑇 𝑐 )
3
𝑇 𝑐 2
) . Within the limit of small temperature gradient ∆𝑇 , the 𝑉 𝑜𝑐
is linearly
proportional to ∆𝑇 , which is consistent with experiment data in this work as shown in
Figure 2.7. For, the theoretical model, we use the barrier height of Φ
𝐵𝑛
=125 meV and
modified Richardson constant for graphene AG of 0.104 Acm
-2
K
-3
.
50
The Richardson
constant for gold AAu of 31.2 Acm
-2
K
-2
is obtained by using the effective mass of h-BN
0.26 as a correction factor multiplied by universal constant A0 = 120 Acm
-2
K
-2
.
51
Note
smooth transition between the two temperature scaling T
3
and T
2
has been developed.
29
In
Figure 6, the experimental data shows that the thermovoltage depends linearly on the
temperature difference across the heterostructure. Our theoretical model also demonstrates
a linear dependence between the thermovoltage and temperature gradient, as represented
by the red dots symbol in the Figure 2.7. Because we have limited information about the
Schottky barrier height at the graphene/h-BN interface and the effective Richardson
constant for the thermionic emission from Au to h-BN, we used typical parameters
( Bn=125 meV and A Au=31.2 Acm
-2
K
-2
) to qualitatively show the general trend. Here, the
linear behavior is in agreement with experiment, and any discrepancy is likely due to
42

inaccuracies in these parameters and/or minor errors in the characterization of the
temperature gradient.

2.4 Conclusion
In summary, we have developed a technique for measuring the thermoelectric
transport across a graphene/h-BN/Au heterostructure. A transparent ITO heater is used to
enable the temperature of the graphene top electrode to be measured optically using Raman
spectroscopy in situ under device operating conditions. We apply an AC voltage at a
frequency of ω to the ITO heater and measure the thermoelectric voltage across the Au/h-
BN/graphene heterostructure at a frequency of 2ω using a lock-in amplifier. We are able
to separate the actual thermovoltage from the second harmonic voltage originating from
the non-linear current-voltage characteristics of Au/h-BN/graphene heterostructure by
performing the experiment in “heating” and “non-heating” configurations. We observe a
Seebeck coefficient of -215 μV/K for this device. This thermoelectric voltage originates
0 2 4 6 8 10
0
1
2
3
4

Experiment
Theory
Thermovoltage (mV)
Temperature Difference (K)
Figure 2.7 Comparison of theoretical model with experimental measurements for
thermo-voltage across Au-hBN-Graphene.
43

from the graphene/h-BN interface due to thermionic emission rather than bulk diffusive
transport. As such, this should be thought of as an interfacial Seebeck coefficient rather
than a Seebeck coefficient of the constituent materials.
2.5 Supplemental Information

Figure S1. Diagram illustrating the Au/MoS2/BN/graphene test structure.
0 2 4 6 8 10
0
20
40
60
80
100

Graphene
MoS
2
Temperature Difference
Temperature (K)
Heater Voltage (V)

Figure S2. Graphene and MoS2 temperatures measured by Raman spectroscopy plotted
as a function of heater voltage. This data shows that there is negligible heating in the MoS2
layer and, hence, the underlying gold layer.
ITO
Al
2
O
3
Graphene
h-BN
MoS
2
44

3 In-plane thermoelectric transport across Graphene
and Graphene-hBN heterojunction
This chapter is similar to our publication in Applied Physics Letters 110(7) 074104.
52

David Choi is the first author of this work.
3.1 Introduction
This project was carried out in collaboration with David Choi, a PhD student at Dr.
Li Shi’s lab at UT Austin. At our lab, we were working on studying thermal transport across
vertical direction in graphene and h-BN heterostructure using Raman thermometry. Our
cross-plane project had certain short comings as we could only probe a temperature across
just one point of our device using a laser spot of diameter 1 μm. Therefore, we proposed
an idea whereby we would quantify the cross-plane heat transport in a graphene/h-BN
device in our lab while the lateral heat transport for the same device would be studied by
David in UT Austin. The idea was to quantify the ratio of the heat transferred in the lateral
direction of graphene to the vertical direction of the graphene/hBN heterostructure. David
Choi, at UT Austin, studied the lateral heat transport using scanning thermal microscopy
and electrostatic force microscopy. He also performed simulation and thorough theoretical
modelling as well as numerical analysis. Scanning thermal microscopy enables imaging
with resolution in sub-100 nm regime.
53
In this technique temperature of surface is mapped
via raster scanning with a sharp tip attached in a cantilever beam. In EFM (Electrostatic
Force Microscopy) the surface is scanned by an AFM tip with a bias voltage. EFM
provides electrical properties and charge density across the surface of a material. Figure
45

3.1 shows the device fabricated for to study in plane thermoelectric transport across
graphene and graphene-hBN structure. The electrodes in these samples are placed
uniformly across graphene-SiO2 and graphene-hBN-SiO2 channels for comparison in heat
dissipation across graphene channels on SiO2 vs hBN.

In the first publication that resulted from this collaboration, Choi et al. probed hot
spot generated in a graphene channel between two gold electrodes using EFM and SThM
and used numerical analysis and simulation to investigate the nature of these hot spots in
regards to in plane thermal conductivity (κ) of graphene and interfacial thermal
1000 1500 2000 2500 3000 3500
0
2000
4000

Counts (a.u.)
Raman Shift (cm
-1
)
Region I
Region II
Region III
Region IV
Region V
10 m
(a) (b)
(c) (d)
Figure 3.1(a) 3D schematic of graphene-hBN device (b) Raman spectra
across various channels (c) and (d) 20X and 100X microscope image of the
active device.
46

conductivity (G) of graphene/SiO2 structure.
52
Figure 3.2 (a) shows the schematic of one
of the graphene channels that Choi et al. studied.
3.2 Results and Discussion

Figure 3.2 (d) shows the experimental results as well as simulated fit of the
temperature distribution across the graphene/SiO2 channel between two gold electrodes.
After thorough analysis, it was concluded that the hot spot temperature is much more
sensitive to the interfacial thermal conductance G for graphene on sub 10 nm oxides while
not sensitive to in plane thermal conductivity  of graphene. Contrary to that, the hot spot
temperature is much sensitive to in plane thermal conductivity  of graphene while
relatively insensitive to interfacial thermal conductance G for graphene on 300 nm oxides.
(a)
(b)
(c)
(d)
Figure 3.2 (a) 3D schematic of the device (b) Numerical calculations of hot spot
temperature distribution across the hot spot between two graphene channels. (c)
Experimental measurement and fit through simulation of the temperature profile across the
channel.

47

Figure 3.3 (a-d) shows the atomic force microscopy (AFM), SThM and EFM
measurement results of a graphene device. The device is comprised of a 12.6 x 10 μm
2

graphene channel contacted by Cr/Pd electrodes, as illustrated in Figure 3.2 (a). The details
of measurement protocol is provided in the paper.
54
Figure 3.2 (b) shows the measured
temperature distribution on the graphene device when a 14 kW cm
-2
dissipated power
density was applied. For operating tear-free graphene devices measured in prior works, the
Figure 3.3 Measured (a) & (b) thermal and electrical profiles (c) EFM image showing a
steep potential drop at the location where the hot spot is shown by the SThM image. (d)
EFM image after the constriction is destroyed by ESD, showing a step change in the
surface potential that spans the entire channel signifying an open circuit. Horizontal scale
bar is 5 μm and the vertical scale bar is 50 nm, 160 K, 4°and 10° phase shift for images
a, b, c, and d, respectively.
48

measured temperature profiles were smooth and diffuse within the channel,

where a
relatively large hot spot can exist because of non-uniformity in the local charge carrier
density due to the variation in the gate field along the channel. Figure 3.3 b, in shows a
confined hot spot that is irregular in shape and concentrated in a very localized area. In
addition, Figure 3.2 (d) shows the experimental thermal profile through the center of the
hot spot. A peak temperature rise of Tmax = 160 ± 40 K was determined, which is more
than one order of magnitude larger than for a defect-free graphene channel with a similar
power density dissipation.
55
Figure 3.3 (c) shows a sharp potential drop down the center
region of the channel with discontinuous steps in potential on either side of a constriction.
The steep gradient within the narrow strip of continuous graphene is coincident with and
geometrically similar to the imaged hot spot, indicating the relation between the two.
The SThM and EFM results suggest the presence of a defect tear in the
graphene, which creates the micro constriction in the channel. After the initial thermal and
EFM scans were completed, the graphene channel was electrically broken by a large
electrostatic discharge (ESD) current. Following this ESD, no current was observed upon
application of a voltage bias to the channel. Figure 3.3 (d) shows a potential discontinuity
spanning the entire channel, confirming breakage of the graphene. To better understand the
experimental results, coupled electro-thermal transport simulation of the device was carried
out using COMSOL Multiphysics and coupled electric and thermal transport equations.
Using basal-plane thermal conductivity and interface thermal conductance values of  =
600 W m
-1
K
-1
and of G = 9.0×10
7
W m
-2
K
-1
reported in the literature for supported
graphene,
38, 43
the simulation predicts a Tmax = 180 K, which is within the uncertainty of
the experimental results. This calculated profile is plotted with the experimental data in
49

Figure 3.2 (d). The agreement suggests that the measured temperature rise can be explained
with the literature  and G values.

Figure 3.4 (a) The calculated maximum temperature rise as a function of the basal-plane
thermal conductivity κ for different interfacial conductance G in increments of
1.5×10
7
W m
−2
K
−1
for a 300-nm-thick dielectric. The inset shows the maximum
temperature rise as a function of increasing G for different κ in the range between
100 W m−1 K−1 (top curve) and 2900 W m
−1
K
−1
in 400 W m
−1
K
−1
increments. (b)
Normalized maximum temperature rise calculated by the analytical model as a function of
the G/Gox ratio (main figure) and the z0 parameter (inset). The solid and dashed lines are
for graphene devices made on a 10- and 300-nm-thick SiO2 dielectric on a high thermal
conductivity substrate, respectively. The graphene basal-plane thermal conductivity is kept
as 600 W m
−1
K
−1
for the main figure, and ranges between 300 and 3000 W m
−1
K
−1
for the
inset, where G is taken as 4 × 107 W m
−2
K
−1
.

An extended numerical study of the effect of thermal conductivity and interfacial
thermal conductance was performed. The calculated maximum temperature rise is plotted
as a function of  in Figure 3.4a for increasing values of interfacial thermal conductance.
Similarly, the inset shows the predicted maximum temperature rise as a function of G for
increasing values of . Several important conclusions can be drawn from Figure 3.4. The
maximum hot spot temperature is very sensitive to the thermal interface conductance when
G is low, and insensitive when the conductance is high. For any given   the gradient
𝜕 𝑇 𝑚𝑎𝑥 𝜕𝐺
|
𝜅 is large for 𝐺 ≤ 4×10
7
W m
-2
K
-1
and drops thereafter. For example, at  = 600
50

W m
-1
K
-1
, increasing G from 1×10
7
to 2×10
7
W m
-2
K
-1
reduces the hot spot
temperature by 40 𝐾 , whereas increasing G from 9×10
7
to 10×10
7
W m
-2
K
-1
only
produces a 0.8 K reduction in maximum temperature. This behavior can be seen more
clearly in the inset of Figure 3.4, where the effect of increasing G on Tmax quickly
saturates, regardless of the graphene thermal conductivity.
In contrast to the interfacial conductance, reductions in Tmax do not saturate
appreciably with increasing thermal conductivities. Within the range 300 <  < 1200 W
m
-1
K
-1
, a maximum drop of ∆𝑇 𝑚𝑎𝑥
from 565 K to 525 K can be attained for an interface
conductance of 1×10
7
W m
-2
K
-1
. Interestingly, increasing or decreasing G has relatively
little effect on these results. For example, for the same range of , but with an order of
magnitude larger interfacial conductance, a 20 K drop in ∆𝑇 𝑚𝑎 𝑥 is still observed.
3.3 Conclusion
These SThM and EFM measurement results have revealed localized hot
spots around a defect introduced in the transfer process of a CVD grown graphene channel
onto a SiO2/Si substrate. The numerical electro-thermal model is able to explain the
measurement results based on reported thermal conductivity and thermal interface
conductance values of supported graphene. The analytical model further clarifies that
increasing the thermal interface conductance G from the level of 4× 10
7
W m
-2
K
-1
, as
measured for non-functionalized graphene, is effective in reducing the hot spot temperature
for devices made with a sub-10 nm gate dielectric on a high thermal conductivity substrate.
However, when the cross-plane thermal conductance Gox of the gate dielectric is not much
higher than G, as is the case for devices made with a relatively thick gate dielectric or on a
51

low-thermal conductivity polymeric substrate, increasing G via surface functionalization
of graphene is ineffective. Furthermore, such functionalization can be counterproductive
if the basal-plane thermal conductivity is reduced as a consequence of the functionalization
process. In comparison, for a graphene device made on a 300 nm SiO2 dielectric layer,
increasing the graphene basal plane thermal conductivity from 300 W m
-1
K
-1
toward 3000
W m
-1
K
-1
can considerably increase the heat spreading length l compared to a micron-
scale localized heat generation spot size, r0, around a defect. This acts to increase the area
for vertical heat transfer through the gate dielectric thereby reducing the peak temperature.
This mechanism is effective even when lateral heat spreading from the hot spot to the metal
electrodes is inefficient, i.e. when the lateral size of the graphene channel is much larger
than l. However, the effect of increasing thermal conductivity becomes ineffective when l
becomes considerably smaller than r0, such as in a device made with a sub-10 nm gate
dielectric on a high-thermal conductivity substrate. These results suggest that the hot spot
52

temperature is sensitive to varying G and  when the G/Gox ratio and the r0/l ratio are
below about 5, respectively.
In the second work, Choi et al. demonstrated that the hot spot temperature in
graphene can be effectively reduced by using h-BN underneath the graphene as shown in
Figure 3.5.
54

(a)

(b)

(c)

(d)

(e)

(f)

Figure 3.5. Figure 3.5 Thermal image obtained using SThM for (a) graphene/SiO2,
graphene/hBN/SiO2, graphene/WillowGlass, graphene/hBN/WillowGlass substrates
63

4 Plasmon Resonant Amplification of a Hot Electron-
Driven Photodiode
This chapter is similar to our publication in Nano Research, 1-5.
56
Lang Shen and I
contributed equally to this work.
4.1 Introduction
Plasmon resonance has been utilized in many applications, including biosensing,
57

surface enhanced Raman spectroscopy (SERS),
58
and photocatalysis,
59-60
through the effect
of local field enhancement. More recently, however, the idea of hot electrons excited in
metals has been utilized in photocatlaysis
61-65
, and solid state devices.
66-67
Brongersma’s
group reported hot-electron photodetection with a plasmonic nanostripe antenna.
66
In this
work, they used a metal/oxide/metal stack, and photocurrent was only generated when the
incident photon energy was larger than the oxide barrier energy. In 2015, Halas’ group
reported similar measurements, which compared the polarization dependence of plasmon-
resonant devices with Ohmic and Schottky contacts with defect-rich TiO2.
67
Several recent
theoretical studies have concluded that plasmon resonant excitations decay into hot
electrons in metal nanostructures,
27, 68-71
which presents the possiblity of engineering useful
devices and structures utilizing this effects despite the extremely short lifetimes of hot
electrons in metals (~10fsec).
72-73

Plasmonic grating structures provide a useful/unique platform for studying
plasmon resonant phenomena. These nanostructures can be excited plasmon-resonantly or
non-resonantly (i.e., bulk metal absorption) by simply varying the polarization of the
64

incident light. These nanostructures enable us to distinguish between plasmon-resonant
excitations (p-polarization) and non-resonant bulk metal absorption (s-polarization), while
maintaining all other variables in the experiment constant (i.e., sample morphology, photon
energy, etc.). In the work presented here, we study the amplification of a hot electron-
driven photodetector using a plasmon resonant grating. The angle dependence of the
65

photocurrent is correlated to the photoreflection in order to verify the conditions for
resonantly exciting the plasmon mode. Electromagnetic simulations are used to further
verify the nature of this amplification.
Figure 1
Au
-5.1 eV
-4.8 eV
Al
2
O
3
-3.3 eV
1.96 eV
e-
Gr
Au
Si
(a)
(b)
(c)
Au grating
Graphene
Al
2
O
3
Figure 4.1 (a) Cross-sectional scanning electron microscope (SEM) image
and (b) schematic diagram of the plasmon resonant grating structure. (c)
Energy band diagram illustrating the mechanism of hot electron injection.
66

4.2 Methods
In this work, metal gratings are fabricated by first etching a silicon wafer
using reactive ion etching. This creates a corrugated surface with a pitch of 500nm. Then,
a 50nm film of gold is deposited on top of this structure. An SEM cross-sectional image of
one of these gratings is shown in Figure 4.1a. Next, a 5nm film of Al2O3 is deposited using
atomic layer deposition (ALD). Monolayer graphene is grown by chemical vapor
deposition (CVD) on copper foil at 1000
o
C in methane gas. After growth, the copper foil
is spin-coated with PMMA-A6 at 2000 rpm for 45s and then baked at 150℃ for 5 minutes.
The copper foil is then etched away in copper etchant and the graphene with PMMA is
“scooped” out and rinsed in 10% HCl and DI water. The monolayer graphene is then
transferred to the target substrate with the same scooping method and then baked at 120℃
for 5 minutes to improve adhesion. The PMMA layer is then removed with a 5-minute
acetone dip.
74
Lastly, the sample is mounted on a rotational stage and illuminated with
columnated 633nm wavelength light. A chopper wheel is used to modulate the light at
200Hz, and the AC photocurrent is measured using a lock-in amplifier.
4.3 Results and Discussion
Figure 4.2a shows the AC photocurrent plotted as a function of the incident angle
for light polarized both parallel and perpendicular to the lines on the grating. Here, we see
two peaks appearing at ±10
o
from normal incidence when the light is polarized parallel to
the plane of incidence (p-polarization) to the grating, but a constant (angle independent)
photocurrent when the light is polarized perpendicular to this plane (s-polarization) to the
grating. This data shows that the hot electrons are, in fact, amplified by a factor 4.6X in
67

metals using a plasmon resonant nanostructure. Figure 4.2b shows the reflectance plotted
as a function of the incident angle, exhibiting sharp dips at ±10
o
from normal. This is a
clear signature of the plasmon resonance, which is achieved when there is wavevector
matching between the incident light and the plasmon resonant modes in the grating.

Figure 2
-20 -10 0 10 20
0
20
40
60
80
100

Reflection (%)
Incident Angle (Degrees)
s polarization
p polarization
-20 -10 0 10 20
10
20
30
40
50
60
70
80
V
AC
(uV)
Incident Angle (Degrees)
s polarization
p polarization
(a)
(b)
(c)
Rotational stage
Chopper LASER
λ/2 plate
Figure 4.2 (a) Photocurrent and (b) photoreflectance plotted as a function incident
angle for light polarized parallel and perpendicular to the grating structure. (c)
Schematic diagram of the experimental measurement configuration
68

In order to further understand the amplification, we are seeing in these
plasmon resonant devices, we performed electromagnetic simulations using the finite-
difference time-domain (FDTD) method. Figure 4.3a shows the calculated reflectance
plotted as a function of the incident angle for both s- and p-polarized light. As in our
experimental measurements, the simulated data exhibits sharp dips at ±10
o
from normal
for p-polarized light and nearly constant reflection for s-polarized light. Figures 4.3b and
5.3c show electric field intensity distribution in the cross section of these gratings when
illuminated at normal incidence. Here, we see relatively low electric field intensities
corresponding to the case where plasmonic modes are being coupled to. At 10
o
incident
angel, however, we can clearly see the plasmon resonant mode excited by p-polarized light,
which produces an electric field enhancement of approximately 10X at the surface of the
metal. With s-polarized light, however, the field profile looks almost the same as the 0
o

incident light.

Figure 4.3(a) Finite difference time domain (FDTD) simulation of the
photoreflectance as a function of the incident angle for s- and p-polarized light. (b,c)
Cross-sectional electric field intensity profiles for illumination at normal and (d,e)
10
o
incidence.
69

4.4 Conclusion
In conclusion, we observe plasmon resonant amplification of hot electrons in a
Au/Al2O3/graphene photodetector. Here, optically-excited hot electrons jump over the
oxide barrier, thus, producing a photocurrent. In this device configuration, the bottom gold
electrode consists of a plasmon resonant grating, in order to increase the photodetection
efficiency. Using 633nm wavelength light, we observe clear peaks in the photocurrent at
an angle of ±10
o
from normal for light polarized perpendicular to the gating, while no
modulation for the photocurrent is observed with light polarized parallel to the grating.
This data shows an amplification factor of 4.6X for hot carrier injection when using a
plasmon resonant grating structure. We also see sharp dips in the photoreflectance at these
same angles, which corresponds to good wavevector matching between the plasmon mode
in the grating and the incident light. Electromagnetic (FDTD) simulations predict the same
reflectance profiles observed experimentally, and show the clear excitation of a plasmon
resonant mode when irradiated under resonant conditions (i.e., p-polarization at ±10
o
).

69

5 Future Direction
5.1 Cross Plane Thermal Conductivity and Seebeck
Coefficient of TMDCs using GERS
5.1.1 Introduction
Raman spectroscopy is a powerful method of material characterization. The low
scattering cross-section of Raman (10
-30
cm
2
molecule
-1
), the weak intensity of Raman
signals results in low sensitivity.
75
Various techniques have been developed for
enhancement for the weak Raman signals.
76-80
These techniques mainly employed are
Resonant Raman Scattering (RRS) or Surface enhanced Raman scattering (SERS).
78, 81-82

There have been two widely accepted mechanisms for Raman signal enhancement, which
are Electromagnetic Mechanism (EM) or Chemical Mechanism (CM).
75, 82
In SERS the
dominant mechanism is the EM, which can enhance the signal by 10
10
times.
83-84
The
enhancement mechanism in EM depends on the electromagnetic frequency resonance
82-83
.
For CM enhancement, the enhancement mechanism depends on the charge transfer process
between the molecule and the substrate.
83
GERS is dominated by chemical mechanism
with enhancement varying from 10 to 100.
75, 83, 85
In this work, we propose using GERS
molecules as temperature probes for characterizing the cross plane thermal conductivity
and Seebeck coefficients of MoS2

70

Figure 5.1 (a) Illustration of GERS thermoelectric measurement structure with ZnPC as
top GERS molecule and CuPC as bottom GERS molecule (b) Proposed GERS Geometry
with Crystal Violet and R6G as GERS temperature calibration molecules

Figure 5.1 illustrates rendering of two proposed structures to perform
thermoelectric characterizations of TMDCs using GERS molecule.
5.1.2 Preliminary Results and Future Work

MX
2
ZnPC on Graphene
CuPC on Graphene
(a) (b)
1400 1500 1600 1700 1800
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1452
1530

Count (a.u.)
Raman Shift (cm
-1
)
300K
320K
340K
360K
380K
G-band
1520 1530 1540 1550
0.0
0.2
0.4
0.6
0.8
1.0
1.2

Count (a.u.)
Raman Shift (cm
-1
)
300K
320K
340K
360K
380K
h-BN
CuPc
633 nm
Graphene
300 320 340 360 380
1530.4
1530.8
1531.2
1531.6
1532.0
1532.4

Cycle 1
Cycle 2
Cycle 3
CuPc Peak (cm
-1
)
Temperature (K)
-0.022 cm
-1
/K
(a) (b)
(c)
(d)
Figure 5.2 (a) CupC in graphene illuminated by 633 nm laser. (b) Temperature
calibration of Raman Peak of CuPC downshift with increasing temperature (c)
Raman spectra of CuPC on graphene (d) Peak downshift with increasing
temperature.
71

Figure 5.2 illustrates the temperature response of CuPC on graphene with
increasing temperature. The downshift is quite linear in the temperature regime of 300 K
to 400K. Moreover, the FWHM of the peaks from CuPC is one third of the FWHM of the
2D peak of graphene. Therefore, the sharper peaks of these GERS molecule can lead to
higher resolution in establishing temperature gradient in the vertical direction across the
TMDCs.

Figure 5.3 (a) Optical Image of the final device. The materials in this case is MoSe2 (b) &
(c) The temperature calibrations of the downshift of CuPC (on bottom graphene) and ZnPC
(on top graphene) GERS spectra (d) Raman downshift of MoSe2 downshift.

Figure 5.3 shows one of the completed devices where GERS measurement was
started. However, the ITO heater burnt during the electrical calibration. Therefore, the
complete thermoelectric measurement could not be completed at the time. Therefore, for
Graphene
ITO
300 320 340 360 380 400
1506.4
1506.8
1507.2
1507.6

Raman Shift (cm
-1
)
Temperature (K)
Slope = -0.013 cm
-1
K
-1

ZnPC
300 320 340 360 380 400
1525.2
1525.6
1526.0
1526.4
1526.8
1527.2
Slope = -0.013 cm
-1
K
-1

Raman Shift (cm
-1
)
Temperature (K)
CuPC
300 320 340 360 380 400
240.9
241.0
241.1
241.2
241.3
241.4
Slope = -0.004 cm
-1
K
-1

Raman Shift (cm
-1
)
Temperature (K)
MoSe
2
(a)
(c)
(b)
(d)
72

the future work to demonstrate thermoelectric characterization like Seebeck coefficient
measurements, a few more steps should be taken. This calibration should be followed by
the measurement of Raman downshift with increasing heater voltage. After the thermal
calibration, the Seebeck voltage induced in graphene/MX2/graphene can be measured
using either DC method or AC lock in method as done in chapter 2.
5.2 Enhancing Thermoelectric Properties of MoS2

using
Remote Oxygen Plasma Treatment
5.2.1 Introduction
Monolayer moly disulfide (MoS2) shows some amazing properties like a
large intrinsic band gap (1.8eV), high in plane mobility and amazing mechanical
properties.
86-87
This project was continuation of work previously carried out in this lab.
Dhall et al. from this lab demonstrated two important effects of remote oxygen plasma
treatment on thin MoS2 flakes. In their first work, they demonstrated enhancement in
photoluminescence by up to 20 times and blue shift for a few layer MoS 2 flakes due to
interlayer decoupling induced by oxygen plasma treatment. In the follow up work, Dhall
et al. showed an average threshold voltage shift of 20 V for a back gated MoS 2 based
transistors using remote oxygen plasma treatment. The photoluminescence enhancement,
73

as shown in Figure 5.4, was observed for 32 out of 35 flakes and the threshold voltage shift
was observed for 7 out of 8 samples.
88-89

At this point, I like to take detour to get insight into Dhall’s work on threshold
voltage shift. Lattice point defects and interstitials are trap centers for electrons, holes and
excitons in semiconductors and affect the transport and optical properties of the materials.
90

While for the most conventional 3D semiconductors defects database are quite broad and
the physics of defects is well understood, there is still a lot of work to be done to
comprehensively understand the science of defects for 2D semiconductors.
Tongay et al. induced Sulfur vacancies in their monolayer samples by irradiating
their exfoliated layers with 3MeV α- particles.
90
The introduction of sulfur vacancies
simultaneously enhanced the direct band gap peak XB at around 1.8 eV and the exciton
peak (X0) as shown in the Figure 5.5. However, the effect on Raman peak is negligible.
Many exfoliated MoS2 samples suffer from unintentional doping induced by sulfur
1.4 1.6 1.8 2.0
0
4000
8000
12000
Photoluminsence Count (a.u.)
Energy (eV)
Before
After
24 X
(a) (b)
Figure 5.4. (a) PL enhancement observed in sample after remote oxygen plasma treatment
Optical Image (c) Illustration of the mechanism underlying the enhancement process
74

vacancies. This sulfur vacancies causes n-type exfoliated MoS2 based FET to have a huge
negative threshold voltage.
5.2.2 Methods
First of all a suitable MoS2 flake is transferred to Si/SiO2 (300 nm oxide) substrate
using viscoelastic dry transfer as shown in Figure 2.2 earlier.
49
The technique to select the
right flake is based on a contrast method, which works quite well for MoS2

flakes. After
the flakes has been transferred to a PDMS stamp, PDMS stamp is placed under the
microscope and the right flake is selected based on the contrast of the flakes with SiO 2 as
the background. The thinner flakes and thicker flakes have characteristic hue, which can
give an idea of the relative thickness of the flake being selected. While this might not give
us a perfectly accurate thickness of flake, it will give us the flake in our desired range of
thickness. After transferring the flake to the substrate, standard Electron Beam Lithography
followed by Electron Beam Evaporation of 5nm of Ti and 50 nm of Au are used to make
heaters, Resistance Temperature Detectors (RTDs) and source and drain electrodes as
Figure 5.5 Illustration of effect of α- particles irradiation in (a) ratio of
Sulpher to Molybdenum atom (b) Raman spectra (c) Photoluminescence of
MoS2.
4

75

shown in the Figure 6.6 The back-gate voltage to the device is provided through the
underlying silicon.
The plasma used for the cleaning process in Dhall’s work was remotely generated
using ambient air with 20W of RF power, and the sample is placed in a chamber (XEI
Scientific Evactron) at 200mTorr, about 10cm upstream from the plasma source, for 2
minutes. The process was performed in XEI Scientific’s facility in Redwood city
California. Currently, we have purchased the instrument and it is in CEMMA at University
of Southern California. Therefore, it was imperative I optimize my recipe for this new
instrument. I observed that while the same parameter of 200mTorr, about 10cm upstream
from the plasma source, for 2 minutes gave a threshold voltage downshift and improved
mobility for Dhall et al’s devices, it didn’t produce the same effects on mine.
(a)
(b)
(c)
(d)
Figure 5.6 (a), (b) & (c) 3D schematic, side view and top view of the device
(d) Optical image of the device
76

The thermoelectric measurement was performed nevertheless before the
oxygen plasma treatment. Before the thermoelectric measurement, a basic electrical
measurement was performed to characterize the device as shown in Figure 5.7.

5.2.3 Preliminary Results and Future Work
After doing the basic FET characterization, we measured the Seebeck Coefficient
of the MoS2 flakes. For our first sample, we used the DC thermoelectric measurement. The
-40 -20 0 20 40
0
100
200
300

Drain Current (nA)
Gate Voltage (V)
V
DS
=1V
-40 -20 0 20 40
1E-10
1E-9
1E-8
1E-7

Drain Current (A)
Gate Voltage (V)
V
DS
=1V
V
g
=-40V
V
g
=-20V
V
g
=0V
V
g
=20V
V
g
=40V
0.0 0.5 1.0 1.5 2.0
0
400
800
1200
1600

Current (nA)
Drain-Source Voltage (V)
(b) (a)
(c)
Figure 5.7: (a) Ids vs Vg for Vds of 1V (b) Log plot of Ids vs Vg for Vds of 1V (c) Ids vs Vds
at various gate voltages
77

heater voltage was swept in both positive and negative direction as shown in the Figure
5.10b. For thermal Voltage we would expect a parabolic behavior as demonstrated in the
Figure 5.10b. The source and the drain electrodes are measured using a Keithley 2000
Digital Multimeter with the input impedance of 20 G . To reduce the overload effect that
arises when the channel resistance is higher than the input impedance of the lockin
amplifier a SRS 560 voltage pre-amplifier (100 M  input impedance) is also incorporated
V
0
sin (2 πft)
3 4
5 6
A
B
A-B mode
V
A-B
Figure 5.8 Configuration of RTD thermometry using a four-probe technique.
(a) (b)
Figure 5.9 (a) Seebeck Coefficient as a function of applied back gate voltage for various
Temperature for monolayer MoS2
1
(b) Seebeck Coefficient as a function of difference
between applied back gate voltage and threshold voltage for various layered of MoS2
samples.
4

78

into the measurement setup. The channel resistance is in the range of a few G  when the
FET is in the off state. For some samples, the channel resistance can be higher or even
comparable to voltmeter’s input impedance at various gate voltages. The two RTD gold
electrodes have been separately calibrated using 4 probe techniques using lockin amplifier.
The basis of measurement technique was burrowed from Kayyalha et al.’s work.
91
After
the complete Thermoelectric measurement, a remote oxygen plasma treatment was
performed on the sample at a chamber pressure of 200 mTorr and RF power of 20 Watts
for 2 minutes. However, we observed that the flake became insulating and lost all its FET
characteristic after the treatment. Therefore, the thermoelectric measurement could not be
performed after plasma treatment.
0.0 0.5 1.0 1.5 2.0
0.0
0.2
0.4
0.6
0.8

Temperature (K)
Heater Voltage (V)
RTD 1
RTD 2
0.0 0.1 0.2 0.3 0.4
-60
-40
-20
0

V
g
=30V
V
g
=20V
V
g
=10V
Voltage ( V)
Temperature Difference (K)
-2 -1 0 1 2
-60
-40
-20
0

Vg=30V
Vg=20V
Vg=10V
Voltage ( V)
Heater Voltage (V)
(a)
(c)
(b)
Figure 5.10 (a) Temperature of RTDs as a function of Heater Voltage (b)
Thermal Voltage as a function of Heater Voltage (c) Thermal Voltage as a
function of Temperature Difference

79

From my initial analysis I estimated the in plane Seebeck coefficient of MoS2 flake
as approximately -155 V/K for gate voltage of 30V and approximately -165 V/K for
gate voltage of 20V and 10 V respectively, as shown in Figure 5.10. The values are
comparable in terms of magnitude with those reported in the literature for monolayer MoS2
as shown in the Figure 5.9.
91-92
The impedance of the channel is comparable to the input
impedance of the voltmeter at 0 gate voltage. Therefore, the measurement here cannot be
reliable. After a few flakes became more insulating after O-plasma treatment with Dhall’s
parameter, I performed a series of experiment with 2 terminal MoS2 based devices to find
the right recipe that induces threshold voltage shift. I found that the remote O-plasma
treatment with parameters of 200 mTorr, 40 second and 15 W RF power in the XEI
Evacitron induces the threshold voltage shift as shown in the Figure 5.11 however it
reduces PL enhancement as shown in Figure 5.12.

-40 -20 0 20 40
1E-11
1E-10
1E-9
1E-8
1E-7
1E-6
V
T
= 12V
Drain Current (A)
Gate Voltage (V)
Before (Forward)
Before(Reverse)
After (Forward)
After (Reverse)
V
d
=0.1V
-40 -20 0 20 40
1E-11
1E-10
1E-9
1E-8
1E-7

Drain Current (A)
Gate Voltage (V)
Before (Forward)
Before (Reverse)
After (Forward)
After (Reverse)
V
T
= 30V
V
d
=0.1V
(a) (b)
Figure 5.11(a) and (b) Threshold voltage shift observed in two different remote oxygen
plasma treated samples
80

This average positive threshold voltage shift of almost 20 V can be attributed to
the curing of unintentional doping induced by sulfur vacancies. Tongay et al.’s work
showed that inducing doping enhances and broadens the direct peak at 1.8 eV. Their optical
measurements were performed at 77K and for a monolayer MoS2 sample. I also performed
before and after optical measurements for my multilayer MoS2 samples. I observe that the
Raman spectra is almost the same while the effect on photoluminescence is opposite to that
of observed Tongay et al. Therefore, I conservatively assume that while their irradiation
treatment induced defects, our remote oxygen plasma treatment cured some of the
unintentional defects and significantly reduced the unintentional doping induced by sulfur
vacancies that I believe comes from the intrinsic nature of the exfoliation process.
In the future, the thermoelectric transport measurement should be performed before
and after the treatment for the same device. I expect there will be enhancement of Seebeck
coefficient in the in-plane direction as the unintentional doping is cured in the MoS 2

as
discussed earlier.
300 350 400 450 500
0
2000
4000
Count (a.u)
Raman Shift (cm
-1
)
Before
After
1.4 1.6 1.8 2.0
0
500
1000
1500
Count (a.u.)
Energy (eV)
Before
After
(a) (b) (c)
Figure 5.12 (a) Optical Image of the sample (b) Photoluminescence spectra (c)
Raman spectra before and after remote oxygen plasma treatment.
81

6 Appendix A
6.1 Fundamental Physics of Plasmonics
The idea of waves propagating on a surface has been explore since 1900s. In 1907,
German physicists demonstrated surface waves between two media with varying
conductivity and dielectric constants.
93
Sommerfeld further studied the phenomenon of
plane waves travelling through a dielectric and ground by dividing the waves into dielectric
wave and surface wave.
94
Understanding surface waves on air and ground case is quite
simple. A technology to transmit surface wave was developed by Georg Goubau in 1950s
and it was called “G-line” .
95-96

∇×𝐻⃗⃗⃗
= 𝑗⃗+
𝜕 𝐷⃗⃗⃗
𝜕𝑡

(A.1)

∇×𝐸⃗⃗
= −
𝜕 𝐵⃗⃗
𝜕𝑡

(A.2)
∇.𝐵⃗⃗
= 0 (A.3)
∇.𝐷⃗⃗⃗
= 𝜌 (A.4)
𝐵⃗⃗
= 𝜇 𝐻⃗⃗⃗
(A.5)
𝐷⃗⃗⃗
= 𝜀 𝐸⃗⃗
(A.6)

Ohm’s law relates current and electric field by the following equation
82

𝑗⃗ = 𝜎 𝐸⃗⃗
(A.7)

If we assume a time varying (harmonic) Electric and Magnetic fields as given by
𝐻⃗⃗⃗
= 𝐻 0
⃗⃗⃗⃗⃗

𝑖 (𝑘𝑥 −𝜔𝑡 )
(A.8)
𝐸⃗⃗
= 𝐸 0
⃗⃗⃗⃗⃗

𝑖 (𝑘𝑥 −𝜔𝑡 )
(A.9)

We can express above equations as
∇×𝐻⃗⃗⃗
= 𝑗⃗−𝑖𝜔𝜀 𝐸⃗⃗
(A.10)
∇×𝐸⃗⃗
= 𝑖𝜔𝜇 𝐻⃗⃗⃗
(A.11)

If we multiply Equation (5.11) from both sides by ∇ and use Ohm’s law (5.7), we get
∇×∇×𝐸⃗⃗
= 𝑖𝜔𝜇 (∇×𝐻⃗⃗⃗
) (A.12)

∇×∇×
𝑗⃗
𝜎 = 𝑖𝜔𝜇 (𝑗⃗−𝑖𝜔𝜀 𝑗⃗
𝜎 ⃗⃗⃗
)
(A.13)

For good conductors, 𝜔 𝜀 ≪ 𝜎 . Therefore,
∇×∇×𝑗⃗ = 𝑖𝜔𝜇𝜎 𝑗⃗ (A.14)
We can express ∇×𝑗⃗ as
83

∇×𝑗⃗ = 𝑥⃗ ×
𝜕 𝑗⃗
𝜕𝑥
+𝑦⃗ ×
𝜕 𝑗⃗
𝜕𝑦
+𝑧⃗ ×
𝜕 𝑗⃗
𝜕𝑧
(A.15)

If we are in a regime, where frequency varies much faster in the x direction than in
any other direction, we can approximate ∇×𝑗⃗ as

∇×𝑗⃗ ≈ 𝑥⃗ ×
𝜕 𝑗⃗
𝜕𝑥

(A.16)

∇×𝑥⃗ ×
𝜕 𝑗⃗
𝜕𝑥

(A.17)

then simplifies to

∇×𝑥⃗ ×
𝜕 𝑗⃗
𝜕𝑥
≈ −
𝜕 2

𝑧 𝜕 𝑥 2

(A.18)

−
𝜕 2

𝑧 𝜕 𝑥 2
= 𝑖𝜔𝜇𝜎
𝑧
(A.19)

The above expression is a second order differential equation with solution

𝑧 ≈ 𝐴
√𝑖𝜔𝜇𝜎 𝑥
(A.20)

By using simple complex number manipulation, we can write √𝑖 as
√𝑖 =
1
√2
(1+𝑖 )
84

Hence,

𝑧 ≈ 𝐴
√
𝜔𝜇𝜎 √2
𝑥
𝑖 √
𝜔𝜇𝜎 √2
𝑥
(A.21)

We define δ =√
2
𝜔𝜇𝜎 as skin depth of a material.
Classically, we view metals as lattice of ions with see of free electrons. German
Physicist Paul Drude developed a model to explain electrical transport in conductors in
1900. Drude model doesn’t consider long range electron-ion electrons as well as ignores
any kind of interactions between electrons. The model is quite simple and provides good
insight for photon-matter interactions in metals.
This model can be viewed as a driven oscillator. A classical damped oscillator
driven externally by a sinusoidal force 𝐹 = 𝐹 0
cos (𝜔𝑡 +𝜑 𝑑 ) is modelled by the equation
97

𝑚 𝑑 2
𝑥 𝑑 𝑡 2
+𝑐 𝑑𝑥 𝑑𝑡 +𝑘𝑥 = 𝐹 0
cos (𝜔𝑡 +𝜑 𝑑 )
(A.22)
97

The solution to the above equations can be divided into a transient solution and
steady state solution and is given by
97

𝑥 (𝑡 )= 𝐴 ℎ

−𝛾𝑡
sin(𝜔 ′
𝑡 +𝜑 ℎ
)+𝐴𝑐𝑜𝑠 (𝜔𝑡 −𝜑 ) (A.23)

𝐴 =
𝐹 0
𝑚 ⁄
√[𝜔 0
2
−𝜔 2
]
2
+4𝛾 2
𝜔 2

(A.24)

𝛾 =
𝑐 2𝑚 (A.25)
85

𝜑 = 𝑇𝑎𝑛 −1
[
𝑐 𝜔 𝑘 −𝑚𝜔
2
]−𝜑 𝑑 (A.26)

𝜔 ′
= √𝜔 0
2
−𝛾 2

(A.27)

In Drude model, the sea of electrons are driven by

𝑚 𝑑 𝑣⃗
𝑑𝑡 +
𝑚 𝑣⃗
𝜏 = 𝐸 0
⃗⃗⃗⃗⃗

−𝑖𝜔𝑡
(A.28)

The solution for the following equation has the form
𝑣⃗ = 𝑣⃗
0

−𝑖𝜔𝑡 (A.29)

By substituting the solution to the equation given and using the Ohm’s law as well as
expression for current density

𝑗⃗ = 𝑛 𝑣⃗
0
= 𝜎 𝐸⃗⃗

(A.30)

We get following expressions for 𝑣⃗
0
and 𝜎

𝑣⃗
0
=
𝐸⃗⃗
0
𝑚 𝜏 −𝑖𝑚𝜔 ⁄

(A.31)

86

𝜎 =
𝑛
2
𝜏 𝑚 (1−𝑖𝜔𝜏 )
(A.32)

The expression for permittivity for a material can also be expressed as
𝜀 = 𝜀 𝑐𝑜𝑟𝑒 (𝜔 )+𝑖 𝜎 𝜔 = 𝜀 1
+𝑖 𝜀 2
= 𝜀 0
(𝑛̃ +𝑖 𝑘̃
)
2
(A.33)

Where 𝑛̃ is the refractive index of the material while 𝑘̃
is the extinction coefficient
of the material.
It is much easier to analyze the above expression for permittivity 𝜀 in two regimes
of low frequency 𝜔𝜏 ≪ 1 and high frequency 𝜔𝜏 1. In the low frequency regime, we
obtain

𝜀 = 𝜀 𝑐𝑜𝑟𝑒 (𝜔 )+
𝑖 𝑛
2
𝜏 𝑚𝜔

(A.34)

After performing algebraic manipulation, we can obtain the following expressions
for refractive index and extinction coefficients

𝑛̃ = 𝑘̃
= √
𝑛
2
𝜏 2𝜀 𝑜 𝑚𝜔

(A.35)

In a typical measurement, we are interested in Reflectivity measurement.
Reflectivity for a low frequency regime is given by
87

𝑅 =
(𝑛̃ −1)
2
+𝑘̃
2
(𝑛̃ +1)
2
+𝑘̃2
≈ 1−
4𝑛̃
𝑛̃
2
+𝑘̃2
≈ 1−
2
𝑛̃
(A.36)

For the high frequency limit,

𝜀 = 𝜀 𝑐𝑜𝑟𝑒 (𝜔 )−
𝑛
2
𝑚 𝜔 2

(A. 37)

And reflectivity is given by,
𝑅 =
(𝑛̃ −1)
2
(𝑛̃ +1)
2
(A.38)
The above analysis of reflection off metal surfaces helps us understands one of the
important concept in light-matter interactions. Every metallic and semiconducting
materials have a characteristic frequency associated with them. At this frequency the
material goes from reflecting and attenuating incident electromagnetic wave to letting the
electromagnetic waves pass through. At this frequency, the material becomes more
dielectric than metallic. At this frequency the real part of dielectric function vanishes too.
To understand the origin of plasma frequency mathematically, we can revisit the equation
𝜀 = 𝜀 𝑐𝑜𝑟𝑒 (𝜔 )+𝑖 𝜎 𝜔 = 𝜀 1
+𝑖 𝜀 2
(A.39)
If we use the Drude Model, the above equation can be re-expressed as

𝜀 = 𝜀 1
+𝑖 𝜀 2
= 𝜀 𝑐𝑜𝑟𝑒 (𝜔 )+
𝑖 𝜔
𝑛
2
𝜏 𝑚 (1−𝑖𝜔𝜏 )
×
(1+𝑖𝜔𝜏 )
(1+𝑖𝜔𝜏 )
(A.40)

88

And hence we can decouple,
𝜀 1
(𝜔 )= 𝜀 𝑐𝑜𝑟𝑒 (𝜔 )−
𝑛
2
𝜏 𝑚 (1+𝜔 2
𝜏 2
)
(A.41)

And
𝜀 2
(𝜔 )=
1
𝜔 𝑛
2
𝜏 𝑚 (1+𝜔 2
𝜏 2
)
(A.42)
Plasma frequency is the frequency at which, the real part of the permittivity
vanishes and is given by the following expression

𝜔 𝑝 = √
𝑛
2
𝑚 𝜀 𝑐𝑜𝑟𝑒
(A.43)

Plasma frequency for metals is usually in the UV frequency. However, Nano
structuring has been observed to shift the frequency closer to the visible range.
6.2 The concept of surface plasmon polariton
The derivation given here is based on the work by Zhang et al.
98
Let us assume that
a p-polarized wave is incident on a dielectric-metal interface as shown in the figure. The
momentum of the incident light is given by ħkd, where 𝑘 𝑑 =
2𝜋 𝑛 𝑑 𝜆 is the wavevector of the
electromagnetic wave going through a media of refractive index 𝑛 𝑑 . Snell’s law gives us
the following relation
𝑛 𝑑 𝑠𝑖𝑛 𝜃 1
= 𝑛 𝑚 𝑠𝑖𝑛 𝜃 2
(A.44)
89

In most cases at visible wavelength , dielectric materials tend to have higher
refractive index than the metals. Therefore, this also provides for a total internal reflection
phenomenon under the following condition.
𝜃 𝑐 = 𝑠𝑖𝑛 −1
(
𝑛 𝑚 𝑛 𝑑 ). (A.45)

Although the wave incident is totally reflected at the critical angle, there are
evanscening wave propagating into metal due to the oscillating charges at the metal. Let
us assume that the waves in the dielectric and metal is given by following expressions. We
assume that there is no s-polarization in our incident wave.
The E field and H field in the dielectric is given by the following expression.

𝐸⃗⃗
𝑑 = (
𝐸 𝑥𝑑
0
𝐸 𝑧𝑑
)
−𝑘 𝑧𝑑
𝑧
𝑖 (𝑘 𝑥 𝑥 −𝜔𝑡 )

𝐻⃗⃗⃗
𝑑 = (
0
𝐻 𝑦𝑑
0
)
−𝑘 𝑧𝑑
𝑧
𝑖 (𝑘 𝑥 𝑥 −𝜔𝑡 )

(A.46)

Similarly, E field and H-field in the metal is given by the following expression

𝐸⃗⃗
𝑚 = (
𝐸 𝑥𝑚
0
𝐸 𝑧𝑚
)
−𝑘 𝑧𝑚
𝑧
𝑖 (𝑘 𝑥 𝑥 −𝜔𝑡 )
(A.47)
90

𝐻⃗⃗⃗
𝑚 = (
0
𝐻 𝑦𝑚
0
)
−𝑘 𝑧𝑚
𝑧
𝑖 (𝑘 𝑥 𝑥 −𝜔𝑡 )

We can now utlize Maxwell equations boundary condtions at the interface.
For this case, Gauss’s law simply becomes
∇.𝐸⃗⃗
= 0
Hence, we have following expression
𝐸 𝑧𝑑
= 𝑖 𝑘 𝑥 𝑘 𝑧𝑑
𝐸 𝑥𝑑
(A.48)
𝐸 𝑧𝑚
= −𝑖 𝑘 𝑥 𝑘 𝑧𝑚
𝐸 𝑥𝑚
(A.49)

We can also utilize the other Maxwell’s equation along with 𝜔 = 𝑘𝑐
∇×𝐸⃗⃗
= −
𝜕 𝐵⃗⃗
𝜕𝑡

To get following relations
−𝑘 𝑧𝑑
𝐸 𝑥𝑑
−𝑖 𝑘 𝑥 𝐸 𝑧𝑑
= 𝑖𝑘 𝐻 𝑦𝑑
(A.50)
𝑘 𝑧𝑚
𝐸 𝑥𝑚
−𝑖 𝑘 𝑥 𝐸 𝑧𝑚
= 𝑖𝑘 𝐻 𝑦𝑚
(A.51)

We get the following expressions
𝜀 𝑑 𝑘 𝐸 𝑥𝑑
= 𝑖 𝑘 𝑧𝑑
𝐻 𝑦𝑑
(A.52)
91

𝜀 𝑚 𝑘 𝐸 𝑥𝑚
= −𝑖 𝑘 𝑧𝑚
𝐻 𝑦𝑚
(A.53)

Where
𝑘 𝑧𝑑
2
= 𝑘 𝑥 2
−𝜀 𝑑 𝑘 2
(A.54)
𝑘 𝑧𝑚
2
= 𝑘 𝑥 2
−𝜀 𝑚 𝑘 2
(A.55)

At the interface, the tangential component of both the electric fields and magnetic
fields at dielectric and metallic medium have to be continuous. Therefore, we have 𝐸 𝑥𝑑
=
𝐸 𝑥𝑚
and 𝐻 𝑦𝑑
= 𝐻 𝑦𝑚
.
We finally obtain the relation

𝑘 𝑧𝑑
𝑘 𝑧𝑚
= −
𝜀 𝑑 𝜀 𝑚

(A.56)
Hence, we finally come up with the relation.
𝑘 𝑥 = 𝑘 𝑠𝑝𝑝 = 𝑘 √
𝜀 𝑑 𝜀 𝑚 𝜀 𝑑 +𝜀 𝑚 (A.57)

The above expression implies that the momentum of the surface plasmon polariton
is larger than the momentum of the light in the free space. This difference in the momentum
must be matched coupling light and SPP at the interface under the following condition
92

𝜀 𝑑 +𝜀 𝑚 = 0 (A.58)

From Drude’s theory of free electrons, the dielectric constant of metal has the
following form

𝜀 𝑚 = 1−
𝜔 𝑝 2
𝜔 2

(A.59)

Moreover, we can express the surface plasmon polariton frequency as

𝜔 𝑠𝑝𝑝 =
𝜔 𝑝 √1+𝜀 𝑑
(A.60)

Coupling light and SPP is an interesting problem and various solutions can be used
to match the momentum difference between SPP and light.

93

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Abstract (if available)

Abstract In this work, I have studied thermoelectric transport across atomically abrupt interfaces. Thermoectric transport in heterostructures of thin film materials haven’t been studied thoroughly in literature because of lack of established tools and techniques for measurements. 2D materials, which are atomically thin provide an excellent platform to study thermoelectric transport across the interfaces. In this thesis I present a novel technique and device structure to study thermoelectric transport across Au/h-BN/graphene heterostructures. In addition to understanding mechanism of cross plane thermoelectric transport in heterostructures composed of 2D materials, I was also interested in understanding thermal transport in lateral direction when these materials are stacked on top of each other. More specifically, I was interested in graphene on BN as BN acts as a perfect substrate for graphene as most of the properties of graphene that gets subdued in silicon based substrate tend to recover when placed in graphene because of similar lattice structure of these two materials. In addition to thermoelectric transport, I also present my work in plasmonically enhanced photodiode based on graphene/insulator/graphene structure which can potentially be used for photocatalysis purposes. Lastly, I present my work in improving optoelectronic properties of GaAs nanowires using simple passivation technique—remote oxygen plasma treatment.

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Asset Metadata

Creator Poudel, Nirakar (author)

Core Title In plane a & cross-plane thermoelectric characterizations of van der Waals heterostructures

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publication Date 10/12/2018

Defense Date 06/22/2018

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag 2D materials,device physics,Energy,OAI-PMH Harvest,photocatalysis,plasmonics,solid state physics,surface passsivation,thermoelectricity,transistors

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Cronin, Stephen Burke (committee chair), Nakano, Aiichiro (committee member), Wang, Han (committee member)

Creator Email nirakar.poudel@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c89-88492

Unique identifier UC11675208

Identifier etd-PoudelNira-6823.pdf (filename),usctheses-c89-88492 (legacy record id)

Legacy Identifier etd-PoudelNira-6823.pdf

Dmrecord 88492

Document Type Dissertation

Format application/pdf (imt)

Rights Poudel, Nirakar

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA