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INTRINSIC AND INTERFACIAL THERMAL
TRANSPORT IN Stanene/Hexagonal Boron Nitride
HETEROSTRUCTURE
by
PRIOM DAS
0422102056
Submitted in partial fulfillment of the requirements for the degree of
Master of Science in Mechanical Engineering
Under the Supervision of
Dr. A.K.M. Monjur Morshed
Professor
Department of Mechanical Engineering
Bangladesh University of Engineering and Technology
Department of Mechanical Engineering
Bangladesh University of Engineering and Technology
Dhaka-1000, Bangladesh
September 2024
Candidate’s Declaration
I, do, hereby, certify that the work presented in this thesis, titled, “INTRINSIC
AND INTERFACIAL THERMAL TRANSPORT IN Stanene/Hexagonal Boron Nitride
HETEROSTRUCTURE”, is the outcome of the investigation and research carried out by me
under the supervision of Dr. A.K.M. Monjur Morshed, Professor, Department of Mechanical
Engineering, BUET.
I also declare that neither this thesis nor any part thereof has been submitted anywhere else for
the award of any degree, diploma or other qualifications.
PRIOM DAS
0422102056
i
The thesis titled “INTRINSIC AND INTERFACIAL THERMAL TRANSPORT IN
Stanene/Hexagonal Boron Nitride HETEROSTRUCTURE”, submitted by PRIOM DAS,
Student ID: 0422102056, Session: APRIL-2022, has been accepted as satisfactory in partial
fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering
on September 29, 2024.
Board of Examiners
1.
Dr. A.K.M. Monjur Morshed Chairman
Professor (Supervisor)
ME, BUET, Dhaka
2.
Dr. Md. Afsar Ali Member
Professor and Head (Ex-Officio)
ME, BUET, Dhaka
3.
Dr. Mohammad Nasim Hasan Member
Professor
ME, BUET, Dhaka
4.
Dr. Kazi Arafat Rahman Member
Associate Professor
ME, BUET, Dhaka
5.
Dr. Md. Shakhawat Hossain Firoz Member
Professor (External)
Chem, BUET, Dhaka
ii
Acknowledgement
I begin by offering my deepest gratitude to the Almighty, whose endless grace and guidance
have sustained me throughout this journey, allowing me to complete this thesis.
I extend my deepest appreciation to my thesis supervisor, Dr. A.K.M. Monjur Morshed, Professor,
Department of Mechanical Engineering, BUET. His insightful guidance, unwavering support,
and thoughtful mentorship have played a pivotal role in the success of this work. His ability
to challenge and inspire me, while also offering constant encouragement, has helped me grow
not only as a researcher but also as a critical thinker. I am truly grateful for his patience, for
always being available to discuss ideas, and for his belief in my potential. His dedication to
fostering my academic and personal development is something I will carry with me throughout
my career. It has been an honor to learn from him, and I am deeply thankful for his influence on
my journey into the field of nanoscale heat transfer. I would also like to extend my gratitude to the
Department of Mechanical Engineering, BUET, for providing me with a supportive environment
and access to the resources necessary for this work. My appreciation also goes to the IICT for
their support in granting access to computational facilities crucial to this research.
Finally, I want to express my heartfelt appreciation to my family—my parents, my sister,
and my late grandmother—for their unwavering love and support. Their constant encouragement
and belief in me have been the cornerstone of my strength, helping me through every step of this
journey.
Dhaka
September 29, 2024
PRIOM DAS
0422102056
iii
Contents
Candidate’s Declaration i
Board of Examiners ii
Acknowledgement iii
List of Figures vii
List of Tables xiii
List of Symbols xiv
Abstract xvi
1 Introduction 1
1.1 Theoretical and Computational Background . . . . . . . . . . . . . . . . . . . 1
1.1.1 Theoretical Background of Thermal Transport Properties . . . . . . . . 1
1.2 Computational Analysis of Transport Properties at the Nanoscale . . . . . . . . 5
1.2.1 MolecularDynamics ........................... 5
1.2.2 LatticeDynamics............................. 7
1.3 Experimentaltechniques ............................. 8
1.3.1 MicrobridgeMethod ........................... 8
1.3.2 3ωMethod ................................ 9
1.3.3 Time-Domain Thermoreflectance (TDTR) . . . . . . . . . . . . . . . . 10
1.3.4 OptothermalRaman............................ 12
1.4 Overviewof2Dmaterials............................. 12
1.5 van der Waals heterostructures . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.5.1 Types of van der Waals heterostructures . . . . . . . . . . . . . . . . . 15
1.5.2 Applications................................ 16
Layout of the Thesis 18
2 Literature Review 19
2.1 Motivation behind the study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
iv
2.2 Objectives with specific aims . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3 PossibleOutcomes ................................ 29
3 Simulation Methodology 30
3.1 Advantages of Molecular dynamics . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2 Limitations of Molecular dynamics . . . . . . . . . . . . . . . . . . . . . . . 32
3.3 Equlibrium Molecular Dynamics and Non-Equilibrium Molecular Dynamics . . 33
3.4 Steps in molecular dynamics simulation . . . . . . . . . . . . . . . . . . . . . 34
3.5 Potentials ..................................... 34
3.6 Cutoffradius ................................... 40
3.7 Time integration algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.7.1 VerletAlgorithm ............................. 41
3.7.2 Leap-Frog Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.7.3 Velocity verlet algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.7.4 Predictor corrector method . . . . . . . . . . . . . . . . . . . . . . . . 42
3.8 Boundaryconditions ............................... 43
3.8.1 Periodic Boundary condition . . . . . . . . . . . . . . . . . . . . . . . 43
3.8.2 Fixed boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . 44
3.8.3 Mirror Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . . 45
3.9 Thermostatting and barostatting . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.10LangevinDynamics................................ 45
3.11Ensembles..................................... 45
3.11.1 NVEEnsemble .............................. 46
3.11.2 NVTEnsemble .............................. 46
3.11.3 NPTEnsemble .............................. 46
3.11.4 NPHEnsemble .............................. 46
3.11.5 µPTensemble............................... 46
3.12Ensembleaverage................................. 47
3.13Timeaverage ................................... 47
3.14Ergodichypothesis ................................ 48
3.15LAMMPS..................................... 48
3.16OVITO ...................................... 49
4 Modelling and Simulation Procedure 50
4.1
Range of Input Parameters for ITR and In-Plane Thermal Conductivity Calculations
51
4.2 Calculation of Interfacial thermal resistance . . . . . . . . . . . . . . . . . . . 52
4.2.1
Validation of Simulation Procedure for Calculation of Interfacial thermal
resistance ................................. 55
4.3 Calculation of In-Plane thermal conductivity . . . . . . . . . . . . . . . . . . . 56
v
4.3.1
Validation of Simulation Procedure for Calculation of In-Plane Thermal
Conductivity ............................... 60
5 Results and Discussion 63
5.1 Interfacial thermal resistance (ITR) . . . . . . . . . . . . . . . . . . . . . . . . 63
5.1.1 Effect of system size and heat flow direction . . . . . . . . . . . . . . . 68
5.1.2 Effect of Contact Pressure and Temperature . . . . . . . . . . . . . . . 70
5.1.3 Effect of Defect Concentration . . . . . . . . . . . . . . . . . . . . . . 75
5.1.4 Effect of In-plane Tensile strain . . . . . . . . . . . . . . . . . . . . . 78
5.2 In-Plane Phonon Thermal Conductivity . . . . . . . . . . . . . . . . . . . . . 82
5.2.1 Effect of System Size . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.2.2 Effect of Vacancy Defect . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.2.3 Effects of temperature on PTC of defective Sn/hBN heterostructure . . 94
5.2.4 Effects of Contact Pressure . . . . . . . . . . . . . . . . . . . . . . . . 97
5.2.5 Effects of In-Plane Tensile Strain . . . . . . . . . . . . . . . . . . . . . 97
6 Conclusions and Scope for Future Work 101
6.1 Summary of the present work . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.2 Scopeforfuturework............................... 102
6.3 FutureScopeoftheWork............................. 102
References 104
vi
List of Figures
1.1
Comparison of transmission using two methods: the Acoustic Modeling Method
(AMM) assumes smooth reflections, while the Diffuse Modeling Method
(DMM) assumes random scattering of acoustic waves . . . . . . . . . . . . . 2
1.2
Figure illustrating the microbridge method, where an integrated heater and
thermometer measure the thermal properties of a sample using alternating current.
9
1.3
Figure illustrating the 3
ω
technique configuration, with the sample on a substrate
and integrated heater and thermometer. The heater is energized by alternating
current, measuring the third harmonic response. . . . . . . . . . . . . . . . . 10
1.4
Figure illustrating the TDTR method, where a pulsed laser heats the sample,
and the resulting temperature change is monitored to measure thermal properties
11
1.5
Figure illustrating the Optothermal Raman method, where laser-induced heating
is combined with Raman spectroscopy to analyze thermal properties and
materialbehavior. ................................ 12
1.6
Illustration of different two-dimensional materials, including graphene,
transition metal dichalcogenides (TMDs), hexagonal boron nitride (h-BN),
andstanene. ................................... 14
1.7 Two-dimensional layered materials and van der Waals heterostructures . . . . 16
1.8 Various applications of van der Waals heterostructures . . . . . . . . . . . . . 17
2.1
Figure illustrating the Top , Front and Isometric view of a monolayer Stanene
layer. “h” is the buckling height of Sn layer (.086 nm). . . . . . . . . . . . . . 20
2.2 Figure illustrating the Top , Front and Isometric view of a monolayer hBN layer. 20
2.3
Thermal conductivity of some typical single-layer 2D materials at room
temperature. ................................... 23
2.4
In-Plane thermal conductivity and Cross Plane thermal transport of Sn/hBN
bilayer....................................... 26
2.5 Thermal Rectification at the Sn/hBN bilayer interface. . . . . . . . . . . . . . 26
2.6
(a) A modern nanoelectronic device (transistor) made of different materials
of nanoscale dimension consists of various interfaces (b) metal- 2D material
interface (c) 2D-2D interface (d) 2D- dielectric material interface (e) metal-
dielectric material interface. . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
vii
3.1 A collection of system of N particles interacting in a box . . . . . . . . . . . 30
3.2 Scheme to Extract property from MD simulation . . . . . . . . . . . . . . . . 31
3.3 Different time and length scale used in simulation . . . . . . . . . . . . . . . 31
3.4 Steps involved in molecular dynamics simulation . . . . . . . . . . . . . . . 34
3.5 Particle’s interaction according to LJ model . . . . . . . . . . . . . . . . . . . 37
3.6 Force calculation according to Lj potential . . . . . . . . . . . . . . . . . . . 38
3.7 Periodic boundary condition (The central box is the Simulation domain) . . . 44
3.8 Fixed boundary condition (rigid wall) . . . . . . . . . . . . . . . . . . . . . . 44
4.1
The diagrammatic arrangement of a pristine Sn/hBN hetero-bilayer nanoribbon
(Green, orange, and blue color indicate Sn, B, and N atoms respectively)
in upper perspective, forward perspective and in isometric perspective. The
symbol ’L’ represents the heterostructure’s length, whereas the ‘W’ represents
the system’s width. ’h’ represents the buckling height of the Sn sheet ( 0.086
nm). Interlayer distance ’d’ between Sn and hBN layer is 0.4 nm . . . . . . . 51
4.2
A 3D view of the initial arrangement of the Sn/h-BN heterostructure is presented.
The uppermost layer corresponds to the h-BN nanosheet, while the lowermost
layer corresponds to the Stanene nanosheet. The red arrow indicates the path of
heat transport from the h-BN layer to the Sn layer. The interlayer spacing of
the van der Waals (vdW) heterostructure is approximately 0.4 nm. . . . . . . . 53
4.3
Variation of the total energy and temperature of the Sn and h-BN layer with
time. The steady-state energy and temperature profile of both the Sn and h-BN
layers demonstrate system equilibrium and stability. . . . . . . . . . . . . . . 53
4.4
Diagram of the Fast Pump Probe technique (FPP) used in MD simulation to
determine the ITR under different conditions. In the Pump section h-BN layer is
heated with a certain intensity heat flux and in the probe section the temperature
difference between the two layers as well as the energy profile of the h-BN
layerismonitored................................. 54
4.5
A comparison between the graphs from our simulation and the published
publication..................................... 56
4.6
Variation of the total energy of the system for both pristine and defective (2%
vacancy concentration) Sn/hBN heterostructure. The steady state energy profile
demonstrates system equilibrium and stability.) . . . . . . . . . . . . . . . . . 57
4.7
Heat flow direction in the NEMD simulation. The red zone represents the
”source” whereas the green portion represents the ”sink”. Heat transfers from a
heatsourcetoaheatsink.)............................ 58
4.8
Steady-state temperature profile of the Sn/hBN heterostructure (a)Pristine (b)
Point defect (2% vacancy concentration) (c) Bivacancy defect (2% vacancy
concentration) (d) Edge defect (2% vacancy concentration) . . . . . . . . . . 59
viii
4.9
Validation of Simulation Procedure. NEMD simulation results closely match
published studies by Tabarraei et al. [76] and Ahammed et al. [61] and Liu et
al.’s theoretical prediction [53] . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.10
Validation of Simulation Procedure. NEMD simulation results closely match
with Liu et al.’s theoretical prediction [53] . . . . . . . . . . . . . . . . . . . 61
5.1
Temporal evolution of Energy of h-BN layer (Right Y axis) and temperature
variations of h-BN and Sn layer (Left Y axis) following a 50 fs heat impulse
for (Pristine, unstrained,
χ= 1
,
T= 300
K,
L= 30
nm) condition. The green
solid line represents the curve obtained from fitting the energy data generated
bytheMDsimulation............................... 64
5.2 Variation of the total energy of h-BN layer with R∆T dt ............ 64
5.3
Variation of ITR between Sn/h-BN heterostructure with time. The blue squares
indicate instantaneous ITR, whereas the solid red line indicates the overall ITR. 65
5.4
PDOS profile of Sn and h-BN layers: (a) PDOS-X, (b) PDOS-Y, (c) PDOS-Z,
and (d) Overall. The inset shows a magnified view of the overlap area between
the Sn and h-BN PDOS profiles. The symbol ‘S’ indicates the overlap factor
between the PDOS profiles of the individual Sn and h-BN layers. . . . . . . . 67
5.5
Variation of the interfacial thermal resistance (ITR) of the Sn/h-BN
heterostructure with system size, varying between 10 nm to 150 nm (T =
300 K,
χ= 1
), is shown for both the h-BN
→
Sn and Sn
→
h-BN heat flow
directions. The system’s width is maintained at a constant value of 10 nm. . . 70
5.6
Interfacial thermal resistance (ITR) of the Sn/h-BN heterostructure at various
temperatures (100 K to 600 K) and coupling strength
χ= 0.5
to
2.5
. ITR
shows a monotonous decline trend with increasing χand temperature. . . . . . 71
5.7
Variation of Interfacial thermal resistance (ITR) of the Sn/h-BN heterostructure
at various coupling strength
χ= 0.5
to
2.5
at 300 K. ITR shows a monotonous
decline trend with increasing
χ
and there is no significant thermal rectification
effectobserved. ................................. 71
5.8
Variation of Interfacial thermal resistance (ITR) of the Sn/h-BN heterostructure
at various temperatures (100 K to 600 K) and coupling strength
χ= 1
. ITR
shows a monotonous decline trend with increasing temperature and there is no
significant thermal rectification effect observed. . . . . . . . . . . . . . . . . 72
5.9 PDOS profiles of Sn and h-BN layers and the overlap factor Sfor χ= 2.5. . . 73
5.10
Lateral and flexural PDOS profiles of the h-BN monolayer for
χ= 1
and
χ= 2.5
.
73
5.11
PDOS profiles and overlap factor
S
of Sn and h-BN layers at a temperature of
600K. ...................................... 74
ix
5.12
Variation of Interfacial Thermal Resistance (ITR) of Sn/h-BN heterostructure
with defect concentrations ranging from 0.25% to 2% for the h-BN
→
Sn heat
flow direction. The system size is 30 ×10 nm2.................. 76
5.13
Variation of Interfacial Thermal Resistance (ITR) of Sn/h-BN heterostructure
with defect concentrations ranging from 0.25% to 2% for both the h-BN
→
Sn
heat flow direction and Sn
→
h-BN heat flow direction . The system size
is
30 ×10 nm2
. A significant thermal rectification effect is observed.Heat
encounters much higher resistance when transferring from the h-BN layer to
theSnlayer.................................... 76
5.14
PDOS profile and overlap factor
S
of Sn and h-BN layers with a defect
concentrationof2.5%............................... 77
5.15
Lateral and flexural PDOS profile of pristine and 2.5% defective h-BN monolayer.
78
5.16
Variation of interfacial thermal resistance (ITR) of Sn/h-BN heterostructure
with various in-plane tensile strains ranging from 1% to 5% for the
h
-BN
→
Sn
heat flow direction. The system size is
30 ×10
nm
2
. ITR is higher for biaxial
tensile strain compared to uniaxial tensile strain applied in the
x
and
y
directions.
79
5.17
Variation of interfacial thermal resistance (ITR) of Sn/h-BN heterostructure
with various in-plane tensile strains ranging from 1% to 5% for the Sn to hBN
direction. The system size is
30 ×10
nm
2
. ITR is higher for biaxial tensile
strain compared to uniaxial tensile strain applied in the xand ydirections. . . 79
5.18
PDOS profile of biaxially strained (5%) and unstrained h-BN monolayers. With
the application of tensile strain, phonon softening and redshift behavior in the
PDOSprofileareobserved. ........................... 80
5.19
PDOS profile and overlap factor (S) for the h-BN and Sn layers when tensile
strain is applied in the biaxial direction. . . . . . . . . . . . . . . . . . . . . . 81
5.20
PDOS profile and overlap factor (S) for the h-BN and Sn layers when tensile
strain is applied in the X direction. . . . . . . . . . . . . . . . . . . . . . . . 81
5.21
PDOS profile and overlap factor (S) for the h-BN and Sn layers when tensile
strain is applied in the Y direction. . . . . . . . . . . . . . . . . . . . . . . . 82
5.22
Variation of phonon thermal conductivity for freestanding Sn, hBN, and Sn/hBN
heterostructure as a function of system size. In the ballistic regime, thermal
conductivity increases rapidly with system size, while the rate of increase slows
as the system transitions to the diffusive regime, reflecting the shift in phonon
transportmechanisms............................... 83
5.23
Variation of the inverse phonon thermal conductivity (PTC) of freestanding
stanene with inverse length, used to calculate length-independent thermal
conductivity by applying the Matthiessen’s rule. . . . . . . . . . . . . . . . . 84
x
5.24
Variation of the inverse phonon thermal conductivity (PTC) of freestanding
hBN as a function of inverse length, employed for determining the length-
independent thermal conductivity using Matthiessen’s rule. . . . . . . . . . . 85
5.25
Variation of the inverse phonon thermal conductivity (PTC) of the stanene/hBN
heterostructure with inverse length, enabling the calculation of length-
independent thermal conductivity through Matthiessen’s rule. . . . . . . . . . 85
5.26
Initial Structure of Sn/hBN heterostructure with Point defect (random removal
ofatomsfromlattice) .............................. 86
5.27 Edge defect (random removal of atoms from system boundary) . . . . . . . . 87
5.28
Bi vacancy defect (constitute of consecutive point vacancies). The black-dotted
portions in the figure represent areas from where atoms have been removed. . 87
5.29
Variation of PTC with defect percentage for a 30 nm x 10 nm Sn/hBN
heterostructure at 300 K with various types of vacancies present in both Sn and
hBNlayers.................................... 88
5.30
Variation of PTC with defect percentage for a 30 nm x 10 nm Sn/hBN
heterostructure at 300 K with various types of vacancies present solely in
theSnlayer ................................... 89
5.31
Variation of PTC with defect percentage for a 30 nm x 10 nm Sn/hBN
heterostructure at 300 K with various types of vacancies present exclusively in
thehBNlayer .................................. 89
5.32
PDOS Profile of pristine Sn and hBN nanosheet. The right Y axis is for hBN,
and the Left Y axis represents data of the Sn layer. . . . . . . . . . . . . . . . 90
5.33
PDOS profile of pristine, point vacancy, edge vacancy, and bivacancy-induced
hBN nanosheets. The vacancy concentration is kept at 2%. The PDOS of the
defective hBN nanosheets undergoes substantial peak suppression. The three
insets show the PDOS profile behavior in low, intermediate, and high-frequency
regions....................................... 91
5.34
Percentage reduction in PTC with increasing defect percentages compared to
a pristine 30 nm
×
10 nm hetero-bilayer structure, highlighting the impact of
various types of vacancies present in both Sn and hBN layers. . . . . . . . . . 93
5.35
Percentage reduction in PTC with increasing defect percentages compared to
a pristine 30 nm
×
10 nm hetero-bilayer structure, highlighting the impact of
various types of vacancies present in exclusively in Sn layer . . . . . . . . . . 93
5.36
Percentage reduction in PTC with increasing defect percentages compared to
a pristine 30 nm
×
10 nm hetero-bilayer structure, highlighting the impact of
various types of vacancies present in solely in hBN layer.In every scenario,
the point vacancy experiences the most significant reduction, followed by the
bivacancy and edge vacancy. . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
xi
5.37
The PTC of pristine and defective (1% defect concentration) 30 nm
×
10
nm Sn/hBN hetero-bilayer at different temperatures shows that the vacancy
is present on both Sn and hBN layers. The pristine Sn/hBN heterostructure
exhibits a faster rate of decrease in PTC compared to its defective counterparts. 95
5.38
Variation of the Phonon Density of States (PDOS) with different temperatures
(100 K and 600 K) in a pristine hBN nanosheet. . . . . . . . . . . . . . . . . 96
5.39
Variation of the Phonon Density of States (PDOS) in 2% point vacancy, edge
vacancy, and bivacancy induced hBN nanosheet at 600 K. . . . . . . . . . . . 96
5.40 Variations of PTC of stanene/hBN as a function of scaling factor at 300 K . . . 97
5.41
Variations of PTC of stanene/hBN as a function of in-plane tensile strain applied
atXdirection................................... 98
5.42
Variations of PTC of stanene/hBN as a function of in-plane tensile strain applied
atYdirection................................... 99
5.43
Variations of PTC of stanene/hBN as a function of in-plane tensile strain applied
atbiaxialdirection................................ 99
xii
List of Tables
4.1
Values of parameters of Lennard-Jones (LJ) potential function for Sn/hBN
interaction[95,97]. ................................ 50
4.2
Range of Input Parameters for ITR and In-Plane Thermal Conductivity Calculations
52
5.1
An overview of the ITR from previous classical MD simulations using the
FPP method of several common 2D heterostructures. The Temperatures and R
values provided in the table represent the lower and upper limits of that quantity
observed in the simulations reported in previous literature. . . . . . . . . . . . 66
5.2
Comparison of the PTC of stanene and h-BN between the present study and
existingliterature.................................. 86
xiii
List of Symbols
Symbols
εEnergy parameter of Lennard-Jones (LJ) potential
εSn-B
Energy parameter of LJ potential for interactions between Stanene and Boron atom
εSn-N
Energy parameter of LJ potential for interactions between Stanene and Nitrogen
atom
σDistance parameter of LJ potential
σSn-B
Distance parameter of LJ potential for interactions between Stanene and Boron atom
σSn-N
Distance parameter of LJ potential for interactions between Stanene and Nitrogen
atom
⟨·⟩ Ensemble average
jHeat flux (W/m2)
kPhonon thermal conductivity (PTC) (W/m/K)
k∞PTC of a system at infinite length
kSn PTC of Sn layer
khBN PTC of hBN layer
kBPTC of Sn/hBN heterostructure
ACross-sectional area of a nanosheet
∆Q
∆tResultant heat current in a specific direction
tSn Thickness of Sn layer
thBN Thickness of hBN layer
tBThickness of Sn/hBN heterostructure
LSystem length
G(ω)Phonon density of states (PDOS)
v⟨0⟩Initial velocity of an atom
v⟨t⟩Velocity of an atom after a specific time interval t
τThermal relaxation time
TSn Temperature of Sn layer
ThBN Temperature of hBN layer
TR Thermal rectification ratio
ArContact area of Sn/hBN nanosheet
EtEnergy profile of hBN layer
E0Initial energy profile of hBN layer
ROverall interfacial thermal resistance
RiInstantaneous interfacial thermal resistance
M F P Mean free path of phonons
lAverage effective mean free path (MFP) of phonons
VPotential function
rInteratomic distance
χvan der Waals interaction strength
Sn Stanene
hBN Hexagonal boron nitride
h buckling height of the Sn sheet ( 0.086 nm)
xiv
NEMD Non-equilibrium molecular dynamics
NVT Canonical ensemble (constant-temperature, constant-volume)
NPT Constant-temperature, constant-pressure ensemble
NVE Microcanonical ensemble (constant-energy, constant-volume)
TTR Time-domain thermoreflectance
FPP Fixed Pump Probe
PDOS Phonon Density of States
˚
A Angstrom
nm Nanometer
ps Picosecond
fs Femtosecond
vdW Van der Waals heterostructures
TMDs Transition Metal Dichalcogenides
MEAM Modified Embedded-Atom Method (MEAM)
xv
Abstract
Recently, the stanene (Sn) / hexagonal boron nitride (h-BN) van der Waals heterostructure
(vdW) has garnered significant attention among the scientific community due to its distinctive
electrical and optical characteristics. Despite the promising potential of this heterostructure,
its in-plane phonon thermal conductivity (PTC) and interfacial thermal resistance (ITR)
remain unexplored. In this study, we employ molecular dynamics (MD) simulations to
explore the thermal characteristics of this heterostructure, revealing an ITR of approximately
7×10−8K·m2/W
and a PTC of about
37.1W/m·K
for a
30×10 nm2
Sn/h-BN nanosheet at
room temperature.We further investigate the influence of several key parameters—including
nanosheet size (ranging from
10 nm
to
400 nm
), temperature (spanning from
100 K
to
600 K
), vacancy concentration (0.25% to 2%), contact pressure (0.5 to 20), and mechanical
tensile strain (1% to 5%) in both uniaxial and biaxial directions—on the modulation of
these thermal properties. Our findings reveal that with increasing nanosheet size, the in-
plane phonon thermal conductivity (PTC) gradually rises, while the interfacial thermal
resistance (ITR) consistently decreases. The results further demonstrate that increasing
temperature, contact pressure, and defect concentration tend to reduce both PTC and ITR,
whereas mechanical strain notably enhances both properties. To elucidate these behaviors,
we calculate the Phonon Density of States (PDOS) profiles of both the h-BN and Sn layers.
All these parameters collectively change the PDOS profiles of the individual Sn and h-BN
monolayers, thereby influencing their thermal properties. This work will provide both
theoretical support and logical guidelines for modulating thermal resistance and in-plane
thermal conductivity across diverse dissimilar material interfaces, which will be necessary
for the development of advanced nanodevices used in next-generation nanoelectronics,
nanophotonic, and optoelectronics applications.
xvi
Chapter 1
Introduction
1.1 Theoretical and Computational Background
Comprehending interfacial thermal resistance (ITR) is essential in the fields of materials science
and nanotechnology to enhance thermal management and efficiency in diverse systems. The
presence of interfacial thermal resistance, which occurs at the boundaries between distinct
materials, has a substantial effect on the total thermal conduction of the system. This, in turn,
affects the transport of heat at the nanoscale. Understanding the processes that control heat
transfer in these systems requires a comprehensive approach that combines theoretical modeling,
computer simulations, and experimental approaches. The objective of this section is to thoroughly
investigate interfacial thermal resistance by utilizing classical theories, quantum mechanical
formalisms, and sophisticated computational approaches. By enhancing our comprehension of
heat transfer phenomena, we aim to facilitate the progress of advanced thermal management
systems and novel nanomaterial-based technologies.
1.1.1 Theoretical Background of Thermal Transport Properties
1.1.1.1 Classical Theories
Following the identification of interfacial thermal resistance, researchers have put forward
multiple macroscopic ideas to elucidate its fundamental origins. An example of such a theory is
the acoustic mismatch model (AMM). This model makes the assumption that the divergence
of acoustic impedances between the constituent materials of the interface is the primary cause
of interfacial thermal resistance. The variation in acoustic propagation qualities, such as sound
velocity, between the two bulk materials is the cause of this mismatch. The AMM oversimplifies
by concentrating only on the bulk materials’ acoustic characteristics, ignoring complex interface
characteristics like geometry, orientation, and chemical bonding. It states that the following
formula can be used to determine the transmission coefficient for an auditory mode traveling via
the interface created by materials 1 and 2 [1]:
1
1.1. THEORETICAL AND COMPUTATIONAL BACKGROUND 2
t12 =4Z1Z2
(Z1+Z2)2(1.1)
where
Z1
and
Z2
represent the acoustic impedance of two different materials forming the
interface respectively. This model indicates that there is symmetry in the transmission of heat
between materials A and B. It suggests that the interfacial thermal conductance remains constant
even when the direction of heat flow is reversed. Nevertheless, it is important to note that
the assumption of symmetry may not always hold true, and interfaces have the potential to
demonstrate thermal rectification.
Figure 1.1: Comparison of transmission using two methods: the Acoustic Modeling Method
(AMM) assumes smooth reflections, while the Diffuse Modeling Method (DMM) assumes
random scattering of acoustic waves
The AMM provides an accurate description of interfacial thermal conductivity at low
temperatures, despite its straightforward nature. At low temperatures, the idea that thermal
conductance scales proportionately with the temperature cube is consistent with preliminary
experimental findings. However, because AMM relies on the assumption that phonon modes
do not experience scattering at the interface, they frequently underestimate interfacial thermal
resistance.
To complement the AMM, the diffusion mismatch theory (DMM) postulates that phonons
experience total scattering at interfaces. Within the context of DMM, phonons completely erase
any recollection of their prior states upon reaching the interface and possess an equal probability
of being redirected into another phonon state on either side [2,3]. Hence, the likelihood of phonon
transmission is determined exclusively by the density of states, resulting in a direct relationship
between the energy of transmitted phonons and the coupling between them. Nevertheless, the
DMM frequently overestimates the interfacial thermal resistance, especially in high-temperature
conditions, because it assumes perfect dispersion. Both the AMM and DMM exclude atomic-
level features and interface topologies, offering solely qualitative approximations. Atomic
scale theoretical models or advanced simulation tools are essential for achieving increased
computational accuracy. In addition, DMM exclusively takes into account elastic scatterings,
1.1. THEORETICAL AND COMPUTATIONAL BACKGROUND 3
disregarding inelastic scatterings that become prominent when there is a high mismatch in
phonon spectra. This can potentially result in phenomena such as heat rectification.
Enhanced models derived from the AMM and DMM strive to reduce pivotal assumptions and
take into account supplementary aspects. Modified models of DMM, for example, take into
account the complete dispersion relation in the lattice, hence broadening its range of applicability.
Additional improvements involve the integration of electron-phonon interactions, disorders,
and various phonon scatterings. Beyond AMM and DMM, numerous macroscopic models
elucidate the intricate mechanisms governing interfacial thermal transport. Analytical equations
derived from surface displacement offer useful understandings into the transfer of heat across
flat interfaces. Furthermore, the study of heat transport at the interface between metals and
nonmetals has included an examination of the role of electron-phonon interactions. Overall,
macroscopic models are constantly developing to encompass the overall understanding of
interfacial thermal transport, even though they have limits in accurately predicting quantitative
outcomes. Additional progress is required to consider intricate contact characteristics and
precisely simulate the resistance to heat transfer at the interface under a broad spectrum of
situations.
Classical theories of thermal transport are insufficient in explaining events that occur at extremely
small scales, such as the nanoscale, or at very low temperatures. Therefore, it is necessary to
employ quantum theories to accurately describe these phenomena.
1.1.1.2 Quantum Theories
Quantum distributions and evolutions play a crucial role in determining interfacial heat transport
properties in the nanoscale or low-temperature zone. Incorporating temperature into quantum
dynamics necessitates the presence of a quantum heat bath, in which particles adhere to either
Bose-Einstein or Fermi-Dirac distributions. Nevertheless, the countless possible configurations
of the heat bath provide difficulties when applying quantum mechanics. The formulation of
Non-Equilibrium Green’s function (NEGF) is a well-developed quantum theory that is often
used to investigate phenomena related to thermal transport [4–6].
This phenomenon involves the process of quantizing lattice dynamics and applying scattering
theories to gain a detailed knowledge of material behavior. Additionally, it is capable of
recovering the Landauer formula in the elastic regime.
I=1
2πZ∞
0
ℏωT (ω)[fL(ω)−fR(ω)] dω (1.2)
Where phonon transport properties are represented by
I
, the transmission coefficient is denoted
by
T(ω)
,
ℏω
represents the phonons’ energy, while
fL(ω)
and
fR(ω)
respectively correspond to
the Bose-Einstein distributions function. Lstands for left, and Rstands for right regimes.
1.1. THEORETICAL AND COMPUTATIONAL BACKGROUND 4
NEGF offers formal representations that go beyond the Landauer picture for inelastic transport.
The NEGF formalism is based on a configuration where heat reservoirs are linked to a central non-
equilibrium zone by junctions. Expanding this configuration to assess the transfer of heat across
interfaces raises the level of computational intricacy, but it presents encouraging possibilities
for practical applications in actual materials. Continual progress is being made in technological
developments that utilize the NEGF approach. These developments involve the integration
of inelastic phonon-phonon and electron-phonon interactions. Additionally, there have been
suggestions to do modal analysis within the NEGF framework and combine it with molecular
dynamics (MD) for intricate contact architectures. In addition to NEGF, many quantum theories
utilizing wave function representations have been developed, providing new methods for studying
thermal transport across interfaces, including the assessment of interfacial heat flux through the
analysis of atom displacement fluctuations.
1.1.1.3 Boltzmann Transport Theory
In the realm of particle-based analysis, the Boltzmann Transport Equation (BTE) serves as the
cornerstone for understanding phonon transport. Under thermal equilibrium, phonons follow
the Bose-Einstein distribution function
n0λ
. However, in the presence of a temperature gradient,
they exhibit deviations from this equilibrium state. Here, the symbol
λ
denotes the phonon
mode, characterized by its wave vector
q
and polarization
s
. The departure from the equilibrium
distribution can be rigorously derived from the BTE [7–9]:
vλ∇T∂nλ
∂T =dnλ
dT
scat
(1.3)
where
vλ
represents the group velocity of the phonon mode. Given a sufficiently small
temperature difference
∇T
in most practical cases,
nλ
can be expanded to first order as
nλ≈n0λ+ ∆nλ
, where
∆nλ
depends linearly on
∇T
. When considering only two- and
three-phonon scattering processes, the BTE simplifies to:
nλ−n0λ≈ −Fλ∇Tzn0λ
zT
(1.4)
where
Fλ
is related to the relaxation time
τ0λ
of mode
λ
through the relaxation time approximation
(RTA):
Fλ=τ0λ(νλ+ ∆ν)(1.5)
The relaxation time
τ0λ
corresponds to the relaxation time approximation (RTA), which considers
both absorption and emission processes.
According to Matthiessen’s rule, which assumes that different scattering processes are not
1.2. COMPUTATIONAL ANALYSIS OF TRANSPORT PROPERTIES AT THE NANOSCALE 5
influenced by each other, the total relaxation time can be determined as follows:
τ−1
λ,t =τ−1
phonon-phonon +τ−1
phonon-electron (1.6)
where the first term,
τ−1
phonon-phonon
, denotes intrinsic phonon-phonon scattering, and the second
term, τ−1
phonon-electron, includes any additional interactions, such as phonon-electron scattering.
Overall, the classical, quantum, and Boltzmann theories offer a range of approaches for analyzing
interfacial thermal resistance, each with its own advantages and limitations. These models and
theories are essential for advancing our understanding of thermal transport in complex systems
and improving the design of materials and devices for various applications.
1.2 Computational Analysis of Transport Properties at the Nanoscale
1.2.1 Molecular Dynamics
Macroscopic theories provide a comprehensive understanding of interfacial thermal transport,
although they often oversimplify the complexity of interfaces. Atomic features, such as
chemical bonds, defects, and atomic species, significantly impact thermal resistance, especially in
nanoscale structures. Classical molecular dynamics (MD) simulations are essential for studying
atomistic many-body problems and capturing anharmonic effects within atomistic interactions.
MD is extensively used to investigate phonon transport phenomena, including structural effects
and various influences such as doping, defects, strain, and substrate interactions. The thermal
properties of 2D materials are assessed using both equilibrium molecular dynamics (EMD) and
non-equilibrium molecular dynamics (NEMD) simulations [10, 11].
NEMD, in particular, offers insights into interface-modulated phonon dynamics, providing a
direct approach to simulate interfacial thermal transfer. Non-equilibrium molecular dynamics
(NEMD) is the most commonly used MD technique for simulating interfacial thermal transfer.
However, accurately estimating temperature in NEMD simulations poses challenges due to
their non-equilibrium nature. In MD simulations, phonon thermal conductivity is calculated by
monitoring the heat flux through the system over time. The evolution of atomic velocities and
locations is tracked by MD by creating a temperature gradient or by coupling the system to hot
and cold baths. This enables the calculation of thermal conductivity using Fourier’s equation
of heat conduction. This approach allows the extraction of thermal properties, such as thermal
conductivity, from the atomic-scale dynamics of the simulated material.
To calculate interfacial thermal resistance in NEMD, the materials forming the interface are
subjected to a direct temperature difference. The system then evolves based on inter-atomic
potentials, with the evolution taking a range of nanoseconds to microseconds, depending on
the system’s size and required accuracy. The heat current
J
is monitored during this process,
allowing the assessment of interfacial thermal conductance σusing the formula:
1.2. COMPUTATIONAL ANALYSIS OF TRANSPORT PROPERTIES AT THE NANOSCALE 6
σ=J
∆T(1.7)
where
∆T
represents the temperature differential across the interface. However, accurately
estimating the temperature in NEMD simulations is challenging due to their non-equilibrium
character. Temperature is often defined by regulating the arrangement of velocities, which
is accomplished by connecting to a heat bath that is in thermal equilibrium. Advanced heat
reservoirs, such as white-noise reservoirs, can be utilized. Quantifying the temperature at specific
points within a material and at its interface is difficult when it is not in a state of equilibrium.
Several approaches address this challenge, such as employing the equal partition formula, fitting
velocity profiles, and utilizing temperature probes.
Another method for extracting interfacial thermal resistance (ITR) involves recording the transient
thermal relaxation process, in addition to direct NEMD simulations. This technique mimics the
Time Domain Thermoreflectance (TDTR) method by applying two laser pulses to a material.
One laser pulse heats the sample, while another laser pulse monitors the temporal temperature
variation. A lumped heat capacity model can be used to calculate the decay constant by tracking
the temperature evolutions of two individual layers. The computation relies on the system’s
effective heat capacity, which can be determined by supplementary simulations or by referring to
existing literature. Alternatively, monitoring the changes in energy over time during relaxation
can offer a straightforward approach to compute the ITR without requiring any supplementary
simulations. This method provides a quick and effective means of calculating interfacial heat
resistance in MD simulations.
Molecular dynamics (MD) offers unique insights into the mechanics of transport, enabling
a thorough examination of local thermal currents and the efficiency of transport channels.
Modal analysis helps assess the impact of inelastic scattering. Techniques such as spectral
energy density (SED) and phonon wave-packet (PWP) methods have been devised to discern
mode-specific thermal conductivity and phonon transmission coefficients, respectively, offering
unique perspectives compared to traditional methods [12]. Additionally, employing spectral
decomposition in NEMD simulations provides intricate insights into interfacial thermal
conductance, delineating inelastic effects and phonon transmission across interface atoms with
precision. These methods enhance our understanding of thermal transport at interfaces and offer
valuable insights into complex phenomena like phonon-interface scattering and Kapitza thermal
resistance.
Although MD simulation is commonly employed to model thermal energy transport at interfaces,
its accuracy is contingent upon the potential quality and the length of the simulation. MD
simulation is a purely classical method that adheres to Newtonian dynamics, thus it cannot
reliably predict quantum events, such as discrete transport quanta or non-classical phonon
distributions, especially under low-temperature conditions. Quantum theories are essential
1.2. COMPUTATIONAL ANALYSIS OF TRANSPORT PROPERTIES AT THE NANOSCALE 7
in situations when quantum phenomena become prominent. Lattice dynamics is a reliable
approach to overcome these limitations and provides a feasible option for assessing the thermal
conductivity at surfaces created by crystals.
1.2.2 Lattice Dynamics
Unlike molecular dynamics (MD), which operates within the realm of classical dynamics and
overlooks quantum phenomena, lattice dynamics offers a distinct approach. It delves into the
subtle movements of atoms from their equilibrium positions and addresses them through the lens
of the Hamiltonian or potential energy associated with these atomic shifts. Lattice dynamics,
often implemented under the harmonic approximation, simplifies the potential energy into a
quadratic expression. This simplification involves a matrix representation that encapsulates the
system’s spring constants. This method allows for a detailed examination of how atoms vibrate
around their equilibrium positions, providing insights into the material’s thermal and mechanical
properties without delving into the complexities of quantum mechanics. This simplification
allows for a more precise depiction of heat transport events at the nanoscale [13, 14].
Within a crystal having periodicity, the displacements of atoms can be broken down into separate
vibration modes using Fourier transformation, referred to as normal modes decomposition. Each
mode is associated with a distinct frequency
ω
and wave-vector
q
in the phonon dispersion
relation. Acoustic modes refer to vibrations that are parallel (LA) and perpendicular (TA) to
q
,
while optical modes involve the relative movements of atoms within a unit cell. Phonon mode
transmission at interfaces can be analyzed using lattice dynamics principles, known as scattering
boundary theories. These theories assess the transfer of energy by solving equations of motion at
the interface boundary, assisting in modal analysis to uncover the mechanisms of transportation.
The wave packet approach is highly advantageous as it generates phonon mode wave packets on
one side of the interface and examines their transmission qualities. The wave packet approach
offers a clear and straightforward understanding of phonon transmission. The data indicates
that LA and TA modes undergo minimal reflection and maintain stable group velocities, but ZA
modes display diverse behavior. For instance, ZA modes with frequencies lower than 6 THz are
reflected due to the mismatched phonon dispersions at the interface, whereas those above 6 THz
are partially reflected due to unequal group velocities. The observed behavior is a result of the
disturbance in translational invariance generated by the substrates, which affects the dispersion
of phonons in the vicinity of the interface.
Classical theories in interfacial thermal resistance research have established a strong foundation
for understanding the intricacies of heat transmission across interfaces. The AMM and DMM
provide useful insights into the interaction of diverse materials at surfaces, elucidating the
underlying causes of thermal resistance. Although classical theories have limitations, such as
their inability to accurately describe atomic-level features and inelastic scattering processes, they
1.3. EXPERIMENTAL TECHNIQUES 8
have significantly advanced our understanding of heat transport. By incorporating computational
methods such as molecular dynamics (MD) and lattice dynamics, in conjunction with theoretical
modeling, we continue to enhance our comprehension of interfacial heat resistance. The future
holds great potential for advancing our understanding of quantum theories and developing
new computing methods to control thermal transport at interfaces, which will have significant
implications for various sectors, including nanotechnology and energy systems.
1.3 Experimental techniques
Theoretical simulations, such as Molecular Dynamics (MD), the Boltzmann Transport Equation
(PBTE), and the Atomistic Green’s Function (AGF), are invaluable for predicting the thermal
conductivity of 2D materials. However, accurately modeling real-world conditions, including
impurities, defects, and surface roughness, poses significant challenges. This makes the
advancement of precise experimental methods essential. Integrating experimental approaches
with theoretical models can significantly enhance measurement accuracy. Presently, the
predominant experimental techniques for assessing thermal conductivity are electro-thermal and
optothermal methods.
1.3.1 Microbridge Method
The thermal conductivity in the suspended micro-bridge method is calculated through a
meticulous process involving heat generation and precise temperature measurement. Initially,
the setup consists of two adjacent silicon nitride (SiN
x
) membranes suspended by five SiN
x
beams, each equipped with a platinum resistance thermometer coil connected to the substrate
through platinum leads. A mixed current, comprising microampere-level DC and nanoampere-
level AC components, is applied to the heating membrane. The DC current generates Joule
heat, while the AC current measures the temperature changes in both the heating and sensing
membranes [15–17].
Heat transfer occurs solely through the sample between these membranes. The temperature
difference between the heating membrane (
∆Th
) and the sensing membrane (
∆Ts
) is indicative of
the heat conducted through the sample. By analyzing this temperature difference and considering
the thermal conductance of the SiN
x
beams, the heat flux is determined. The thermal conductivity
(
κ
) of the sample is then calculated using this heat flux along with the sample’s dimensions—its
length and cross-sectional area.
1.3. EXPERIMENTAL TECHNIQUES 9
Figure 1.2: Figure illustrating the microbridge method, where an integrated heater and
thermometer measure the thermal properties of a sample using alternating current.
To ensure accuracy, the method addresses the interface thermal resistance (
Rc
) between the
sample and the membranes, which can affect the measurements. This is mitigated by using
numerical simulations to estimate the temperature rise at the interfaces and by incorporating
high thermal conductivity materials to the membranes to improve temperature uniformity. These
steps help reduce the impact of
Rc
, thereby enhancing the precision of the thermal conductivity
measurements.
1.3.2 3ωMethod
The 3
ω
method calculates the thermal conductivity of thin films by leveraging the frequency-
dependent response of a heating resistor. This technique involves preparing a metal electrode,
typically platinum, on the surface of the thin film sample using photolithography and thermal
evaporation. This electrode serves both as a heater and a thermometer. The thin film is
often deposited on a substrate via chemical vapor deposition (CVD) or high-temperature
oxidation [18–20].
An AC power supply with a frequency of
1ω
is connected to the metal electrode, causing its
1.3. EXPERIMENTAL TECHNIQUES 10
internal resistance to oscillate at a frequency of
2ω
due to the temperature changes induced by
the AC current. This results in a voltage signal with a
3ω
frequency variation, which is extracted
using a lock-in amplifier.
To measure the thermal conductivity of the thin film, two structures are prepared: one with just
the substrate and one with the thin film deposited on the substrate. The metal electrodes on both
structures measure the corresponding temperature changes (
∆Ts
for the substrate and
∆Ts+f
for
the film-substrate structure). The temperature change caused by the film (
∆Tf
) is determined by
subtracting ∆Tsfrom ∆Ts+f.
Figure 1.3: Figure illustrating the 3
ω
technique configuration, with the sample on a substrate and
integrated heater and thermometer. The heater is energized by alternating current, measuring the
third harmonic response.
The film’s thermal conductivity (
κf
) is then calculated using the measured temperature change
(
∆Tf
), the heating power (
P
), and the thickness of the film (
t
). This method is advantageous
because it eliminates the effects of thermal contact resistance between the sample and substrate,
ensuring that the measured thermal conductivity is accurate.
Additionally, the small surface area of the metal electrode minimizes the impact of heat radiation.
However, the 3
ω
method does have limitations, such as requiring a substrate with significantly
higher thermal conductivity than the film and needing a smooth sample surface to prevent damage
to the thin metal wires. Despite these challenges, the 3
ω
method remains a reliable technique for
measuring the thermal conductivity of thin films.
1.3.3 Time-Domain Thermoreflectance (TDTR)
The time-domain thermoreflectance (TDTR) method is a powerful technique for determining
the thermal conductivity of thin films. It involves using an ultrafast laser pulse to rapidly heat
1.3. EXPERIMENTAL TECHNIQUES 11
the surface of the sample, inducing a transient temperature change. A second laser beam, often
referred to as the probe laser, is then directed onto the sample’s surface to measure changes in
reflectance caused by the temperature variations [21–23].
Figure 1.4: Figure illustrating the TDTR method, where a pulsed laser heats the sample, and the
resulting temperature change is monitored to measure thermal properties
By precisely controlling the timing between the pump and probe laser pulses and analyzing
the resulting reflectance signal, researchers can extract valuable information about the thermal
properties of the sample. This includes not only the thermal conductivity but also other parameters
such as thermal boundary conductance and specific heat.One of the key advantages of TDTR is
its high accuracy and sensitivity, particularly in measuring the thermal properties of nanoscale
materials and thin films. Additionally, TDTR offers the ability to study thermal transport
in a wide range of materials and applications.However, TDTR does have its limitations. It
requires specialized equipment and expertise for implementation, and sample preparation can be
challenging. Furthermore, interpreting the data obtained from TDTR experiments may require
sophisticated analysis techniques.Overall, despite its challenges, TDTR remains a valuable tool
for researchers studying thermal transport phenomena and advancing our understanding of heat
transfer in various materials and systems.
1.4. OVERVIEW OF 2D MATERIALS 12
1.3.4 Optothermal Raman
The optothermal Raman method is a sophisticated technique used to determine the thermal
conductivity of materials, particularly in thin films and nanostructures. This method combines
optical excitation with Raman spectroscopy to probe the thermal properties of the sample.
In the optothermal Raman method, a laser beam is used to heat the sample locally, inducing
a temperature gradient. The Raman spectrum of the sample is then measured before and after
heating. The change in the Raman spectrum due to heating provides valuable information
about the thermal properties of the material, including its thermal conductivity.By analyzing the
temperature-dependent shifts and intensities of Raman peaks, researchers can extract quantitative
information about the thermal conductivity of the sample. This is achieved through sophisticated
modeling and analysis techniques that relate the observed changes in the Raman spectrum to
the underlying thermal properties of the material [24
–
26].One of the key advantages of the
optothermal Raman method is its ability to provide spatially resolved measurements of thermal
conductivity, allowing researchers to study variations in thermal properties across the sample
surface. Additionally, the method is non-destructive and can be performed at room temperature,
making it suitable for a wide range of materials and applications. However, like any experimental
technique, the optothermal Raman method has its limitations. It requires careful calibration
and analysis to ensure accurate measurements, and the interpretation of Raman spectra can be
complex. Furthermore, the technique may be limited to materials with well-defined Raman-
active modes.Overall, the optothermal Raman method is a valuable tool for studying the thermal
properties of materials, offering insights into heat transport mechanisms and facilitating the
development of advanced thermal management technologies.
Figure 1.5: Figure illustrating the Optothermal Raman method, where laser-induced heating is
combined with Raman spectroscopy to analyze thermal properties and material behavior.
1.4 Overview of 2D materials
Two-dimensional (2D) materials have emerged as a fascinating category of materials due to
their distinctive properties and the unique physics that arise from their reduced dimensionality.
1.4. OVERVIEW OF 2D MATERIALS 13
Graphene, a single layer of carbon atoms arranged in a hexagonal lattice, is one of the most
extensively studied 2D materials. It boasts remarkable electrical and thermal conductivity,
exceptional mechanical strength, and flexibility, making it ideal for applications in electronics,
energy storage, and composites.
Transition Metal Dichalcogenides (TMDs) form another significant class of 2D materials. These
materials typically consist of a transition metal (like molybdenum or tungsten) bonded to
chalcogen elements (such as sulfur, selenium, or tellurium). TMDs like MoS
2
and WS
2
are
especially noteworthy for their direct bandgap in monolayer form, which facilitates efficient light
absorption and emission, thus enabling applications in photodetectors, solar cells, and field-effect
transistors.Black phosphorus, known for its puckered structure, is another important 2D material.
It exhibits a tunable bandgap that can be adjusted by varying the number of layers, making it
suitable for a wide range of electronic and optoelectronic applications. Its high charge carrier
mobility and distinct optical properties further enhance its utility in photonic devices. Hexagonal
boron nitride (h-BN) is often referred to as “white graphene” due to its structural similarity to
graphene. This 2D material serves as an excellent electrical insulator while also possessing
high thermal stability and mechanical strength. Its use as a substrate for graphene and other
2D materials is crucial for enhancing device performance and stability.Metallic 2D materials
include graphene and certain TMDs, such as NbSe
2
and TaS
2
. These materials retain high
electrical conductivity, akin to metals, and can exhibit superconductivity at low temperatures,
making them attractive for advanced electronic circuits and quantum computing applications.2D
metal-organic frameworks (MOFs) represent a hybrid class of materials formed by metal ions or
clusters coordinated to organic ligands, creating a porous two-dimensional structure. Their high
surface area, tunable pore sizes, and chemical functionality make them suitable for applications
in gas storage, separation, catalysis, and drug delivery.
1.4. OVERVIEW OF 2D MATERIALS 14
Figure 1.6: Illustration of different two-dimensional materials, including graphene, transition
metal dichalcogenides (TMDs), hexagonal boron nitride (h-BN), and stanene.
2D hybrid materials, which integrate organic and inorganic components, showcase tailored
properties that can be engineered for specific applications. These hybrids often combine the
best attributes of each component, leading to innovations in areas such as photovoltaics, light-
emitting diodes (LEDs), and sensors. Stanene, a single layer of tin arranged in a honeycomb
lattice, is another exciting 2D material. It has garnered significant interest due to its theoretical
prediction to behave as a topological insulator, conducting electricity along its edges while
remaining insulating in its bulk form. This unique property, along with its high intrinsic carrier
mobility and strong spin-orbit coupling, makes stanene a potential candidate for applications
in spintronics and next-generation electronic devices. Research on stanene is still in its early
stages, but initial studies have shown promise in fabricating high-quality samples that exhibit
the desired properties. Additionally, other emerging 2D materials such as phosphorene (a
single layer of black phosphorus) and various 2D oxides (like MoO
3
and SnO
2
) are being
explored for their unique electronic and optical properties, expanding the potential applications
in electronics and optoelectronics.In summary, the diverse types of 2D materials, ranging from
conductive to insulating, and from semiconducting to metallic, offer a rich landscape for research
and application across multiple fields. Their unique properties not only enable innovative
technologies but also provide insights into fundamental physics, making them a key focus area
1.5. VAN DER WAALS HETEROSTRUCTURES 15
in materials science and engineering.
1.5 van der Waals heterostructures
Van der Waals (vdW) heterostructures are advanced materials formed by stacking different
two-dimensional (2D) materials on top of each other through weak van der Waals forces. Unlike
traditional materials that require lattice matching, vdW heterostructures exploit these weak
interactions to combine materials with diverse properties, leading to novel physical phenomena
and applications. There are different types of van der Waals heterostructure as shown in Fig. 1.7,
1.5.1 Types of van der Waals heterostructures
•
2D-0D (Two-Dimensional to Zero-Dimensional): These heterostructures involve
stacking 2D materials with quantum dots or nanoparticles. They are used in applications
such as quantum computing and highly sensitive photodetectors.
•
2D-1D (Two-Dimensional to One-Dimensional): These structures combine 2D materials
with nanowires or nanotubes, enhancing properties like electrical conductivity and
mechanical strength. They find uses in nanoscale transistors and energy storage devices.
•
2D-2D (Two-Dimensional to Two-Dimensional): Stacking different 2D materials on
top of each other, often vertically. They are crucial for creating novel electronic and
optoelectronic devices, including flexible electronics and advanced sensors.
•
2D-1.5D (Two-Dimensional to One-and-a-Half-Dimensional): These include 2D
materials interfacing with materials that have quasi-one-dimensional properties, such
as certain polymers or molecular wires. They are used in creating hybrid devices with
unique charge transport characteristics.
•
2D-3D (Two-Dimensional to Three-Dimensional): Combining 2D materials with 3D
bulk materials. This approach enhances the performance of devices in energy conversion
and storage applications, such as batteries and fuel cells.
•
In-Plane Heterostructures: These involve lateral stacking of different 2D materials
within the same plane. They are used to fabricate novel electronic and optoelectronic
components with improved performance and flexibility.
•
Vertical Heterostructures: Involves stacking different 2D materials in a vertical sequence.
These are essential for high-performance transistors, photodetectors, and memory devices.
1.5. VAN DER WAALS HETEROSTRUCTURES 16
Figure 1.7: Two-dimensional layered materials and van der Waals heterostructures
1.5.2 Applications
Van der Waals heterostructures are pivotal in developing next-generation electronic and
optoelectronic devices. Their applications include flexible and transparent electronics, high-
performance transistors, advanced sensors, photodetectors, and energy storage systems. Their
ability to combine disparate materials allows for the creation of devices with tailored properties,
pushing the boundaries of current technology and enabling new functionalities. The diverse
applications of van der Waals heterostructures are illustrated in Fig. 1.8 below.
1.5. VAN DER WAALS HETEROSTRUCTURES 17
Figure 1.8: Various applications of van der Waals heterostructures
1.5. VAN DER WAALS HETEROSTRUCTURES 18
Layout of the Thesis
This thesis investigates the interfacial thermal transport and in-plane thermal conductivity of the
Sn/hBN van der Waals heterostructure. The introductory chapter presents the motivation for the
research, outlines the study’s objectives, and provides an overview of theoretical, computational,
and experimental approaches used to measure thermal properties. In the literature review, past
studies on interlayer thermal resistance and in-plane thermal conductivity are examined, forming
the basis for designing the simulation methodology. Chapter three discusses the molecular
dynamics simulation technique in detail, covering potentials for atomic interactions, time
integration algorithms, boundary conditions, and ensemble selection for extracting properties,
along with the advantages and limitations of the simulation approach. Chapter four focuses on
the simulation procedures for both interfacial thermal resistance and in-plane thermal conductivity
calculations, outlining the steps for code development, validation, and system equilibration over
time. The results, discussed in chapter five, report the interfacial thermal resistance between
Sn and hBN layers, followed by a detailed analysis of the in-plane thermal conductivity of
the Sn/hBN heterostructure. The effects of system size, temperature, contact pressure, defects,
and tensile strain on interfacial thermal resistance and phonon transport behavior are explored.
Finally, chapter six concludes the research by summarizing the key findings and offering
recommendations for future work, emphasizing the potential for further exploration of the
thermal properties of van der Waals heterostructures.
Chapter 2
Literature Review
The advent of two-dimensional (2D) materials has revolutionized the landscape of materials
science, revealing a spectrum of exceptional electrical, optical, and thermal properties . Following
the groundbreaking synthesis of graphene in 2004, renowned for its outstanding electrical and
thermal conductivity as well as remarkable mechanical strength, the scientific community has
embarked on an enthusiastic quest to discover and synthesize a myriad of 2D materials [27].
Among these, hexagonal boron nitride (h-BN) and stanene stand out as exceptionally significant
for future nanoscale devices. The h-BN, also referred to as White Graphene due to its comparable
honeycomb structure to the Graphene monolayer, possesses a band gap of 5.8 eV, and exhibits
favorable chemical, mechanical, and thermal properties [28
–
31]. Similarly, Stanene, a buckled
structure composed of tin atoms, exhibits distinctive quantum behavior, such as the Quantum Spin
Hall Effect. Additionally, its increased atomic mass and comparatively low Debye temperature
result in decreased lattice thermal conductivity at room temperature, making it a favorable option
for future thermoelectric applications [32
–
34]. Combining Stanene with Hexagonal boron nitride
to create a van der Waals heterostructure reveals a range of fascinating phenomena poised to
drive significant advancements in next-generation nanoscale technologies.
19
20
Figure 2.1: Figure illustrating the Top , Front and Isometric view of a monolayer Stanene layer.
“h” is the buckling height of Sn layer (.086 nm).
Figure 2.2: Figure illustrating the Top , Front and Isometric view of a monolayer hBN layer.
21
Recently Van der Waals heterostructuring has gained significant attention among the scientific
community [35
–
37]. This approach combines two distinct 2D materials through van der Waals
forces, unveiling novel physical properties that surpass those found in individual monolayers.
Moreover, vdW heterostructures provide novel methods for controlling the electrical, optical,
and thermal characteristics of emerging nanoscale devices by adjusting the arrangement of layers,
the relative degree of rotation between adjoining layers, and the spacing between layers [38
–
40].
The development of numerous nanomanufacturing techniques has enabled the production of
these high-quality vdW heterostructures [36, 37]. This facilitates the development of a range of
ultrathin devices, including tunneling and field-effect transistors, wearable flexible electronics,
photodiodes, and solar cells [36]. However, the functionality and effectiveness of these devices
depend completely on the efficient dissipation of heat from them. The primary bottleneck in
achieving effective heat dissipation from these devices lies in the presence of interfacial thermal
resistance between various interfaces. Consequently, it is imperative to understand and quantify
the interfacial thermal resistance across various interfaces in 2D van der Waals heterostructures.
While the electrical and in-plane thermal properties of Sn/h-BN heterostructure have been
extensively investigated, the thermal transport behavior across the Sn/h-BN interface has yet to
be explored [41,42]. According to the authors’ concern, there are currently no reported values
for the interfacial thermal resistance (ITR) between the Sn and h-BN bilayer. This study will
present the interfacial thermal resistance between Sn and h-BN layers, filling a crucial gap in
current research. Furthermore, this research aims to reveal the complex intricacies of interfacial
thermal transport in Sn/h-BN heterostructures, to offer novel perspectives that could lead to
substantial progress in the field of materials science and technology.
Since the discovery of interfacial thermal resistance (ITR), also known as Kapitza resistance,
scientists have developed substantial theoretical models, computational methods, and
experimental techniques to understand and control heat transfer across solid-solid, solid-liquid,
and solid-gas interfaces [43]. Advancements in nanotechnology have led to the development of
a wide range of technologies for measuring the interfacial thermal properties of nanomaterials.
However, these measurements remain both expensive and challenging. Researchers commonly
utilize Molecular Dynamics (MD) to comprehend phonon transmission at different interfaces
[44
–
47]. Ren et al. conducted molecular dynamics simulations to examine the impact of
interlayer rotation on interfacial thermal conductance in graphene/h-BN heterostructures. They
found that increasing the rotation angle between the 2D layers results in a monotonic decrease
in low-frequency phonon transmission, thereby reducing the interfacial thermal transport [48].
In another study, the same research group demonstrated that, in addition to interfacial thermal
conductance, the in-plane thermal conductivity of graphene/h-BN Moir
´
e superlattices also
decreases monotonically with increasing interlayer rotation [49]. Wang et al. investigated the
impact of interfacial roughness on thermal contact resistance (TCR) in VO2/Si interfaces, finding
that increased roughness significantly elevates TCR and uncertainty in heat transport, both
22
experimentally and through molecular dynamics simulations [50]. Peng et al. demonstrated
a significant reduction of up to 40% in the interfacial thermal resistance between graphene
and water interface by introducing a periodic superlattice structure in the graphene layer.
This notable improvement is attributed to enhanced spectral interfacial thermal conductance,
highlighting the effectiveness of superlattice structures in optimizing heat transfer across solid-
liquid interfaces [51].
Recently, transient molecular dynamics modeling approaches have been employed to analyze
interfacial thermal resistance between various two-dimensional material interfaces including
GR/silicene, GR/MoS
2
, GR/phosphorene, GR/stanene, GR/SiC, MoS
2
/MoSe
2
, GR/MoSe
2
,
phosphorene/h-BN [52
–
63]. Typically, 2D materials are deposited onto a 3D substrate. At
these 2D-3D interfaces, there exists a substantial thermal impedance between the substrate
and the 2D layer. These transient Molecular dynamics technique is also employed to assess
the thermal resistance between distinct 2D layers and their substrates. For instance, Zhang
et al. conducted the MD simulation to determine the interfacial thermal resistance between
phosphorene and silicon substrates, as well as the thermal resistance between silicene and other
substrates [63, 64]. Similarly, Farahani et al. investigated the thermal resistance between MoS2
and silica substrates [65]. The transient molecular dynamics simulation technique replicates the
Time Domain Thermoreflectance (TDTR) method by applying two laser pulses to a material.
One laser pulse heats the sample, while another laser pulse monitors the temporal temperature
variation. By employing these sophisticated modeling tools, researchers can reveal intricate
information regarding the interfacial thermal transport occurring at various interfaces on a
nanoscale level.
Current studies have shown that factors such as temperature, contact pressure, vacancy defects,
chemical functionalization, strain effects, and twisting angles are crucial for understanding
interfacial thermal resistance (ITR) between various 2D materials [54, 66, 67]. This study
therefore focuses on analyzing the impact of various factors including system temperature,
contact pressure, defects, strain, and system size on ITR between the Sn and h-BN layers. In
addition, utilizing Transient Molecular Dynamics Simulation the interfacial thermal resistance
(ITR) value between Sn and h-BN layers is being reported for the first time in this study.
Understanding Phonon transmission at interfaces and utilizing engineering methods to control it
will pave the way for the innovation of modern nanoelectronics, optoelectronics, photonics, and
other nanoscale devices.
The rapid advancement of nanotechnology has spurred significant interest in two-dimensional
materials due to their unique electrical, optical, thermal, and mechanical properties [68
–
71].
Stanene is a material composed of a single layer of tin atoms organized in a hexagonal lattice
structure. It has a distinct buckled topology, a significant bandgap, and exhibits amazing quantum
behavior. These characteristics make it a highly intriguing contender for nanometer-scale
applications in the future [32–34, 72]. Similarly, hexagonal boron nitride (hBN) is a monolayer
23
comprised of boron and nitrogen atoms arranged asymmetrically. It has a wide bandgap and
exceptional thermal and mechanical properties, which makes it well-suited for electrical devices
that need accurate energy barriers and effective control of charge carriers. In addition, hBN is a
highly suitable foundation for creating vertical heterostructures by combining other 2D materials,
such as graphene and transition metal dichalcogenides (TMDs) [28
–
31, 73]. Integrating these
two wonderful 2D materials, Stanene and hBN, together with van der Waals forces to form a
heterostructure may generate interesting properties propelling advancements in various fields of
nanotechnology.
Figure 2.3: Thermal conductivity of some typical single-layer 2D materials at room temperature.
The performance and reliability of microelectronic devices may deteriorate due to heat generation
arising from their continued miniaturization. Researchers are actively seeking 2D materials that
exhibit high thermal conductivity for use in electronics, as well as low thermal conductivity
for applications in thermoelectric devices. Phonons are the primary carriers of heat in 2D
materials [74]. However, vacancy-induced defects such as point vacancies, bivacancies, and edge
vacancies are commonly formed during the experimental production and incorporation of 2D
materials. These imperfections intricately disrupt phonon transport through scattering, imparting
profound and intricate influences on the material’s thermal behavior. Therefore, understanding
the intricate interplay between these defects and phonon thermal conductivity is pivotal for
harnessing the full potential of 2D materials in diverse applications. Our main goal in this work is
24
to examine how different vacancy-induced defects affect the Sn/hBN bilayer’s thermal behavior
in order to gain a better understanding of how these defects affect the phonon dynamics of the
system.
Conducting experiments at the nanoscale presents challenges due to the limited availability
of high-precision instruments, difficulties in sample preparation, and the significant impact of
quantum phenomena on material behavior at this scale. Molecular dynamics modeling is a
highly powerful method for determining the thermal conductivity of 2D materials, providing
an exceptional understanding of their thermal behavior. Sevik et al. [75] utilized equilibrium
molecular dynamics simulations (EMD) along with a Tersoff-type empirical interatomic potential
to assess the thermal conductivity of boron nitride (BN) nanostructures. Tabarraei et al. [76]
employed reverse nonequilibrium molecular dynamics simulations (RNEMD) to determine the
thermal conductivity of bulk armchair and zigzag monolayer nanoribbons of hexagonal boron
nitride (hBN), reporting values of 277.78 W/m K for armchair nanoribbons and 588.24 W/m K
for zigzag nanoribbons. Furthermore, researchers demonstrated that the thermal conductivity
characteristics of hBN ribbons were significantly influenced by the presence of Stone-Wales
defects. Later, Khan et al. [77] investigated the thermal conductivity of hexagonal boron nitride
nanoribbons (hBNNRs), building upon previous research, to explore the effects of temperature,
length, breadth, and various types of vacancies. It is evident from previous research that the
presence of defects results in a significant reduction in the phonon thermal conductivity of
hBN nanostructures. Similarly, numerous molecular dynamics studies have been conducted
on stanene nanostructures. Khan et al. [78] investigated the mechanical and thermal properties
of stanene nanoribbons type nanostructures (STNRs) using equilibrium molecular dynamics
(EMD) simulations, employing the modified embedded-atom method (MEAM) interatomic
potential. Another study by the same research group demonstrated the significant influence of
doping on the thermal characteristics of materials [79]. Specifically, doping stanene nanoribbons
(STNR) with carbon and silicon resulted in an increase in thermal conductivity compared
to undoped stanene nanoribbons. This work was further corroborated in a subsequent study
using the non-equilibrium molecular dynamics (NEMD) method, revealing a strong correlation
between stanene’s phonon thermal conductivity and defect concentrations [80]. Although a
lot of studies have been conducted on the thermal conductivity of