Thermal Boundary
Conductance
Acoustic and Diffuse Mismatch Models
Introduction
Acoustic Mismatch
Model
Diffuse Mismatch Model
01
02
03
TABLE OF
CONTENTS
04
Literature on TBC
05
Conclusion
06
Questions ?
Introduction
The gross heat current density across interface is the
Efficient heat dissipation from devices
We could just use a substrate with high thermal
conductivity like diamond?
sum over all frequencies and incident angles of
the #phonons * ph energy *transmission
probability.
But TBR!!
TBC help in heat dissipation from devices
For small (T2 —T1)/T2, the thermal boundary
conductivity can be defined as
Which is essential for the performance and reliability of
devices
Heat flow in semiconductors is mainly through phonons
A phonon incident at an interface may or may not be
transmitted.
Swartz, E. T. & Pohl, R. O. Thermal boundary resistance. Rev. Mod.
Phys. 61, 605–668 (1989).
The problem is in principle solved if the
transmission probabilities, ɑ, are known
(AMM and DMM).
Acoustic Mismatch Model (AMM)
Critical
angle
Main assumption:
❖ the phonons are governed by continuum
acoustics and the interface is treated as a
plane.
❖ That is, phonons are treated as plane waves,
and the materials are treated as continua (no
lattice). Continuum acoustics govern the
phonon system.
❖ Scattering of phonons at interface is ignored
Given this assumption, if the incident angle is
smaller than critical angle, the phonons are
(specularly transmitted)
❖ refracted, or
❖ refracted and mode converted.
θin
,
ctran>cin
θtran
The relationship between the angles and phonon
velocities can be determined by the acoustic analog
of Snell's law for electromagnetic waves.
Swartz, E. T. & Pohl, R. O. Thermal boundary resistance. Rev. Mod. Phys. 61, 605–668
(1989).
If the incident angle is greater than critical
angle, the phonon is
❖ specularly reflected ,
❖ reflected and mode converted
The transmission probability from m1 to m2 for a
phonon with normal incidence (acoustic analog of
the Fresnel equation)
Critical
angle
θin
ctran>cin
,
Z - is an acoustic impedance.
Transmission probabilities are independent of
phonon frequencies.
If both sides of the interface are identical, then
transmission probability = 1.
Thermal boundary conductivity can be written
as
There is no critical angle on the side
with greater phonon velocities
Swartz, E. T. & Pohl, R. O. Thermal boundary resistance. Rev. Mod. Phys. 61, 605–668
(1989).
At low temperature, the upper limit of
frequency is set to infinity and the integral
can be done to get Ram
Average transmission coefficient
➢ Ram > 0 for similar materials
Swartz, E. T. & Pohl, R. O. Thermal boundary resistance. Rev. Mod. Phys. 61, 605–668 (1989).
Diffuse Mismatch Model (DMM)
High frequency phonons are scattered at
interface.
Main assumption:
❖ all the phonons are diffusely scattered
at the interface
❖ No acoustic correlations at the
interface
The transmission probabilities are
determined by,
❖ Relative phonon density of states
(mismatch) and
❖ the principle of detailed balance
m1
m2
Relative phonon densities of states influence transmission
probability, e.g., If phDOS2 > phDOS1 transmission
probability is high
Swartz, E. T. & Pohl, R. O. Thermal boundary resistance. Rev. Mod. Phys. 61, 605–668 (1989).
The average transmission coefficients become
➢
Only the transmission probability is modified
➢ The wave vector k and mode j of the diffusely
scattered phonon are independent of that of
the incident phonon
➢ Thus the transmission probability is
independent of wave vector and phonon mode
In the low-temperature limit, the thermal
boundary resistance in the limit of diffuse
mismatch can therefore be written as
Swartz, E. T. & Pohl, R. O. Thermal boundary resistance. Rev. Mod. Phys. 61, 605–668 (1989).
The ratio of the diffuse thermal boundary resistance.
In the case of Kapitza boundary,
❖ According to AMM the phonon is likely to reflect due to the
large acoustic mismatch between m1 and m2.
❖ According to DMM the phonon is likely to forward scatter
TBRamm>TBRdmm
If the two materials have identical acoustic properties, then the
transmission probability are
❖ AMM ~ 1
❖ DMM ~ 0.5
➢ Rdm=2Ram, Diffuse scattering increase TBR Swartz, E. T. & Pohl, R. O. Thermal boundary resistance. Rev. Mod. Phys. 61,
605–668 (1989).
Interface thermal conductance between β-Ga2O3 and different substrates
For β-Ga2O3 devices, heat dissipation is limited by
❖ low thermal conductivity of the channel and
❖ TBR
The TBC between β-Ga2O3 and different substrates
(SiO2 , 4H-SiC, α-Al2O3 and Si) is investigated by
combining Landauer formula with the AMM and DMM.
AMM and DMM have been applied to
calculate the phonon transmission
probability
Landauer formula for TBC (G)
Ma, D., Zhang, G. & Zhang, L. Interface thermal conductance between
β-Ga2O3and different substrates. J. Phys. D. Appl. Phys. 53, (2020).
The change in TBC with T is due to Bose–Einstein
distribution
phDOS, group velocity and transmission probability
remain the same
The TBC predicted by AMM is an upper limit while that
predicted by DMM can be regarded as cases when the
interface roughness or disorder is large.
The transmission function:
describes the number of phonon modes
transmitted across the interface
Ma, D., Zhang, G. & Zhang, L. Interface thermal conductance between βGa2O3and different substrates. J. Phys. D. Appl. Phys. 53, (2020).
The AMM assumes no scattering takes place thus the
transmission probability is governed by impedance
mismatch
The DMM assumes that all phonons are difusely
scattered.
At low temperatures the two models predict almost
same TBC
❖ The phonon transmission function predicted by
AMM is the same as that by DMM at low
frequencies.
❖ the TBC will be insensitive to interfacial defects
or disorder at low temperature.
Ma, D., Zhang, G. & Zhang, L. Interface thermal conductance between βGa2O3and different substrates. J. Phys. D. Appl. Phys. 53, (2020).
Thermal conductance across β-Ga2O3-diamond van der Waals heterogeneous interfaces
Thermal conductivity of β-Ga2O3 (10-30 W/mK) is significantly
lower than those of other wide bandgap semiconductors (such
as AlN, SiC (490 W/mK), GaN (230 W/mK), and diamond
(>2000 W/mK)) and decreases with decreasing film thickness
due to phonon scattering with the boundaries.
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In this work, the TBC across β-Ga2O3/Diamond
a (100) oriented β-Ga2O3 nano-membrane is mechanically
exfoliated onto a single crystal (100) CVD diamond
substrate.
Time-domain thermoreflectance (TDTR) was used to
measure the TBC
Landauer approach with transmission function from DMM
is used.
The transmission function from DMM
M is the phonon number of modes
Cheng, Z. et al. Thermal conductance across β-Ga2O3-diamond van
der Waals heterogeneous interfaces. APL Mater. 7, 0–7 (2019).
Because the transmission function from DMM does not depend
on the angle of incidence, the Landauer formula can be
simplified as
The phonon properties of diamond are obtained from first
principles calculation with VASP, and that of Ga 2O3 from
Materials Project
Cheng, Z. et al. Thermal conductance across β-Ga2O3-diamond van
der Waals heterogeneous interfaces. APL Mater. 7, 0–7 (2019).
Observed TBC
TBC is measured to be ~ 17MW/m2K.
The calculated TBC from the Landauer approach and DMM
is 312 MW/m2K at the Ga2O3 cutoff frequency
TBC is influenced by;
➢ the weak van der Waals force between Ga 2O3 and
diamond; phonon transmission is low
➢ the small contact area at the interface due to
roughness, and
➢ The large phonon density of state (DOS) mismatch
between Ga2O3 and diamond.
Acoustic phonons of Ga2O3 only contribute to about
8% of the total TBC, while optical phonons contribute to
about 92%
Cheng, Z. et al. Thermal conductance across β-Ga2O3-diamond van
der Waals heterogeneous interfaces. APL Mater. 7, 0–7 (2019).
❖ Due to the large difference between the
acoustic group velocities, the phonon
transmission coefficient is very low at
low frequencies;
❖ group velocity of diamond is 2.58 times
that of Ga2O3
Conclusion
❖ Both the AMM and DMM assumes perfectly paring of two materials
(without cross section mismatch) and perfectly strong interface bonding
In a real interface
❖ Weak bonding (e.g. vdW) lead to lower TBC.
❖ Lattice mismatch due to different lattice constants
➢ create lattice distortion at the interface which cause stress and
strain resulting in scattering of high frequency phonons hence
lower TBC.
❖ Thermal transport across van der Waals interfaces is limited by the real
contact area and low phonon transmission due to weak adhesion
energy.
❖ Thermal transport across these interfaces remains an open issue.
Thank you
Questions ?