Effects of phonon interference through long range interatomic bonds on thermal interface conductance
We investigate the role of two-path destructive phonon interference induced by interatomic bonds beyond the
 nearest neighbor in the thermal conductance of a silicon-germanium-like metasurface. Controlled by the ratio between
 the second and first nearest-neighbor harmonic force cons...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Effects of phonon interference through long range interatomic bonds on thermal interface conductance / Haoxue Han, Lei Feng, Shiyun Xiong, Takuma Shiga, Junichiro Shiomi, Sebastian Volz, Yuriy A. Kosevich // Физика низких температур. — 2016. — Т. 42, № 8. — С. 902-908. — Бібліогр.: 33 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860073776292560896 |
|---|---|
| author | Haoxue Han Lei Feng Shiyun Xiong Takuma Shiga Junichiro Shiomi Volz, Sebastian Kosevich, Yu.A. |
| author_facet | Haoxue Han Lei Feng Shiyun Xiong Takuma Shiga Junichiro Shiomi Volz, Sebastian Kosevich, Yu.A. |
| citation_txt | Effects of phonon interference through long range interatomic bonds on thermal interface conductance / Haoxue Han, Lei Feng, Shiyun Xiong, Takuma Shiga, Junichiro Shiomi, Sebastian Volz, Yuriy A. Kosevich // Физика низких температур. — 2016. — Т. 42, № 8. — С. 902-908. — Бібліогр.: 33 назв. — англ. |
| collection | DSpace DC |
| container_title | Физика низких температур |
| description | We investigate the role of two-path destructive phonon interference induced by interatomic bonds beyond the
nearest neighbor in the thermal conductance of a silicon-germanium-like metasurface. Controlled by the ratio between
the second and first nearest-neighbor harmonic force constants, the thermal conductance across a germanium
atomic plane in the silicon lattice exhibits a trend switch induced by the destructive interference of the
nearest-neighbor phonon path with a direct path bypassing the defect atoms. We show that bypassing of the
heavy isotope impurity is crucial to the realization of the local minimum in the thermal conductance. We highlight
the effect of the second phonon path on the distinct behaviors of the dependence of the thermal conductance
on the impurity mass ratio. All our conclusions are confirmed both by Green’s Function calculations for the
equivalent quasi-1D lattice models and by molecular dynamics simulations.
|
| first_indexed | 2025-12-07T17:12:05Z |
| format | Article |
| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 8, pp. 902–908
Effects of phonon interference through long range
interatomic bonds on thermal interface conductance
Haoxue Han1,2, Lei Feng3, Shiyun Xiong2, Takuma Shiga3, Junichiro Shiomi3,
Sebastian Volz2, and Yuriy A. Kosevich4
1Theoretische Physikalische Chemie, Eduard-Zintl-Institut für Anorganische und Physikalische Chemie, Technische
Universität Darmstadt. 4 Alarich-Weiss-Straße, Darmstadt 64287, Germany
E-mail: haoxue.han@ecp.fr
2Laboratoire EM2C, CNRS, CentraleSupélec, Université Paris-Saclay, Grande Voie des Vignes
92295 Châtenay-Malabry cedex, France
3Department of Mechanical Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo, Japan
4Semenov Institute of Chemical Physics, Russian Academy of Sciences, 4 Kosygin Str., Moscow 119991, Russia
E-mail: yukosevich@gmail.com
Received April 12, 2016, published online June 24, 2016
We investigate the role of two-path destructive phonon interference induced by interatomic bonds beyond the
nearest neighbor in the thermal conductance of a silicon-germanium-like metasurface. Controlled by the ratio be-
tween the second and first nearest-neighbor harmonic force constants, the thermal conductance across a germa-
nium atomic plane in the silicon lattice exhibits a trend switch induced by the destructive interference of the
nearest-neighbor phonon path with a direct path bypassing the defect atoms. We show that bypassing of the
heavy isotope impurity is crucial to the realization of the local minimum in the thermal conductance. We high-
light the effect of the second phonon path on the distinct behaviors of the dependence of the thermal conductance
on the impurity mass ratio. All our conclusions are confirmed both by Green’s Function calculations for the
equivalent quasi-1D lattice models and by molecular dynamics simulations.
PACS: 65.80.–g Thermal properties of small particles, nanocrystals, nanotubes, and other related systems;
43.40.+s Structural acoustics and vibration;
63.20.–e Phonons in crystal lattices;
66.70.Lm Other systems such as ionic crystals, molecular crystals, nanotubes, etc.
Keywords: phonon interference, thermal conductance, metasurface, molecular dynamics.
1. Introduction
Kapitza measured the thermal boundary resistance at
the interface of superfluid helium and a solid [1]. Kapitza
resistance R , more generally referred to as thermal inter-
face resistance, is defined as the ratio between the tempera-
ture difference T∆ at the interface over the heat flux per
unit area Q flowing across the interface,
= .TR
Q
∆ (1)
To explain such thermal resistance at the boundary to helium,
Khalatnikov proposed a model relating the transmission prob-
ability of phonon waves to their acoustic impedance of each
medium, which is now known as the acoustic mismatch mo-
del (AMM) [2]. AMM predicts well the experimental meas-
urements of the thermal boundary resistances for materials
with relative small acoustic mismatch. However, the
Khalatnikov formula overpredicts the Kapitza resistance at the
solid-helium interface by two orders of magnitude. To allevi-
ate this discrepancy between AMM and experiments, a dif-
fuse mismatch model (DMM) was proposed to consider the
density of vibrational (phonon) states in the calculation of
transmission probability, since the model assumes a complete
scattering of the incoming phonon waves [3]. Such scattering
opens additional phonon channels for heat transfer and hence
reduces the Kapitza resistance.
© Haoxue Han, Lei Feng, Shiyun Xiong, Takuma Shiga, Junichiro Shiomi, Sebastian Volz, and Yuriy A. Kosevich, 2016
Effects of phonon interference through long range interatomic bonds on thermal interface conductance
Due to the wave nature of phonons, interference could
play an important role in the thermal transport and hence
impact the thermal interface resistance [4–9]. Destructive
interference effects with total phonon reflection as a signa-
ture were discovered in acoustic systems [10,11]. Such an
enhanced acoustic reflection was first described theoreti-
cally in Refs. 10 and 11 independently. Fellay et al. [10]
studied the asymmetric profile in the phonon transmission
through one-dimensional chains, in analogy with electron
scattering. Reference 11 interpreted the reflection of an
acoustic wave by using two-dimensional crystal defects
for the destructive interference between two phonon
paths. Phonon interference effects can be employed in
manipulating the thermal transport in nanomaterials,
hence understanding of phonon wave dynamics is helpful
in improving the thermoelectric efficiency. The figure of
merit zT for thermoelectric conversion efficiency can be
expressed as 2= /zT S Tσ κ, where S , T , σ , and κ are the
Seebeck coefficient, temperature, electrical, and thermal
conductivities, respectively. Thus, a low thermal conduc-
tivity is favorable for good thermoelectric performance. Re-
cent efforts concentrated on reducing the thermal conduc-
tivity κ via nanostructured materials with grain boundaries
[12–14] and embedded nanoparticles [15–17]. Reducing κ
is often achieved by enhancing phonon scattering rate and
thus diminishing the mean free path (MFP), which belongs
to a particle description. Nevertheless, the role of destructive
phonon-wave interferences remains to be well understood in
the tailoring of thermal transport in a wave picture.
In the paper we investigate the role of the two-path de-
structive phonon interference induced by interatomic forc-
es beyond the nearest neighbor on the thermal conductance
of a silicon-germanium-like metasurface and a quasi-1D
harmonic chain model. Controlled by the ratio between the
second and first nearest-neighbor harmonic force con-
stants, the thermal conductance across a germanium atomic
plane in the silicon lattice exhibits a trend switch induced
by the destructive interference of the nearest-neighbor
phonon path with a direct path bypassing the defect atoms.
We show that the heavy isotope impurity is crucial in the
realization of the local minimum in the thermal conduct-
ance. We highlight the effect of the second phonon path on
the distinct behaviors of the dependence of the thermal
conductance on the impurity mass ratio.
2. Phonon interference in thermal conductance
The atomistic scheme of two-path phonon interference
is illustrated in the insets of Fig. 1. In most of the lattices
where the atoms are mainly coupled to each other through
the nearest-neighbor bonds, the propagation path of pho-
non is restricted to the host-guest atom bonds when en-
countering an impurity atom. An additional phonon path
becomes available when considering the second-nearest-
neighbor bonds between the host atoms on the two sides of
the impurity atom in addition to the first path through the
nearest-neighbor bonds linking the host and adjacent impu-
rity atoms, as shown in Fig. 1. Destructive interferences
will emerge due to the opening of the second phonon path
that couples directly crystal layers adjacent to the defect
atoms. We first investigate the intriguing role of such pho-
non interferences in the interfacial thermal conductance by
using molecular dynamics (MD) modeling of the transmis-
sion of phonon wave-packets (WP) propagating in a silicon
(Si) host lattice through a defect atomic plane of germani-
um (Ge), as shown in inset (i) of Fig. 1(b). Such WP mod-
eling provides the per-phonon-mode energy transmission
coefficient [9] (see Appendix 4). The spatial width l (co-
herence length) of the WP is taken much larger than the
wavelength cλ of the WP central frequency, corresponding
to the plane-wave approximation [7,8]. The MD simula-
tions were performed with the LAMMPS code package
Fig. 1. (Color online) (a) Spectral transmission coefficients
( , )α ω ρ predicted by MD simulations (open circles) for a Si host
crystal with a single atomic layer of Ge atoms and by Green's
Function calculations for an equivalent quasi-1D model (solid
lines). Only the longitudinal polarization is shown for the MD
prediction. (b) Thermal conductance ( )G ρ versus the relative
strength of the second nearest-neighbor bond ρ at = 300T K.
Open squares linked by a dashed line represent the thermal
conductances corresponding to the bond ratios ρ . Inset ( i ): host
silicon lattice with a single 001〈 〉 atomic layer of guest Ge atoms.
Inset ( ii ): quasi-1D tight-binding model which incorporates the
second nearest-neighbor bonds 11C bypassing the nearest-
neighbor bonds 12C between the host atom with mass 1m and the
guest atom with mass 2m . The host atoms are coupled through
the nearest-neighbor bonds 0C . Black (red) sticks represent the
NN (second NN) bonds. The region inside the red dashed rectan-
gle is the scattering region.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 8 903
Haoxue Han, Lei Feng, Shiyun Xiong, Takuma Shiga, Junichiro Shiomi, Sebastian Volz, and Yuriy A. Kosevich
[18]. The covalent Si:Si/Ge:Ge/Si:Ge interactions are
modeled by the Stillinger–Weber (SW) potential [19].
The spectral transmission coefficients ( , )α ω ρ retrieved
from MD simulations of a Si host crystal with a single
atomic layer of Ge atoms are illustrated in Fig. 1(a), where
the mass ratio of Ge and Si atoms Ge Si/m m = 2.6 was used.
ρ is defined as the ratio of the second nearest-neighbor
(NN) force constant 11C of Si atoms and the first-NN force
constant 12C between Si and Ge atoms, 11 12= /C Cρ . ρ
measures the relative strength of the second NN interaction
versus the NN force. Our First Principle calculations show
that 0.057ρ ≈ in natural SiGe alloys where the nearest
neighbor (NN) bond is 12 = 3.21C eV/Å2 (3.51 eV/Å2) for
the Si–Si (Si–Ge) atom pair, and the second-NN bond
strength is 11 = 0.187C eV/Å2 (0.192 eV/Å2) for the Si–Si
(Ge–Ge) pair (see Appendix 5). Empirical SW potential
reproduces relatively precise NN bonds but its second
NN bonds are negligible. When a single phonon path is
available for phonons crossing the Ge atomic plane, i.e.,
= 0ρ , the transmission coefficient ( , = 0)α ω ρ monoto-
nously decays as frequency increases with ( = 0) = 1α ω
and max( = ) = 0α ω ω . Relatively weak second NN interac-
tion = 0.16ρ reduces the transmission on the whole spec-
trum but the effect is especially strong at short wavelengths
when ( 15α ω ≥ THz) 0≈ . Such a remarkable drop in the
transmission coefficient is due to the destructive interfer-
ence between the two phonon paths: through the nearest-
neighbor Si–Ge bonds and through the non-nearest-neighbor
Si–Si bonds which couple directly atomic layers adjacent to
the defect plane [7]. For a larger = 0.28ρ , a total transmis-
sion antiresonance emerges ( = 13α ω THz) = 0 followed by
a local transmission maximum. When ρ further strength-
ens, the total reflection shifts to longer wavelengths and
the local maximum finally becomes a total transmission.
Figure 1(b) shows the thermal conductance G vs ρ at
T = 300 K. ( )G ρ is calculated by following the Landauer-
like formalism [2]:
,
3( ) = ( , ) ( , ) ,
(2 )
g
BE
dG v n T
T
⊥
ν
∂
ρ ω α ω ρ
∂ π
∑∫ q q
kq (2)
where q denotes the phonon mode ( , )νk , k is the wave
vector and ν is the phonon polarization. ,g
qv ⊥ is the group
velocity component perpendicular to the Ge atomic plane.
1= [exp ( / ) 1]BE Bn k T −ω −q is the Bose–Einstein distribu-
tion of phonons, where T refers to the mean temperature of
the system, Bk and represent the Boltzmann and the re-
duced Planck's constants, respectively. The thermal con-
ductance G first decreases as the second phonon path
strengthens, since the opening of the second phonon path
through the second NN bonds 11C interferes destructively
with the first path through the NN bonds 12C , which induc-
es a maximum reduction in G of 33% at = 0.28ρ . G then
starts to increase as the second NN bond further strength-
ens, due to the raised phonon transmission at high frequen-
cies, c.f. Fig. 1(a). The destructive interference is compen-
sated by the very strong second phonon path at large ρ. To
further understand the phonon antiresonances caused by
the destructive interference between two phonon channels,
we use an equivalent tight-binding model of a monatomic
quasi-1D lattice of coupled harmonic oscillators, depicted
in the inset (ii) of Fig. 1(b). The interatomic bonds could
be adjusted to the same as in the Si–Ge system to allow for
a direct comparison. The phonon transmission is calculated
by atomistic Green’s Function (AGF) and the thermal con-
ductance can be obtained by following the Landauer–
Büttiker formula:
max
0
= ( ) ,
2BE
dG n
T
ω
∂ ω
Ξ ω ω
∂ π∫ (3)
where ω and maxω are the energy and the Debye frequen-
cies. The transmission ( )Ξ ω is obtained from a
nonequilibrium Green’s function approach [20] (see Ap-
pendix 6 for the derivation of the dynamical matrices). As
shown in Fig. 1(a), the AGF prediction of the phonon
transmission from the quasi-1D model agrees relatively
well with that of the longitudinal phonons transmission
from MD simulations in a Si–Ge lattice.
We turn to the study of the phonon transport in the qua-
si-1D chain model in Fig. 1(b) with the second phonon
path but without mass defect. The mass of the guest atom
is the same as that of the host atoms, 2 1 Si= =m m m . The
phonon transmission through such a lattice is shown in
Fig. 2(a) and the associated thermal conductance at T =
= 300 K is shown in Fig. 2(b). When there is only a single
phonon path, i.e., 11 12= / = 0C Cρ , the total transmission
is recovered in the whole spectrum, max( [0, )) = 1α ω∈ ω .
When a weak second NN bond is available getting around
the guest atom, a local minimum arises in the spectrum of
high frequency phonons, whereas the transmission remains
perfect both at the long-wave limit and close to the maxi-
mum frequency, while we always have max( = ) = 0α ω ω .
As ρ increases, such local minimum shifts to higher fre-
quencies and finally turns to a transmission antiresonance
(complete reflection) at maxω for = 0.5ρ . As ρ increases,
the total reflection remains and shifts to lower frequencies,
as is found in Fig. 1(a). The thermal conductance in Fig.
4(b) shows a monotonous reduction as the second NN
bond strengthens. The minimum in the thermal conduct-
ance, which was found in Fig. 1(b), has disappeared.
Therefore, the presence of the heavy mass defect is crucial
to realize such a minimal thermal conductance in the mod-
el incorporating the second phonon path, cf. Figs. 2(b) and
2(b), in contrast to the realization of the transmission
antiresonance, which occurs in 2D metasurfaces with both
light and heavy impurities, cf. Figs. 1(a) and 2(a).
904 Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 8
Effects of phonon interference through long range interatomic bonds on thermal interface conductance
So far, we have investigated the strength variation of
second phonon path on the phonon transmission and ther-
mal conductance. In a real experiment, it can be tricky to
tune such second nearest neighbor forces. Nevertheless, it
is rather easy to change the mass of the impurity atoms by
choosing different isotopes of the host atoms. Moreover,
we have shown that the minimum of thermal conductance
can only be realized with heavy impurities. Hence it is in-
formative to discuss the dependence on impurity mass of
the phonon transport properties of a lattice incorporating
additional phonon paths. To this purpose, we investigate
the phonon transport in the quasi-1D chain model shown in
Fig. 1(b) with the second phonon path by highlighting the
role of the mass defect. We first study the case of a lattice
with an isotope with the relative strength of the second NN
bond = 0.058ρ which corresponds to the force constant
obtained from our first principle calculations. In Fig. 3(a),
we plot the phonon transmission in the quasi-1D model
with an impurity atom for different impurity mass ratio
[0.1, 3]mρ ∈ , and the related thermal conductance versus
mρ in Fig. 3(b). It is clear that on the whole spectrum the
phonon transmission first increases with mρ in the range
[0.1, 0.93]mρ ∈ and then decreases as the impurity mass
further increases. The thermal conductance follows the
same trend and reaches a maximum at 0.93mρ ≈ . Such a
behavior is rather straight ward to understand since for a
weak second NN bond, the phonon transmission is recov-
ered the best when the isotopic scattering is the least. A
perfect transmission would be achieved for = 1mρ when
only NN interaction is possible.
When the second NN bond 11C is comparably strong
with the NN bond 12 0=C C , the thermal conductance ver-
sus the impurity mass ratio mρ cannot be continued from
the limit of 11 0C ≈ . In Figs. 4(a) and 4(b), we compare the
phonon transmission and thermal conductance for impurity
mass ratios = 0.9,1.2mρ and for the second NN bond
strengths of = 0.5,2ρ . The dependence of the thermal
conductance on the impurity mass ratio can exhibit distinct
behavior for different second NN bonds, as can be seen
from Fig. 4(b). As the relative second NN bond strength
increases from = 0.5ρ to = 2ρ , the impurity-mass de-
pendent thermal conductance ( )mG ρ changes from a mo-
notonous reduction to a more complicated curve with a
local maximum at = 0.9mρ and a minimum at = 1.2mρ .
Fig. 2. (a) (Color online) Phonon transmission α of the quasi-1D model with the impurity mass 2 1 Si= =m m m in Fig. 1 vs frequency for dif-
ferent relative second NN bond strength ρ . (b) Thermal conductance G vs the relative strength of the second NN bond ρ at T = 300 K.
Fig. 3. (Color online) (a) Phonon transmission α of the quasi-1D model with an impurity atom in Fig. 1 vs frequency for different impu-
rity mass ratio mρ . The force constant ratio 0.058.ρ = The isotope of mass 2m is coupled to the host atoms of mass 1m and
2 1= /m m mρ with 1 Si=m m . (b) Thermal conductance ( )mG ρ vs the impurity mass ratio mρ at = 300T K.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 8 905
Haoxue Han, Lei Feng, Shiyun Xiong, Takuma Shiga, Junichiro Shiomi, Sebastian Volz, and Yuriy A. Kosevich
The corresponding phonon transmission spectra are shown
in Fig. 4(a). The transition of thermal conductance m( )G ρ
from its local minimum to the maximum is due to the in-
creased transmission of high frequency phonons, similar to
what was found in Fig. 1(a).
3. Conclusion
In conclusion, we investigate the role of second nearest
neighbor interatomic forces on the thermal conductance of
a silicon-germanium-like 2D metasurface. Controlled by
the ratio between the second and first nearest-neighbor
harmonic force constants, the thermal conductance across a
germanium atomic plane in the silicon lattice exhibits a
trend switch induced by the destructive interference of the
nearest-neighbor phonon path with a direct path bypassing
the defect atoms. We show that the heavy isotope impurity
bypassed by long range interatomic bonds is crucial for the
realization of the local minimum in the thermal conduct-
ance. We highlight the effect of the second phonon path on
the distinct behaviors of the dependence of the thermal
conductance on the impurity mass ratio. All our conclu-
sions are confirmed both by Green’s Function calculations
for the equivalent quasi-1D lattice models and by molecu-
lar dynamics simulations.
Appendix A: Phonon wave-packet technique using
molecular dynamics
To probe the phonon transmission, MD with the phonon
wave packet method was used to provide the per-phonon-
mode energy transmission coefficient ( , )lα ω . We excited a
realistic 3D Gaussian wave packet centered at the frequen-
cy ω and wave vector k in the reciprocal space and at 0r in
the real space, with the spatial width (coherence length) l
in the direction of k . The generation of the WP was per-
formed by assigning the displacement iu for the atom i as:
[ ]( )
2
0
0 2= ( )exp i ( ) exp ,
4
i g
i i i
t
A t
l
− − ⋅ − −ω −
r r v
u e k k r r
(A.1)
where A is the wave packet amplitude, ( )ie k is the phonon
polarization vector, ω is the eigenfrequency for the wave
vector k within a single branch of the phonon dispersion
curve, gv is the phonon group velocity along the wave
vector k at the wave packet center frequency ω. Wave
amplitude A of the generated phonon wave packets was
taken sufficiently small such that the anharmonic coupling
to other lattice modes is kept weak. Hence the wave pack-
ets propagate in an effectively harmonic crystal without
any perceptible spreading or scattering. The wave packet
was set to propagate normally to the defect layer, where an
elastic scattering results in transmitted and reflected waves.
The wave packet energy transmission coefficient ( , )lα ω is
defined as the ratio between the energy carried by the
transmitted and initial wave packets, centered at the given
phonon mode ( , )ω k with the spatial extent l . The plane-
wave limit is reproduced by the wave packets with the spa-
tial width l much larger than the wavelength cλ of the
wave packet central frequency. All the MD simulations
were performed with the LAMMPS code package.
Appendix B: Harmonic force constants determination
from First Principle calculations
We calculate the harmonic interatomic force constants
(IFCs) of silicon by using a real-space displacement meth-
od combined with First Principle calculations [31]. The
First Principle calculations were carried out using the
quantum chemistry DFT code Quantum ESPRESSO [32].
The calculation was conducted for a 2×2×2 supercells of
bulk silicon containing 64 atoms by the plane-wave basis
method implemented in the Quantum Espresso package.
We adopted both generalized gradient approximation
Fig. 4. (Color online) (a) Phonon transmission α of the quasi-1D model with a heavy impurity atom in Fig. 1(b) vs frequency for impuri-
ty mass ratio = 0.9,1.2mρ and for second NN bond strengths of = 0.5, 2.ρ (b) Thermal conductance G vs the impurity mass ratio
mρ at = 300T K for for second NN bond strength of = 0.5ρ and 2.
906 Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 8
Effects of phonon interference through long range interatomic bonds on thermal interface conductance
(GGA) for the exchange-correlation functional to study
pseudopotential-dependent phonon properties. The pa-
rameterizations of Perdew–Burke–Ernzerhof were select-
ed for GGA. The cutoff energies for plane-wave expan-
sion was set to 60 Ry (≈ 816 eV) for GGA. A
Monkhorst–Pack k-point grid of 4×4×4 was used to
achieve the desired accuracy. Constraints due to transla-
tional invariance and other symmetries were employed to
identify the minimum number of independent IFCs that
needed to be computed. We have set the range of har-
monic IFCs to five nearest neighbor shells. This results in
17 independent harmonic IFCs. The IFCs were fitted with
the ALAMODE package [33].
Appendix C:Thermal conductance and phonon
transmission in a 1D model using atomistic Green’s
function
In the scattering theory, the system contains three cou-
pled subsystems: two semi-infinite leads connected
through the scattering region. The heat flux flowing in
along the system axis writes
3
, 3= ( )
(2 )
g k L Rz
BZ
dJ v n n tω −
π∫ k k
k
(C.1)
where ωk is the energy quantum of the phonon mode k ,
,g kz
v is the phonon group velocity of the phonon mode zk ,
,L Rn is the phonon number on the left and right reservoir
following the Bose–Einstein distribution
1
= e 1k TBn
−ω
−
,
tk is the normalized transmission probability of the phonon
mode k and [ 0, 1 ]t ∈k . The integration goes through all the
phonon modes in the irreducible Brillouin Zone (BZ). In the
linear regime, the phonon population undergoes small per-
turbations and thus the thermal conductance writes
3
, 3= / =
(2 )
g kz
BZ
n dG J T v t
T
∂
∆ ω
∂ π∫ k k
k
. (C.2)
We note that 3
x y zd dk dk dk=k and ,g k zz
v dk =
/ z zk dk d= ∂ω ∂ = ω . The Eq. (C.2) reduces to
1
3/ e 1 .
(2 )
k TB x y
BZ
dG J T t dk dk
T
−ω
ω
∂ ω = ∆ = ω − ∂ π
∫
(C.3)
Hence we identify the spectral phonon transmission func-
tion ( ) = ( )t gωΞ ω ω where ( ) x yg dk dkω = is the projected
phonon density of states in the nonperiodic directions of
the system.
We probe the spectral phonon transmission function
( )Ξ ω by atomistic Green’s Function (AGF) and the ther-
mal conductance can be obtained by following the
Landauer formula:
1
max
0
= ( ) e 1
2
k TB dG
T
−ωω
∂ ω Ξ ω − ω ∂ π
∫
(C.4)
where ω and maxω are the energy and the Debye frequen-
cies. T refers to the mean temperature of the system, Bk
and represent the Boltzmann and the reduced Planck’s
constants, respectively. The transmission ( )Ξ ω is obtained
from a nonequilibrium Green’s function approach as
Tr[ ]L s R sG G+Γ Γ . The advanced and retarded Green func-
tions sG+ and sG can be deduced from
12= ( )s s L RG i I K
−
ω+ ∆ − −Σ −Σ (C.5)
where ∆ is an infinitesimal imaginary part that maintains
the causality of the Green’s function and =L ab L abK g K +Σ ,
=R ab R abK g K +Σ are the self-energies of the left and right
leads, the “+” exponent indicating the Hermitian conjuga-
tion. Finally, Lg and Rg refer to the surface Green’s func-
tions of the left and the right leads, while sK and abK are
the force constant matrices derived from the potential, for
the scattering region and between neighboring atoms in the
lead, respectively. The expression of the transmission also
includes = ( )L L Li +Γ Σ −Σ and = ( )R R Ri +Γ Σ −Σ .
In the quasi-1D tight-binding model of the Fig. 1(b) the
self-coupling matrix of the scattering region sK writes,
0 11 12 12 11
1 11 2
12 12 12
22 1 2 1
0 11 1211 12
1 11 2
2
= .s
C C C C C
m mm m
C C C
mm m m mK
C C CC C
m mm m
+ + − −
− −
+ + − −
The matrix L
abK coupling the scattering region to the left
lead writes,
0
0
= 0 0 .L
ab
C
K
m
−
The matrix R
abK coupling the scattering region to the right
lead writes,
0
0
= 0 0 .R
ab
C
K
m
−
In the calculations, the NN bonds in the host lead 0C
equals to that in the scattering region 12C .
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1. Introduction
2. Phonon interference in thermal conductance
3. Conclusion
Appendix A: Phonon wave-packet technique using molecular dynamics
Appendix B: Harmonic force constants determination from First Principle calculations
Appendix C:Thermal conductance and phonon transmission in a 1D model using atomistic Green’s function
|
| id | nasplib_isofts_kiev_ua-123456789-129285 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0132-6414 |
| language | English |
| last_indexed | 2025-12-07T17:12:05Z |
| publishDate | 2016 |
| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| record_format | dspace |
| spelling | Haoxue Han Lei Feng Shiyun Xiong Takuma Shiga Junichiro Shiomi Volz, Sebastian Kosevich, Yu.A. 2018-01-18T17:24:47Z 2018-01-18T17:24:47Z 2016 Effects of phonon interference through long range interatomic bonds on thermal interface conductance / Haoxue Han, Lei Feng, Shiyun Xiong, Takuma Shiga, Junichiro Shiomi, Sebastian Volz, Yuriy A. Kosevich // Физика низких температур. — 2016. — Т. 42, № 8. — С. 902-908. — Бібліогр.: 33 назв. — англ. 0132-6414 PACS: 65.80.–g, 43.40.+s, 63.20.–e, 66.70.Lm, https://nasplib.isofts.kiev.ua/handle/123456789/129285 We investigate the role of two-path destructive phonon interference induced by interatomic bonds beyond the
 nearest neighbor in the thermal conductance of a silicon-germanium-like metasurface. Controlled by the ratio between
 the second and first nearest-neighbor harmonic force constants, the thermal conductance across a germanium
 atomic plane in the silicon lattice exhibits a trend switch induced by the destructive interference of the
 nearest-neighbor phonon path with a direct path bypassing the defect atoms. We show that bypassing of the
 heavy isotope impurity is crucial to the realization of the local minimum in the thermal conductance. We highlight
 the effect of the second phonon path on the distinct behaviors of the dependence of the thermal conductance
 on the impurity mass ratio. All our conclusions are confirmed both by Green’s Function calculations for the
 equivalent quasi-1D lattice models and by molecular dynamics simulations. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур К 75-летию открытия теплового сопротивления Капицы Effects of phonon interference through long range interatomic bonds on thermal interface conductance Article published earlier |
| spellingShingle | Effects of phonon interference through long range interatomic bonds on thermal interface conductance Haoxue Han Lei Feng Shiyun Xiong Takuma Shiga Junichiro Shiomi Volz, Sebastian Kosevich, Yu.A. К 75-летию открытия теплового сопротивления Капицы |
| title | Effects of phonon interference through long range interatomic bonds on thermal interface conductance |
| title_full | Effects of phonon interference through long range interatomic bonds on thermal interface conductance |
| title_fullStr | Effects of phonon interference through long range interatomic bonds on thermal interface conductance |
| title_full_unstemmed | Effects of phonon interference through long range interatomic bonds on thermal interface conductance |
| title_short | Effects of phonon interference through long range interatomic bonds on thermal interface conductance |
| title_sort | effects of phonon interference through long range interatomic bonds on thermal interface conductance |
| topic | К 75-летию открытия теплового сопротивления Капицы |
| topic_facet | К 75-летию открытия теплового сопротивления Капицы |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/129285 |
| work_keys_str_mv | AT haoxuehan effectsofphononinterferencethroughlongrangeinteratomicbondsonthermalinterfaceconductance AT leifeng effectsofphononinterferencethroughlongrangeinteratomicbondsonthermalinterfaceconductance AT shiyunxiong effectsofphononinterferencethroughlongrangeinteratomicbondsonthermalinterfaceconductance AT takumashiga effectsofphononinterferencethroughlongrangeinteratomicbondsonthermalinterfaceconductance AT junichiroshiomi effectsofphononinterferencethroughlongrangeinteratomicbondsonthermalinterfaceconductance AT volzsebastian effectsofphononinterferencethroughlongrangeinteratomicbondsonthermalinterfaceconductance AT kosevichyua effectsofphononinterferencethroughlongrangeinteratomicbondsonthermalinterfaceconductance |