Phonon coherence and minimum thermal conductivity in disordered superlattice

Abstract

Phonon coherence elucidates the propagation and interaction of phonon quantum states within superlattice, unveiling the wave-like nature and collective behaviors of phonons. Taking MoSe2_2/WSe2_2 lateral heterostructures as a model system, we demonstrate that the intricate interplay between wave-like and particle-like phonons, previously observed in perfect superlattice only, also occurs in disordered superlattice. By employing molecular dynamics simulation based on a highly accurate and efficient machine-learned potential constructed herein, we observe a non-monotonic dependence of the lattice thermal conductivity on the interface density in both perfect and disordered superlattice, with a global minimum occurring at relatively higher interface density for disordered superlattice. The counter-intuitive phonon coherence contribution can be characterized by the lagged self-similarity of the structural sequences in the disordered superlattice. Our findings extend the realm of coherent phonon transport from perfect superlattice to more general structures, which offers more flexibility in tuning thermal transport in superlattices.

Full text

Phonon coherence and minimum thermal conductivity in disordered superlattice
Xin Wu,1Zhang Wu,2Ting Liang,3Zheyong Fan,4, ∗Jianbin Xu,3Masahiro Nomura,1, †and Penghua Ying5, ‡
1Institute of Industrial Science, The University of Tokyo, Tokyo 153-8505, Japan
2AVIC Jiangxi Hongdu Aviation Industry Group Company Ltd., Nanchang 330024, P. R. China
3Department of Electronic Engineering and Materials Science and Technology Research Center,
The Chinese University of Hong Kong, Shatin, N.T., Hong Kong SAR, 999077, P. R. China
4College of Physical Science and Technology, Bohai University, Jinzhou 121013, P. R. China
5Department of Physical Chemistry, School of Chemistry, Tel Aviv University, Tel Aviv, 6997801, Israel
(Dated: October 3, 2024)
Phonon coherence elucidates the propagation and interaction of phonon quantum states within su-
perlattice, unveiling the wave-like nature and collective behaviors of phonons. Taking MoSe2/WSe2
lateral heterostructures as a model system, we demonstrate that the intricate interplay between
wave-like and particle-like phonons, previously observed in perfect superlattice only, also occurs in
disordered superlattice. By employing molecular dynamics simulation based on a highly accurate
and efficient machine-learned potential constructed herein, we observe a non-monotonic dependence
of the lattice thermal conductivity on the interface density in both perfect and disordered super-
lattice, with a global minimum occurring at relatively higher interface density for disordered su-
perlattice. The counter-intuitive phonon coherence contribution can be characterized by the lagged
self-similarity of the structural sequences in the disordered superlattice. Our findings extend the
realm of coherent phonon transport from perfect superlattice to more general structures, which
offers more flexibility in tuning thermal transport in superlattices.
I. INTRODUCTION
Phonon thermal transport exhibits significant poten-
tial in nanoscale thermal physics, especially in semicon-
ductors and insulators where the lattice thermal conduc-
tivity (LTC) is almost entirely derived from lattice vibra-
tions [1]. As the quanta of lattice vibrations, phonons
exhibit particle-like behavior through energy quantiza-
tion and quasi-particle interactions, playing a crucial role
in thermal transport by affecting LTC through scatter-
ing and collision processes. However, the wave nature
of phonons is equally significant [2,3]. It describes the
characteristics of lattice vibrations through the relation-
ship between wave vector and frequency, manifesting
as dispersion relations in wave equations. By control-
ling phonon wave vectors and frequencies, one can in-
fluence phonon propagation paths and scattering mech-
anisms. For instance, when the phonon wavelength is
comparable to the characteristic dimensions of a mate-
rial, phonon transport exhibits significant coherence [4–
10]. This means phonons can travel coherently without
losing phase information, thereby significantly altering
the LTC and thermal transport properties of the mate-
rial.
In periodically alternating nanostructures known as su-
perlattices, composed of two or more different materials,
phonons experience multiple reflections and interference
at periodic interfaces, leading to the formation of new
phonon spectra and coherent transport [4,11]. The wave
nature of phonons becomes critically important. Perfect
∗brucenju@gmail.com
†nomura@iis.u-tokyo.ac.jp
‡hityingph@tauex.tau,ac.il
superlattices provide an ideal platform to explore and
utilize the coherent transport properties of phonons [12–
14]. This behavior has been theoretically predicted and
experimentally verified in various systems, from semicon-
ductors such as GaAs/AlAs [6,12] to two-dimensional
(2D) materials like graphene/h-BN [8,9,15]. However,
disordered superlattices offer complex interface scatter-
ing and localization effects, making them ideal systems
to contrast with perfect superlattices. These structures
are often used to verify phonon coherence in superlattices
but have not received equivalent attention or treatment
in their own right and remain elusive.
For predicting the properties of new structures and
materials, molecular dynamics (MD) simulation is an ex-
cellent choice, which can implicitly include all orders of
lattice anharmonicity and phonon scattering and provide
detailed atomic-level information [16]. It circumvents the
limitations of Boltzmann transport equation (BTE) in
describing higher-order phonon scattering and in han-
dling complex systems with large unitcells. Addition-
ally, machine-learned potential (MLP) [17] offers high-
precision descriptions of interatomic interactions, provid-
ing excellent descriptions for thermal transport in many
materials [18–22].
In this work, we developed a MLP based on the neu-
roevolution potential (NEP) framework [23–25] and used
nearly lattice-matched 2D MoSe2/WSe2(2H-phase) lat-
eral heterostructures as the model systems. We investi-
gated the LTC of perfect and disordered superlattices at
room temperature with various interface densities. De-
tailed analyses of the spectral LTC elucidate the origins
of two distinct minimum thermal conductivities in per-
fect and disordered superlattices. We also established
a factor of disorder to describe the LTC trends in dis-
ordered superlattices and revealed the LTC variations
arXiv:2410.01311v1 [cond-mat.mtrl-sci] 2 Oct 2024

2
FIG. 1. The construction of NEP model. (a) The cells of the reference structures, including pure MoSe2, pure WSe2, and their
heterostructures. (b) The energy, virial, and force calculated from the NEP model and the SW potential compared to the DFT
reference data.
in the coherent transport regime of perfect superlattices
through phonon band folding.
II. RESULTS AND DISCUSSIONS
A. NEP model evaluation
We employed the NEP approach [23–25] (see sec-
tion IV C for details) to train a MLP for MoSe2/WSe2
lateral heterostructures against the total energy, atomic
forces, and virials of a training dataset from density func-
tional theory (DFT) calculations (see section IV B for
details). As shown in Figure 2(a), the training dataset
includes pure MX2structures, their lateral heterostruc-
tures with both flat and embedded interfaces, and ran-
dom ternary MoxW1−xSe structures (see section IV A for
details).
As depicted in Figure 1(b), the NEP model achieves
high accuracy in both training and test datasets, with
root mean square errors (RMSEs) of energy, virial,
and force below 0.5 meV/atom, 3.3 meV/atom, and
23.7 meV/˚
A, respectively. For comparison, we evaluated
the atomic forces predicted by a Stillinger-Weber (SW)
potential [39] against DFT results and found that its
RMSE (384.2 meV/˚
A) is an order of magnitude higher.
In terms of computational efficiency, on a single GeForce
RTX 4090 graphic processing units (GPU), the present
NEP model can achieve 2.4 ×107atom-steps per sec-
ond as implemented in GPUMD.[25] This performance
is comparable to the speed of the traditional SW poten-
tial implemented in LAMMPS [40], which achieves 5.8 ×
107atom-step per second using 64 Xeon Platinum 9242
CPU cores.
After obtaining the NEP model, we verified its accu-
racy by comparing the predicted phonon dispersion and
LTC of MoSe2and WSe2with results from DFT, SW,
as well as experimental measurement. As shown in Fig-
ure 2(a), the phonon dispersion curves predicted by the
NEP model align perfectly with DFT, providing a much
more accurate band gap for both MoSe2and WSe2com-
pared to the SW potential [41]. For the LTC predic-
tions for monolayer MoSe2and WSe2(see Figure 2(d)),
our NEP-based homogeneous non-equilibrium molecu-
lar dynamics (HNEMD) results (see section IV D for
details) closely match experimental measurements [27–
29], whereas SW-driven MD simulations [35–38] signifi-
cantly underestimate the LTCs, aligning more with mea-
sured LTC of bulk MoSe2and WSe2[26]. The BTE-
anharmonic lattice dynamics (ALD) predictions [30–34]
based on DFT force constants (labeled as “DFT” in Fig-
ure 2(b)) exhibit large variations and less satisfactory
alignment to experimental results than the predictions
from NEP.
B. The model lateral heterostructures
As shown in Figure 3, we chose MoSe2and WSe2to
form a binary model system, both in the 2H phase. The

3
FIG. 2. Benchmark of the accuracy of the NEP model for MoSe2and WSe2. (a) The phonon dispersion relations of MoSe2
and WSe2calculated by DFT, the NEP model, and the SW potential. (b) Comparison of LTC of monolayer MoSe2and WSe2
with respect to other experimental [26–29], DFT-based [30–34], and SW-based MD [35–38] results at the room temperature.
The two horizontal bands are the results with error bars from this work by NEP-based MD, which show good agreement with
the experiments.
FIG. 3. Illustration of MoSe2/WSe2lateral heterostructures. (a) Two pairs of MoSe2/WSe2lateral heterostructures, each
consisting of a perfect (P) and a disordered (D) superlattice with equal interface density: P3paired with D2, and P5paired
with D4.Ldenotes the periodic length of the superlattice. (b) Local atomic-level zoomed-in view of the MoSe2/WSe2
heterostructure interface. (c) The unit cell of MoSe2and WSe2and their lattice constants.
optimized lattice constant from DFT (NEP) are 3.323 ˚
A
and 3.325 ˚
A(3.321 ˚
A and 3.321 ˚
A) for MoSe2and WSe2,
respectively, resulting in a negligible lattice mismatch.
Notably, a monolayer MoSe2/WSe2lateral heterostruc-
ture was recently synthesized and found to exhibit elec-
trical rectification [42].
The lateral heterostructures here include two types:
eight perfect superlattices (labeled as P1∼P8) and ten
disordered superlattices (labeled as D1∼D10 ). For each
superlattice, we define the interface density as the num-
ber of heterointerfaces per unit lateral length. This defi-
nition allows for a clear correspondence between perfect
and disordered superlattices. The structural parameters
of considered superlattices are listed inTable I, and Fig-
ure 3(a) presents the sequences of two pairs of perfect and
disordered superlattices (P3and D2,P5and D4) with
identical interface density. For a disordered superlattice
with the specified interface density, we generate the cor-
responding configurations using genetic algorithms [43]
under a constraint of MoSe2:WSe2= 1:1. In this case,

4
TABLE I. The MoSe2/WSe2perfect and disordered superlat-
tice with different interface densities considered in this work.
The fourth column, Ldenotes the periodic length of the su-
perlattice (see Figure 3(a))
Interface density
(nm−1)Perfect Disordered L(nm)
0.0351 P1N/A 56.9500
0.0702 P2D128.4750
0.1405 P3D214.2370
0.3512 P4D35.6950
0.7024 P5D42.8470
0.8780 P6D52.2780
1.2291 N/A D6N/A
1.7559 P7D71.1390
2.1071 N/A D8N/A
2.6339 N/A D9N/A
3.1607 N/A D10 N/A
3.5119 P8N/A 0.5695
the difference between perfect and disordered superlat-
tices with the same interface density lies solely in the
varying positions of the heterointerfaces.
One should note that for perfect superlattices, P1and
P8, which have the largest and smallest periodic lengths
(see Table I) respectively, there are no corresponding dis-
ordered counterparts. In the case of P1, with minimal
interface density and only a single heterointerface, alter-
ing the position of the heterointerface would inevitably
break the constraint of MoSe2:WSe2= 1:1. For P8with
maximum interface density, the periodic length equal to
the lattice constant of component orthogonal unit cells.
Thus, its all possible hetero-interface locations are fully
occupied, leaving no room for further adjustment. There-
fore, P1and P8represent the boundary limits for disor-
dered superlattices.
C. Thermal transport properties
Among MD methods for predicting LTC of 2D materi-
als, the HNEMD method is widely used for its efficiency
over other approaches such as equilibrium molecular dy-
namics (EMD) and non-equilibrium molecular dynamics
(NEMD) [20,44–47]. Therefore, we apply the HNEMD
approach (see section IV D for details) to calculate the
LTC of MoSe2/WSe2lateral heterostructures.
Figure 4 presents the LTC of MoSe2/WSe2lateral
heterostructures with different interface densities at the
room temperature of 300 K. As the interface density
increases, the HNEMD-predicted LTC of the perfect
superlattice exhibits a non-monotonic trend, first de-
creasing and then increasing, with a global minimum
at P2with an interface density of around 0.07 nm−1.
This behavior reflects the transition of phonon trans-
port from the incoherent to the coherent regime, a phe-
nomenon previously reported in various superlattice sys-
tems [5,6,8,9,12], which can be understood from the
FIG. 4. Thermal conductivity of MoSe2/WSe2lateral het-
erostructures, including perfect (P1∼P8, filled circles) and
disordered (D1∼D10, filled triangles) superlattices, as a func-
tion of interface density at 300K. The two horizontal solid
lines at the top represent the LTC of monolayer MoSe2and
WSe2, respectively. The dashed line for disordered superlat-
tices corresponds to the disordering factor R-based predic-
tions using Eq. (1) and Eq. (2). Groups marked with num-
bers will be selected as representative groups for subsequent
phonon analysis in Figure 6.
competition between LTC reduction due to enhanced in-
terface scattering and the LTC increase driven by co-
herent phonon transport. In the incoherent transport
regime of low interface density (below 0.07 nm−1), the
LTC of the perfect superlattice is primarily influenced
by interface scattering, with the lower interface density
of P1resulting in a significantly higher LTC compared
to P2. Conversely, when the interface density is high
(above 0.07 nm−1), phonon transport of perfect superlat-
tices transitions the coherent regime, resulting in a mono-
tonic increase in LTC with interface density. Notably, at
the maximum interface density of 3.51 nm−1, the super-
lattice P8, with its smallest indivisible unit, reaches a
maximum LTC of 58.32 W/(m K), approaching that of
MoSe2(61.75 W/(m K)) and WSe2(69.65 W/(m K)), in-
dicating that phonon coherence dominates LTC in this
region.
To further understand the phonon coherence observed
in perfect superlattices, in Figure 5 we examined the
phonon dispersion for P4, P6, and P8in the coherent
transport regime. The clear band folding of acoustic
phonons propagating through superlattice is observed
here, allowing phonons to propagate in homogeneous ma-
terials and free of interfaces. However, this folding intro-
duces two additional stop bands: One is the anti-crossing
point, which refers to the intra-mode stop bands gener-
ated by the crossing of folded phonon branches with other
phonons; another one is the band gap that occurred at

5
FIG. 5. Phonon dispersion relation of MoSe2/WSe2lateral
superlattice P4, P6, and P8within the coherent transport
regime. The first Brillouin zone is defined by the smallest
repeating unit of the superlattice, with the Γ-M direction
is along the periodicity of its components, also the thermal
transport direction. The phonon dispersion results were cal-
culated by the calorine [48] together with phonopy pack-
ages [49] based on the finite difference method, where the
force constants were calculated from the NEP model.
the center and boundary of the folded Brillouin zone. As
interface density increases (P4→P8), fewer bands in the
perfect superlattice fold, reducing the number of stop
bands, which may result in higher group velocity and
phonon mean free path (MFP), thereby enhancing the
LTC.
In disordered superlattices, phonon coherence is ex-
pected to be nearly absent, as shown in previous studies
[6,12]. Thus, the difference in LTC between the per-
fect and disordered superlattices with identical interface
densities reflects the contribution of phonon coherence
(see the vertical arrow in Figure 4). In this case, one
may intuitively expect the LTC of disordered superlat-
tices to decrease monotonically with increasing interface
density due to enhanced interface scattering. However,
we observed a similar non-monotonic trend in LTC for
disordered superlattices, as seen in perfect superlattices,
with the minimum LTC occurring near the maximum in-
terface density limit. This indicates that at very high
interface densities (above 1.76 nm−1), wave-like phonons
contribute significantly more to LTC than particle-like
phonons. We attribute this to the fact that, as the in-
terface density approaches the maximum limit, the dis-
ordered superlattice begins to resemble a perfect super-
lattice, exhibiting a high degree of lagged self-similarity
in the structural sequence.
To quantify the lagged self-similarity for disordered su-
perlattices, we can define a disordering factor that mea-
sures system disorder using the auto-correlation function
of parameterized sequences as follows:
R(τ) = 1
N−τ
N−τ−1
X
t=0 x(t)−µx(t+τ)−µ,(1)
where τis the lag parameter, indicating the step number
of sequence move backward, and µis the average of pa-
rameterized sequences. The disordering factor Rcan be
further obtained by considering the average of different
lag parameters R(τ):
R=1
n
n
X
τ=1
R(τ),(2)
Here we chose n= 4 to quantify R. Furthermore, one
can predict the LTC using a single scaling factor αand a
basis β:κ=αR+β. Remarkably, the αR +bprofiles are
in excellent agreement with the LTC of disordered super-
lattices predicted by HNEMD using α= 20.72 W/(m K)
and β= 6.2 W/(m K) (see Figure 4). This strong cor-
relation suggests that the disordering factor serves as a
simple and cost-effective predictor for interface density in
disordered superlattices with minimum LTC, and can be
adapted to describe other aperiodic heterostructures[12].
The LTCs of both perfect and disordered superlattices
span a wide range, from approximately 8 W/(m K) to
58 W/(m K). This demonstrates the promising potential
for LTC modulation in lateral heterostructures through
ordered- or disordered-sequence engineering, particularly
as the LTC range of order- and disorder-sequence het-
erostructures is distinct (see Figure 4).
To gain microscale insights into the interplay be-
tween particle-like and wave-like phonons in perfect and
disordered superlattices, we spectrally decomposed the
LTCs within the HNEMD framework using the spec-
tral heat current decomposition method (see section IV E
for details)[44]. Figure 6(a-b) present the spectral LTC
(κ(ω)) results for perfect (P1, P2, P3, P5, and P8)
and disordered superlattices (D1, D3, D7, D9, and
D10), respectively, each with five representative pairs
of perfect and disordered superlattices that have iden-
tical interface densities (see Figure 4), effectively cap-
turing the changing trends. For perfect superlattices,
the LTCs are primarily contributed by low-frequency in-
plane phonons below 4 THz. In the incoherent transport
regime (P1→P2), enhanced phonon-interface scattering
significantly weakens LTC from low-frequency phonons
below 4 THz. Upon entering a coherent transport regime
(P2→P8), increasing phonon coherence enables more
low-frequency, long-wavelength phonons to participate in
transport, primarily concentrated in the in-plane part,
also as shown in Figure Figure 6(c). This is in contrast to
single-atom-thickness graphene/C3N lateral heterostruc-
ture, where the LTC is much higher and dominated by
out-of-plane phonon modes [50]. This difference is at-
tributed to the sandwich structure of MoSe2and WSe2
makes their flexural phonon modes no longer the pure
out-of-plane vibration. As a result, the symmetry selec-
tion rules are broken, allowing three-phonon scattering
processes that involve an odd number of flexural phonons
[51].
As shown in Figure 6 (b), the κ(ω) of disordered super-
lattices, D1and D3, with low interface density differs sig-
nificantly from that of perfect superlattices, with peaks
appearing around 2 THz, rather than near the 0 THz

6
FIG. 6. Spectral LTCs of selected MoSe2/WSe2lateral heterostructures. (a-b) Total spectral LTC (left column) and corre-
sponding in-plane (middle column) and out-of-plane (right column) phonon contributions for (a) perfect and (b) disordered
superlattices. (c) Comparison of cumulative in-plane and out-of-plane phonon contributions to LTC for each heterostructure.
(d) Spectral phonon MFP for five representative structures of the MoSe2/WSe2lateral superlattice. In (a-b), the dashed
lines with arrows indicate the trend of spectral LTC peak that dominates the LTC with the increasing interface density. The
horizontal dashed lines are the reference lines for each κ(ω) value around 0 THz. Colored shaded areas represent error bars
from multiple independent simulations.
as in superlattices. Compared with D1, the enhanced
phonon-interface scattering in D3results in a weakened
κ(ω) in the entire frequency range, with peaks attenu-
ated but still present. However, when the interface den-
sity increases from around 0.35 nm−1(D3) to 1.76 nm−1
(D7), the κ(ω) profile undergoes a complete transfor-
mation, with the peak shifting towards the 0 THz, re-
sembling the peak shape observed in perfect superlat-
tices. As indicated by the dashed arrows labeled in Fig-
ure 6(a-b), disordered superlattices show spectral peak
shifts that are absent in superlattices. In the high inter-
face density region (D7→D10) where LTC increases with
interface density, in-plane phonons below 2 THz domi-
nants the increase and exhibit coherent characteristics
similar to those observed in perfect superlattices (also
see P2→P8in Figure 6(c)). These results suggest that
phonon coherence contributes to the increase in LTC in
disordered superlattices at high interface densities, as the
smaller differences between adjacent units in these con-
figurations facilitate the transmission of low-frequency,
long-wavelength wave-like phonons.
As defined in Eq. (8) (see section IV D), one can pre-
dict frequency-dependent phonon MFP, λ(ω), using the
following equation:[44]
λ(ω) = κ(ω)
G(ω),(3)
where G(ω) is the spectrally thermal conductance in the
ballistic regime. Figure 6(d) depicts the λ(ω) profiles for
representative perfect superlattices below 6 THz, the fre-
quency range that contributes most to LTC. The phonon
MFPs align with their LTCs, especially for phonons be-
low 2 THz. In the coherent transport regime, enhanced
phonon coherence reduces scattering and extends the
phonon MFPs. Coherent phonon wave packets maintain
their phase relationships, minimizing incoherent scatter-
ing, particularly for low-frequency phonons, whose longer
wavelengths favor coherent states.
III. SUMMARY AND CONCLUSIONS
In summary, we develop an accurate and efficient
machine-learned NEP model for lateral MoSe2/WSe2
heterostructure, and then perform extensive HNEMD
simulations to investigate thermal transport in both per-
fect and disordered MoSe2/WSe2superlattices. Surpris-
ingly, both perfect and disordered superlattices show

7
a minimum LTC with varying interface density, driven
by the transition from incoherent to coherent transport
regimes. The counter-intuitive thermal transport behav-
ior of disorder superlattices is well captured by the dis-
ordering factor, defined by the auto-correlation function
of the parameterized sequence. For the first time, spec-
tral LTC decomposition reveals wave-like low-frequency
(long-wavelength) phonons contribute to the increase in
LTC of disordered superlattices with high interface densi-
ties. We also explained the intrinsic mechanism of coher-
ent phonons enhancing LTC in the perfect superlattices
through phonon MFP and stop bands analysis. These
findings provide novel physical insights into tuning ther-
mal transport of 2D lateral heterostructures through se-
quence ordering and disordering, offering a framework
applicable to other analogous systems.
IV. METHODS
A. The training dataset
As shown in Figure 1(a), our training dataset con-
sists of nine different configurations, i.e., two pure MX2
structures (MoSe2and WSe2), two MoSe2/WSe2het-
erostructures with the ideal flat interface and mutually
embedded interface, and five random ternary transitional
MoxW1−xSe structures with xincreasing from 1
6to 5
6in
increments of 1
6. For each configuration, we performed
constant volume MD simulation driven by the SW po-
tential developed by Jiang et al. [39] for 150 ps with the
target temperature linearly increasing from 100 to 800
K and sampled 15 structures. In addition to this, we
generated 5 structures by applying random cell deforma-
tions (-3 to 3%) and atomic displacements (within 0.1 ˚
A)
starting from each initial configuration. In total, we ob-
tained 9 ×(15 + 5) = 180 structures, which were further
randomly divided to a training dataset (150 structures)
and a test dataset (30 structures).
B. DFT calculations
To obtain the total energy, atomic forces, and virial
for selected structures in the reference dataset, we use
the VASP code [52–54] to perform single-point DFT cal-
culations at PBE [55] level. A plane-wave basis set was
employed with the energy cutoff of 650 eV and a dense
Γ-centered grid with a k-point density of 0.25/˚
A was sam-
pled in the Brillouin zone. We set a threshold of 10−7eV
for the electronic self-consistent loop.
C. NEP model training
Using the provided training and test datasets, we
trained the NEP model for the MoSe2/WSe2het-
erosystem with the NEP3 architecture, implemented in
GPUMD.[25] The NEP approach [23–25] uses a simple
feedforward artificial neural network (ANN) to represent
the site energy Ui[56] of atom ias a function of a de-
scriptor vector with Ndes components:
Ui=
Nneu
X
µ=1
w(1)
µtanh Ndes
X
ν=1
w(0)
µν qi
ν−b(0)
µ!−b(1),(4)
where tanh(x) is the activation function, w(0),w(1),b(0) ,
and b(1) are the trainable weight and bias parameters in
the ANN, and qi
νis the descriptor vector constructed sim-
ilarly to the atomic cluster expansion (ACE) approach
[57], but treating radial and angular components sepa-
rately. Based on the separable natural evolution strategy
(SNES) method [58], the optimization of NEP model pa-
rameters involves minimizing a loss function that incor-
porates a weighted sum of the RMSEs for energy, force,
and virial, along with regularization terms.
After extensive testing, we determined the following
hyper-parameters for NEP training: the cutoffs of radial
and angular descriptors were both set to 5 ˚
A and with 8
radial functions. For angular terms, we focused on three-
body and four-body interactions, with maximum expan-
sion orders of 4 and 2, respectively. A single hidden layer
with 50 neurons was used for the ANN. The optimization
was performed with a population size of 50 for 5 ×105
steps. In the loss function, we assigned weights of 1.0 to
the RMSEs of energy and force, and 0.1 to the RMSE of
virial.
D. The HNEMD method
Based on the linear response theory, the HNEMD
method simulates the thermal gradient effect in a solid
by applying a directional driving force Fe
ion each atom
i[44,59]:
Fe
i=Fe·Wi,(5)
where Feis the external driving force parameter with
the dimension of inverse length, and Wiis virial ten-
sor of atom i. The running LTC, κ(t), along the lateral
transport direction can be calculated as:
κ(t) = ⟨J(t)⟩ne
T V Fe
,(6)
where kBis the Boltzmann constant, Tis the temper-
ature, Vis the volume of the system, and ⟨J⟩ne is
non-equilibrium ensemble average of the heat current
J=PiWi·vi, with vibeing the velocity of atom
i. A thickness of 0.65 nm is used to calculate the vol-
ume Vin Eq. (6) for all structures. To observe the
convergence of the calculated LTC results, Eq. (6) can
be further redefined as the following cumulative average:
κ(t) = 1
tRt
0κ(τ)dτ.
We set the magnitude of the driving force parameter to
be Fe=1 ×10−5/˚
A along lateral transport directions for

8
all heterostructures, which is small enough to maintain
the linear response regime and is large enough to achieve
a sufficiently large signal-to-noise ratio. All heterostruc-
tures were set to a length of 56.95 nm along the thermal
transport direction, with a width of 9.86 nm and a to-
tal of 18,000 atoms. Periodic boundary conditions were
applied in both in-plane directions. For the superlattice
group, five independent runs were conducted to enhance
the statistical accuracy and obtain an error estimate. For
the disordered group, three different configurations were
created for each interface density, with three independent
simulations performed for each configuration. All MD
simulations were performed using gpumd package.[25].
In all MD simulations, the system was first equilibrated
at 300 K and zero pressure for 1 ns in the NPT ensemble.
In the production stage, the Nose-Hoover chain thermo-
stat [60] was used to maintain the overall temperature
and heat current data were collected for 10ns. A time
step of 1 fs was used in all the MD simulations.
E. Spectral heat current decomposition
In the framework of the HNEMD method, one can cal-
culate spectrally decomposed thermal conductivity with
the following formula [44]:
κ=Z∞
0
dω
2πκ(ω),(7)
where
κ(ω) = 2
V T FeZ∞
−∞
dteiωtK(t).(8)
Here, K(t) is the x-component (along lateral direction)
of the virial-velocity correlation function [61]:
K(t) = X
i
⟨Wi(0) ·vi(t)⟩.(9)
For 2D materials considered here, the spectral LTC κ(ω)
in Eq. (8) can be further decomposed into in-plane and
out-of-plane (flexural) phonon contributions [62].
ACKNOWLEDGMENTS
X. Wu is the JSPS Postdoctoral Fellow for Research
in Japan (No. P24058). T. Liang and J. Xu acknowl-
edge support from the National Key R&D Project from
the Ministry of Science and Technology of China (Grant
No. 2022YFA1203100), the Research Grants Council of
Hong Kong (Grant No. AoE/P-701/20), and RGC GRF
(Grant No. 14220022). X. Wu and M. Nomura acknowl-
edge support from the JSPS Grants-in-Aid for Scien-
tific Research (Grant Nos. 21H04635) and JST SICORP
EIG CONCERT-Japan (Grant No. JPMJSC22C6). P.
Ying is supported by the Israel Academy of Sciences and
Humanities & Council for Higher Education Excellence
Fellowship Program for International Postdoctoral Re-
searchers.
Conflict of Interest
The authors have no conflicts to disclose.
Data availability
The source code and documentation for gpumd
are available at https://github.com/brucefan1983/
GPUMD and https://gpumd.org, respectively. The doc-
umentation for calorine is available at https://
calorine.materialsmodeling.org. The documenta-
tion for phonopy is available at https://phonopy.
github.io/phonopy/. The inputs and outputs related
to the NEP model training are freely available at the
Gitlab repository https://gitlab.com/brucefan1983/
nep-data.
[1] Gang Chen, Nanoscale Energy Transport and Conver-
sion: A Parallel Treatment of Electrons, Molecules,
Phonons, and Photons (Oxford University Press, 2005).
[2] Zhongwei Zhang, Yangyu Guo, Marc Bescond, Jie Chen,
Masahiro Nomura, and Sebastian Volz, “How coherence
is governing diffuson heat transfer in amorphous solids,”
npj Computational Materials 8, 96 (2022,12).
[3] Zhongwei Zhang, Yangyu Guo, Marc Bescond, Jie Chen,
Masahiro Nomura, and Sebastian Volz, “Generalized de-
cay law for particlelike and wavelike thermal phonons,”
Physical Review B 103, 184307 (2021).
[4] S. Tamura, D. C. Hurley, and J. P. Wolfe, “Acoustic-
phonon propagation in superlattices,” Physical Review
B38, 1427–1449 (1988).
[5] M. V. Simkin and G. D. Mahan, “Minimum thermal con-
ductivity of superlattices,” Physical Review Letters 84,
927–930 (2000).
[6] Maria N. Luckyanova, Jivtesh Garg, Keivan Esfarjani,
Adam Jandl, Mayank T. Bulsara, Aaron J. Schmidt,
Austin J. Minnich, Shuo Chen, Mildred S. Dresselhaus,
Zhifeng Ren, Eugene A. Fitzgerald, and Gang Chen,
“Coherent phonon heat conduction in superlattices,” Sci-
ence 338, 936–939 (2012).
[7] B. Latour, S. Volz, and Y. Chalopin, “Microscopic de-
scription of thermal-phonon coherence: From coherent
transport to diffuse interface scattering in superlattices,”
Physical Review B 90, 014307 (2014).
[8] Isaac M. Felix and Luiz Felipe C. Pereira, “Thermal con-
ductivity of graphene-hbn superlattice ribbons,” Scien-
tific Reports 8, 2737 (2018).
[9] Xin Wu and Qiang Han, “Transition from incoherent to
coherent phonon thermal transport across graphene/h-
bn van der waals superlattices,” International Journal of
Heat and Mass Transfer 184, 122390 (2022).

9
[10] Masahiro Nomura, Roman Anufriev, Zhongwei Zhang,
Jeremie Maire, Yangyu Guo, Ryoto Yanagisawa, and Se-
bastian Volz, “Review of thermal transport in phononic
crystals,” Materials Today Physics 22, 100613 (2022).
[11] C. Colvard, T. A. Gant, M. V. Klein, R. Merlin, R. Fis-
cher, H. Morkoc, and A. C. Gossard, “Folded acoustic
and quantized optic phonons in (gaal)as superlattices,”
Physical Review B 31, 2080–2091 (1985).
[12] Run Hu, Sotaro Iwamoto, Lei Feng, Shenghong Ju,
Shiqian Hu, Masato Ohnishi, Naomi Nagai, Kazuhiko
Hirakawa, and Junichiro Shiomi, “Machine-learning-
optimized aperiodic superlattice minimizes coherent
phonon heat conduction,” Physical Review X 10, 021050
(2020).
[13] Yan Wang, Haoxiang Huang, and Xiulin Ruan, “Decom-
position of coherent and incoherent phonon conduction in
superlattices and random multilayers,” Physical Review
B90, 165406 (2014).
[14] Han Wei, Yue Hu, Hua Bao, and Xiulin Ruan, “Quan-
tifying the diverse wave effects in thermal transport of
nanoporous graphene,” Carbon 197, 18–26 (2022).
[15] Zhang Wu, Rumeng Liu, Ning Wei, and Lifeng Wang,
“Unexpected reduction in thermal conductivity observed
in graphene/h-bn heterostructures,” Physical Chemistry
Chemical Physics 26, 3823–3831 (2024).
[16] Hua Bao, Jie Chen, Xiaokun Gu, and Bingyang Cao,
“A review of simulation methods in micro/nanoscale
heat conduction,” ES Energy & Environment (2018),
10.30919/esee8c149.
[17] J¨org Behler, “Perspective: Machine learning potentials
for atomistic simulations,” The Journal of Chemical
Physics 145, 170901 (2016).
[18] Yanzhou Wang, Zheyong Fan, Ping Qian, Miguel A.
Caro, and Tapio Ala-Nissila, “Quantum-corrected
thickness-dependent thermal conductivity in amorphous
silicon predicted by machine learning molecular dynam-
ics simulations,” Physical Review B 107, 054303 (2023).
[19] Ting Liang, Penghua Ying, Ke Xu, Zhenqiang Ye, Chao
Ling, Zheyong Fan, and Jianbin Xu, “Mechanisms of
temperature-dependent thermal transport in amorphous
silica from machine-learning molecular dynamics,” Phys-
ical Review B 108, 184203 (2023).
[20] Penghua Ying, Ting Liang, Ke Xu, Jin Zhang, Jianbin
Xu, Zheng Zhong, and Zheyong Fan, “Sub-micrometer
phonon mean free paths in metal–organic frameworks re-
vealed by machine learning molecular dynamics simula-
tions,” ACS Applied Materials & Interfaces 15, 36412–
36422 (2023).
[21] Ke Xu, Yongchao Hao, Ting Liang, Penghua Ying, Jian-
bin Xu, Jianyang Wu, and Zheyong Fan, “Accurate pre-
diction of heat conductivity of water by a neuroevolution
potential,” The Journal of Chemical Physics 158, 204114
(2023).
[22] Haikuan Dong, Yongbo Shi, Penghua Ying, Ke Xu, Ting
Liang, Yanzhou Wang, Zezhu Zeng, Xin Wu, Wenjiang
Zhou, Shiyun Xiong, Shunda Chen, and Zheyong Fan,
“Molecular dynamics simulations of heat transport using
machine-learned potentials: A mini-review and tutorial
on gpumd with neuroevolution potentials,” Journal of
Applied Physics 135, 161101 (2024).
[23] Zheyong Fan, Zezhu Zeng, Cunzhi Zhang, Yanzhou
Wang, Keke Song, Haikuan Dong, Yue Chen, and Tapio
Ala-Nissila, “Neuroevolution machine learning poten-
tials: Combining high accuracy and low cost in atomistic
simulations and application to heat transport,” Physical
Review B 104, 104309 (2021).
[24] Zheyong Fan, “Improving the accuracy of the neuroevolu-
tion machine learning potential for multi-component sys-
tems,” Journal of Physics: Condensed Matter 34, 125902
(2022).
[25] Zheyong Fan, Yanzhou Wang, Penghua Ying, Keke Song,
Junjie Wang, Yong Wang, Zezhu Zeng, Ke Xu, Eric
Lindgren, J. Magnus Rahm, Alexander J. Gabourie, Ji-
ahui Liu, Haikuan Dong, Jianyang Wu, Yue Chen, Zheng
Zhong, Jian Sun, Paul Erhart, Yanjing Su, and Tapio
Ala-Nissila, “Gpumd: A package for constructing accu-
rate machine-learned potentials and performing highly
efficient atomistic simulations,” The Journal of Chemi-
cal Physics 157, 114801 (2022).
[26] Puqing Jiang, Xin Qian, Xiaokun Gu, and Ronggui
Yang, “Probing anisotropic thermal conductivity of tran-
sition metal dichalcogenides mx2 (m = mo, w and x =
s, se) using time-domain thermoreflectance,” Advanced
Materials 29, 1701068 (2017).
[27] Xian Zhang, Dezheng Sun, Yilei Li, Gwan-Hyoung Lee,
Xu Cui, Daniel Chenet, Yumeng You, Tony F. Heinz,
and James C. Hone, “Measurement of lateral and interfa-
cial thermal conductivity of single- and bilayer mos2 and
mose2 using refined optothermal raman technique,” ACS
Applied Materials & Interfaces 7, 25923–25929 (2015).
[28] Yifei Yu, Tamzid Minhaj, Lujun Huang, Yiling Yu,
and Linyou Cao, “In-plane and interfacial thermal con-
duction of two-dimensional transition-metal dichalco-
genides,” Physical Review Applied 13, 034059 (2020).
[29] Elham Easy, Yuan Gao, Yingtao Wang, Dingkai Yan,
Seyed M. Goushehgir, Eui-Hyeok Yang, Baoxing Xu, and
Xian Zhang, “Experimental and computational investiga-
tion of layer-dependent thermal conductivities and inter-
facial thermal conductance of one- to three-layer wse2,”
ACS Applied Materials & Interfaces 13, 13063–13071
(2021).
[30] Xiaokun Gu and Ronggui Yang, “Phonon transport in
single-layer transition metal dichalcogenides: A first-
principles study,” Applied Physics Letters 105, 131903
(2014).
[31] Bo Peng, Hao Zhang, Hezhu Shao, Yuchen Xu, Xi-
angchao Zhang, and Heyuan Zhu, “Thermal conductiv-
ity of monolayer mos 2, mose 2, and ws 2: Interplay
of mass effect, interatomic bonding and anharmonicity,”
RSC Advances 6, 5767–5773 (2016).
[32] Roberta Farris, Olle Hellman, Zeila Zanolli, David
Saleta Reig, Sebin Varghese, Pablo Ordej´on, Klaas-Jan
Tielrooij, and Matthieu Jean Verstraete, “Microscopic
understanding of the in-plane thermal transport prop-
erties of 2 h transition metal dichalcogenides,” Physical
Review B 109, 125422 (2024).
[33] Kunpeng Yuan, Xiaoliang Zhang, Lin Li, and Dawei
Tang, “Effects of tensile strain and finite size on ther-
mal conductivity in monolayer wse2,” Physical Chem-
istry Chemical Physics 21, 468–477 (2018).
[34] Yingtao Wang and Xian Zhang, “On the role of crystal
defects on the lattice thermal conductivity of monolayer
wse2 (p63/mmc) thermoelectric materials by dft calcu-
lation,” Superlattices and Microstructures 160, 107057
(2021).
[35] Yang Hong, Jingchao Zhang, and Xiao Cheng Zeng,
“Thermal conductivity of monolayer mose2 and mos2,”
The Journal of Physical Chemistry C 120, 26067–26075

10
(2016).
[36] Ali Kandemir, Haluk Yapicioglu, Alper Kinaci, Tahir
C¸a˘gın, and Cem Sevik, “Thermal transport properties
of mos 2and mose 2monolayers,” Nanotechnology 27,
055703 (2016).
[37] Jingchao Zhang, Yang Hong, Xinyu Wang, Yanan
Yue, Danmei Xie, Jin Jiang, Yangheng Xiong, and
Peisheng Li, “Phonon thermal properties of transition-
metal dichalcogenides mos 2and mose 2heterostructure,”
The Journal of Physical Chemistry C 121, 10336–10344
(2017).
[38] Arash Mobaraki, Cem Sevik, Haluk Yapicioglu, Deniz
C¸akır, and O˘guz G¨ulseren, “Temperature-dependent
phonon spectrum of transition metal dichalcogenides cal-
culated from the spectral energy density: Lattice thermal
conductivity as an application,” Physical Review B 100,
035402 (2019).
[39] Jin-Wu Jiang, “Misfit strain-induced buckling for
transition-metal dichalcogenide lateral heterostructures:
A molecular dynamics study,” Acta Mechanica Solida
Sinica 32, 17–28 (2019).
[40] Aidan P. Thompson, H. Metin Aktulga, Richard Berger,
Dan S. Bolintineanu, W. Michael Brown, Paul S. Crozier,
Pieter J. in ’t Veld, Axel Kohlmeyer, Stan G. Moore,
Trung Dac Nguyen, Ray Shan, Mark J. Stevens, Julien
Tranchida, Christian Trott, and Steven J. Plimp-
ton, “LAMMPS-A Flexible Simulation Tool for Particle-
Based Materials Modeling at the Atomic, Meso, and Con-
tinuum Scales,” Computer Physics Communications 271,
108171 (2022).
[41] Jinwu Jiang and Yuping Zhou, eds., Handbook
of Stillinger-Weber Potential Parameters for Two-
Dimensional Atomic Crystals (InTech, 2017).
[42] Yufeng Zhang, Qian Lv, Haidong Wang, Shuaiyi Zhao,
Qihua Xiong, Ruitao Lv, and Xing Zhang, “Simultane-
ous electrical and thermal rectification in a monolayer
lateral heterojunction,” Science 378, 169–175 (2022).
[43] Darrell Whitley, “A genetic algorithm tutorial,” Statis-
tics and computing 4, 65–85 (1994).
[44] Zheyong Fan, Haikuan Dong, Ari Harju, and Tapio
Ala-Nissila, “Homogeneous nonequilibrium molecular dy-
namics method for heat transport and spectral decom-
position with many-body potentials,” Physical Review B
99, 064308 (2019).
[45] Xiaokun Gu, Zheyong Fan, and Hua Bao, “Thermal
conductivity prediction by atomistic simulation methods:
Recent advances and detailed comparison,” Journal of
Applied Physics 130, 210902 (2021).
[46] Xin Wu and Qiang Han, “Thermal transport in pristine
and defective two-dimensional polyaniline (c3n),” Inter-
national Journal of Heat and Mass Transfer 173, 121235
(2021).
[47] Haikuan Dong, Yongbo Shi, Penghua Ying, Ke Xu, Ting
Liang, Yanzhou Wang, Zezhu Zeng, Xin Wu, Wenjiang
Zhou, Shiyun Xiong, et al., “Molecular dynamics simula-
tions of heat transport using machine-learned potentials:
A mini-review and tutorial on gpumd with neuroevolu-
tion potentials,” Journal of Applied Physics 135, 161101
(2024).
[48] Eric Lindgren, Magnus Rahm, Erik Fransson, Fredrik
Eriksson, Nicklas ¨
Osterbacka, Zheyong Fan, and Paul
Erhart, “Calorine: A python package for constructing
and sampling neuroevolution potential models,” Journal
of Open Source Software 9, 6264 (2024).
[49] Atsushi Togo, “First-principles phonon calculations with
phonopy and phono3py,” Journal of the Physical Society
of Japan 92, 012001 (2023).
[50] Xin Wu, Penghua Ying, Chunlei Li, and Qiang Han,
“Dual effects of hetero-interfaces on phonon thermal
transport across graphene/c3n lateral superlattices,” In-
ternational Journal of Heat and Mass Transfer 201,
123643 (2023).
[51] Xiaokun Gu, Yujie Wei, Xiaobo Yin, Baowen Li, and
Ronggui Yang, “Colloquium: Phononic thermal proper-
ties of two-dimensional materials,” Reviews of Modern
Physics 90, 041002 (2018).
[52] Georg Kresse and J¨urgen Furthm¨uller, “Efficiency of ab-
initio total energy calculations for metals and semicon-
ductors using a plane-wave basis set,” Computational
materials science 6, 15–50 (1996).
[53] Georg Kresse and J¨urgen Furthm¨uller, “Efficient Iter-
ative Schemes for Ab Initio Total-Energy Calculations
Using a Plane-Wave Basis Set,” Physical Review B 54,
11169 (1996).
[54] Georg Kresse and Daniel Joubert, “From Ultrasoft
Pseudopotentials to the Projector Augmented-Wave
Method,” Physical Review B 59, 1758 (1999).
[55] Peter E Bl¨ochl, “Projector augmented-wave method,”
Physical review B 50, 17953 (1994).
[56] J¨org Behler and Michele Parrinello, “Generalized Neural-
Network Representation of High-Dimensional Potential-
Energy Surfaces,” Physical Review Letters 98, 146401
(2007).
[57] Ralf Drautz, “Atomic cluster expansion for accurate and
transferable interatomic potentials,” Physical Review B
99, 014104 (2019).
[58] Tom Schaul, Tobias Glasmachers, and J¨urgen Schmid-
huber, “High Dimensions and Heavy Tails for Natural
Evolution Strategies,” in Proceedings of the 13th Annual
Conference on Genetic and Evolutionary Computation,
GECCO ’11 (Association for Computing Machinery, New
York, NY, USA, 2011) pp. 845–852.
[59] Denis J. Evans, “Homogeneous nemd algorithm for ther-
mal conductivity—application of non-canonical linear re-
sponse theory,” Physics Letters A 91, 457–460 (1982).
[60] Mark E Tuckerman, Statistical mechanics: theory and
molecular simulation (Oxford university press, 2023).
[61] Alexander J Gabourie, Zheyong Fan, Tapio Ala-Nissila,
and Eric Pop, “Spectral decomposition of thermal con-
ductivity: Comparing velocity decomposition methods in
homogeneous molecular dynamics simulations,” Physical
Review B 103, 205421 (2021).
[62] Zheyong Fan, Luiz Felipe C Pereira, Petri Hirvonen,
Mikko M Ervasti, Ken R Elder, Davide Donadio, Tapio
Ala-Nissila, and Ari Harju, “Thermal conductivity de-
composition in two-dimensional materials: Application
to graphene,” Physical Review B 95, 144309 (2017).