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Nanoscale
MINIREVIEW
Cite this: DOI: 10.1039/d4nr04385e
Received 24th October 2024,
Accepted 28th November 2024
DOI: 10.1039/d4nr04385e
rsc.li/nanoscale
Mathematically inspired structure design in
nanoscale thermal transport
Xin Wu * and Masahiro Nomura *
Mathematically inspired structure design has emerged as a powerful approach for tailoring material pro-
perties, especially in nanoscale thermal transport, with promising applications both within this field and
beyond. By employing mathematical principles, based on number theory, such as periodicity and quasi-
periodic organizations, researchers have developed advanced structures with unique thermal behaviours.
Although periodic phononic crystals have been extensively explored, various structural design methods
based on alternative mathematical sequences have gained attention in recent years. This review provides
an in-depth overview of these mathematical frameworks, focusing on nanoscale thermal transport. We
examine key mathematical sequences, their foundational principles, and analyze the influence of thermal
behavior, highlighting recent advancements in this field. Looking ahead, further exploration of mathemat-
ical sequences offers significant potential for the development of next-generation materials with tailored,
multi-functional properties suited to diverse technological applications.
1 Introduction
The design of advanced materials has become increasingly
crucial in the quest to enhance performance in a wide range of
technological applications, from energy storage and electronics
to thermal management systems.
1–8
A particularly critical chal-
lenge is the ability to control and optimize thermal transport pro-
perties, as effective thermal management directly impacts the
efficiency and longevity of modern devices.
9–11
Beyond thermal
transport, precise control of physical properties is also crucial in
fields such as acoustic metamaterials,
12–14
where sound manipu-
lation is essential, and photonic crystals,
15,16
which are funda-
mental to light propagation technologies. In this broader context,
researchers have explored various approaches to tailor the
material properties, with one emerging strategy being the inte-
gration of mathematical principles into material design.
The design of mathematics-inspired structures provides a
versatile framework for manipulating various physical pro-
Xin Wu
Xin Wu is a JSPS postdoctoral
research fellow at the Institute of
Industrial Science, The
University of Tokyo. He received
his Ph.D. degree in Solid
Mechanics from the South China
University of Technology in
2023. His research focuses on
the phonon thermal transport in
nanomaterials, particularly 2D
materials and their hetero-
structures. He primarily focuses
on computational research,
including machine learned-based
molecular dynamics simulations
and first-principles calculations.
Masahiro Nomura
Masahiro Nomura is a Professor
at the Institute of Industrial
Science, The University of Tokyo.
He received his Ph.D. degree in
Applied Physics from the
University of Tokyo in 2005. His
current research interests include
hybrid quantum science, physics
and control technology of
phonon/heat transport in semi-
conductor nanostructures, radia-
tive heat transfer, and thermo-
electric energy harvesting. The
concept of his current research is
"from photonics to phononics" based on his research experience in
quantum electronics. He received 17 awards including The 16th
JSPS Prize (2019) for the top 25 scientists below 45 in Japan.
Institute of Industrial Science, The University of Tokyo, Tokyo 153-8505, Japan.
E-mail: xinwu@iis.u-tokyo.ac.jp, nomura@iis.u-tokyo.ac.jp
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perties of materials. One of the most well-established
approaches in this field is structural periodicity, which has
been proven effective in modulating thermal and acoustic be-
havior. In the context of thermal transport, phononic crystals,
which rely on periodic arrangements to influence phonon
propagation, serve as a foundational model with a significant
research history.
6,17
These structures have laid the groundwork
for understanding how structure design can control thermal
properties. Recently, however, researchers have begun explor-
ing alternative mathematical sequences beyond simple period-
icity, such as quasi-periodic and disordered structures. There
is also some work based on the mathematical algorithms and
graphic design,
18
such as cyclotomic rule.
19
These emerging
designs offer new possibilities for precisely tuning the behav-
ior of materials at the nanoscale, opening up avenues for more
sophisticated control over thermal and other physical
properties.
Moving beyond simple periodic patterns, more intricate
mathematical sequences, such as the Golomb ruler and
Fibonacci sequences, introduce novel strategies for modulating
thermal transport and various other physical properties. The
non-redundant spacing of the Golomb ruler reduces phonon
coherence, leading to phonon confinement and increased
interface phonon scattering, thereby reducing thermal
conductivity.
20,21
Similarly, the Fibonacci sequence, with its
self-similar structure, facilitates the design of fractal-like archi-
tectures that reduce phonon coherence and manage heat dissi-
pation across multiple scales, making it ideal for hierarchical
thermal management in applications such as microelec-
tronics.
22
Binary sequences composed of 0 s and 1 s, such as
Thue-Morse and Double-Period,
23
break translational sym-
metry while maintaining long-range order, effectively scatter-
ing phonons across a wide frequency range. This results in sig-
nificantly reduced thermal conductivity, making these struc-
tures well-suited for thermal barrier coatings in high-tempera-
ture environments such as aerospace and power generation,
where decoupling thermal conductivity from mechanical
strength is crucial. In addition, graded materials and hierarch-
ical structures, inspired by natural systems, optimize thermal
transport by creating gradual transitions in thermal conduc-
tivity and complex heat dissipation pathways, making them
ideal for robust thermal management in energy systems and
high-performance electronics.
24
This review aims to provide a comprehensive overview of
the various mathematical frameworks based on number theory
used in structure design, their effect on the thermal transport
mechanisms of phonons, and the latest advances in their prac-
tical applications (Fig. 1). By integrating mathematical con-
cepts with physical properties, we can achieve a level of control
over material behavior that was previously unattainable,
opening up new possibilities for the design of advanced
materials with tailored functionalities. These advancements
promise not only to boost fields like thermal management and
energy conversion but also to pave the way for multifunctional
materials that can meet the increasingly complex demands of
modern technology.
2 Mathematical sequences for
structural design
In this section, we list six typical mathematical sequences,
respectively explaining their mathematical principles and
characteristics used in structure design. To facilitate under-
standing, Fig. 2 uses simple two-dimensional graphene/hexag-
onal boron nitride,
22,23,25
graphene with isotope interfaces,
21
and Si/Ge
24
systems as the model to show the structure design
based on the various sequences mentioned above.
2.1 Period and superlattice
In solid-state physics and materials science, crystal and super-
lattice structures are essential concepts used to describe the
periodic arrangement of atoms or molecules in solids. These
structures can be mathematically characterized by their trans-
lation symmetry, exemplified by binary sequences such as 0, 1,
0, 1, 0, 1, 0, 1, and so forth, which dictate their periodicity and
spatial configuration.
In a crystal, the position of each atom can be described by a
combination of the primitive lattice vectors a
1
,a
2
,a
3
and a set
of integers n
1
,n
2
,n
3
, as given by the lattice vector: R=n
1
a
1
+
n
2
a
2
+n
3
a
3
. The periodicity implies that any physical property
f(r) of the system satisfies the condition: f(r+R)=f(r), for all
lattice vectors Rand positions rin the crystal. This translation
invariance is the cornerstone of the Bloch theorem, which
asserts that the electronic wave functions in a periodic poten-
tial can be expressed as Bloch functions, a product of a plane
wave and a function with the same periodicity as the lattice.
This underlying periodic order is responsible for many funda-
mental properties of materials, such as the formation of elec-
tronic band structures and the propagation of phonons.
A superlattice structure is a more complex form of period-
icity resulting from the combination of two or more different
Fig. 1 Mathematics sequences for structure design in this review, from
their generation principles to potential applications. The detail infor-
mation of each sequence can be found in section 2.
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materials, discovered by Johansson and Linde
26
in 1925. It can
be thought of as a modulation of the original lattice structure,
often achieved by alternating layers of different materials or
varying the composition periodically. The superlattice is
characterized by a longer periodicity, called the superlattice
vector R
SL
, which can be expressed as: R
SL
=m
1
a
1
+m
2
a
2
+
m
3
a
3
, where m
1
,m
2
, and m
3
are integers that define the super-
lattice periodicity, typically larger than the primitive lattice
vectors a
i
. This results in a larger unit cell for the superlattice
compared to the underlying periodic structure.
Furthermore, this concept of introducing periodic modu-
lation to manipulate material properties extends to other types
of periodic structures such as phononic
12,27,28
and
photonic
29–31
crystals. They expand the concept of periodic
materials beyond superlattices, as they are not limited to two
alternating materials, but can include a variety of structures
such as arrays of holes, voids, or embedded inclusions.
32–35
These periodic configurations allow for precise control over
the propagation of sound and light waves, creating band gaps
that block specific frequencies. This broader versatility enables
a wide range of applications, from acoustic insulation to
advanced optical devices.
36–39
Periodic structures improve material properties by introdu-
cing an additional periodicity that modifies the fundamental
characteristics of materials, such as the thermal,
40–43
electronic,
44,45
optical,
46,47
and mechanical properties.
48,49
This structural complexity enables fine-tuning of material be-
havior beyond what is achievable in traditional crystals,
making superlattices a versatile platform for exploring new
physical phenomena and engineering advanced materials.
2.2 Golomb ruler
The Golomb ruler sequence was discovered by Sidon
50
and
Babcock
51
independently and named for Solomon W. Golomb
in first half of 20th century. Golomb rulers have since gained
prominence in various mathematical and practical contexts
due to their unique properties of non-redundant spacing.
A Golomb ruler is a sequence of marks at different posi-
tions along a ruler such that no two pairs of marks are the
same distance apart.
52–55
Mathematically, an nth-order
Golomb ruler with marks at positions m
1
,m
2
,…,m
n
satisfies
the condition that all pairwise distances |m
i
−m
j
| for 1 ≤i<j
≤nare unique. The length of the Golomb ruler is defined as
the distance between the smallest and largest mark, i.e.,m
n
−
m
1
. A Golomb ruler is not required to measure all distances up
to its length. However, if it can do so, it is referred to as a
perfect Golomb ruler. A Golomb ruler is considered optimal if
no shorter ruler of the same order can be found. It has been
proven that no perfect Golomb ruler exists for five or more
marks, which means that all Golomb rulers of the fifth order
and above are optimal.
Golomb rulers have versatile applications across various
fields due to their unique property of non-redundant spacing.
In communication systems, they are used to design error cor-
rection codes and synchronization sequences, minimizing
signal interference.
53
In radio astronomy, the Golomb rulers
Fig. 2 Structure design cases based on two-dimensional mathematical sequence applied in the field of nanoscale thermal transport. (a) Unitcell of
graphene/hexagonal boron nitride superlattices with increasing period length
25
(Copyright 2018, Springer Nature). (b) Graphene with isotope inter-
faces based on Golomb ruler sequences
21
(Copyright 2024, Elsevier). Graphene/hexagonal boron nitride lateral heterostructures following the (c)
Fibonacci
22
(Copyright 2020, Elsevier), (d) Thue-Morse
23
(Copyright 2022, Elsevier), and (e) Double Period
23
(Copyright 2022, Elsevier) sequences. (f )
Schematic of the graded Si/Ge heterostructures
24
(Copyright 2021, AIP).
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optimize antenna array configurations, ensuring unique base-
line measurements for better image resolution.
51,56
They also
play a role in sensor placement for X-ray crystallography and
seismic arrays, improving measurement accuracy by preventing
redundant data collection.
57
Additionally, Golomb rulers are
useful in network design and audio signal processing, where
they help avoid interference and overlapping echoes, respect-
ively.
57
Their broad applicability makes them valuable in any
scenario requiring distinct and non-overlapping intervals.
2.3 Fibonacci
The Fibonacci sequence, named after the Italian mathemati-
cian Leonardo of Pisa (commonly known as Fibonacci), holds
substantial historical and mathematical importance. Fibonacci
introduced this sequence to the Western world in his 1202
book, Liber Abaci,
58
where he used it to model the growth of
an idealized rabbit population, illustrating the concept of
exponential growth. However, this sequence was already well
known in Indian mathematics long before Fibonacci’s work,
having been documented as early as 200 BCE in the context of
Sanskrit prosody, where it was used to describe rhythmic pat-
terns in poetry.
59
The Fibonacci sequence is defined by the recurrence
relation F(n)=F(n−1) + F(n−2) with initial conditions F(0) =
0 and F(1) = 1, generating a sequence that starts 0, 1, 1, 2, 3, 5,
8, 13, 21, and so on.
60
This simple yet powerful sequence is
deeply intertwined with the golden ratio, as the ratio of succes-
sive Fibonacci numbers gradually converges to ϕ¼1þ
ffiffi
5
p
2as n
approaches infinity. This convergence highlights the intrinsic
connection between Fibonacci numbers and the golden ratio,
reflecting the underlying harmony and aesthetic appeal that
these mathematical concepts share.
The Fibonacci sequence finds diverse applications in
several fields. In nature, it appears in the arrangement of
leaves, flower petals, and seed heads, optimizing growth pat-
terns and packing density.
61
In the realm of computer science,
it serves as a fundamental example of recursion and dynamic
programming, providing valuable insights for algorithm
design.
62
Furthermore, the connection of the sequence to the
golden ratio is of significance in art and architecture, where it
supports aesthetically pleasing proportions.
63
In general, the
Fibonacci sequence is a remarkable mathematical construct
that bridges the abstract and the concrete, finding expression
in natural phenomena, human creativity, and sophisticated
problem-solving frameworks. Its simplicity, coupled with its
universality, renders it a persistent subject of scholarly inter-
est, underscoring the profound interconnectedness of math-
ematics with the world around us.
2.4 Thue-Morse
The Thue-Morse sequence was first studied by Axel Thue in
1906 as part of his work on sequences that avoid repetitive pat-
terns in the study of combinatorics on words. Later, in 1921,
Marston Morse explored it further in the context of differential
geometry and brought it to worldwide attention. It was orig-
inally developed to investigate sequences with complex struc-
tures that do not repeat in simple ways, laying the groundwork
for areas like formal language theory and combinatorics.
The Thue-Morse sequence is an infinite binary sequence
generated through a recursive process. It begins with t
0
=0
and is constructed by successively appending the bitwise
complement of the sequence obtained so far. The first few
terms of this sequence are: 0, 01, 0110, 01101001, etc.
Formally, the n-th term t
n
of the sequence can be determined
by examining the number of 1 s in the binary representation of
n: if the count of 1 s is even, then t
n
= 0; if odd, t
n
=1.
Alternatively, the Thue-Morse sequence can also be
described using substitution rules, which reflect its inflation-
ary nature. Starting with an initial element, 0, the sequence
evolves by applying two simple substitution rules: ϕ(0) = 01
and ϕ(1) = 10. Thus, each occurrence of 0 in the sequence is
replaced by “01”and each occurrence of 1 by “10”. This recur-
sive substitution process ensures that the sequence lacks
repeating patterns and exhibits self-similarity, attributes that
render it particularly useful for studies in combinatorics, com-
puter science, and even physics.
Due to its non-repetitive, self-similar structure, Thue-
Morse sequences have a range of applications. In computer
science, it is used in algorithm design, digital signal proces-
sing, and error correction, as it helps minimize redundancy
and interference.
64
In physics, the sequence is applied in
the study of quasi-crystals and aperiodic tiling, providing
insights into non-periodic structures.
65–67
These applications
leverage the unique property of the sequence of avoiding
consecutive identical patterns, which makes it valuable in
various fields.
2.5 Double-Period
The Double-Period sequence represents a modified variation
of the Thue-Morse sequence, distinguished by an alteration in
its inflation rules. Rather than adhering to the standard Thue-
Morse substitution rules, the Double-Period sequence is gener-
ated by the substitutions ψ(0) = 01 and ψ(1) = 00. Initiating
from an initial element of 0, this process yields a sequence
that evolves as follows: 0, 01, 0100, 01000101, and so forth.
The modified substitution rules produce a sequence with a
more intricate and less balanced structure compared to the
Thue-Morse sequence. Specifically, while the Thue-Morse
sequence is known to avoid consecutive repetitions and main-
tain a high degree of self-similarity, the Double-Period
sequence introduces frequent occurrences of ‘00’pairs. This
alteration results in a sequence that diverges from the purely
self-similar and non-repetitive characteristics of the Thue-
Morse sequence, leading to a unique and less predictable
pattern formation.
Due to these distinctive properties, the Double-Period
sequence offers a valuable framework for exploring various
nonperiodic patterns. Its irregular structure makes it an intri-
guing subject for research in fields such as coding theory,
where its potential applications may include the design of
error-correcting codes. In addition, the aperiodic nature of the
sequence also holds promise for studies related to aperiodic
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structures and quasi-crystals, where non-repeating patterns
play a significant role in understanding material properties
and phenomena.
2.6 Graded
The graded sequence refers to a systematically controlled pro-
gression of material compositions in a specific direction,
allowing deliberate adjustment of properties. These sequences
can be engineered to follow various mathematical patterns,
such as linear, exponential, or more complex functions,
depending on the desired performance characteristics. This
level of control offers a precise means of optimizing material
properties across different regions of a structure.
A notable variant of graded sequences is found in function-
ally graded materials. Functionally graded materials represent
a class of advanced materials in which properties, such as
composition, microstructure, mechanical strength, or thermal
conductivity, vary continuously or in discrete steps over a par-
ticular dimension. This gradual variation enables these
materials to provide a smooth and seamless transition
between distinct phases or properties, effectively reducing
issues related to material discontinuities, such as stress con-
centrations or thermal mismatches.
68,69
Using this unique
capability, functionally graded materials offers the potential
for enhanced performance and customized functionalities that
exceed those of traditional homogeneous materials. As a
result, they are widely applied in industries that require high-
performance materials with specialized properties, such as
aerospace, electronics, and biomedical fields, where specific
gradients in material properties can optimize their perform-
ance under various operating conditions.
70,71
3 Applications in nanoscale thermal
transport
Mathematically inspired structure design offers a novel
approach to optimizing material properties for various appli-
cations, particularly in nanoscale thermal transport. By apply-
ing mathematical principles, materials can be engineered with
tailored functionalities that enhance performance in targeted
areas. This section explores the application of these designs in
nanoscale thermal transport within heterostructures, high-
lighting the potential for broader future applications across
diverse fields.
3.1 Periodic
The periodic structure design in nanoscale thermal transport
plays a crucial role in revealing the wave-particle crossover of
phonon transport, where the phonon behavior transits
between incoherent and coherent regime. The concept of
minimum thermal conductivity in superlattices was first intro-
duced by Simkin and Mahan
78
in 2000 through theoretical cal-
culations, providing a foundation for understanding thermal
behavior in these systems. Subsequent experimental investi-
gations have validated the theoretical prediction. In GaAs/AlAs
superlattices, the coherent phonon thermal transport process
was experimentally confirmed for the first time by Luckyanova
et al.,
72
with measured thermal conductivity showing a linear
increase as the total superlattice thickness increased over a
temperature range of 30 K to 150 K (Fig. 3a). Similar obser-
vations have been made in epitaxial oxide SrTiO
3
/CaTiO
3
superlattices, as reported by Ravichandran et al.,
79
which pro-
vides experimental evidence of the transition from diffuse
(particle-like) to specular (wave-like) phonon scattering. They
directly observed the minimum in lattice thermal conductivity
when varying the interface density. Moreover, Saha et al.
73
also
demonstrated the phenomenon of phonon wave-particle cross-
over in (Ti,W) N/(Al,Sc) N metal/semiconductor superlattices,
grown epitaxially with periodicities from 1 to 240 nm. It
revealed a delicate balance between long-wavelength coherent
phonon modes and incoherent phonon scattering from heavy-
tungsten atomic sites and superlattice interfaces. The ability to
manipulate the transition from particle-like to wave-like
phonon scattering by altering the superlattice design enables
the optimization of thermal conductivity in various materials,
offering a powerful platform for exploring phonon interference
effects in superlattices with wide-ranging applications in ther-
moelectronics and thermal management.
Expanding on the foundational knowledge of phonon trans-
port in periodic structures, recent studies on two-dimensional
materials and their heterostructures provide more profound
insight into the wave-particle crossover phenomenon. For the
two-dimensional lateral heterostructures, in the graphene/hex-
agonal boron nitride case, molecular dynamics simulations
conducted by Felix and Pereira
23
clearly demonstrated this
crossover, as phonon transport evolves from a particle-to-wave-
like mechanism (Fig. 3c). This phenomenon is similarly
observed in graphene/2D polyaniline (C
3
N) lateral superlat-
tices, where the dual effect of the heterointerface mechanism
identified by Wu et al.
74
further supports the understanding of
the behavior of phonons at nanoscale interfaces (Fig. 3d). The
gradual transition from incoherent scattering to coherent wave
propagation in these systems is governed by the interplay of
superlattice periodicity and material properties, which directly
affects thermal conductivity. For the two-dimensional van der
Waals heterostructures, Wu and Han
42
extended this analysis
by showcasing how the layered nature of graphene/hexagonal
boron nitride van der Waals superlattices leads to a transition
in phonon transport regimes (Fig. 3e). As the period thickness
decreases, the phonon transport shifts toward coherence, sig-
nificantly affecting the thermal conductivity. These insights
underscore the potential for the tuning of thermal properties
in layered two-dimensional systems, opening possibilities for
advanced thermal management in nano devices. Additionally,
Cheng et al.
75
demonstrated that in twisted bilayer graphene
moiré superlattices, which are unique periodic structures
formed by the slight rotational misalignment of two graphene
layers, the twist angle plays a critical role in determining
phonon transport (Fig. 3f ). This specific arrangement induces
a moiré pattern that creates periodic potential landscapes, sig-
nificantly affecting both electronic and phononic properties.
80–82
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At specific magic angles, a minimum in thermal conductivity
was observed as a result of enhanced phonon scattering and
interference, further reinforcing the intricate relationship
between lattice structure and phonon dynamics.
In addition to two-dimensional materials, silicon-based
structures represent a significant area of interest in thermal
transport research. Silicon, widely used in electronics and ther-
moelectric devices, serves as an ideal platform for exploring
phonon transport manipulation. The work of Maire et al.
76
demonstrated the successful fabrication of one-dimensional
and two-dimensional phononic crystals in silicon, with
ordered and disordered arrays of holes (Fig. 3g). The compari-
son of these structures shows how phonon transport can be
tailored through periodicity and disorder, offering valuable
insights for thermal management strategies in silicon-based
devices. Furthermore, phononic crystals also offers promising
approaches to improve thermoelectric performance, as shown
by Yanagisawa et al.
77
(Fig. 3h). By introducing periodic nano-
Fig. 3 The periodic structure design in the nanoscale thermal transport field. Experimental researches of phonon wave-particle crossover phenom-
enon in the (a) GaAs/AlAs
72
(Copyright 2012, AAAS) and (b) Ti
0.7
W
0.3
N/Al
0.72
Sc
0.28
N
73
(Copyright 2016, APS) three-dimensional semiconductor-based
superlattices. Wave-particle crossover of phonon transport in the (c) graphene/hexagonal boron nitride
25
(Copyright 2018, Springer Nature) and (d)
graphene/2D polyaniline (C
3
N) lateral superlattices
74
(Copyright 2023, Elsevier), and (e) graphene/hexagonal boron nitride vdW superlattices
42
(Copyright 2022, Elsevier) via molecular dynamics simulations, which is the indication of transition in the phonon transport mechanism from the
incoherent to coherent regime. (f) Thermal conductivity minimum of twisted bilayer graphene moiré superlattices
75
(Copyright 2023, Elsevier). (g)
Schematic and SEM images show fabricated samples of one-dimensional and two-dimensional phononic crystals with ordered and disordered
arrays of holes in silicon
76
(Copyright 2017, AAAS). (h) Enhancement of thermoelectric figure of merit by phononic nanostructure in silicon
77
(Copyright 2024, Elsevier).
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structures in silicon, phonon scattering is significantly
enhanced, resulting in a reduction of thermal conductivity
while preserving electrical properties. This modification
boosts the thermoelectric figure of merit, facilitating a ten-fold
improvement in power generation efficiency. Consequently,
this advancement enables once-a-day sensing applications to
become feasible in practical environments, marking a critical
step forward in energy harvesting technologies.
Fig. 4 The quasi-periodic and disordered structure design in the nanoscale thermal transport. (a) Quasi-periodic generations for Fibonacci, Thue-
Morse, and Double-Period sequences, and the intrinsic thermal conductivity versus domain sizes of sequenced graphene/hexagonal boron nitride
lateral heterostructures
22,23
(Copyright 2020, Elsevier). (b) Graphene/hexagonal boron nitride lateral heterostructures based on the 5th-order
Golomb ruler sequence and their thermal conductivity with different interface densities
20
(Copyright 2023, Elsevier). (c) Graphene nanoribbon with
13
C isotope interfaces based on the 9th-order Golomb ruler sequence, and its thermal conductivity compared with other mathematical sequences
at 20 K
21
(Copyright 2024, Elsevier). (d) Phonon localization in graded Si/Ge superlattices.
24
(Copyright 2021, AIP).
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This comprehensive exploration of phonon transport
mechanisms across different material systems underscores the
importance of periodic structure design in manipulating the
thermal properties at the nanoscale. The combined insights
from simulation, theoretical analysis, and experimental vali-
dation highlight the potential to advance thermal manage-
ment and thermoelectric applications through controlled
phonon transport.
3.2 Quasi-periodic and disordered
In the field of nanoscale thermal transport, quasiperiodic and
disordered structure designs, particularly in two-dimensional
materials, have shown promising potential to modulate
thermal conductivity. The exploration of such designs,
inspired by mathematical sequences, aims to understand and
control phonon transport behaviors within these materials.
This is especially pertinent for applications requiring precise
thermal management, such as thermoelectric devices and heat
dissipation technologies.
One prominent approach is the quasi-periodic arrangement
of graphene/hexagonal boron nitride lateral heterostructures,
utilizing sequences such as Fibonacci, Thue-Morse, and
Double-Period to create a variety of configurations (Fig. 4a).
Specifically, the research from Felix and Pereira
22
indicated
that quasiperiodicity of Fibonacci sequences in graphene/hex-
agonal boron nitride heterostructures suppresses coherent
phonon transport by increasing interface density as Fibonacci
generation increases, disrupting phonon coherence. This
effect enhances nanoscale control over thermal transport,
suggesting potential for advanced thermal management device
applications. They also explored phonon transport in gra-
phene/hexagonal boron nitride superlattices with Thue-Morse
and Double-Period sequences,
23
finding that high generations
suppress coherent thermal transport similarly to Fibonacci-
based structures. The suppression is caused by an increase in
the superlattice period (or quasi-periodic generation), which
can lead to phonon localization with some certain wave-
lengths, reducing the lattice thermal conductivity. However,
whether it is the Fibonacci, Thue-Morse, or Double-Period
sequences, their effectiveness in suppressing phonon thermal
transport remains limited because of their inherent
periodicity.
To achieve more effective suppression of thermal conduc-
tivity, Wu et al.
20
reported a more disordered structure design
based on the fifth order Golomb ruler sequence in graphene/
hexagonal boron nitride heterostructures (Fig. 4b). This design
introduces greater diversity and irregularity in the interface
spacing, enabling a wider range of phonon wavelengths to be
suppressed. This sequence has shown potential for reducing
the thermal conductivity by increasing the interface density,
which is a valuable feature for applications requiring thermal
insulation. Furthermore, they extended it to graphene nano-
ribbons with
13
C isotope interfaces, utilizing a 9th-order
Golomb ruler sequence to manage thermal properties at cryo-
genic temperatures
21
(Fig. 4c). They found that the specific
arrangement of these isotope interfaces notably suppresses
thermal transport at 20 K, illustrating how isotope engineering
in combination with mathematical sequences can effectively
modulate phonon transport over a broader range of wave-
lengths in cryogenic environments. The Golomb ruler
sequence stands out for its ability to disrupt phonon coher-
ence through irregular spacing, enabling the suppression of a
wide range of phonon wavelengths. Unlike random sequences,
the Golomb ruler sequence provides a structured form of
strong disorder that is both predictable and mathematically
defined. This sequence offers a controlled irregularity, allow-
ing researchers to precisely design systems with known
spacing irregularities. By employing the Golomb ruler
sequence, one can achieve a consistent, repeatable pattern of
disorder, offering a unique approach to modulating thermal
conductivity and other transport properties with greater
control than purely random arrangements allow.
Fig. 5 The timeline of the discovery of the mathematical sequences in this review and their applications in nanoscale thermal transport field.
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Additionally, recent studies have also explored innovative
methods to manipulate phonon behavior in Si/Ge superlat-
tices, aiming to enhance thermal control through graded-
based interfaces. Guo et al.
24
introduced a novel approach to
phonon localization in graded Si/Ge superlattices, showing
that a sufficient level of long-range disorder can lead to a
thermal conductivity minimum by partially localizing moder-
ate-frequency phonons (Fig. 4d). By analyzing length-depen-
dent transmission and participation ratios, as well as phonon
density distributions, the work highlights a unique regime
where phonon transmission decays exponentially, offering new
insights into phonon localization and thermal conduction
engineering through wave-based phonon control. Ma et al.
83
studied the thermal conduction phenomena and phonon spec-
trum characteristics of interfaces with graded Si/Ge system.
Compared with linearly varying mass gradient layers, as well
as uniform mass layers and pure Si/Ge interfaces, the exponen-
tially varying mass gradient model has more phonons that can
be transmitted across adjacent layers, resulting in significant
improvement of interface thermal conductance.
As above, by incorporating non-periodic sequences,
researchers have developed materials that effectively modulate
heat conduction through tailored phonon scattering mecha-
nisms. These approaches demonstrate the potential of math-
ematical sequences to enhance thermal management in
advanced material applications, offering new pathways for
modulating nanoscale thermal transport.
4 Conclusions and outlook
This review has summarized recent advancements in control-
ling nanoscale thermal transport through the design of math-
ematically inspired structures, such as superlattices and quasi-
periodic frameworks. While classical mathematical sequences
have been studied for nearly a century, it is only in the 21st
century that they have inspired novel approaches to physical
structure optimization, marking a fresh perspective in the field
(Fig. 5). These designs deepen our understanding of nanoscale
thermal transport physics, showing substantial promise for
manipulating thermal conductivity and more applications by
enabling precise control over thermal transport mechanisms.
With this in mind, we propose the following outlook:
While recent machine learning-driven approaches have pro-
duced innovative designs for nanoscale thermal transport,
84–88
they often rely on data-intensive optimization and rule-based
selection processes. However, although these methods are
highly effective, their results often lack strong predictability
and clear definition. Mathematical principles, on the contrary,
offer clearer and direct insights, though they may not achieve
the same high optimization level. Looking ahead, integrating
mathematical approaches with machine learning could unlock
new possibilities in thermal management, energy harvesting,
and advanced material development. Furthermore, mathemat-
ical methods can be considered to guide intrinsic thermal
nonlinearity for more efficient heat harvesting, as discussed by
Zhou et al.
89
By combining the intuitive clarity of mathemat-
ical designs with the optimization strength of machine learn-
ing, we could harness the strengths of both methods to refine
and enhance the thermal transport properties.
Mathematical sequence-inspired designs offer promising
guidance for experimental studies in thermal transport, pro-
viding a structured approach to controlling heat flow at the
nanoscale. Although simulations reveal the potential of these
designs, directly translating them to experimental scales is
challenging. Despite this, simulations still offer valuable
insight that can guide and inspire the creation of similar
designs in experimental settings. By bridging simulation
insights with practical applications, these frameworks can
deepen our understanding of complex heat conduction beha-
viors and drive advancements in advanced thermal manage-
ment and thermal design systems.
Furthermore, mathematically inspired structure design
methods hold potential for broader applications beyond
thermal transport control. These principles could drive
advances in fields such as energy harvesting, signal proces-
sing, and wave manipulation, where precise control over
material properties is crucial. By enabling adaptable, scale-sen-
sitive designs, mathematical sequences provide pathways for
creating materials with tunable behaviors to meet diverse func-
tional requirements. As computational methods continue to
evolve, these designs may foster breakthroughs in the optimiz-
ation of complex material systems in various scientific and
engineering disciplines.
Data availability
No new data were generated or analyzed as part of this review.
Conflicts of interest
There are no conflicts to declare.
Acknowledgements
This work was supported by the JSPS Grants-in-Aid for
Scientific Research (Grant No. 24KF0027, 20H05649, and
21H04635) and JST SICORP EIG CONCERT-Japan (Grant No.
JPMJSC22C6). X. W. is the JSPS Postdoctoral Fellow for
Research in Japan (No. P24058).
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