Thermal transport in mechanically deformed two-dimensional materials and designed structures with their applications

Abstract

Two-dimensional (2D) materials have garnered notable research interest due to their extraordinary properties. Assembling two or more 2D materials into heterostructures introduces properties that are not present in any individual components, leading to a spectrum of nanodevices and applications. The lifetime and performance of nanodevices can be largely dictated by the working temperatures, and the heat dissipation in 2D materials and heterostructures is vital to the reliability and functionality of devices. However, mechanical effects encountered can potentially impact thermal transport. A comprehensive understanding of the interplay between mechanical loadings and thermal transport in 2D materials and their heterostructures is fundamental to devising effective cooling strategies for devices operating under complex conditions. The tunable thermal properties of these materials offer a platform for designing mechanically adjustable devices and reversible performance optimization. This review starts with a summary of the thermal conductivities (TCs) in various 2D materials adjusted by mechanical loadings. A brief overview of the underlying tuning mechanism is provided, followed by a discussion on the effect of structural designs. Several potential applications based on the thermo-mechanical correlation are mentioned. Finally, the current limitations and challenges in the field are included, and several suggestions for future research directions are discussed.

---

1. Introduction

2D materials refer to crystalline materials composed of a single or a few layers of atoms, which have attracted tremendous research attention since the first successful exfoliation of graphene.¹ The single-layer structure of graphene results in outstanding mechanical,² thermal,³ electric,⁴ magnetic,⁵ and optical properties⁶ that exceed their bulk phases, leading to a wide variety of applications in sensing,⁷ electronics devices,⁸ and energy conversion and storage.⁹ Other 2D materials in the graphene family¹⁰ such as graphene¹¹ and graphyne,¹² and structurally similar 2D materials such as hexagonal boron nitride¹³ (hBN) and boron-carbon-nitrogen¹⁴ (BCN), mono-element 2D materials (Xenes) such as borophene,¹⁵ silicene,¹⁶ and germanene,¹⁷ transition metal dichalcogenides (TMDs) such as MoS₂¹⁸ and WSe₂,¹⁹ 2D oxides such as MnO₂²⁰ and TiO₂,²¹ and transition metal carbides/nitrides²² (MXenes) are also explored with miscellaneous properties and functions.

The expansion of the 2D-materials family provides an escalating platform for composing heterostructures with either van der Waals²³ (vdW) or covalently bonded interfaces.²⁴ The heterostructures usually possess not only properties of all constituents but also novel properties that are not present in any component layers due to the presence of heterojunctions. For example, placing graphene on an hBN layer can reduce the surface effects on the charge-carrier mobility of graphene due to the ultra-flat and chemically clean nature of the hBN layer. In addition, the hBN layer can protect graphene from environmental chemical effects.²⁵ Stacking MoSe₂–WS₂ heterostructures can result in an ultrafast hole transfer rate of approximately 50 fs, which exceeds that in both monolayers.²⁶ The weak absorption characteristics of graphene can be improved in graphene–MoS₂ heterostructures, which possess strong optical absorption and a visible-range bandgap.²⁷ These artificial yet extraordinary properties have led to applications of sensors,²⁸ transistors,²⁹ diodes,³⁰ light-harvesting devices,³¹ and light-emitting devices.³²

The shrinking feature size of 2D materials enables high-density and three-dimensional integration, boosting the performance of devices based on 2D materials over traditional devices.³³ For example, the carrier mobility of graphene devices can exceed 10³ cm² V⁻¹ s⁻¹,³⁴ while the mobility of hBN-graphene heterostructures can reach 10⁵ cm² V⁻¹ s⁻¹, two orders of magnitude higher than that of the silicon devices.³⁵ The high degree of integration amplifies the Joule heating, and the local heat flux density can exceed ∼10³ W cm⁻².³⁶ The temperature largely limits the performance, lifetime, and reliability of devices. For example, the hall mobility of hBN-encapsulated MoS₂ downscales with temperature according to a power law.³⁷ The temperature-dependent performance poses a long-term challenge in the heat dissipation in 2D electronic devices.

Understanding the thermal transport in 2D materials and their heterostructures is critical for the design and development of 2D functional devices,³⁸ and a notable amount of work has been conducted with theoretical and/or experimental approaches.³⁹ In semiconductor 2D materials, the main heat carriers are phonons.⁴⁰ The phononic TC in some of the 2D materials is exceptional, leading to a spectrum of 2D thermal devices.⁴¹ The TCs of pristine graphene and hBN can reach over 1000 and 750 W m⁻¹ K⁻¹,^42,43 respectively, making them ideal candidates for the reinforcing phase of thermal interface materials.³⁹ Graphene has also been utilized to elevate the lateral and cross-interface heat dissipation.^44,45 MXenes can be used in thermal storage devices.⁴⁶

Despite notable research efforts on the intrinsic thermal transport properties of 2D materials and heterostructures, the effects of mechanical stimuli on the thermal transport properties are usually overlooked. The mechanical effects include both the internal and external stress/strain in the working environment or during manufacturing processes, such as packing pressure,⁴⁷ residual stress after thermal processing,⁴⁸ and lattice structure or thermal mismatch-induced stress.^49,50 For instance, in addition to the mechanical deformation due to the lattice mismatch of 2D materials and the substrate, a substantial residual strain of ∼0.22% is found in 2D MoS₂ during transfer onto devices.⁵¹ Besides, the effective contact between the thermal device and mating surfaces is conventionally achieved with a packing pressure of ∼10 MPa.^47,52 Significant mechanical deformations will alter the phonon transport or the heat transfer path, leading to a notable discrepancy between the actual thermal transport and the expectation based on the pristine materials or structures.^53–55 The fundamental mechanisms underlying mechanically modulated thermal transport can be categorized into three main aspects: intrinsic material properties, mechanical loading response, and phonon behavior (Fig. 1(a)). These mechanisms jointly determine how 2D systems respond thermally under both internal and external stress or strain, providing the physical foundation for subsequent structural designs and functional applications.

Fig. 1 Overview of mechanisms, structural designs, and applications of mechanically modulated thermal transport in 2D materials and heterostructures. (a) Fundamental mechanisms of mechanically modulated thermal transport, including intrinsic material properties, loading response, and phonon behavior. (b) Structural design approaches to optimize the modulation range of thermal transport, including vdW heterostructures, in-plane junctions, and multilayer configurations. (c) Current and potential applications based on mechanically adjustable thermal properties, including thermal switches, thermal diodes, and flexible thermoelectric generators (F-TEGs).

Investigating the heat transfer of mechanically deformed 2D materials and heterostructures is fundamentally crucial to harnessing heat in thermal management for functional devices. Given the strong coupling between mechanical stimuli and thermal transport, structural design plays a key role in enabling and controlling such modulation. To this end, various approaches have been proposed, including vdW heterostructures, in-plane heterostructures, and multilayer configurations, which allow precise tuning of thermal transport through interfacial engineering and dimensional control (Fig. 1(b)). Moreover, the correlation between the thermal transport properties and mechanical loadings offers a possibility of designing devices with adjustable properties and performance, which are ideal in thermal applications of thermal switches and diodes,^56,57 thermoelectric devices,⁵⁸ strain-gated phononic devices, etc.⁵⁹ (Fig. 1(c)). With the growing refinement in the structural design of 2D materials, other advanced applications such as flexible electronics⁶⁰ and thermo-mechanical sensors^61,62 have emerged as prominent application targets.

In this review, the current research on the phononic thermal transport properties in various mechanically deformed 2D materials and their heterostructures is summarized. We categorize previous works according to the mode of mechanical loadings, given that various loading modes affect thermal transport properties via contrasting mechanisms. Next, the effects of structures and interfaces on the thermo-mechanical correlation are systematically discussed. Several applications based on the mechanically adjustable thermal transport in 2D materials are summarized. Finally, the current challenges and opportunities in the field are discussed, and some suggestions on future research directions are proposed.

2. Thermal conductivity of 2D materials adjusted by various mechanical loading modes

Previous studies have demonstrated that the TC of 2D materials is not only influenced by defects, size, and boundary effects,^63,64 but is also highly sensitive to external mechanical loadings.⁶⁵ These mechanical deformations can alter the lattice structure and interatomic bond lengths, thereby affecting stress distribution within the material, which influences the phonon dispersion relations and scattering mechanisms,⁶⁶ and consequently altering the thermal transport properties. The effect of strain on the TC of 2D materials was first investigated by Wei et al.¹¹ The current research focuses on the influence of typical loading methods, such as tension, compression, torsion, and bending, on the TC of 2D materials. The trend of the TC with mechanical strain depends on the loading mode. Meanwhile, 2D materials from different families exhibit divergent changes in TCs under the same mechanical loading due to various atomic structures. Based on these considerations, we summarize recent works according to two aspects: (i) the loading modes, including tension, compression, torsion, and bending; (ii) different families of 2D materials, such as the graphene family, and structurally similar 2D materials: TMDs, Xenes, and MXenes.

We retrieved relevant articles from Web of Science (WOS) core collection from 2010 to 2025 (data were obtained on July 1, 2025), and the language was limited to English. The search keywords of TCs under tensile loadings are TS = (2D material AND thermal conduct* AND tensi*), the keywords of compressive loadings are TS = (2D material AND thermal conduct* AND compressi*), the keywords of torsional loadings are TS = (2D material AND thermal conduct* AND (torsi* OR twis*)), and the keywords of bent loadings are TS = (2D material AND thermal conduct* AND (bend* OR bent)). The total number of relevant publications is 902. The proportion of different loading modes is presented in Fig. 2(a). Tensile loading is the current research hotspot, accounting for 51.1% of the overall dataset. The proportion of compression is 27.5%, followed by torsion, which is 15.6%. The effect of bent loadings on the TC has attracted increasing attention; however, related studies remain limited, with only 5.8% of total publications. Moreover, we compiled the tuning range of typical 2D materials’ TC under different mechanical loading, as shown in the bar chart of Fig. 2(a). It is obvious that traditional 2D materials in the graphene family and structurally similar 2D materials, such as graphene, exhibit a particularly high tuning range of TC under mechanical loading, while their novel counterparts, such as hBCN, exhibit a very limited tuning range. The TC tuning range of most MXenes is slightly lower than that of the graphene family and structurally similar 2D materials, with TMDs marginally behind. Among the typical 2D materials, Xenes possess the narrowest tuning range of TCs affected by mechanical loading.

Fig. 2 Overview of literature focusing on the TCs influenced by mechanical strain. (a) The tuning range of the TC of typical 2D materials under different mechanical loadings. The inset pie chart presents the proportion of scientific publications on the effect of various loading modes on 2D materials’ TC, κ. (b) Relationship between mechanical strain and TC variation (κ − κ₀)/κ₀ of diverse 2D materials under different types of loadings, where κ₀ denotes the TC without mechanical loadings.^67–80

Fig. 2(b) illustrates the effects of different types of mechanical loadings, including tension, compression, torsion, and bending, on the TC of typical 2D materials. In general, all four loading modes lead to a reduction in TCs of 2D materials. Tensile and compressive loading are the most intensively studied forms of mechanical deformation due to their relative ease of application on the 2D-material models. These two types of strain are defined as ΔL/L, in which ΔL is the change in length along the loading direction and L is the length of the 2D materials. Many studies have proved that tensile loading offers the widest strain range on 2D materials and enables the greatest tunability on their TCs (blue symbols). Compared to tension, the strain range of compression is relatively small (orange symbols) due to the structural instability of compressive loading, which leads to a narrower tuning range. Torsion in Fig. 2 refers to in-plane torsional loading on a 2D material ring system. To ensure alignment with tension and compression, mechanical strain of torsional loading is defined as (R_i/(R_o − R_i))·θ,⁷⁶ where R_i and R_o refer to internal and external radii of the tested 2D material ring, and θ is the torsion angle. Torsional strain introduces non-uniform local stress fields, which lead to a narrower TC tuning range (green symbols) compared to tensile loading. The effect of bending strain on the TC is of considerable academic value, yet current studies on 2D materials remain limited. The strain of bending can be defined as (h/2)·C, in which h is the thickness of 2D materials and C is the curvature of bending 2D materials.⁸¹ According to previous 2D material studies based on existing research results, the bent strain suppresses TCs in 2D materials (purple symbols).

2.1. Effect of tensile strain on the thermal conductivity of 2D materials

For the graphene family and structurally similar 2D materials, their TCs generally decrease with the increase of uniaxial tensile strain. Zhan et al.⁶⁷ adopted non-equilibrium molecular dynamics (NEMD) simulations to study the thermal transport properties of graphene under different tensile strains (Fig. 3(a)). The selected strain magnitudes ε were 2%, 5%, 8%, and 10%, and the relative TC, κ(ε)/κ(0), was used for characterization, where ε and κ(0) (same as κ₀) denote the strain and TC without loading, respectively. The normalization accounts for the size effects on the calculation results. Simulational results suggest that the relative TC of graphene monotonically decreases with the increase of tensile strain, and the reduction in κ is close to 50% under 10% of tensile strain (Fig. 3(b)). In addition, a cubic polynomial fitting could match this changing trend. Song et al.⁶⁹ discovered through reverse NEMD calculations that the relative TC of C₂N is subjected to uniaxial tensile strain. The TC κ slightly increases in the low strain stage (<12%), but is significantly decreased by further stretching. At 20% tensile strain, the TC of C₂N is reduced by approximately 30%. The main reason for this is that C₂N does not maintain an ideal planar shape at finite temperatures and, under the influence of structural relaxation, forms slight wrinkles. When the tensile strain reaches approximately 12%, the wrinkles are effectively unfolded, and the material morphology approaches the ideal 2D planar form, thereby reducing the phonon scattering caused by the wrinkles. With a larger strain, phonon modes are softened, and the enhanced phonon scattering suppresses the TC. Additionally, the study found that biaxial tensile strain can decrease TC more rapidly. Specifically, a biaxial tensile strain of 14% can decrease the TC of C₂N by more than 50%. The superposition of strain fields intensifies phonon scatterings, thus hindering the phononic thermal transport of the material to a larger extent.

Fig. 3 TCs in 2D materials in response to tensile strain. (a) The diagram of NEMD simulations for a graphene sheet under uniaxial tensile strain. (b) Relative TC of strained graphene. The red line is the cubic fitting of the MD results.⁶⁷ Reproduced from ref. 67 with permission from Institute of Physics Publishing, copyright 2019. (c) Schematic diagram of Raman measurement. (d) Relative TC of the MoS₂ monolayer as a function of tensile strain and comparison with computational model predictions.^{65,72,82–86} Reproduced from ref. 65 with permission from Springer Nature, copyright 2025. (e) Relaxation times of short- and long-wavelength phonons within the strain range of [0, 0.1].⁸⁷ Reproduced from ref. 87 with permission from Oxford University Press, copyright 2021. (f) Comparison of phonon spectra for Ti₃C₂O₂ nanosheets under 8% tensile strain along the heat flux direction with strain-free spectra.⁸⁸ Reproduced from ref. 88 with permission from Springer Nature, copyright 2025.

As for the TMD family, such as MoS₂, both experimental and simulational results have revealed that their TCs decrease monotonically with external tensile strain. As shown in Fig. 3(c), Liu et al.⁶⁵ measured the in-plane TC of single-layer MoS₂ using photothermal Raman technology. The in-plane TC of single-layer MoS₂ declines by 60%, from 34.3 W m⁻¹ K⁻¹ to 12.9 W m⁻¹ K⁻¹ at a 6.3% strain (Fig. 3(d)). The TC exhibits an upward trend in the strain range of 0–1.6%, mainly because the initial small strain can effectively alleviate the local wrinkles and interface scattering of MoS₂ on the polydimethylsiloxane (PDMS) substrate. Jiang et al.⁸² also pointed out through molecular dynamics (MD) simulations that tensile strain induces the elongation of the Mo–S bond, leading to an increase in phonon scattering rate and a decrease in specific heat, thereby causing a decline in the TC.

In low-buckling Xene materials like silicene, the TC changes nonmonotonically with tensile strain. Ding et al.⁸⁷ found through equilibrium molecular dynamics (EMD) research that the TC of silicene increases by more than 10 times within the strain range of 0–10%, the TC monotonically increases from 3.97 W m⁻¹ K⁻¹ at ε = 0 to 43.26 W m⁻¹ K⁻¹ at ε = 10%, then tends to saturate and fluctuates. As illustrated in Fig. 3(e), within this strain range, the relaxation time of long-wavelength phonons increases from 0.76 ps to 6.44 ps, an increase of approximately 8 times, indicating that geometric flattening helps to enhance the phonon lifetime and propagation ability. However, in the high-strain stage, the enhanced bond length fluctuations bring additional phonon scattering, which suppresses thermal transport. Boltzmann transport equation (BTE) simulations further validate this phenomenon: within the strain range of 4–8%, the out-of-plane acoustic phonon mode (ZA) scattering channel is suppressed, and the TC rapidly increases. When the strain exceeds this range, structural distortion intensifies phonon scattering, and the TC drops by 10% to 50%.⁸⁹

The TC of MXene materials is highly sensitive to the tensile strain. Jiang et al.⁷⁵ conducted reactive force field MD simulation and found that the TC of Ti₃C₂T_x (including Ti₃C₂O₂, Ti₃C₂(OH)₂, and Ti₃C₂O₁(OH)₁) nanosheets decreases by more than 30% at 8% tensile strain (Fig. 3(f)). The strain-induced peak splitting and broadening of the phonon spectrum dominated by carbon atoms (∼22.5 THz and 30–35 THz), which enhance phonon–phonon scattering and significantly shorten phonon lifetimes, thereby reducing the overall thermal transport efficiency.⁸⁸

2.2. Effect of compressive strain on the thermal conductivity of 2D materials

Li et al.⁹⁰ used EMD simulation combined with spectroscopic analysis to study the thermal transport behavior of single-layer graphene under compressive strain. The results suggest that the TC of graphene decreases significantly with the compressive strain. The TC drops by 46% at 8% strain. The mechanism is that the structural deformation caused by compression significantly enhances phonon scattering, shortens the average free path of phonons, and thereby inhibits effective heat transport. Zhao et al.⁷⁸ performed numerical studies based on density functional theory (DFT) and phonon BTE, focusing on the correlation between compressive strain and the TC of hBN. The study found that an out-of-plane compressive strain (−6%) can reduce the TC of hBN by approximately 50%, while a larger compressive strain (−18%) led to a recovery in TCs (Fig. 4(a)). Fig. 4(b) shows the phonon band structure of the hBN with AC′(B–B) configuration under different compressive strains, and the results suggest that the frequency of the softer layer-breathing mode continuously increases with the increase in strain. Fig. 4(c) demonstrates the anharmonic scattering rate (ASR) under different strains, consistent with the change in TCs. Compressive strain enhances the non-harmonic scattering of ZA phonons in hBN, thereby reducing the TC; while excessive compressive strain inhibits non-harmonic scattering, indicating that non-harmonic scattering plays a dominant role in the TC control under compressive strain.

Fig. 4 TCs in 2D materials influenced by compressive strain. (a) The TC of boron nitride under different strains. The red and blue points represent the data calculated based on the iterative way and relaxation time approximation (RTA), respectively. (b) Phonon band structures of hexagonal boron nitride under 0%, −6% and −18% out-of-plane compressive strain. The low-frequency part is enlarged. (c) Anharmonic scattering rate of boron nitride under 0%, −6% and −18%.⁷⁸ Reproduced from ref. 78 with permission from Royal Society of Chemistry, copyright 2023. (d) Phonon group velocities of the WS₂ monolayer under the representative biaxial compressive strains. (e) Phonon lifetimes of the WS₂ monolayer under the representative biaxial compressive strains at 300 K.⁶⁸ Reproduced from ref. 68 with permission from Elsevier, copyright 2020.

For TMD materials, such as tungsten disulfide, WS₂, the range of strain-induced regulation of its TC is relatively small. Han et al.⁶⁸ conducted a computational study on the thermal transport properties of single-layer WS₂ under compressive strain based on DFT and the iterative solution of the phonon BTE. The results illustrate that when the biaxial compressive strain reached −4%, the TC of WS₂ decreased from 262.78 W m⁻¹ K⁻¹ without strain to 217.40 W m⁻¹ K⁻¹, a decrease of approximately 17% (Fig. 2(b)). Fig. 4(d) presents the changes in phonon group velocities of WS₂ single layer under different compressive strains. The compressive strain increased the highest phonon group velocities of the transverse acoustic mode (TA) and longitudinal acoustic mode (LA) from 3.51 km s⁻¹ to 3.94 km s⁻¹ and from 5.52 km s⁻¹ to 6.16 km s⁻¹, respectively. Fig. 4(e) presents the phonon lifetime distribution of WS₂ under different compressive strains. The compressive strain significantly reduces the lifetime of phonons in the low-frequency region, reflecting a significant enhancement in phonon scattering rate under compression conditions, which in turn inhibits effective heat transport.

For Xene materials, such as phosphorene, the range of TC regulation achievable via compressive strain is further narrowed. Zhang et al.⁹¹ used the NEMD simulation method to study the effect of compressive strain on the TC of phosphorene. The results demonstrate that when the compressive strain increases from 0 to 5%, the decrease in the TC is less than 10%. Under the action of compressive strain, phosphorene undergoes unstable deformation, thereby enhancing phonon scattering and resulting in a decrease in the TC.

Although the thermal transport properties of MXene materials have received considerable attention in recent years, there is currently a lack of research on the mechanism by which compressive strain regulates their TC.

2.3. Effect of torsional strain on the thermal conductivity of 2D materials

Torsional strain can be classified into two modes: in-plane torsion and out-of-plane torsion. These two modes have different effects on the atomic arrangement and interaction within and outside the plane of 2D materials, thereby exerting differential control effects on their TC.

Arabha et al.⁷⁶ studied the influence of in-plane torsional strain on the thermal transport properties of 2D nanomaterials using the NEMD simulations. The atomistic model is shown in Fig. 5(a). For the graphene family and structurally similar 2D materials, the research results indicate that it is not sensitive to torsional strain, and the TCs of hBN and graphene at 7% in-plane torsional strain decrease by approximately 10% and 17%, respectively (Fig. 5(b)). Fig. 5(c) presents the phonon spectrum of graphene under different in-plane torsional strains. The results prove that torsional strain causes phonon softening, thereby leading to a decrease in the TC. For typical TMD materials, such as MoS₂, 7% torsional strain causes a decrease in TC of approximately 30% (Fig. 5(b)). It was found that the wrinkles induced by in-plane torsional strain significantly enhance the phonon scattering rate, thereby reducing the TC. Among them, the wrinkles of MoS₂ have a larger amplitude, resulting in the largest decrease in the TC, and this decrease is proportional to the square of the amplitude of the wrinkles. Currently, the research on the influence of in-plane torsional strain on the TC of Xene and MXene materials is still lacking and needs further exploration.

Fig. 5 TCs in 2D materials subjected to torsional strain. (a) MD simulation setup for TC prediction of graphene, MoS₂, and hBN under torsional deformation applied on the inner boundary. (b) In-plane TCs of graphene, hBN and MoS₂ under torsional deformations with respect to the applied strain. Results were normalized with the TCs of pristine forms. (c) Vibrational power spectra of graphene under torsional deformations.⁷⁶ Reproduced from ref. 76 with permission from Elsevier, copyright 2020. (d) TCs predicted by EMD simulations of twisted multi-layer graphene. (e) Phonon dispersions of the Γ–M direction for different compressive forces.⁹² Reproduced from ref. 92 with permission from American Institute of Physics, copyright 2017.

To be consistent with tensile and compressive strains, the out-of-plane torsion strain is defined as ε = θ W/(2h), where θ is the torsion angle (measured in radians), W is the width of the 2D material, and h is the thickness of the 2D material. Savin et al.⁹³ studied the effect of uniform out-of-plane torsion strain on the TC of single-layer carbon nanotubes through MD simulation systems. The results demonstrate that with the increase of the torsional strain, the TC of the carbon nanotube first increases and then tends to stabilize, reaching a peak at approximately 34% strain and approaching saturation. The TC increased by about 10%. Si et al.⁹² used the EMD method to explore the regulation rules of out-of-plane torsional strain on the TC of multilayer graphene. The TC of multilayer graphene slightly decreases first and then increases (Fig. 5(d)). At an out-of-plane strain of 58% (θ = 22.5°), the TC reached the lowest value of 4692.40 W m⁻¹ K⁻¹, a decrease of about 8% compared to the unstrained state; as the out-of-plane strain further increased to 310% (θ = 112.5°), the TC increased to the maximum value, with an increase of about 33%, and gradually approached saturation. Based on the spectral energy density (SED) analysis (Fig. 5(e)), the study found that the main influence of torsion strain on phonon dispersion is concentrated on the curved ZA branch: on the one hand, the torsion strain reduces the lifetime of ZA phonons; on the other hand, it increases the group velocity and phonon specific heat capacity of ZA phonons. When the torsion angle is small, the effect of reduced phonon lifetime dominates, leading to a decrease in TC; while at larger torsion angles, the increase in phonon specific heat capacity has a more significant effect on the increase of TC, causing the overall TC to rise.

2.4. Effect of bent loading on the thermal conductivity of 2D materials

Compared with the aforementioned loading methods, such as tension, compression, and torsion, the bending strain has the narrowest range of control over the TC of 2D materials, and the TC generally exhibits a slow decreasing trend with the bent loading. Taking the graphene family and structurally similar 2D materials as an example, existing studies suggest that the decrease in TC typically does not exceed 5%. Liu et al.⁷⁹ used NEMD to simulate the effect of bent loading on the TC of graphene; the atomistic model is depicted in Fig. 6(a), and the results indicate that at a bent strain of 1.7%, the TC of graphene decreases by 5%. Fig. 6(b) demonstrates the phonon density of states (DOS) of graphene, and the results suggest that the bending deformation significantly reduces the amplitude of the low-frequency phonon modes, triggers the red shift of the high-frequency phonon modes, decreases the group velocity, and thus leads to a decrease in the TC of graphene. The phonon mean free path (MFP) results are given in Fig. 6(c). The low-frequency phonons play a dominant role in heat transport, while the bent strain limits the effective propagation path of low-frequency phonons, thereby suppressing the overall TC. In addition, Qiao et al.⁸⁰ studied the effect of bent strain on the TC of hBN through MD simulation, and the results demonstrate that at a bent strain of 10%, the TC of hBN decreases by only about 3%.

Fig. 6 TCs in 2D materials under bending strain. (a) Schematic diagram of the bending loading of two-dimensional materials. (b) Total phonon DOS of deformed graphene. (c) Graphene phonon MFP.⁷⁹ Reproduced from ref. 79 with permission from American Chemical Society, copyright 2022.

For TMD materials, Qiao et al.⁸⁰ in the aforementioned study investigated the effect of bent strain on the TC of MoS₂. The TC of MoS₂ decreased by 9% at a 10% bent strain. The research indicates that as the bent strain increases, the peak of low-frequency phonon modes (such as the ZA mode) shifts, resulting in a broadening of the phonon spectrum and an enhancement of phonon scattering, thereby reducing the TC. Currently, there is still a lack of research on the effect of bent strain on the TC of materials such as Xenes and MXenes, and the related mechanisms need to be further explored and revealed.

2.5. Theoretical and computational frameworks for multiscale thermal transport

The previous subsections have reviewed the trend of TC of typical 2D materials under different mechanical loadings (such as tension, compression, torsion, and bending). Various numerical methods have been adopted to model the strain-induced variations in phononic thermal transport, depending on the time and length scales of the material system (Fig. 7). At the atomic scale (<10⁻⁹ m), the first-principles calculation based on DFT can reveal the lattice dynamics and phonon dispersion under strain.⁹⁴ At the nanoscale (10⁻⁹−10⁻⁷ m), MD simulations, including EMD and NEMD methods, can characterize the phonon scattering and interface thermal resistance under mechanical strain.^95,96 At the mesoscopic scale (10⁻⁸–10⁻⁵ m), kinetic theory provides an analytical framework for connecting atomic scale information with macroscopic observability. BTE is often used to calculate the effective thermal conductivity and describe the non-equilibrium distribution behavior of phonons.⁹⁷ The acoustic mismatch model (AMM) and diffusive mismatch model (DMM) can quantitatively describe the phonon transmission and reflection behavior at interfaces.^98,99 At the macro scale (>10⁻⁶ m), continuum theory and finite element method (FEM) are used for the distribution of heat flux and thermal stress at the device level.¹⁰⁰

Fig. 7 Multiscale theoretical and computational frameworks for thermal transport in 2D materials.

These methods form a multi-level cognitive system ranging from atomic interactions to macroscopic thermal responses, providing theoretical support and computational basis for the numerical analyses of strain-induced thermal transport in 2D materials and structures. The computational methods also lay a theoretical foundation for the structural design and optimization for enhanced thermal performance in 2D systems.

3. Optimizing the tuning range by structural design in 2D heterostructures or multi-layered 2D materials

Structural design plays an increasingly important role in developing structures and devices with tunable thermal transport properties. By adjusting the interface configuration, enhancing the interlayer interaction, and reconstructing the geometric structure, the controllable range of TC or thermal conductance across the interface within the structure can be significantly expanded.^101,102 Compared with intrinsic 2D materials, these structures introduce more complex phonon propagation paths and stronger anharmonic interactions in both in-plane and out-of-plane directions, making the thermal conduction mechanism more diverse and highly engineering-controllable.¹⁰³ The thermo-mechanical correlation depends on the types of structures, which correspond to different phonon transport mechanisms subject to mechanical loadings. The 2D structure types can be classified into three categories based on the composition dimension and connection mode: vertically stacked vdW heterojunctions, lateral heterojunctions connected by intramolecular chemical bonds, and multi-layer vertically stacked structures. Therefore, this section will summarize current research according to structural configuration. Both the TC and interfacial thermal conductance (ITC) of the structures under various loading modes will be discussed.

3.1. Thermal transport in vdW heterostructures influenced by mechanical loadings

The vdW heterostructures comprise more than two vertically stacking 2D structures. The interlayer coupling is solely based on vdW forces without covalent bonds. It possesses engineering-friendly interfaces, controllable physical properties, and highly flexible structural characteristics.^104,105 Previous studies have reported that external mechanical loading can alter both in-plane and cross-plane thermal transport properties. Gao et al.⁵⁴ employed the NEMD method to investigate the thermal conduction behavior of graphene-MoS₂ bilayer vdW heterostructures under different tensile strains. The computational model is presented in Fig. 8(a). The results indicate that under no strain conditions, the TCs of the heterostructure, bilayer graphene, and bilayer MoS₂ are 34.05, 100.79, and 8.50 W m⁻¹ K⁻¹, respectively. As the tensile strain increases, the TCs of all three materials decrease monotonically (Fig. 8(b)). Among them, the tensile strain significantly weakens the out-of-plane warping caused by lattice mismatch, thereby intensifying interfacial scattering and reducing the overall TC (Fig. 8(c)). Additionally, tensile strain causes the high-frequency modes in the graphene phonon spectrum to shift towards lower frequencies (Fig. 8(d)), reflecting a phonon softening effect, which further suppresses thermal energy transport. Wu et al.¹⁰⁶ also utilized NEMD calculations to explore the changes in ITC of a graphene–MoS₂ heterostructure under cross-plane compressive strain. The results indicate that a 10% compressive strain can increase the ITC by approximately 100%, while the same magnitude of tensile strain leads to a reduction of about 70%.¹⁰⁷

Fig. 8 Thermal transport properties of vdW heterostructures subjected to mechanical loadings. (a) Schematic of bilayer graphene–MoS₂ heterostructure with loading direction of the tensile strain. (b) TC of the graphene–MoS₂ heterostructure (GM), bilayer graphene (BGE), and bilayer MoS₂ (BMS) as functions of the applied tensile strain ε. (c) Lattice mismatch-induced out-of-plane displacement, d₀, of atomic positions in graphene and MoS₂ layer in the heterostructure at equilibrium under tensile strain of 0% and 15%. (d) Total phonon spectrum G(ω) of graphene in the heterostructure under tensile strain ε of 0%, 5%, and 15%.⁵⁴ Reproduced from ref. 54 with permission from American Chemical Society, copyright 2016. (e) Varying range of the thermal conductance G along with cross-plane compressive strain. (f) Normalized overlap parameter S/S₀ of vibration spectra at interfaces along with cross-plane compressive strain. S₀ is the overlap parameter of strain-free values, which are 0.034 and 0.317 for in-plane and out-of-plane, respectively.¹⁰⁸ Reproduced from ref. 108 with permission from Elsevier, copyright 2017.

Chen et al.¹⁰⁸ adopted MD simulations to investigate the influence of cross-sectional compressive strain on the ITC of graphene-black phosphorus vdW heterojunctions. The simulational results are shown in Fig. 8(e), where the ITC increases significantly with compressive strain: when the strain increases from 0% to 11.93%, the normalized ITC increases by nearly an order of magnitude; when the strain approaches ∼18%, the enhancement factor reaches as high as 16.13. Further analysis of Fig. 8(f) reveals that compressive strain enhances the interfacial phonon coupling strength (characterized by the overlap parameter S), thereby effectively improving the ITC.

Zhao et al.¹⁰⁹ employed the NEMD calculations to investigate the influence of biaxial strain on the thermal transport properties of the MoS₂–WS₂ vdW heterostructure. The results showed that biaxial strain could significantly suppress the lattice TC, with the tensile strain effect being particularly significant. At 300 K and a 3% tensile strain, the TC decreased by approximately 22%. This phenomenon is mainly attributed to the enhanced lattice asymmetry induced by the strain, which increased phonon scattering and thereby weakened the TC.

In addition to normal and in-plane loads, interlayer rotation can be regarded as a special form of torsional strain, and the interface configuration can be changed mechanically, thus effectively regulating the heat transfer behavior of the vdW heterostructure. Ren et al.¹¹⁰ studied the effect of twist angle on the ITC in graphene-hBN heterostructures by conducting NEMD simulations. The results show that ITC changes non-monotonically with the torsion angle: when the torsion angle increases from 0 to about 30, the TC drops sharply and then tends to be stable. Without rotation (θ = 0), the ITC of the graphene-hBN interface layer can reach 509 MW m⁻² K⁻¹ at 500 K, whereas at θ = 26.33°, ITC decreases by about 50%. This is due to the mismatch of phonon spectra between layers and the weakening of lattice registration, which hinders the transmission of coherent phonons. Further analysis shows that interlayer rotation destroys the potential field registration between graphene and hBN layer, weakens coherent coupling and enhances phonon scattering, thus significantly suppressing cross-layer heat conduction.

3.2. Thermal transport in lateral heterostructures subjected to mechanical loadings

2D lateral heterojunctions are formed by horizontally bonding two different 2D structures through covalent bonds on the same atomic plane. The interface presents a one-dimensional line defect structure.^111,112 Different interface configurations exhibit complex and controllable TC characteristics under external mechanical loading. Gao et al.¹¹³ utilized NEMD simulations to investigate the interface thermal transport across the graphene-boron nitride (BN) lateral heterojunction under tensile strain. The computational model is presented in Fig. 9(a). The results show that the interfacial heat flow vary with tensile strain in a non-monotonic fashion (Fig. 9(b), left): In the small strain range (ε < 5%), the heat flux from graphene to boron nitride (J_G→BN) first decreases and then recovers, while the heat flow in the reverse direction (J_BN→G) shows a mirror-like trend. As shown in Fig. 9(b), right, the structure has a significant thermal rectification effect, quantified by the thermal rectification ratio η = (J(_BN→G) − (J_G→BN)/J_(G→BN)), which can also be adjusted by tensile strain. The mechanism is shown in Fig. 9(c): the stress concentration factor P monotonically decreases with the strain, consistent with the later decline of the thermal rectification ratio, indicating that interface stress relaxation in the large strain range is the dominant factor for the change in thermal rectification; while in the small strain range (Fig. 9(d)), the nonlinear changes of the interface phonon spectrum overlap parameters S₀ and H₀ are synchronized with the peak of the thermal rectification ratio, indicating that phonon resonance and phonon localization effects dominate the thermal rectification behavior.

Fig. 9 Thermal transport properties of in-plane heterojunctions under mechanical loadings. (a) Schematic of a graphene-boron nitride lateral heterostructure with thermal rectification phenomenon. (b) Heat flow J and variation of thermal rectification η in heterostructures as functions of tensile strain ε. (c) Comparison of relative stress concentration P in different heterostructures as functions of tensile strain ε. (d) Overlap of out-of-plane phonon spectra S₀ and relative overlap H₀ in the heterostructure under different strains.¹¹³ Reproduced from ref. 113 with permission from American Chemical Society, copyright 2017. (e) Variation of ITC under tensile strain. (f) Representative phonon DOS of the boron nitride-carbon (BN-C) heterostructure system with different x-axial tensile strains. The higher frequency peaks denote the G-band. (g) Variations of ITC under compressive strain. The insets demonstrate the representative schematic of model deformation under different compressive strains. (h) Shear strain effects on the ITC of heterostructure systems.¹¹⁴ Reproduced from ref. 114 with permission from Elsevier, copyright 2022. (i) Temperature distribution of the MoSSe–WSSe heterostructure with armchair edge interface with the heat flux along the x-direction, the insets represent the temperature profile of the MoSSe–WSSe system. (j) The schematic of spatial distribution for heat flux of arm-1 and arm-2 heterostructures (vector arrows on all atoms, steady-state average over 4 ns). (k) The initial configurations of the planar MoSSe–WSSe heterostructures before and after deformation. The structure predicted by the MD simulation fits well with that from the Euler beam model.¹¹⁹ Reproduced from ref. 119 with permission from Wiley-VCH Verlag, copyright 2022.

For the graphene family heterostructure, Liu et al.¹¹⁴ discovered through the NEMD simulations that the interfacial thermal conductance of graphene-hBN heterostructures significantly decreased under a large tensile strain (Fig. 9(e)). The mechanism is shown in Fig. 9(f). As the tensile strain increases, the redshift of the G-band is obvious, indicating that the phonon group velocity decreases, resulting in a reduction in thermal conduction efficiency.^115,116 Slight compression and shear strain enhance atomic coupling, improving the propagation of phonons across the interface, thereby enhancing TC (Fig. 9(g)). Additionally, different connection configurations have different responses to strain, and the direction sensitivity of shear strain is significant. Excessive strain causes bending deformation, which instead inhibits heat transport (Fig. 9(h)). Moreover, Gao et al.¹¹⁷ designed a pre-patterned heterostructure with auxetic graphene and contractile graphene. When arranged in a stripe-like periodic pattern, the Poisson's ratio of the heterostructure can be extended into a negative domain, which contributes to broadening the tunable range of TC up to about 75%. By embedding a highly stretchable unit design¹¹⁸ on the loading direction, the tunable range is expected to be further enhanced.

Ren et al.¹¹⁹ conducted the NEMD simulations on the ITC of the MoSSe–WSSe junction, which is a Janus structure. The result suggests that the ITC across the MoSSe–WSSe junction depends on the connection methods (arm-1 and arm-2, Fig. 9(i)) due to the contrasting stress states. This difference stems from the more significant interface curvature and the structural discontinuity induced by the prestress in the reverse connection (Fig. 9(j)). The DFT calculation results further confirm that the local deformation caused by interface prestress significantly changed the phonon propagation path (Fig. 9(k)), thereby regulating the temperature response behavior of interfacial thermal conduction. Moreover, the thermal rectification ratio of this structure is 11%, indicating that geometric asymmetry is a major factor leading to asymmetric heat transfer.

3.3. Thermal transport in multilayer 2D materials affected by mechanical strain

Multilayer 2D materials are formed by stacking three or more 2D structures vertically and coupling them through vdW forces. Under the combined effect of interlayer phonon coupling, interface scattering, and cross-layer phonon tunneling, their TC and cross-plane thermal conductance exhibit significant anisotropy and layer-dependent characteristics.¹²⁰ Under mechanical strain, the lattice constant and phonon energy spectrum undergo reconfiguration, thereby regulating their thermal transport properties.

Nakagawa et al.¹²¹ utilized Raman spectroscopy and atomic force microscopy to probe the effect of tensile strain on the TC of multilayer graphene. The study revealed that applying a 0.1% tensile strain alone can reduce the TC by nearly 70%. This significant decrease was mainly attributed to the tensile strain weakening the stiffness tensor and enhancing the lattice anharmonicity, thereby reducing the group velocity and MFP of phonons. Chen et al.¹²² employed NEMD simulations to explore the regulatory mechanism of strain on the interface thermal transport of multilayer graphene. As shown in Fig. 10(a), the Kapitza thermal resistance decreased gradually with the increase in the number of graphene layers and reached a saturated value (approximately 6 × 10⁻¹⁰ m² K W⁻¹) at approximately 16 nm thickness. When a out-of-plane single-axis strain is applied, the Kapitza thermal resistance exhibits a monotonic increasing trend from compression to tension: at a 5% compression strain, the thermal resistance decreases by approximately 50%; while at a 7% tension strain, the thermal resistance increases to approximately three times the original value. According to Fig. 10(b), this difference originated from the compressive strain shortening the interlayer distance, enhancing interlayer coupling, and raising the frequency of low-frequency phonons, thereby increasing the group velocity of phonons,¹²³ enhancing the interface thermal transport capacity, and reducing the thermal resistance; while the tensile strain weakened the interlayer interaction, resulting in the opposite effect.

Fig. 10 Thermal transport properties of multilayer structures under mechanical loadings. (a) Room-temperature Kapitza resistance, R, in multilayer graphene versus uniaxial cross-plane strain, e_c. (b) DOS of multilayer graphene under different uniaxial cross-plane strain, e_c.¹²² Reproduced from ref. 122 with permission from American Chemical Society, copyright 2014. (c) Schematic of TC measurements with a diamond anvil cell integrated with a picosecond transient thermos reflectance (ps-TTR) system. (d) Extracted cross-plane TC (both lattice and electronic) as a function of pressure. The red curve is included only as a guide to the eye. (e) first-principles calculations of the pressure-induced change of phonon dispersion curves and phonon group velocities in multilayer MoS₂ along both cross-plane (Γ–A) and in-plane (Γ–M) directions. (f) Schematic of CPS measurements. (g) Pressure-dependent LAP frequencies extracted from CPS measurements (black circles) and LAP group velocities from first-principles calculations (red squares).¹²⁴ Reproduced from ref. 124 with permission from American Physical Society, copyright 2019.

Meng et al.¹²⁴ conducted picosecond time-resolved thermal reflection (TDTR) experiments in conjunction with coherent acoustic spectroscopy (CPS) to investigate the out-of-plane TC of multilayer MoS₂. The experimental setup is illustrated in Fig. 10(c). The experiments demonstrated that as the equivalent pressure gradually increased from atmospheric pressure to 30 GPa, the out-of-plane TC of the MoS₂ bulk material increased from approximately 3.5 W m⁻¹ K⁻¹ to approximately 25 W m⁻¹ K⁻¹, more than seven times the original value (Fig. 10(d)). This increase was mainly attributed to the enhancement of lattice rigidity under high pressure, leading to an increase in the group velocity and frequency of phonons. However, further pressure increase led to a saturation in the TC increase, and even a downward trend. The main reason for this was the enhancement of non-harmonic scattering induced by high pressure: although the frequency and group velocity of longitudinal acoustic phonons (LAP) continued to increase (Fig. 10(e)), expanding the phonon dispersion bandwidth, it also triggered more scattering channels between phonons, resulting in an enhanced non-harmonic effect, shortened phonon lifetime.^125,126 The pressure-dependent LAP mode frequency obtained by CPS measurements is shown in Fig. 10(f and g). It presents the comparison results of the extracted modal frequencies from the experiments and the LAP group velocities calculated by first principles, further verifying the trend of significant increase in LAP group velocity under high pressure, indicating that the non-harmonic scattering mechanism gradually becomes the dominant limiting factor in high-pressure heat transfer.

In highly anisotropic phosphorene, Zhang et al.⁹¹ discovered through NEMD that the in-plane TC of multilayer phosphorene is almost unaffected by the number of layers. This is attributed to the wrinkled structure of phosphorene, which hinders inter-layer interactions and keeps the in-plane TC relatively stable in the multilayer structure. Tensile strain can enhance the in-plane TC of phosphorene. Tensile strain can increase the rigidity of phonons, improve the group velocity of phonons, and thereby increase the TC. In contrast, compressive strain reduces the TC.¹²⁴

4. Applications based on mechanically adjustable thermal properties in 2D material systems with structural design

In the previous sections, we have discussed the effect of mechanical loading on thermal transport in 2D materials and structures. Given the wide tuning range of thermal transport properties, several applications as thermal devices have been proposed by previous research,^127,128 such as thermal switches, thermal diodes, phononic devices, and thermal energy conversion/storage systems. This section will introduce these applications based on the mechanically tunable thermal transport properties and the associated mechanisms.

4.1. Thermal switch based on 2D materials and structures

Thermal switches are thermal management devices that can regulate heat flow between two distinct states: the “on” state (high TC) and the “off” state (low TC). The regulation of thermal switches is controlled by non-thermal parameters reversibly and rapidly.^129–132 Therefore, 2D material systems with great tunability on the TC and mechanical reversibility hold promising potential in the application of thermal switches.

Chen et al.²³ proposed an electromechanically actuated thermal switch based on a suspended graphene membrane, demonstrating the practical potential of 2D materials in low-power dynamic thermal regulation. The operating principle is illustrated in Fig. 11(a). In the “off” state, the graphene microstrip is suspended between top and bottom electrodes, isolating thermal conduction. Upon applying an applied voltage, V_A, electrostatic forces deflect the membrane to contact the bottom silicon electrode, establishing a new thermal pathway and transitioning the device to the “on” state. The thermal resistance of “off” state and “on” state are denoted as R_th,off and R_th,on, and the author defined the thermal switching ratio as R_ratio = R_th,off/R_th,on to describe the characteristic of thermal switch. In this 2D material thermal switch, the R_ratio is about 1.1, which demonstrates a notable improvement in the thermal conductance. Moreover, Chen et al. carried out layer designs on the 2D graphene membrane by adding the layer number of graphene or patterning serpentine Cr structures over the graphene. The result is shown in Fig. 11(b). With more graphene layers, the R_ratio is slightly enhanced, while with the serpentine Cr, the R_ratio increases to 2.5 at the same V_A. It proves that the structural design can significantly improve the tunability of the 2D material thermal switch with lower cost on mechanical loading.

Fig. 11 Thermal switch enabled by mechanically tunable thermal transport properties. (a) Schematic of the electrostatic thermal switch based on a graphene membrane. The arrows indicate parallel heat flow paths through the supporting pillar and through the graphene membrane when this is collapsed. (b) Relationship between R_ratio and applied voltage V_A with different structural designs.²³ Reproduced from ref. 23 with permission from Institute of Physics Publishing, copyright 2021. (c) Graphene kirigami models with two and three sectors. (d) The relationship between the interfacial thermal resistance, R, of two kinds of thermal switch models under different switch states. (e) Thermal conductance, T_C, as a function of the twisting angle of the thermal switch.¹³³ Reproduced from ref. 133 with permission from TAYLOR & FRANCIS LTD, copyright 2020.

Wei et al.¹³³ designed a graphene kirigami thermal switch using MD simulation. As presented in Fig. 11(c), the authors designed thermal switches as bilayer graphene, which includes a 2-sector kirigami model or a 3-sector kirigami model. When the thermal switch is on, the two layers of graphene overlap, while when the thermal switch is off, these two layers are spatially separated. Fig. 11(d) shows the interfacial thermal resistance R of the “on” state and the “off” state under various sizes of the model. In sizes below 800 nm², when the state turns to “on”, the values R approach 0, which is much lower than those in the “off” state, elucidating a well-enhanced thermal conductance. Moreover, Wei et al. calculated a series of thermal conductances, T_C, under different twisting angles of the thermal switch, shown in Fig. 11(e). T_C gradually increases when the overlap area ratio continuously increases, indicating that the thermal conductance of the kirigami thermal switch can be tuned by precise output twisting.

4.2. Thermal diodes with adjustable performance

Thermal diodes are a type of functional device that can achieve efficient heat transfer in a specified direction and significantly suppress heat flow in the reverse direction, similar to the rectification effect of electronic diodes.¹³⁴ Their working mechanism stems from the structural and property asymmetry during the heat transfer process.^135–137 Therefore, 2D materials with high rectification efficiency and mechanical tunability have opened up broad prospects for the development of thermal diodes.¹³⁸

Zhang et al.¹³⁹ constructed a single-layer MoSe₂–WSe₂ lateral heterojunction and used the suspended H-type gold nano-sensor/thermometer integrated platform to accurately measure its thermal transport properties in a vacuum environment ranging from 273 to 378 K. The measurement principle is shown in Fig. 12(a). The system employs two gold sensors (acting as the heating end and the detection end), obtaining the TC of the heterojunction under two opposite heat flow directions. As shown in Fig. 12(b), when the interface is perpendicular to the heat flow direction (θ = 0°), the TC in the direction from MoSe₂ to WSe₂ (J⁺) is significantly higher than the reverse direction (J⁻), and the thermal rectification factor reaches up to 96%. As θ increases, the rectification effect gradually weakens and disappears completely at θ = 90°. Fig. 12(c) shows the variation laws of the normalized thermal rectification ratio η and the phonon spectrum overlap ratio H with the interface heat-flow angle θ. Both decrease monotonically with θ and tend to zero at θ = 90°, verifying the angle dependence of thermal rectification. This result indicates that by simply rotating the interface direction, the thermal rectification factor can be continuously adjusted from the maximum value to zero, achieving angle-programmable 2D on-chip thermal diodes.

Fig. 12 Thermal diode with mechanically tunable performance. (a) Schematic diagram of the thermal measurement circuit based on the H-type sensing device. (b) The measured TCs of sample 1, in two opposite heat flow directions within a temperature range of 273 to 378 K. The insets show the atomic diagrams to demonstrate the interface morphology features at different angles. (c) Normalized TR ratio (η) and spectra overlap ratio (H) plotted with angle (θ).¹³⁹ Reproduced from ref. 139 with permission from American Association for the Advancement of Science, copyright 2022.

In vdW heterostructures, strain-induced interfacial phonon coupling modulation also provides an effective rectification strategy.¹⁴⁰ For example, MoS₂–WSe₂ heterostructures exhibit significant modulation of phonon matching at the interface under biaxial strain, thereby altering the heat transmission probability in forward versus reverse directions.¹⁴¹ At 6% biaxial strain, the rectification ratio increased from under 100% in the unstrained state to over 250%, emphasizing the sensitivity of interface thermal transport to strain.^142,143 In addition, strain-induced phase transitions represent a key mechanism in thermal diode engineering. Studies have shown that applying shear strain to materials such as MoS₂ can trigger a phase transition, which alters both phonon scattering mechanisms and electronic structure, thereby enabling phase-change-based thermal rectification.^144,145 This approach offers good reversibility and integration compatibility, making it well-suited for the realization of switchable thermal diodes and thermal logic gates.

4.3. Phononic devices based on mechanically tunable thermal transport

Strain engineering is not only a powerful tool for tuning the electronic properties of 2D materials, but also holds great promise for modulating phonon dynamics and controlling thermal transport. It provides a rich physical foundation and structural design strategies for the development of next-generation thermal functional devices.^146,147 In recent years, extensive research has been devoted to strain-induced phonon spectrum modulation, interface thermal resistance optimization, and the construction of non-uniform strain fields, laying the groundwork for mechanically reconfigurable phononic devices.

Studies have shown that strain can significantly alter the phonon spectral characteristics of 2D materials.¹²⁸ In transition metal dichalcogenides, such as MoS₂, the application of strain leads to shifts and splitting of Raman peaks, reflecting changes in phonon frequencies and modes.^148,149 In the Raman spectra of graphene, strain induces the splitting of the G peak into G⁺ and G⁻ modes, indicating the lifting of phonon degeneracy. This splitting is highly controllable by both strain magnitude and direction, forming the physical core of “phonon engineering” for dynamically regulating thermal carrier transport.¹⁵⁰

Researchers have found that as the strain increases, the corresponding band gap decreases, thereby altering the electronic band structure at the nanoscale; before the exciton undergoes recombination, it migrates hundreds of nanometers towards the low band gap region at the top of the fold, a phenomenon known as the “funnel effect”. The device is for guiding phonons directionally toward high-strain regions, thereby controlling heat flow paths and enabling devices like thermal waveguides and thermal concentrators.¹⁵¹ As illustrated in Fig. 13(a and b), monolayer MoS₂ under non-uniform strain exhibits an exciton funnel effect, with excitons migrating toward regions with smaller band gaps. Fig. 13(c) presents the variation of exciton wavelength distribution in four-layer MoS₂, revealing that strain fields can regulate the strength and spatial distribution of excitonic states. This suggests that thermally excited phonons can also be modulated by local strain, enabling spatially controlled heat concentration or dispersion.

Fig. 13 Functional phononic devices enabled by mechanically tunable properties. (a) Calculated band structures for nonuniformly strained monolayer MoS₂ under 0% (left panel), 2% (middle panel), and 4% (right panel) strain. (b) Schematic of the exciton funnel effect induced by the inhomogeneous strain in a wrinkled MoS₂ region. (c) Spatial distribution of the A exciton wavelength in a four-layer thick MoS₂ flake with four wrinkles.¹⁵¹ Reproduced from ref. 151 with permission from American Chemical Society, copyright 2013. (d) Schematic illustration of the phonon dispersion modulation subjected to a strain gradient. (e) Left, calculated phonon dispersion of Si under different states of elastic strain. Right, strain-gradient-induced phonon scattering rate τ_sg⁻¹, at 300 K of each phonon mode under a strain gradient of 0.118% nm⁻¹ (10% strain difference over 85 nm width). Right y-axis represents the phonon frequency in meV. The characteristic phonon frequency at 300 K is estimated as f_p = (K_BT) = 2πħ, where K_B and ħ are the Boltzmann and reduced Planck constants, respectively. Inset, atomic interaction cut-off distance calculation using density functional theory.¹⁵⁶ Reproduced from ref. 156 with permission from Springer Nature, copyright 2024.

In vdW heterostructures, mechanical tuning strategies offer additional structural degrees of freedom for thermal regulation. Qi et al.¹⁵² demonstrated that dry transfer stacking techniques can create clean interfaces with significantly reduced thermal boundary resistance and enhanced phonon transmission. Moreover, adjusting the interlayer twist angle modifies the moiré superlattice, which in turn alters phonon propagation paths and band structures, thereby influencing TC and phonon scattering.¹⁵³ For instance, in the case of magic-angle bilayer graphene (∼1.1°), the emergence of flat phonon modes leads to a substantial reduction in TC.^154,155 In WSe₂–MoSe₂ heterostructures, exciton–phonon resonances have also been observed, indicating coupling between phononic, electronic, and optical processes and expanding the potential for 2D materials in quantum thermal devices.

Stress regulation also provides a viable route for thermal control in vdW heterostructures. For example, applying bending stress to BP–MoS₂ heterostructures has enabled reversible modulation of the TC due to the strain-induced reconstruction of phonon dispersion and scattering channels. Such strategies are highly adaptable for use in flexible electronics and reconfigurable thermal logic devices.

In nanostructured systems, the phonon spectral broadening effect induced by non-uniform strain offers a novel pathway for thermal transport regulation.¹⁵⁷ Yang et al.¹⁵⁶ introduced strain gradients into bent nano-meter-thin silicon nanoribbons (SiNRs), experimentally demonstrating a reduction in TC by up to 34% ± 5%. First-principles calculations revealed that the strain gradient broadens the phonon frequency distribution and enlarges the phase space for three-phonon scattering, thereby reducing phonon lifetimes and effectively suppressing heat transport. This effect was validated through both scanning transmission electron microscopy-electron energy loss spectroscopy experiments and ab initio simulations. Fig. 13(d and e) schematically illustrates the changes in phonon dispersion and enhanced scattering rates under strain gradients, highlighting the potential of this mechanism for phonon filtering and localized thermal modulation.

4.4. Thermal energy conversion and storage devices with mechanically tunable performance

2D materials and structures demonstrate significant advantages in regulating the phase transformation strain of phase change materials (PCM) and enhancing the performance of thermal energy conversion and storage, enabling the design of high-performance thermal energy devices.^158–161 During the phase transformation process, PCM experiences volume changes due to the state transitions from solid to liquid or from solid to solid, which generate phase transformation strain. Without constraints, this strain is prone to causing material structure damage and a decline in cycle stability, becoming a key bottleneck hindering its practical application.^162,163 However, 2D materials can effectively suppress the phase transformation strain through framework support, interlayer constraints, or cross-linking network design. They can also simultaneously enhance the TC efficiency and energy conversion performance of the materials.

MXenes serve as the supporting framework, their nanosheets form an ordered constrained structure through physical encapsulation. In the layered phase-change material prepared by Fan et al.,¹⁶⁴ Ti₃C₂T_X nanosheets tightly encapsulate polyethylene glycol (PEG) molecules (Fig. 14(a)). The vdW forces and mechanical constraints between the sheet layers significantly suppress the expansion strain of PEG during melting, enabling the material to maintain excellent shape stability before and after the phase transition. The latent heat of the phase transition is as high as 164.9 J g⁻¹ (Fig. 14(b)), strongly confirming the efficient regulation effect of the layered structure on the phase transition strain. Similarly, Wu et al.¹⁶⁵ compressed graphite sheets and paraffin to form an electronic aggregate (Fig. 14(c)). The high orientation of the graphite sheets builds a rigid heat-conducting network, not only increasing the TC to 4.4–35.0 W m⁻¹ K⁻¹ (Fig. 14(d)), but also effectively constraining the flow strain of paraffin during melting through sheet stacking. When this aggregate is applied to the thermal management of the commercial 18 650 lithium-ion batteries, it can control the maximum surface temperature of the battery at 47.8 °C at a discharge rate of 2.3C (1C = rated capacity current), significantly outperforming the 61.0 °C temperature of the battery without encapsulation.^166,167

Fig. 14 Energy conversion devices enabled by 2D materials and structures with mechanically adjustable thermal transport properties. (a) SEM images of the multilayered Ti₃C₂T_X. (b) Differential scanning calorimetry and thermogravimetric (TG) curves of PEG and Ti₃C₂T_X/PEG composites with different Ti₃C₂T_X contents.¹⁶⁴ Reproduced from ref. 164 with permission from the Royal Society of Chemistry, copyright 2019. (c) A schematic illustration of the thermal energy storage device. (d) The thermal cycle stability of the thermal energy storage device. 165 Reproduced from ref. 165 with permission from Wiley-Blackwell, copyright 2019. (e) Infrared images of BN (left) and phase change composites (right) during different heating and cooling times. (f) Differential scanning calorimetry curves.¹⁶⁸ Reproduced from ref. 168 with permission from the Royal Society of Chemistry, copyright 2016. (g) Schematic illustration for the process of solar-thermal-electric energy conversion. (h) Schematic thermoelectric measurement system of phase change aerogel-loaded thermoelectric generators.¹⁶⁹ Reproduced from ref. 169 with permission from the Royal Society of Chemistry, copyright 2020. (i) The temperature curves of the thermal control device obtained at different currents.¹⁷² Reproduced from ref. 172 with permission from Elsevier, copyright 2021.

For applications that require both insulation and strain stability, the modification strategies of BN nanosheets demonstrate unique value. Yang et al.¹⁶⁸ designed a biomimetic structure to enable the BN layers to form a stable framework through the bridging effect of polydopamine, preserving the insulation property of BN while also inhibiting the volume strain of phase transformation through the interaction between the layers, combined with the solar energy absorption ability of polydopamine, achieving a solar-to-thermal energy conversion and storage efficiency of 73.1% (Fig. 14(e and f)). Cao et al.¹⁶⁹ prepared a based on graphene oxide (GO) phase change aerogel through a one-step hydrothermal method (Fig. 14(g)), where the porous network formed by GO nanosheets can effectively buffer the volume changes of the phase transformation components, thereby alleviating the strain. Under simulated light of 200 mW cm⁻², the output power of the loaded thermoelectric generator reached 2.13 mW (Fig. 14(h)), fully demonstrating the role of strain stabilization in ensuring energy conversion performance.^170,171

In the solid–solid phase transition system, the core of strain control lies in eliminating the irreversible strain caused by liquid flow through chemical cross-linking. In the flexible graphene/PEG composite film developed by Kou et al.,¹⁷² the PEG molecules cross-link with the graphene film to form a network structure, converting the solid–liquid phase transition into a solid–solid phase transition, completely suppressing the flow strain during melting, enabling the material to stably cycle 1000 times within the phase transition temperature range of 5–60 °C, and maintaining excellent flexibility (foldable and cuttable). Based on this, the flexible thermal management device was constructed, and the electrical-to-thermal conversion efficiency under electric heating conditions reached 94% (Fig. 14(i)), fully demonstrating the decisive influence of strain control on the long-term service performance of phase change materials.^162,173

5. Summary and outlook

We have summarized the TC of 2D materials and their nanostructures influenced by mechanical effects. Mechanical stimuli such as tension, compression, torsion, and bending can lead to the deformation of the lattice, which alters the phonon transport properties and the phononic TCs of 2D materials. Although the correlation between TC and mechanical strain depends on the type of 2D material and the loading mode, mechanical strain generally suppresses TC due to the reduced phonon group velocity and MFP in stretched lattices. There are some exceptions due to the atomic structures. The compressive loading is expected to stiffen the phonon modes, which are difficult to exert due to the low flexural rigidity. Consequently, the 2D material is buckled, and the TC is reduced by a shortened geometric heat transfer path. The response of thermal transport to mechanical stimuli can be adjusted by structural design, such as vdW (vertical) heterostructures, in-plane heterostructures (heterojunctions), and multi-layered structures.

Different from the pristine materials, the mechanical loading mainly impacts the heat transfer at the interfaces and the heat transfer paths instead of the intrinsic phonon properties of materials, which leads to a broader tuning range of TC than the pristine materials. The mechanically tunable thermal transport properties in 2D materials and their nanostructures can be adopted in multiple thermal and energy applications, including thermal diodes, thermal switches, tunable phononic devices, and energy conversion and harvesting devices. Future studies in this field are of critical importance to the thermal management of electronic devices that frequently experience mechanical loading and the design and development of next-generation functional thermal devices with tunable performance. Here, we also propose some directions for future studies.

(i) Novel structure-dependent thermal tuning in unconventional 2D materials.

Current studies have primarily focused on conventional two-dimensional materials with planar and high-symmetry structures, such as graphene, hBN, and monolayer transition metal dichalcogenides. In these systems, mechanical deformation-particularly tensile strain-typically leads to reduced TC due to phonon softening and enhanced scattering. However, future research could explore emerging 2D materials with unconventional spatial configurations. These materials may exhibit nontrivial thermal responses under strain, potentially even enhancing heat transport. Investigating such cases could uncover exceptions to the prevailing trends and deepen our understanding of thermal transport in mechanically modulated low-dimensional systems.

(ii) From forward to inverse design of mechanically tunable thermal transport.

Beyond these structural-focused explorations, much of the existing research has concentrated on forward problems, in which the influence of mechanical effects on thermal transport is first observed and then explained. A promising future direction is to pursue the inverse problem. In this approach, the design process begins with a targeted thermal response, such as a specific modulation amplitude, directional rectification, or anisotropic transport, and then seeks suitable material structures and deformation strategies to achieve it. Realizing this approach requires predictive frameworks linking microscopic phonon dynamics to macroscopic functionality, along with optimization strategies that can efficiently navigate the vast configuration space. This perspective transforms thermal transport studies from descriptive analysis toward design-oriented guidance, facilitating the rapid development of reconfigurable thermal components and broadening the design paradigm for low-dimensional thermal management.

(iii) Thermal transport under realistic and complex mechanical loading.

Most existing studies have concentrated on the effects of simple, idealized mechanical loading-such as uniaxial or biaxial strain-on thermal transport in 2D materials. However, in practical applications, devices often undergo hybrid or complex deformations, including bending, torsion, and localized compression. These coupled or dynamic mechanical fields may induce nonlinear phonon behavior and offer greater tunability of thermal properties than uniform strain alone. Future research should therefore investigate heat transport under such realistic mechanical conditions, which is essential for the development of reconfigurable and adaptive thermal systems.

(iv) Experimental challenges and integrated platforms for coupled thermo-mechanical measurements.

Despite significant theoretical progress, experimental validation remains challenging due to the difficulty of applying controlled mechanical loading and accurately measuring TCs at the nanoscale. Current techniques often face limitations in spatial resolution, force precision, or thermal sensitivity. Future efforts should focus on integrating micro-actuation platforms (e.g., MEMS-based strain applicators) with high-resolution thermal metrology methods such as Raman thermometry and scanning thermal microscopy. Achieving precise coupling between mechanical inputs and thermal responses will be crucial for experimentally verifying predictions and enabling device-level thermal tuning.

(v) Machine learning and data-driven strategies for thermo-mechanical design.

Building on the physical insights and design principles discussed above, a key frontier lies in methodological innovation. The strong coupling between structure, mechanical deformation, and thermal transport forms a high-dimensional, nonlinear optimization problem that often exceeds the capabilities of conventional theoretical and computational methods. Data-driven and machine learning (ML) frameworks offer a promising alternative by revealing hidden correlations, predicting material responses, and guiding the design of optimal structures and deformation strategies. Looking ahead, ML-driven approaches are expected to accelerate the development of reconfigurable and adaptive thermal systems, while also representing a broader trend in computational materials science toward predictive and design-oriented discovery.

Author contributions

B. Xu and Y. Gao proposed the project. B. Xu, R. Chen and Y. Gao supervised the project K. Chen and Z. Chen wrote the first draft of the manuscript with comments from X. Yu, B. Xu, R. Chen, and Y. Gao. All authors discussed the results and commented on the manuscript.

Conflicts of interest

The authors declare no competing interests.

Data availability

No primary research results, software or code have been included and no new data were generated or analysed as part of this review.

Acknowledgements

This work was financially supported by the National Natural Science Foundation of China (Grant No. 0214100234). R. C. acknowledges the support from the National Key R&D Program of China (Grant No. 2022YFF1500400) and the New Cornerstone Science Foundation through the XPLORER PRIZE.

Notes and references

1. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva and A. A. J. S. Firsov, Science, 2004, 306, 666–669.
2. C. Lee, X. Wei, J. W. Kysar and J. Hone, Science, 2008, 321, 385–388.
3. A. A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan, F. Miao and C. N. Lau, Nano Lett., 2008, 8, 902–907.
4. K. Xu, P. Cao and J. R. Heath, Nano Lett., 2009, 9, 4446–4451.
5. N. Levy, S. Burke, K. Meaker, Ms Panlasigui, A. Zettl, F. Guinea, A. C. Neto and M. F. Crommie, Science, 2010, 329, 544–547.
6. F. Wang, Y. Zhang, C. Tian, C. Girit, A. Zettl, M. Crommie and Y. R. Shen, Science, 2008, 320, 206–209.
7. M. Pumera, A. Ambrosi, A. Bonanni, E. L. K. Chng and H. L. Poh, TrAC, Trends Anal. Chem., 2010, 29, 954–965.
8. T.-H. Han, H. Kim, S.-J. Kwon and T.-W. Lee, Mater. Sci. Eng., R, 2017, 118, 1–43.
9. F. Bonaccorso, L. Colombo, G. Yu, M. Stoller, V. Tozzini, A. C. Ferrari, R. S. Ruoff and V. Pellegrini, Science, 2015, 347, 1246501.
10. A. I. Cooper and M. J. Bojdys, Mater. Today, 2014, 17, 468–469.
11. N. Wei, L. Xu, H.-Q. Wang and J.-C. Zheng, Nanotechnology, 2011, 22, 105705.
12. Y. Zhang, J. Ma, N. Wei, J. Yang and Q.-X. Pei, Phys. Chem. Chem. Phys., 2021, 23, 753–776.
13. G. Cassabois, P. Valvin and B. Gil, Nat. Photonics, 2016, 10, 262–266.
14. S. Beniwal, J. Hooper, D. P. Miller, P. S. Costa, G. Chen, S.-Y. Liu, P. A. Dowben, E. C. H. Sykes, E. Zurek and A. Enders, ACS Nano, 2017, 11, 2486–2493.
15. A. J. Mannix, X.-F. Zhou, B. Kiraly, J. D. Wood, D. Alducin, B. D. Myers, X. Liu, B. L. Fisher, U. Santiago and J. Guest, Science, 2015, 350, 1513–1516.
16. P. Vogt, P. De Padova, C. Quaresima, J. Avila, E. Frantzeskakis, M. C. Asensio, A. Resta, B. Ealet and G. Le Lay, Phys. Rev. Lett., 2012, 108, 155501.
17. N. Liu, G. Bo, Y. Liu, X. Xu, Y. Du and S. X. Dou, Small, 2019, 15, 1805147.
18. R. Kappera, D. Voiry, S. E. Yalcin, B. Branch, G. Gupta, A. D. Mohite and M. Chhowalla, Nat. Mater., 2014, 13, 1128–1134.
19. H. Li, J. Wu, Z. Yin and H. Zhang, Acc. Chem. Res., 2014, 47, 1067–1075.
20. R. Deng, X. Xie, M. Vendrell, Y.-T. Chang and X. Liu, J. Am. Chem. Soc., 2011, 133, 20168–20171.
21. W. Guo, F. Zhang, C. Lin and Z. L. Wang, Adv. Mater., 2012, 24, 4761–4764.
22. K. R. G. Lim, M. Shekhirev, B. C. Wyatt, B. Anasori, Y. Gogotsi and Z. W. Seh, Nat. Synth., 2022, 1, 601–614.
23. M. E. Chen, M. M. Rojo, F. F. Lian, J. Koeln, A. Sood, S. M. Bohaichuk, C. M. Neumann, S. G. Garrow, K. E. Goodson, A. G. Alleyne and E. Pop, 2D Mater., 2021, 8, 035055.
24. Z. Liu, L. Ma, G. Shi, W. Zhou, Y. Gong, S. Lei, X. Yang, J. Zhang, J. Yu and K. P. Hackenberg, Nat. Nanotechnol., 2013, 8, 119–124.
25. A. S. Mayorov, R. V. Gorbachev, S. V. Morozov, L. Britnell, R. Jalil, L. A. Ponomarenko, P. Blake, K. S. Novoselov, K. Watanabe and T. Taniguchi, Nano Lett., 2011, 11, 2396–2399.
26. P. Rivera, J. R. Schaibley, A. M. Jones, J. S. Ross, S. Wu, G. Aivazian, P. Klement, K. Seyler, G. Clark and N. J. Ghimire, Nat. Comm., 2015, 6, 1–6.
27. Y. Liu, R. Cheng, L. Liao, H. Zhou, J. Bai, G. Liu, L. Liu, Y. Huang and X. Duan, Nat. Comm., 2011, 2, 1–7.
28. Z. Li, Z. Yao, A. A. Haidry, Y. Luan, Y. Chen, B. Y. Zhang, K. Xu, R. Deng, N. D. Hoa and J. Zhou, Nano Today, 2021, 40, 101287.
29. B. Radisavljevic, A. Radenovic, J. Brivio, V. Giacometti and A. Kis, Nat. Nanotechnol., 2011, 6, 147–150.
30. J. S. Ross, P. Klement, A. M. Jones, N. J. Ghimire, J. Yan, D. Mandrus, T. Taniguchi, K. Watanabe, K. Kitamura and W. Yao, Nat. Nanotechnol., 2014, 9, 268–272.
31. M. Massicotte, P. Schmidt, F. Vialla, K. G. Schädler, A. Reserbat-Plantey, K. Watanabe, T. Taniguchi, K.-J. Tielrooij and F. H. Koppens, Nat. Nanotechnol., 2016, 11, 42–46.
32. F. Withers, O. Del Pozo-Zamudio, S. Schwarz, S. Dufferwiel, P. Walker, T. Godde, A. Rooney, A. Gholinia, C. Woods and P. Blake, Nano Lett., 2015, 15, 8223–8228.
33. D. Jayachandran, N. U. Sakib and S. Das, Nat. Rev. Electr. Eng., 2024, 1, 300–316.
34. S. Wang, X. Liu, M. Xu, L. Liu, D. Yang and P. Zhou, Nat. Mater., 2022, 21, 1225–1239.
35. A. Pal, T. Chavan, J. Jabbour, W. Cao and K. Banerjee, Nat. Electron., 2024, 7, 1147–1157.
36. Y. Fu, J. Hansson, Y. Liu, S. Chen, A. Zehri, M. K. Samani, N. Wang, Y. Ni, Y. Zhang and Z.-B. J. D. M. Zhang, 2D Mater., 2020, 7, 012001.
37. X. Cui, G.-H. Lee, Y. D. Kim, G. Arefe, P. Y. Huang, C.-H. Lee, D. A. Chenet, X. Zhang, L. Wang, F. Ye, F. Pizzocchero, B. S. Jessen, K. Watanabe, T. Taniguchi, D. A. Muller, T. Low, P. Kim and J. Hone, Nat. Nanotechnol., 2015, 10, 534–540.
38. X.-K. Chen, Y.-J. Zeng and K.-Q. Chen, Front. Mater., 2020, 427.
39. X. Gu, Y. Wei, X. Yin, B. Li and R. Yang, Rev. Mod. Phys., 2018, 90, 041002.
40. R. Wang, H. Zobeiri, Y. Xie, X. Wang, X. Zhang and Y. Yue, Adv. Sci., 2020, 7, 2000097.
41. Y. Zhao, Y. Cai, L. Zhang, B. Li, G. Zhang and J. T. L. Thong, Adv. Func. Mater., 2020, 30, 1903929.
42. A. A. Balandin, Nat. Mater., 2011, 10, 569–581.
43. Q. R. Cai, D. Scullion, W. Gan, A. Falin, S. Y. Zhang, K. Watanabe, T. Taniguchi, Y. Chen, E. J. G. Santos and L. H. Li, Sci. Adv., 2019, 5, eaav0129.
44. Z. Zhang, S. Hu, J. Chen and B. Li, Nanotechnology, 2017, 28, 225704.
45. Z. Liu, X. Yu, F. Wang, X. Wang, R. Zhang, J. Huang, Y. Yue, R. Dai, K. Li and X. J. D. Yang, Device, 2024, 2, 100520.
46. X. Hu, Q. Fan, S. Wang, Y. Chen, D. Wang, K. Chen, F. Ge, W. Zhou and K. Liang, Research, 2024, 7, 0542.
47. R. Dou, T. Ge, X. Liu and Z. Wen, Int. J. Heat Mass Transfer, 2016, 94, 156–163.
48. D. Estrada, Z. Li, G.-M. Choi, S. N. Dunham, A. Serov, J. Lee, Y. Meng, F. Lian, N. C. Wang and A. Perez, npj 2D Mater. Appl., 2019, 3, 1–7.
49. W. Cai, A. L. Moore, Y. Zhu, X. Li, S. Chen, L. Shi and R. S. Ruoff, Nano Lett., 2010, 10, 1645–1651.
50. J. Wang, L. He, Y. Zhang, H. Nong, S. Li, Q. Wu, J. Tan and B. Liu, Adv. Mater., 2024, 36, 2314145.
51. A. Jain, P. Bharadwaj, S. Heeg, M. Parzefall, T. Taniguchi, K. Watanabe and L. Novotny, Nanotechnology, 2018, 29, 265203.
52. A. Uppal, J. Peterson, J.-Y. Chang, X. Guo, F. Liang and W. Tang, J. Electron. Packag., 2019, 141, 010803.
53. B. Xu, M. An, S. Masubuchi, Y. Z. Li, R. L. Guo, T. Machida and J. Shiomi, Adv. Funct. Mater., 2025, 24, 1–9.
54. Y. Gao, Q. Liu and B. Xu, ACS Nano, 2016, 10, 5431–5439.
55. X. Yu, Z. Peng, L. Xu, W. Shi, Z. Li, X. Meng, X. He, Z. Wang, S. Duan and L. Tong, Small, 2024, 20, 2402561.
56. X. Yu, Y. Li, R. He, Y. Wen, R. Chen, B. Xu and Y. Gao, Nanoscale Horiz., 2024, 9, 1557–1567.
57. M. Y. Wong, C. Y. Tso, T. C. Ho and H. H. Lee, Int. J. Heat Mass Trans., 2021, 164, 120607.
58. Y. Wu, Z. Chen, P. Nan, F. Xiong, S. Lin, X. Zhang, Y. Chen, L. Chen, B. Ge and Y. Pei, Joule, 2019, 3, 1276–1288.
59. Z. Yin, W. Panaccione, A. Hu, O. R. T. Douglas, M. R.-E. Tanjil, Y. Jeong, H. Zhao and M. C. Wang, Nano Lett., 2023, 23, 11763–11770.
60. Q. Liu, B. Tian, J. Liang and W. Wu, Mater. Horiz., 2021, 8, 1634–1656.
61. Y. Gao and B. Xu, ACS Appl. Mater. Interfaces, 2018, 10, 14221–14229.
62. Y. Gao and B. Xu, ACS Nano, 2018, 12, 11254–11262.
63. Q. Liang, Q. Zhang, X. Zhao, M. Liu and A. T. S. Wee, ACS Nano, 2021, 15, 2165–2181.
64. J. H. Hong, C. H. Jin, J. Yuan and Z. Zhang, Adv. Mater., 2017, 29, 1606434.
65. J. Liu, M. Fang, E. H. Yang and X. Zhang, Sci. Rep., 2025, 15, 1976.
66. Z. Peng, X. Chen, Y. Fan, D. J. Srolovitz and D. Lei, Light: Sci. Appl., 2020, 9, 190.
67. N. W. Zhan, B. Chen, C. Z. Li and P. K. Shen, Nanotechnology, 2019, 30, 025705.
68. D. Han, H. Y. Sun, W. Y. Ding, Y. Chen, X. Y. Wang and L. Cheng, Phys. E, 2020, 124, 114312.
69. J. R. Song, Z. H. Xu, X. D. He and X. G. Liang, Phys. Chem. Chem. Phys., 2022, 24, 9648–9658.
70. Y. Zhang, Q. Pei and C. Wang, Comput. Mater. Sci., 2012, 65, 406–410.
71. Y. Y. Zhang, Q. X. Pei, H. Y. Liu and N. Wei, Phys. Chem. Chem. Phys., 2017, 19, 27326–27331.
72. Z. W. Ding, Q. X. Pei, J. W. Jiang and Y. W. Zhang, J. Phys. Chem. C, 2015, 119, 16358–16365.
73. P. Mirchi, C. Adessi, S. Merabia and A. Rajabpour, Phys. Chem. Chem. Phys., 2024, 26, 14216–14227.
74. G. Z. Qin and M. Hu, Small, 2018, 14, 1702465.
75. J. Jiang, H. Jin, Z. Y. Xia, Y. T. Zong, J. W. Sun and B. Liu, Comput. Mater. Sci., 2025, 253, 113890.
76. S. Arabha and A. Rajabpour, Mater. Today Commun., 2020, 22, 100706.
77. X. B. Li, K. Maute, M. L. Dunn and R. G. Yang, Phys. Rev. B, 2010, 81, 245318.
78. Y. M. Zhao, C. Zhang, S. Shin and L. Shen, J. Mater. Chem. C, 2023, 11, 11082–11090.
79. Q. C. Liu and B. X. Xu, ACS Appl. Nano Mater., 2022, 5, 13180–13186.
80. S. Qiao, Y. Tao and L. Yang, Phys. Rev. B, 2024, 110, 245411.
81. R. Roldán, A. Castellanos-Gomez, E. Cappelluti and F. Guinea, J. Phys.-Condes. Matter, 2015, 27, 313201.
82. J. W. Jiang, H. S. Park and T. Rabczuk, J. Appl. Phys, 2013, 114, 064307.
83. J. X. Xiang, R. N. Ali, Y. F. Yang, Z. Zheng, B. Xiang and X. D. Cui, Phys. E, 2019, 109, 248–252.
84. L. Y. Zhu, T. T. Zhang, Z. M. Sun, J. H. Li, G. B. Chen and S. Y. A. Yang, Nanotechnology, 2015, 26, 465707.
85. X. N. Wang and A. Tabarraei, Appl. Phys. Lett, 2016, 108, 191905.
86. C. Zhang, C. X. Wang and T. Rabczuk, Phys. E, 2018, 103, 294–299.
87. B. Ding, X. Y. Li, W. X. Zhou, G. Zhang and H. J. Gao, Natl. Sci. Rev., 2021, 8, nwaa220.
88. W. T. Riffe, S. Zare, K. D. Ardrey, V. K. Champagne, M. Milich, K. Y. T. Lee, M. Jassas, S. Makarem, E. J. Opila, D. R. Clarke, P. V. Balachandran and P. E. Hopkins, Nat. Commun., 2025, 16, 3333.
89. H. Xie, T. Ouyang, É. Germaneau, G. Z. Qin, M. Hu and H. Bao, Phys. Rev. B, 2016, 93, 075404.
90. X. Li, K. Maute, M. L. Dunn and R. Yang, Phys. Rev. B, 2010, 81, 2366–2373.
91. Y. Y. Zhang, Q. X. Pei, J. W. Jiang, N. Wei and Y. W. Zhang, Nanoscale, 2016, 8, 483–491.
92. C. Si, G. Lu, B.-Y. Cao, X.-D. Wang, Z. Fan and Z.-H. Feng, J. Appl. Phys, 2017, 121, 205102.
93. A. V. Savin, E. A. Korznikova, A. M. Krivtsov and S. V. Dmitriev, J. Mec. Theor. Appl., 2020, 80, 1393–1399.
94. T. Kocabaş, M. Keçeli, Á. Vázquez-Mayagoitia and C. Sevik, Nanoscale, 2023, 15, 8772–8780.
95. W. Bao, Z. Wang, B. Hu and D. Tang, Int. J. Heat Mass Transfer, 2023, 201, 123569.
96. Y. He, H. Dai, H. Wei and R. Wang, ACS Appl. Nano Mater., 2025, 8, 11297–11308.
97. F. Duan, D. Wei, A. Chen, X. Zheng, H. Wang and G. Qin, Nanoscale, 2023, 15, 1459–1483.
98. C. Foss and Z. Aksamija, Appl. Phys. Lett, 2023, 122, 062201.
99. J. Park, M. Pala and J. Saint-Matin, in The 2023 IEEE International Conference on Simulation of Semiconductor Processes and Devices, 2023, 21–24.
100. W. Dai, Y. Wang, M. Li, L. Chen, Q. Yan, J. Yu, N. Jiang and C. T. Lin, Adv. Mater., 2024, 36, 2311335.
101. M. J. Lee, J. H. Ahn, J. H. Sung, H. Heo, S. G. Jeon, W. Lee, J. Y. Song, K. H. Hong, B. Choi, S. H. Lee and M. H. Jo, Nat. Commun., 2016, 7, 12011.
102. H. F. Song, J. M. Liu, B. L. Liu, J. Q. Wu, H. M. Cheng and F. Y. Kang, Joule, 2018, 2, 442–463.
103. F. C. Liu, R. L. Mao, Z. Q. Liu, J. L. Du and P. Gao, Nature, 2025, 642, 941–946.
104. J. S. He, C. Wang, B. Zhou, Y. Zhao, L. L. Tao and H. Zhang, Mater. Horizons, 2020, 7, 2903–2921.
105. K. S. Novoselov, A. Mishchenko, A. Carvalho and A. H. C. Neto, Science, 2016, 353, aac9439.
106. Y. Wu, S. Lin and P. Yang, Therm. Sci. Eng. Prog., 2024, 47, 252–259.
107. Q. C. Zhan, Z. Fu, X. L. Zhong, H. J. Song and J. B. Wang, Phys. Status Solidi B, 2024, 261, 2300381.
108. Y. Chen, Y. Zhang, K. Cai, J. Jiang, J.-C. Zheng, J. Zhao and N. Wei, Carbon, 2017, 117, 399–410.
109. X. Zhao, G. Tang, Y. Li, M. Zhang and Y. Nie, ACS Appl. Electron. Mater., 2021, 3, 2995–3004.
110. W. Ren, Y. Ouyang, P. Jiang, C. Yu, J. He and J. Chen, Nano Lett., 2021, 21, 2634–2641.
111. F. F. Wang, Z. X. Liu, J. F. Li, J. Huang, L. Fang, X. F. Wang, R. W. Dai, K. Y. Li, R. Zhang, X. R. Yang, Y. Yue, Z. Q. Wang, Y. Gao, K. Yang, L. F. Zhang and G. Q. Xin, Adv. Sci., 2024, 11, 2401586.
112. Y. F. Sheng, H. X. Zhu, S. Q. Xie, Q. Lv, H. Q. Xie, H. D. Wang, R. T. Lv and H. Bao, Nat. Commun., 2025, 16, 1970.
113. Y. Gao and B. Xu, ACS Appl. Mater. Interfaces, 2017, 9, 34506–34517.
114. F. Liu, Y. Gong, R. Zou, H. Ning, N. Hu, Y. Liu, L. Wu, F. Mo, S. Fu and C. Yan, Nano Mater. Sci., 2022, 4, 227–234.
115. F. Liu, R. Zou, N. Hu, H. Ning, C. Yan, Y. Liu, L. Wu, F. Mo and S. Fu, Nanoscale, 2019, 11, 4067–4072.
116. Y. Hong, J. Zhang and X. C. Zeng, Phys. Chem. Chem. Phys., 2016, 18, 24164–24170.
117. Y. Gao, W. Yang and B. Xu, Adv. Mater. Interfaces, 2017, 4, 1700278.
118. Y. Gao, W. Yang and B. Xu, Carbon, 2016, 96, 513–521.
119. K. Ren, H. S. Qin, H. C. Liu, Y. Chen, X. J. Liu and G. Zhang, Adv. Funct. Mater., 2022, 32, 2110846.
120. Y. X. Wang, N. Xu, D. Y. Li and J. Zhu, Adv. Funct. Mater., 2017, 27, 1604134.
121. K. Nakagawa, K. Satoh, S. Murakami, K. Takei, S. Akita and T. Arie, Sci. Rep., 2021, 11, 19533.
122. J. Chen, J. H. Walther and P. Koumoutsakos, Nano Lett., 2014, 14, 819–825.
123. Q. Zhan, Z. Fu, X. Zhong, H. Song and J. Wang, Phys. Status Solidi B, 2023, 9, 326–330.
124. X. Meng, T. Pandey, J. Jeong, S. Fu, J. Yang, K. Chen, A. Singh, F. He, X. Xu, J. Zhou, W. P. Hsieh, A. K. Singh, J. F. Lin and Y. Wang, Phys. Rev. Lett., 2019, 122, 155901.
125. Y. Zhou, Z. Y. Dong, W. P. Hsieh, A. F. Goncharov and X. J. Chen, Nat. Rev. Phys., 2022, 4, 319–335.
126. K. Wilczynski, A. P. Gertych, K. Czerniak-Losiewicz, J. Sitek and M. Zdrojek, Acta Mater., 2022, 240, 118299.
127. Y. Han, J. Z. Zhou, H. Y. Wang, L. B. Gao, S. Z. Feng, K. Cao, Z. P. Xu and Y. Lu, Appl. Nanosci., 2021, 11, 1075–1091.
128. T. Zhu and E. J. P. R. B. Ertekin, Phys. Rev. B, 2015, 91, 205429.
129. G. Wehmeyer, T. Yabuki, C. Monachon, J. Wu and C. Dames, Appl. Phys. Rev., 2017, 4, 041304.
130. T. Du, Z. Xiong, L. Delgado, W. Liao, J. Peoples, R. Kantharaj, P. R. Chowdhury, A. Marconnet and X. J. N. C. Ruan, Nat. Commun., 2021, 12, 4915.
131. M. Li, H. Wu, E. M. Avery, Z. H. Qin, D. P. Goronzy, H. D. Nguyen, T. H. Liu, P. S. Weiss and Y. J. Hu, Science, 2023, 382, 585–589.
132. J. Wang, X. Feng, Y. Yu, H. Huang, M. Zheng, Y. Xu, J. Wu, Y. Yang and J. J. N. E. Lu, Nat. Energy, 2024, 9, 939–946.
133. N. Wei, Y. Zhang, Y. Chen and Y. Zhang, Int. J. Smart Nano Mater., 2020, 11, 310–323.
134. H. Kang, F. Yang and J. J. Urban, Phys. Rev. Appl., 2018, 10, 024034.
135. H. D. Wang, S. Q. Hu, K. Takahashi, X. Zhang, H. Takamatsu and J. Chen, Nat. Commun., 2017, 8, 15843.
136. R. Shrestha, Y. X. Luan, X. Luo, S. M. Shin, T. Zhang, P. Smith, W. Gong, M. Bockstaller, T. F. Luo, R. K. Chen, K. Hippalgaonkar and S. Shen, Nat. Commun., 2020, 11, 4346.
137. L. Li and Q. Cheng, Interdiscip. Mater., 2024, 3, 847–864.
138. Y. P. Qi, M. A. Sadi, D. Hu, M. Zheng, Z. P. Wu, Y. C. Jiang and Y. P. Chen, Adv. Mater., 2023, 35, 2205714.
139. Y. Zhang, Q. Lv, H. Wang, S. Zhao, Q. Xiong, R. Lv and X. Zhang, Science, 2022, 378, 169–175.
140. Y. Y. Li, Z. X. Hu, S. H. Lin, S. K. Lai, W. Ji and S. P. Lau, Adv. Funct. Mater., 2017, 27, 1600986.
141. A. H. Liu, X. W. Zhang, Z. Y. Liu, Y. N. Li, X. Y. Peng, X. Li, Y. Qin, C. Hu, Y. Q. Qiu, H. Jiang, Y. Wang, Y. F. Li, J. Tang, J. Liu, H. Guo, T. Deng, S. A. Peng, H. Tian and T. L. Ren, Nano-Micro Lett., 2024, 16, 119.
142. P. Manchanda, V. Sharma, H. B. Yu, D. J. Sellmyer and R. Skomski, Appl. Phys. Lett, 2015, 107, 032402.
143. J. Bera and S. Sahu, RSC Adv., 2019, 9, 25216–25224.
144. Z. C. Zhang, L. K. Li, J. Horng, N. Z. Wang, F. Y. Yang, Y. J. Yu, Y. Zhang, G. R. Chen, K. Watanabe, T. Taniguchi, X. H. Chen, F. Wang and Y. B. Zhang, Nano Lett., 2017, 17, 6097–6103.
145. J. Q. Zhu, Z. C. Wang, H. Yu, N. Li, J. Zhang, J. L. Meng, M. Z. Liao, J. Zhao, X. B. Lu, L. J. Du, R. Yang, D. Shi, Y. Jiang and G. Y. Zhang, J. Am. Chem. Soc., 2017, 139, 10216–10219.
146. J. L. Du, H. H. Yu, B. S. Liu, M. Y. Hong, Q. L. Liao, Z. Zhang and Y. Zhang, Small Methods, 2021, 5, 2000919.
147. S. X. Yang, Y. J. Chen and C. B. Jiang, InfoMat, 2021, 3, 397–420.
148. T. Peña, S. A. Chowdhury, A. Azizimanesh, A. Sewaket, H. Askari and S. M. J. D. M. Wu, 2D Mater., 2021, 8, 045001.
149. H. J. Conley, B. Wang, J. I. Ziegler, R. F. Haglund Jr, S. T. Pantelides and K. I. J. N. L. Bolotin, Nano Lett., 2013, 13, 3626–3630.
150. C. D. Zhang, A. Johnson, C. L. Hsu, L. J. Li and C. K. Shih, Nano Lett., 2014, 14, 2443–2447.
151. A. Castellanos-Gomez, R. Roldán, E. Cappelluti, M. Buscema, F. Guinea, H. S. J. van der Zant and G. A. Steele, Nano Lett., 2013, 13, 5361–5366.
152. J. L. Qi, Z. X. Wu, W. B. Wang, K. Bao, L. Z. Wang, J. K. Wu, C. X. Ke, Y. Xu and Q. Y. He, Int. J. Extreme Manuf., 2023, 5, 022007.
153. B. You, Y. H. Du, S. X. Yuan, Y. Tian, X. H. Meng, W. B. Wu, Z. P. Shi, G. Y. Wang, X. Chen, X. Yuan and X. L. Zhu, Nano Lett., 2024, 24, 15837–15844.
154. C. Chen, K. P. Nuckolls, S. H. Ding, W. Q. Miao, D. Wong, M. Oh, R. L. Lee, S. M. He, C. Peng, D. Pei, Y. W. Li, C. Y. Hao, H. R. Yan, H. B. Xiao, H. Gao, Q. Li, S. H. Zhang, J. P. Liu, L. He, K. Watanabe, T. Taniguchi, C. Jozwiak, A. Bostwick, E. Rotenberg, C. Li, X. Han, D. Pan, Z. K. Liu, X. Dai, C. X. Liu, B. A. Bernevig, Y. Wang, A. Yazdani and Y. L. Chen, Nature, 2024, 636, 342–347.
155. Y. J. Cheng, Z. Y. Fan, T. Zhang, M. Nomura, S. Volz, G. M. Zhu, B. W. Li and S. Y. Xiong, Mater. Today Phys., 2023, 35, 8.
156. L. Yang, S. Yue, Y. Tao, S. Qiao, H. Li, Z. Dai, B. Song, Y. Chen, J. Du, D. Li and P. Gao, Nature, 2024, 629, 1021–1026.
157. M. Ohnishi and J. Shiomi, Phys. Rev. B, 2021, 104, 6.
158. H. Chen, J. W. Zhang, D. X. Kan, J. B. He, M. S. Song, J. H. Pang, S. R. Wei and K. Y. Chen, Crystals, 2022, 12, 20.
159. X. Zhang, M. Li, Q. Wang, Y. Liang, J. Wei, H. Li and F. Liu, Nanomaterials, 2025, 15, 929.
160. R. X. Fei and L. Yang, Nano Lett., 2014, 14, 2884–2889.
161. Z. H. Dai, L. Q. Liu and Z. Zhang, Adv. Mater., 2019, 31, 11.
162. P. Kang, M. C. Wang, P. M. Knapp and S. Nam, Adv. Mater., 2016, 28, 4639–4645.
163. B. Radatovic, O. Çakiroglu, V. Jadrisko, R. Frisenda, A. Senkic, N. Vujicic, M. Kralj, M. Petrovic and A. Castellanos-Gomez, ACS Appl. Mater. Interfaces, 2024, 16, 15596–15604.
164. X. Fan, L. Liu, X. Jin, W. Wang, S. Zhang and B. J. Tang, J. Mater. Chem. A, 2019, 7, 14319–14327.
165. S. Wu, T. Li, Z. Tong, J. Chao, T. Zhai, J. Xu, T. Yan, M. Wu, Z. Xu and H. J. A. M. Bao, Adv. Mater., 2019, 31, 1905099.
166. S. Y. Lei, J. X. Feng, Y. C. Chen, D. Zheng, W. X. Liu, W. H. Shi, F. F. Wu and X. H. Cao, Rare Met., 2024, 43, 1350–1369.
167. X. Y. Zhang, L. L. Hou, A. Ciesielski and P. Samorì, Adv. Energy Mater., 2016, 6, 21.
168. J. Yang, G.-Q. Qi, L.-S. Tang, R.-Y. Bao, L. Bai, Z.-Y. Liu, W. Yang, B.-H. Xie and M.-B. J. Yang, J. Mater. Chem. A, 2016, 4, 9625–9634.
169. R. Cao, D. Sun, L. Wang, Z. Yan, W. Liu, X. Wang and X. J. Zhang, J. Mater. Chem. A, 2020, 8, 13207–13217.
170. P. Xiong, F. Zhang, X. Y. Zhang, S. J. Wang, H. Liu, B. Sun, J. Q. Zhang, Y. Sun, R. Z. Ma, Y. Bando, C. F. Zhou, Z. W. Liu, T. Sasaki and G. X. Wang, Nat. Commun., 2020, 11, 12.
171. J. Qi, Z. Wu, W. Wang, K. Bao, L. Wang, J. Wu, C. Ke, Y. Xu and Q. He, Int. J. Extreme Manuf., 2023, 5, 022007.
172. Y. Kou, K. Sun, J. Luo, F. Zhou, H. Huang, Z.-S. Wu and Q. J. E. S. M. Shi, Energy Storage Mater., 2021, 34, 508–514.
173. Z. K. Yuan, J. Li, M. J. Yang, Z. S. Fang, J. H. Jian, D. S. Yu, X. D. Chen and L. M. Dai, J. Am. Chem. Soc., 2019, 141, 4972–4979.

---

Footnote

† These authors contributed equally.

---