Regulation of near-field radiative heat transfer between two-dimensional borophene sheets under uniaxial strain based on first-principles calculations

This study investigates the near field radiative heat transfer (NFRHT) regulation behavior of χ 3 -phase two-dimensional borophene (Bχ 3) sheets under uniaxial strain, based on first-principles calculations and fluctuation-dissipation theory. By analyzing the dielectric functions and spectral heat transfer coefficients of borophene under different strain conditions, the near-field radiative heat transfer properties of single-layer and bilayer borophene under x / y -axis strain are explored, where the x -direction corresponds to the Bχ 3 lattice direction a, and the y -axis is perpendicular to the x -axis. Results indicate that strain significantly alters the optical properties of borophene. Under x -axis strain, the normalized heat transfer coefficient first decreases then increases with strain, reaching a maximum increase of 3.6 times compared to the unstrained state. In contrast, y -axis strain exhibits a monotonic increase, with a maximum enhancement of approximately 126 times. Furthermore, bilayer borophene exhibited more pronounced near-field radiative heat transfer modulation effects than monolayer borophene, particularly under tensile strain. The study demonstrates that carefully designing the direction and magnitude of strain enables precise control over near-field heat transport, providing a theoretical basis for developing efficient thermal management devices based on borophene.

Under far-field conditions, radiative energy exchange between objects is achieved through the emission and absorption of photons, which can be described by classical blackbody radiation theory and follows the Stefan-Boltzmann law (Polder and Van Hove, 1971; Volokitin and Persson, 2007). However, this theory is only applicable when the characteristic scale of the system is much larger than the characteristic wavelength of thermal radiation. As the research scale advances to the nanoscale, the coupling of evanescent waves and the induced photon tunneling effect enable energy exchange to surpass the blackbody radiation limit. When the separation distance between two objects is reduced to a range comparable to or even smaller than the characteristic wavelength of thermal radiation, near-field radiative heat transfer (NFRHT) exhibits an enhancement by several orders of magnitude, and its heat flux can significantly exceed the blackbody limit (Song et al., 2021; Zhang et al., 2022).

The enhancement effect of near-field radiative heat transfer demonstrates broad potential in localized heat dissipation (Otey et al., 2010; Guha et al., 2012), thermal rectification (Feng et al., 2021; Hu et al., 2023), near-field thermophotovoltaics (Fiorino et al., 2018; Mittapally et al., 2023), near-field thermal microscopy (De Wilde et al., 2006; Kittel et al., 2008), and various energy conversion technologies (Basu et al., 2009; Pascale et al., 2023; Zhang et al., 2023). In recent years, functional devices such as thermal switches (Gu et al., 2015; Fang et al., 2022), thermal diodes (Ghanekar et al., 2016; Liu et al., 2022), and thermal transistors (Joulain et al., 2015; Zhang et al., 2025a) have enabled precise heat flow management at the micro- and nanoscale. Among these approaches, external magnetic fields (Zhang et al., 2024b; Zhang et al., 2024a), multilayer heterostructures (Zhang et al., 2025b), and electric fields (Yuan et al., 2022) have been extensively investigated. In addition, mechanical strain can directly alter the atomic positions within a material, thereby affecting its band structure and endowing its electronic and optical properties with a high degree of tunability. For example, an NFRHT modulation of up to 73% can be achieved in two-dimensional black phosphorus (BP) by applying a specific uniaxial strain, which disrupts the surface plasmon coupling between the unstrained and strained regions (Wang et al., 2020). Similarly, in bilayer graphene, a reduction of more than 60% in thermal conductance can be achieved when strain is applied, indicating a significant modulation effect (Habibzadeh et al., 2023).

Meanwhile, two-dimensional materials have continued to attract widespread attention due to their unique structures and properties. Boron, which is adjacent to carbon in the periodic table, can form planar clusters (Wang, 2016; Wang, 2024; Zergani et al., 2024), one-dimensional nanotubes, and other low-dimensional structures (Wu et al., 2021; Gupta et al., 2024; Sarma et al., 2025), similar to carbon-based materials. In addition, two-dimensional boron (borophene) has achieved significant progress in both theoretical and experimental studies in recent years (Wu et al., 2012; Zhang et al., 2016; Zhang et al., 2017), particularly with its successful synthesis on silver substrates (Mannix et al., 2015). The excellent electronic properties of borophene endow it with promising application potential in metal-ion batteries (Rao et al., 2017; Lang et al., 2020; Adekoya et al., 2024), gas sensing (Hou et al., 2021; Shen et al., 2021; Zergani and Tavangar, 2022), and micro/nanoelectronics (Zhang et al., 2021). At the same time, its outstanding optical properties enable its near-field thermal radiation capability to surpass that of conventional metals and metal oxides (Zhou et al., 2022). However, owing to the complex anisotropic optical response of borophene, the regulation of its near-field radiative behavior by mechanical strain still requires systematic investigation.

Based on the above background, this study proposes the use of mechanical strain to regulate near-field radiative heat transfer between borophene sheets separated by a nanoscale gap. First-principles calculations are employed in this work to reveal the evolution of the anisotropic optical properties of borophene under strain. Furthermore, the effects of uniaxial strain applied along different directions on near-field heat transfer are examined, and the variation patterns under single-layer and bilayer strain are systematically analyzed. These results can provide theoretical references and design guidelines for the development of mechanically tunable thermal management and energy conversion devices based on borophene.

In this work, the near-field radiative heat transfer between two-dimensional χ 3 -phase borophene sheets (Bχ 3) is investigated based on the computational framework of fluctuational electrodynamics and fluctuation-dissipation theory (Rodriguez et al., 2013; Li et al., 2022; Zhang et al., 2024c). As shown in Figure 1, two suspended Bχ 3 sheets are assumed to extend infinitely in the x - y plane, with temperatures of T 1 and T 2, respectively, and are separated by a vacuum gap of d = 10 nm. Each Bχ 3 sheet is assumed to be in its own internal thermal equilibrium. The x -axis denotes the lattice direction a, while the y -axis represents the direction perpendicular to the lattice direction a. Uniaxial strain is applied along either the x -direction or the y -direction.

The magnitude of the strain applied to Bχ 3 can be defined by the following Equations 1, 2:

Here, a and b denote the lattice constants along the lattice directions a 1 and a 2 in the unstrained state, respectively, whereas a ′ and b ′ represent the corresponding lattice constants after strain is applied, and α denotes the lattice angle. A positive value of ε x or ε y indicates tensile strain applied along the corresponding direction, whereas a negative value corresponds to compressive strain.

The dielectric function can be obtained from first-principles calculations within the framework of Kohn–Sham density functional theory (Yang et al., 2014). In the first-principles calculation of the complex dielectric function, the imaginary part of the dielectric function is first determined by calculating the transition probability derived from the electronic band structure, as expressed in Equation 3 (Sun and Mu, 2023) where u c k + e α q | u v k u v k | u c k + e β q denotes the probability of interband transitions, and the integration over k is performed by summation over specific k-points, each associated with a weighting factor w k. The real part of the dielectric function, ε 1, can be determined from the Kramers–Kronig relation as given by Equation 4, as follows: where P denotes the Cauchy principal value and η is the complex shift parameter. In the specific calculations, a plane-wave basis set was employed with a kinetic-energy cutoff of 400 eV, and non-spin-polarized calculations were performed. The electronic self-consistent-field convergence criterion was set to 10 −6 eV, with a maximum of 60 electronic steps. Gaussian smearing was adopted with a smearing width of 0.05 eV. For the density-of-states and optical-response calculations, 5,000 energy grid points were used to ensure sufficient spectral resolution. To obtain more pronounced structural responses under uniaxial strain applied along the x -axis or y -axis, different supercell models are adopted for different strain directions: a 5 × 2 × 1 supercell is used for x -axis strain, whereas a 3 × 3 × 1 supercell is used for y -axis strain. For Brillouin zone integration, a 4 × 6 × 1 k-point mesh centered at the Γ point is used for the x -axis strain calculations, while a 6 × 6 × 1 Γ-centered k-point mesh is employed for the y -axis strain calculations. In addition, a vacuum layer of approximately 20 Å is introduced along the z -axis in the DFT supercell calculations to avoid artificial interactions between periodic images. This vacuum thickness is different from the vacuum gap distance d = 10 nm used in the near-field radiative heat transfer calculations between the two suspended Bχ 3 sheets.

The net near-field radiative heat flux between two Bχ 3 sheets separated by a distance d and maintained at temperatures T 1 and T 2, respectively, can be derived within the framework of the fluctuation-dissipation theorem and dyadic Green’s functions, as described by Equations 5, 6 (Moncada-Villa et al., 2015) where q (ω) denotes the spectral radiative heat flux, Θ(ω, T) is the average energy of the Planck oscillator at temperature T, and Φ(ω) represents the spectral energy transmission function, which is given by Equation 7, as follows:

Here, ξ (ω, k; d) denotes the energy transmission coefficient, which is given by Equation 8, as follows: where k 0 = ω / c is the wave vector in vacuum, k z 0 = k 0 2 − k 2 is the wave-vector component perpendicular to the interface, and k = k x 2 + k y 2 denotes the in-plane wave vector parallel to the interface, with k x and k y being the x - and y -components of the wave vector k (Lussange et al., 2012; Chen et al., 2018). D = I − R 1 R 2 e 2 i k z 0 d − 1 as well as R 1 and R 2 represent the Fresnel reflection coefficients obtained from the scattering matrix formalism, as defined in Equation 9 (Li, 1996; Chen et al., 2018) where r i m n i = 1, 2 denotes the Fresnel reflection coefficient of the i -th body, and m and n represent the polarization states of the incident and reflected waves, respectively.

It should be noted that the uniaxial tensile and compressive strains applied to Bχ 3 in this study are both restricted within the range of ±4%. Previous first-principles studies have demonstrated that free-standing χ 3 borophene possesses good dynamical and mechanical stability, with relatively high in-plane Young’s moduli in the range of 180–210 N/m along both the a and b directions (Peng et al., 2017). In addition, previous strain-engineering studies on borophene have shown that strain ranges up to approximately ±6% can still remain within the mechanically sustainable deformation regime (Adamska and Sharifzadeh, 2017). Therefore, the strain range considered in this work is intended to remain within the elastic deformation regime and to avoid irreversible structural instability. For all compressive strain cases investigated in this work, the geometry optimization calculations were successfully converged without obvious structural collapse or severe out-of-plane instability within the investigated strain range. Figure 2 presents the real and imaginary parts of the dielectric function of Bχ 3 calculated using a 3 × 3 × 1 supercell over the frequency range of 0–4 eV. The dielectric function ε xx along the x -direction differs significantly from ε yy along the y -direction, clearly demonstrating the intrinsic optical anisotropy of Bχ 3 in the x and y directions.

Dielectric response of Bχ 3 sheets under uniaxial strain

To reveal the regulatory mechanism of mechanical strain on the optical response of χ 3 -phase borophene, the evolution characteristics of the complex dielectric function under uniaxial strain applied along the x - and y -axes were systematically analyzed. The results indicate that the mechanisms by which strain in different directions affects the dielectric response are distinctly different and exhibit pronounced anisotropic characteristics.

To further understand the strain-dependent optical response of Bχ 3, the electronic band structures under representative strain conditions were calculated, as shown in Figure 3. The results indicate that Bχ 3 maintains metallic characteristics under all investigated strain conditions, with several bands continuously crossing the Fermi level. Which is consistent with previous theoretical studies on χ 3 -borophene (Gao et al., 2017).

Under x -axis strain, the electronic structure near the Fermi level exhibits relatively moderate variations, indicating limited reconstruction of the low-energy electronic states. In contrast, y -axis tensile strain induces more pronounced band flattening and denser electronic states near the Fermi level, especially under +4% tensile strain. Such strain-induced electronic-structure modification is expected to enhance low-energy electronic transitions and strengthen the low-frequency dielectric response. As a result, stronger evanescent-wave coupling and significantly enhanced near-field radiative heat transfer are observed under y -axis tensile strain.

Figure 4 shows the dielectric function under uniaxial strain applied along the x -axis. Re (ε xx) consistently maintains a pronounced resonance peak around 0.8–1.0 eV, and the peak intensity continuously increases with increasing compression, accompanied by a slight redshift, indicating that compressive strain enhances the low-energy polarization capability along the x -direction and strengthens the electronic response in the loading direction. After the main peak, Re (ε xx) rapidly changes from positive to negative values, forming a distinct valley, whose depth and position vary synchronously with strain, reflecting that compressive strain not only enhances the resonance intensity but also significantly reconstructs the dispersion behavior in the low-energy region. Correspondingly, Im (ε xx) exhibits a stable absorption peak in the range of 1.1–1.3 eV, whose intensity increases significantly with enhanced compression, while the peak position remains nearly unchanged. This indicates that x -axis compression primarily enhances the transition intensity, with only a limited effect on the energy levels of the dominant transitions. In contrast, both the real and imaginary parts of ε yy exhibit only slight spectral adjustments within the same strain range, with overall weak variations, indicating that the modulation effect of compressive strain is mainly confined to the loading direction, while the perpendicular direction remains highly stable.

Under tensile strain, the dielectric response along the x -direction exhibits an evolution trend opposite to that under compression. As the strain magnitude increases from +1% to +4%, the main resonance peak of Re (ε xx) in the low-energy region continuously weakens and is accompanied by a certain redshift, indicating that tensile strain reduces the polarization response capability along the loading direction. The corresponding main absorption peak of Im (ε xx) decreases simultaneously, further demonstrating that tensile strain weakens the optical transition intensity in this direction. Throughout this process, the variation in ε yy remains small, indicating its insensitivity to tensile strain along the x -direction.

Figure 5 presents the dielectric function under uniaxial strain applied along the y -axis. Within the compressive strain range, the main peak structure of Re (ε xx) near 0.8–0.9 eV remains stable, with only slight line-shape adjustments and minor peak shifts, indicating that y -direction compression does not significantly reconstruct the main resonance characteristics along the x -direction. Re (ε yy) also exhibits only slight local fluctuations, while the overall spectral profile remains essentially stable. The imaginary-part spectra likewise demonstrate strong robustness. The main absorption peak of Im (ε xx) within 1.1–1.3 eV remains nearly unchanged in position, with only limited fluctuations in peak intensity. The magnitude of Im (ε yy) remains consistently low and responds weakly to strain variation. This suggests that within the compressive range, y -direction strain mainly induces slight local intensity modulation without significantly altering the primary transition channels.

As the tensile strain along the y -direction increases, the dielectric function begins to exhibit more pronounced low-frequency response characteristics. When the strain increases to +3% and +4%, Re (ε xx) shows a distinct overall uplift in the low-frequency region near 0 eV, while the main peak structure in the mid-energy region remains stable. This indicates that tensile strain along the y -direction does not destroy the original main resonance mode but significantly enhances the polarization capability in the low-frequency region. Re (ε yy) also exhibits a similar low-frequency uplift trend under larger tensile strain, indicating that this modulation effect acts on both directional components rather than being confined to a single direction. Compared with the variation in the real part, the response of the imaginary part remains relatively moderate. The peak position of the main absorption peak of Im (ε xx) remains essentially unchanged, with only slight fluctuations in peak intensity. Im (ε yy) overall remains at a relatively low level. This implies that tensile strain along the y -direction mainly regulates the optical response of the material by enhancing the low-frequency polarization background rather than significantly altering the structure of the main transition energy levels.

This difference further reveals the distinct modulation mechanisms of the two types of uniaxial strain. Strain along the x -axis directly affects the main resonance and absorption peaks in the loading direction, exhibiting a stronger capability to reconstruct the optical anisotropy of the material and making it more suitable for directional enhancement or suppression. In contrast, strain along the y-axis is more manifested as a continuous regulation of the low-frequency background dielectric response, especially under large tensile conditions, where it significantly enhances the polarization level in the low-energy region and provides more favorable conditions for low-frequency evanescent-wave coupling.

From the perspective of near-field radiative heat transfer, the strain-induced dielectric variation affects the spectral heat transfer coefficient through the Fresnel reflection coefficients and the energy transmission coefficient of evanescent waves. In the evanescent-wave dominated region, the heat-transfer spectrum is strongly related to the surface electromagnetic response of the two borophene sheets. The enhancement of the low-frequency dielectric response under large y -axis tensile strain modifies the resonance condition of surface modes and increases their coupling probability within the dominant heat-transfer frequency window. Therefore, the uplift of the low-frequency dielectric background does not merely indicate a stronger optical polarization response, but also provides a more favorable electromagnetic condition for enhancing the spectral near-field radiative heat transfer peak.

Near-field radiative heat transfer under single-layer strain

To systematically reveal the influence of uniaxial mechanical strain on the near-field radiative heat transfer behavior of Bχ 3 sheets, the spectral heat transfer coefficient h (ω) and the normalized heat transfer coefficient under x - and y -axis strains were investigated. In the calculations, the two borophene sheets were maintained in a parallel configuration with a fixed vacuum gap of 10 nm, and the strain range was set from −4% to +4%. Figure 6 presents the spectral distribution of h (ω) under different strain conditions.

Figures 6a,b show the spectral heat transfer coefficient of monolayer borophene under compressive and tensile strain applied along the x -axis, respectively. When strain is applied along the x -direction, the spectral heat transfer coefficient exhibits pronounced directional dependence. Within the compressive strain range, h (ω) consistently exhibits a single-peak distribution, with the main peak located around ω ≈ 2 × 10 14 rad·s -1. The peak position changes only slightly, indicating that the dominant frequency window for heat transfer remains essentially stable. In comparison, the peak intensity is more sensitive to strain and continuously increases with increasing compression. Taking ε x = −4% as an example, the peak value of h (ω) increases from 4.6 × 10 −16 J·rad -1 ·K −1 ·m -2 in the unstrained state to 5.5 × 10 −16 J·rad -1 ·K −1 ·m -2. This result indicates that compression along the x-direction mainly enhances near-field radiative heat transfer by strengthening the tunneling contribution of evanescent waves within the dominant frequency band, without significantly shifting the main energy exchange interval.

When the strain enters the tensile regime, the system exhibits a nonlinear response. Within the range of ε x = +1% to +3%, the variation in the spectral peak remains relatively moderate, and the peak value of h (ω) is slightly lower than or close to that in the unstrained state. This indicating that small tensile strain along the x -direction has a limited influence on the dominant coupling mode and may even slightly weaken the coupling efficiency. When the strain is further increased to +4%, the intensity of the main spectral peak rises markedly, with the peak value reaching 8.42 × 10 −16 J·rad -1 ·K −1 ·m -2. Meanwhile, the spectral width near the main peak becomes broader, suggesting that larger tensile strain may increase the density of coupling-accessible states in this frequency band and enable surface-mode coupling over a wider frequency range.

Figures 6c,d correspond to the spectral heat transfer coefficient distributions under strain applied along the y -axis. Compared with x -direction strain, y -direction strain produces a stronger nonlinear modulation response. Within the compressive strain range, the main spectral peak gradually weakens with increasing strain, decreasing from approximately 7.8 × 10 −16 J·rad −1 ·K −1 ·m -2 in the unstrained state to 6.3 × 10 −16 J·rad −1 ·K −1 ·m -2 at ε y = −4%. This result indicates that compression along the y -direction weakens near-field coupling in the dominant frequency band, thereby reducing the evanescent-wave transmission contribution. In contrast, tensile strain along the y -direction exhibits a staged increase in h (ω). Within the range of +1% to +2%, the variation remains limited, indicating that the system is still in a gradual adjustment stage. When the strain increases to +3%, the main peak intensity begins to rise rapidly. Upon further increasing the strain to +4%, the peak value of h (ω) reaches 9.56 × 10 −15 J·rad −1 ·K −1 ·m -2, which is much higher than that under the corresponding x -direction condition. This enhancement can be attributed to the strain-induced increase in the low-frequency dielectric response, which improves the surface electromagnetic response and strengthens evanescent-wave coupling within the dominant heat-transfer frequency range.

To further quantitatively evaluate the effect of strain on the overall near-field radiative heat transfer intensity, Figures 7a,b present the normalized heat transfer coefficient q (ε)/ q (ε = 0) under uniaxial strain applied along the x - and y -axes, respectively. For x -direction strain, the normalized coefficient exhibits a nonlinear variation with strain. Within the compressive range (ε x < 0), this ratio remains consistently greater than 1 and gradually increases with the degree of compression, reaching approximately 1.20 at ε x = −4%. This indicates that x -direction compression is favorable for enhancing near-field radiative heat transfer. Upon entering the small tensile range (ε x = +1% to +2%), the normalized coefficient decreases to around 1, with a minimum value of approximately 0.98, indicating that weak tensile strain has a limited modulation effect and may even produce slight suppression. When the strain is further increased to +4%, the normalized coefficient rises to 1.72, showing an enhanced coupling effect under larger tensile deformation. In comparison, y -direction strain exhibits a wider modulation range. Within the compressive range, the normalized coefficient remains below 1 and gradually decreases from 0.94 to 0.80 with increasing compression, indicating that y -direction compression continuously weakens the near-field coupling strength. During the small tensile stage (ε y = +1% to +2%), only limited enhancement is observed, with the normalized coefficient being 1.07 and 1.15, respectively. When the tensile strain increases to +3% and +4%, the normalized coefficient rapidly increases to 4.14 and 8.96, respectively, which are far higher than the corresponding values under x -direction strain.

These results indicate that x -direction strain is more suitable for achieving continuously tunable near-field radiative heat-transfer modulation, whereas large tensile strain along the y -direction can trigger a stronger switch-like response. This difference is closely related to the strain-dependent dielectric response discussed above. Compared with x -axis strain, large y -axis tensile strain more effectively enhances the low-frequency dielectric background, which increases the energy transmission probability of evanescent waves and leads to a rapid increase in the normalized heat-transfer coefficient. Therefore, y -axis tensile strain shows greater heat-transfer regulation depth and potential application value in thermal switching devices.

Near-field heat transfer under bilayer co-directional strain

Based on the single-layer strain study, the regulation behavior of near-field radiative heat transfer between Bχ 3 sheets under bilayer co-directional strain was further investigated. In this configuration, equal-magnitude uniaxial strain with the same direction is simultaneously applied to the upper and lower borophene sheets along either the x -axis or the y -axis, with the strain range still set from −4% to +4% and the vacuum gap fixed at 10 nm. Figure 8 presents the spectral heat transfer coefficient h (ω) distributions under different bilayer co-directional strain conditions.

Figures 8a,b show the h (ω) under bilayer co-directional compressive and tensile strain applied along the x -axis, respectively. Overall, under bilayer strain along the x -direction, h (ω) maintains a single-peak feature, with the main peak located around ω ≈ 2.2 × 1,014 rad·s −1, indicating that the dominant heat-transfer frequency window remains essentially unchanged. Within the compressive strain range, the main peak intensity continuously increases with increasing strain. When ε x = −4%, peak value of h (ω) increases from 4.58 × 10 −16 J·rad −1 ·K −1 ·m −2 in the unstrained state to 6.55 × 10 −16 J·rad −1 ·K −1 ·m −2, indicating that synchronous compression of both layers enhances the evanescent-wave coupling contribution in the dominant frequency band. Upon entering the tensile regime, the system shows a staged response. Within the range of +1% to +3%, the variation in the spectral peak remains relatively moderate. When the strain is further increased to +4%, the main peak intensity rises markedly, reaching 2.10 × 10 −15 J·rad −1 ·K −1 ·m −2, approximately 4.6 times that of the unstrained state. Meanwhile, the spectral peak position undergoes a certain redshift, suggesting that large bilayer tensile strain along the x-direction strengthens interfacial mode coupling and modifies the dispersion characteristics of the dominant frequency band.

Figures 8c,d correspond to the spectral near-field radiative heat transfer coefficient distributions under bilayer co-directional strain applied along the y -axis. Similar to the x -direction case, h (ω) still maintains a single-peak structure, with the main peak located around ω ≈ 2.15 × 10 14 rad·s −1. Within the compressive range, the main peak intensity gradually weakens with increasing compression, decreasing from 7.83 × 10 −16 J·rad −1 ·K −1 ·m −2 in the unstrained state to 5.11 × 10 −16 J·rad −1 ·K −1 ·m −2 at ε y = −4%, indicating that bilayer compression along the y-direction suppresses near-field coupling in the dominant frequency band. In contrast, tensile strain along the y-direction produces a much stronger nonlinear response. Within the range of 0% to +2%, the variation in h (ω) remains relatively limited. When the strain increases to +3% and +4%, the main peak intensity increases sharply. Especially under ε y = +4%, the peak value of h (ω) reaches 1.72 × 10 −13 J·rad −1 ·K −1 ·m −2, approximately 220 times that of the unstrained state. This much stronger enhancement compared with the single-layer strain case arises because both borophene sheets undergo similar y-axis tensile-strain-induced dielectric modulation. The simultaneous enhancement of the low-frequency dielectric response at the two interfaces improves the matching of surface modes, thereby increasing the evanescent-wave transmission coefficient and producing a much higher spectral peak.

To further quantitatively evaluate the effect of bilayer co-directional strain on the overall near-field radiative heat transfer intensity, Figure 9 presents the normalized heat transfer coefficient q (ε)/ q (ε = 0) under strain applied along the x - and y -axes. For bilayer co-directional strain along the x -direction, the normalized coefficient remains consistently greater than 1 within the compressive range and gradually increases with increasing compression, reaching approximately 1.43 at ε x = −4%. This indicates that synchronous compression of both layers further enhances near-field radiative heat transfer. After entering the tensile regime, the variation remains relatively limited during the small-strain stage. When the strain increases to +4%, the normalized coefficient rises significantly to 3.64, showing a clear enhancement effect. In contrast, bilayer co-directional strain along the y -direction exhibits a stronger switch-like modulation capability. Within the range from compression to small tensile strain (−4% to +2%), the normalized coefficient gradually increases from 0.65 to 1.31, with an overall moderate variation. When the tensile strain is further increased to +3% and +4%, the normalized coefficient rapidly increases to 28.11 and 125.92, respectively, far exceeding the corresponding values under x -direction strain.

These results indicate that bilayer co-directional strain significantly broadens the tunable range of near-field radiative heat transfer in Bχ 3 sheets and further strengthens its anisotropic characteristics. Among them, x -direction strain is more suitable for achieving continuously tunable modulation, whereas large tensile strain along the y -direction enables order-of-magnitude switching behavior.

Near-field heat transfer under bilayer orthogonal strain

To further elucidate the near-field radiative heat transfer behavior of Bχ 3 sheets under asymmetric strain modulation, this section investigates a bilayer orthogonal-strain configuration, in which the upper and lower borophene sheets are simultaneously subjected to uniaxial strain of the same magnitude but along mutually perpendicular directions: one layer along the x -direction and the other along the y -direction, with the strain magnitude set as ε (−4% to +4%). Figure 10 presents the spectral heat transfer coefficient h (ω) under different strain conditions, while Figure 11 shows the normalized heat transfer coefficient q (ε)/ q (ε = 0).

Figures 10a,b correspond to the compressive and tensile conditions under bilayer orthogonal strain, respectively. Overall, this configuration exhibits a segmented nonlinear response. Within the compressive range (ε < 0), h (ω) shows low sensitivity to the applied strain, with the main peak positions of all spectra remaining around ω ≈ 2.2 × 10 14 rad·s −1 and only minor variations in both peak position and intensity. The curves under different strain conditions highly overlap, indicating that orthogonal compression has a limited effect on near-field coupling within the dominant frequency band. Correspondingly, in Figure 11, the normalized coefficient exhibits only slight fluctuations within the range of ε = −4%–0%, remaining close to 1 with a variation amplitude of approximately 0.08. This indicates that strain modulation in orthogonal directions has a certain mutual cancellation effect during the compression stage, thereby keeping the overall near-field radiative heat transfer essentially stable.

Upon entering the tensile regime, the system response becomes more sensitive to strain. Within the range of ε = +1% to +2%, the main spectral peak exhibits only a limited uplift, and the normalized coefficient remains around 1, reaching approximately 1.12 at +2%. This indicates that small orthogonal tensile strain is still insufficient to substantially alter the near-field coupling condition. When the strain is further increased to +3%, the main spectral peak begins to rise, with the normalized coefficient increasing to 4.13. Upon further increasing the strain to +4%, the main peak intensity increases further, with the peak value of h (ω) reaching 1.83 × 10 −14 J·rad −1 ·K −1 ·m −2, and the normalized heat transfer coefficient further increasing to 20.58. These results indicate that the system enters a strong-coupling regime under large orthogonal tensile strain.

Combined with the results of single-layer strain and bilayer co-directional strain, the enhancement under bilayer orthogonal strain can be mainly attributed to the y -axis tensile layer, whose enhanced low-frequency dielectric response strengthens coupling within the dominant heat-transfer frequency band. However, because the other layer is strained along the x -direction, its dielectric response and surface-mode dispersion do not fully match those of the y -axis tensile layer. As a result, the energy transmission coefficient is enhanced but does not reach the level of the bilayer co-directional y -axis tensile configuration. Therefore, the overall enhancement under orthogonal strain is higher than that of the single-layer case but lower than that of the bilayer co-directional y -axis tensile case, indicating that the mismatch of strain-modified dielectric responses limits the synergistic coupling between the two interfaces.

In summary, our numerical simulations demonstrate that the application of uniaxial strain to Bχ 3 can significantly regulate near-field radiative heat transfer. The results indicate that this system exhibits pronounced anisotropic and nonlinear strain-dependent heat transfer characteristics. In terms of dielectric response, strain along the x -direction mainly modulates the resonance and absorption peaks along the loading direction, whereas strain along the y -direction predominantly enhances the low-frequency dielectric background under large tensile strain, thereby providing more favorable conditions for evanescent-wave coupling. For the single-layer strain configuration, strain along the x -direction enables relatively moderate and continuous near-field radiative heat transfer modulation, with the normalized heat transfer coefficient reaching 1.72 at ε = +4%. In contrast, large tensile strain along the y -direction induces a much stronger enhancement effect, with the normalized heat transfer coefficient increasing to 8.96. Under bilayer co-directional strain, the modulation effect is significantly amplified due to the synchronous variation of the dielectric responses at both interfaces. In particular, bilayer tensile strain along the y -direction exhibits the strongest enhancement effect, where the normalized heat transfer coefficient reaches 125.92 at ε = +4%. For the bilayer orthogonal strain configuration, the system remains relatively stable under low strain but exhibits significant enhancement under large tensile strain, with the normalized heat transfer coefficient increasing to 20.58. These findings demonstrate that mechanical strain provides an effective strategy for actively tuning near-field radiative heat transfer in borophene and offers theoretical guidance for the design of nanoscale thermal switches and energy conversion devices.

It should also be noted that the present work is based on an ideal freestanding Bχ 3 borophene model. In practical applications, the near-field radiative heat transfer behavior may be affected by substrates, defects, thermal stability, and fabrication conditions. For example, substrates may alter the dielectric response and surface-mode dispersion of borophene, while defects, vacancies, and surface contamination may introduce additional scattering or damping. Moreover, the mechanical and thermal stability of borophene under large tensile strain, as well as the experimental control of strain and nanoscale gaps, should be considered in future studies. Therefore, further investigations based on substrate-supported or defect-containing borophene models would be valuable for evaluating the practical feasibility of strain-tunable borophene-based near-field thermal devices.