Modulation of near-field radiative heat transfer between graphene sheets by strain engineering

Lixin Ge; Zijun Xu; Yuping Cang; Ke Gong

doi:10.1364/OE.27.0A1109

1. Introduction

The radiative heat transfer across a vacuum gap between two closely separated objects is known as near-field radiative heat transfer(NFRHT) [1,2]. The radiative heat flux of NFRHT can exceed the blackbody limit by several orders of magnitude [3], due to the tunneling of evanescent waves (e.g., surface plasmon polaritions (SPPs)[4], surface phonon polaritions (SPhPs) [5,6]. Information processing on the nanoscale is one promising application of NFRHT[7], compared with its counterpart of the slow phonon (heat conduction)[8]. In order to manipulate the flow of radiative heat flux, several fundamental circuit elements have been proposed, including thermal diodes [9,10], thermal memory [11,12], thermal transistors [13], et al. In addition, thermal modulators based on NFRHT have been investigated in a variety of bulk materials [14–17]. However, the thicknesses of the bulk materials are generally large. Two-dimensional (2D) materials are good candidates for thermal modulators regarding its ultra-thin characteristic. In the past few years, the modulations of NFRHT based on 2D materials were reported, where the strategies included electric gating [18–20], optical pump [21] and magnetic field [22], etc. To the best of we knowledge, the highest modulation with reduction of radiative heat flux over 98% is theoretically revealed for electric gating [20], whereas the high-quality electrical connections are required. For some anisotropic 2D materials, the modulation can even be realized by rotation of principal axis [23–25], where electrical connections are not necessary.

Recently, strain engineering of 2D materials has been received considerable interest [26–28], and the word $``straintronics"$ appropriately describes this new field. The electrical, chemical and optical properties of 2D materials can be substantially modified with strain-induced deformation. Graphene, being one atomic layer of graphite, is the first 2D materials isolated by mechanical exfoliation in 2004 [29]. The mechanical property of graphene is outstanding. It is the strongest 2D material ever known [30]. Moreover, the elastic tensile strains for graphene sheets are reversible even when the deformation is as large as 25 % [30]. Due to the Dirac fermion-like band structure and lattice symmetry of graphene, enormous pseudo-magnetic fields can exist under nonuniform deformations [31,32]. For uniaxial strain, many remarkable phenomena were reported, such as optical Hall effect [33], dichroism [34], tunable Casimir effect [35], the splitting and redshift of Raman bands [36]. Owning to its unique mechanical properties and high carrier mobilities, graphene is an excellent platform for mechanical-thermal applications.

In this work, we study the NFRHT between two suspended graphene sheets under uniform uniaxial strain. Two types of strain configurations are considered. For type I strain, one graphene sheet is strained while the other one is unstrained. As the strain modulus increases, the spectra of NFRHT undergo a redshift and the corresponding magnitudes drop remarkably. On the other hand, the stretching direction has a rather weak effect in type I strain. For type II strain, i.e., two graphene sheets have the same strain modulus while the stretching directions could be arbitrary, the stretching directions play a key role in the modulation of NFRHT. Under proper choices of stretching angles, a large modulation of NFRHT with over 60% reduction is revealed. Finally, the modulation of NFRHT due to the mobilities, separation distance and chemical potential are discussed at the end.

2. Theoretical model

The system under study is depicted in Fig. 1. Two graphene sheets are suspended in the vacuum with separation distance $d$. The temperature for the upper sheet (heat source) and the bottom one (heat sink) are $T_1$ and $T_2$, respectively. The in-plane tension forces $F_{i}$ ($i$=1, 2) are tunable through external control. The stretching direction is defined by an angle $\theta$, namely, the tension force with respect to the zigzag crystalline direction of graphene. By assuming the zigzag crystalline direction parallels to the $x$-direction, the strain tensor is written as [37]:

(1)$$\overline{\kappa }=\kappa \left( \begin{array}{cc} \cos ^{2}\theta -\rho \sin ^{2}\theta & (1+\rho )\cos \theta \sin \theta \\ (1+\rho )\cos \theta \sin \theta & \sin ^{2}\theta -\rho \cos ^{2}\theta \end{array} \right),$$

where $\kappa$ is the strain modulus, $\rho$= 0.14 is the Poisson ratio of graphene according to the $ab$ $initio$ calculations [38], which is smaller than the experimental value for graphite with 0.165 by Blakslee et al [39]. The optical conductivity tensor of graphene under uniform strain is given as [40]:

(2)$$\overline{\sigma }(\omega )=\sigma _{0}(\omega )(\mathbf{I}-2\tilde{\beta} \overline{\kappa } +\tilde{\beta} \mathrm{Tr}(\overline{\kappa })\mathbf{I}),$$

where $\tilde {\beta }$=$\beta -1$ and $\beta \simeq 3$ is the Grüneisen parameter, $\omega$ is the angular frequency, $\mathbf {I}$ is the $2 \times 2$ identity matrix. $\sigma _{0}(\omega )$ is the optical conductivity of the unstrained graphene, and it is given by the well-known random phase approximation [41,42]:

(3)$$\begin{aligned}\sigma _{0}(\omega ) &=i\frac{e^{2}}{\pi \hbar ^{2}}\frac{2k_{B}T}{(\omega +i/\tau )}\ln \left[ 2\cosh \frac{E_{F}}{2k_{B}T}\right] \nonumber\\ &\quad+\frac{e^{2}}{4\hbar }\left[ G(\hbar \omega /2)+i\frac{4\hbar \omega }{\pi }\int_{0}^{\infty }\frac{G(x)-G(\hbar \omega /2)}{(\hbar \omega )^{2}-4x^{2}} dx\right], \end{aligned}$$

where

(4)$$G(x)=\frac{\sinh (x/k_{B}T)}{\cosh (E_{F}/k_{B}T)+\cosh (x/k_{B}T)},$$

where $e$ is the charge of an electron, $k_{B}$ is the Boltzmann factor, $T$ is the temperature, $\hbar$ is the reduced Planck constant. $\tau =\mu E_{F}/ev_{F}^{2}$ is the intrinsic relaxation time, $\mu$ is the electron mobility, $E_{F}$ is the Fermi energy, and $v_{F}\approx 1.0\times 10^6$ m/s is the Fermi velocity.

Fig. 1. Schematic view of NFRHT between two graphene sheets. The in-plane tension forces $F_1$ and $F_2$ are tunable. The stretching direction $\theta$ is defined by the angle of tension force with respect to the zigzag crystalline direction of graphene ($x$-axis).

Download Full Size | PDF

Theoretically, the radiative heat flux between two graphene sheets $\left \langle S(T_{1},T_{2})\right \rangle$ can be calculated based on the fluctuation-dissipation theory [43], and the heat transfer coefficient(HTC) is defined as follows:

(5)$$H(T)=\lim \ _{T_{1},T_{2}\rightarrow T}\frac{\left\langle S(T_{1},T_{2})\right\rangle }{T_{1}-T_{2}}=\int_{0}^{\infty }h(\omega ,T) \textrm{d}\omega,$$

where $h(\omega ,T)$ is the spectral HTC, expressed as

(6)$$h(\omega ,T)=\frac{1}{(2\pi )^{3}}\left[ \int_{-\infty }^{\infty }\int_{-\infty }^{\infty }\xi (\omega ,k_{x},k_{y})\textrm{d}k_{x}\textrm{d}k_{y} \right] \frac{\partial \Theta (\omega ,T)}{\partial T},$$

where $\Theta (\omega ,T)=\hbar \omega /(\exp (\hbar \omega /k_{B}T)-1)$ is the average energy of a Planck’s oscillator, $\xi (\omega ,k_{x},k_{y})$ is the energy transmission coefficient, standing for the probability of photon tunneling. The energy transmission coefficient $\xi (\omega ,k_{x},k_{y})$ is contributed from propagating and evanescent modes, expressed as follows:

(7)$$\xi (\omega ,k_{x},k_{y})=\left\{ \begin{array}{c} \textrm{Tr}[(\mathbf{I}-\mathbf{R}_{2}^{{\dagger} }\mathbf{R}_{2})\mathbf{D} ^{12}(\mathbf{I}-\mathbf{R}_{1}^{{\dagger} }\mathbf{R}_{1})\mathbf{D} ^{12\dagger }], \\ \textrm{Tr}[(\mathbf{R}_{2}^{{\dagger} }-\mathbf{R}_{2})\mathbf{D}^{12}(\mathbf{R }_{1}-\mathbf{R}_{1}^{{\dagger} })\mathbf{D}^{12\dagger }]e^{{-}2|\gamma |d}, \end{array} \right. \begin{array}{c} k_{\Vert }<\omega /c \\ k_{\Vert }>\omega /c \end{array}$$

where $k_{\Vert }=\sqrt {k_{x}^{2}+k_{y}^{2}}$ and $\gamma =\sqrt { k_{0}^{2}-k_{\Vert }^{2}}$ are respectively the in-plan and vertical wavevector with $k_{0}=\omega /c$ being the wavevector in the vacuum, $\mathbf {D} ^{12}=(\mathbf {I}-\mathbf {R}_{1}\mathbf {R}_{2}e^{2i\gamma d})^{-1}$ and $\mathbf {R}_{j}$ ($j$=1, 2) is the $2\times 2$ reflection matrix for the $j$-th graphene sheet, having the form:

(8)$$\mathbf{R}_{j}=\left( \begin{array}{cc} r_{j}^{ss} & r_{j}^{sp} \\ r_{j}^{ps} & r_{j}^{pp} \end{array} \right) ,$$

where the superscripts $s$ and $p$ represent the polarizations of transverse electric ($\mathbf {TE}$) and transverse magnetic ($\mathbf {TM}$) modes, respectively. The matrix element $r^{\alpha \beta }$ ($\alpha ,\beta$ = $s,p$) represents the reflection coefficient for an incoming $\alpha$-polarized plane wave turns out to be an outgoing $\beta$-polarized wave. Consider an incident plane wave with in-plane wavevector $k_{\Vert }=k_{x} \vec {e}_{x}+k_{y}\vec {e}_{y}$, the reflection coefficients for a strained graphene sheet can be given as follows [44]:

(9)$$\begin{aligned}r^{ss} &=-\frac{2k_{0}^{2}\overline{\sigma }_{yy}^{{\prime }}+k_{0}\gamma ( \overline{\sigma }_{xx}^{{\prime }}\overline{\sigma }_{yy}^{{\prime }}- \overline{\sigma }_{xy}^{{\prime }}\overline{\sigma }_{yx}^{{\prime }})}{ 4k_{0}\gamma +2\gamma ^{2}\overline{\sigma }_{xx}^{{\prime }}+2k_{0}^{2} \overline{\sigma }_{yy}^{{\prime }}+k_{0}\gamma (\overline{\sigma } _{xx}^{{\prime }}\overline{\sigma }_{yy}^{{\prime }}-\overline{\sigma } _{xy}^{{\prime }}\overline{\sigma }_{yx}^{{\prime }})}, \nonumber\\ r^{pp} &=-\frac{2\gamma ^{2}\overline{\sigma }_{xx}^{{\prime }}+k_{0}\gamma (\overline{\sigma }_{xx}^{{\prime }}\overline{\sigma } _{yy}^{{\prime }}-\overline{\sigma }_{xy}^{{\prime }}\overline{\sigma } _{yx}^{{\prime }})}{4k_{0}\gamma +2\gamma ^{2}\overline{\sigma } _{xx}^{{\prime }}+2k_{0}^{2}\overline{\sigma }_{yy}^{{\prime }}+k_{0}\gamma (\overline{\sigma }_{xx}^{{\prime }}\overline{\sigma } _{yy}^{{\prime }}-\overline{\sigma }_{xy}^{{\prime }}\overline{\sigma } _{yx}^{{\prime }})}, \\ r^{sp} &= r^{ps}=\frac{2k_{0}\gamma \overline{\sigma }_{yx}^{{\prime }}}{ 4k_{0}\gamma +2\gamma ^{2}\overline{\sigma }_{xx}^{{\prime }}+2k_{0}^{2} \overline{\sigma }_{yy}^{{\prime }}+k_{0}\gamma (\overline{\sigma } _{xx}^{{\prime }}\overline{\sigma }_{yy}^{{\prime }}-\overline{\sigma } _{xy}^{{\prime }}\overline{\sigma }_{yx}^{{\prime }})},\end{aligned}$$

where the elements of conductivity tensor are normalized by the free-space impedance $\sqrt {\mu _{0}/\varepsilon _{0}}$. Note that the denominator in all reflection coefficients is the same. By setting this denominator to be zero, the dispersion of SPPs for a strained graphene sheet can be obtained. The primes in superscripts stand for the rotation of conductivity tensor with respect to the unit vector of $k_{\Vert }$, having the form [24,44]:

(10)$$\left[ \begin{array}{cc} \overline{\sigma }_{xx}^{{\prime }} & \overline{\sigma }_{xy}^{{\prime }} \\ \overline{\sigma }_{yx}^{{\prime }} & \overline{\sigma }_{yy}^{{\prime }} \end{array} \right] =\left[ \begin{array}{cc} \overline{\sigma }_{xx}k_{x}^{2}/k_{\Vert }^{2}+\overline{\sigma } _{yy}k_{y}^{2}/k_{\Vert }^{2} & (\overline{\sigma }_{xx}-\overline{\sigma } _{yy})k_{x}k_{y}/k_{\Vert }^{2} \\ (\overline{\sigma }_{xx}-\overline{\sigma }_{yy})k_{x}k_{y}/k_{\Vert }^{2} & \overline{\sigma }_{xx}k_{y}^{2}/k_{\Vert }^{2}+\overline{\sigma } _{yy}k_{x}^{2}/k_{\Vert }^{2} \end{array} \right].$$

3. Results and discussions

The numerical results for spectral HTC can be obtained with above Eqs.(1)–(10). Two types of strain configurations are considered as depicted in the insets of Figs. 2(a) and 2(c). For type I strain, the spectral HTC has a red shift and the magnitudes drop remarkably as the strain modulus $\kappa$ increases from 0 to 0.2. However, the stretching direction has a relative weak influence on spectral HTC as shown in Fig. 2(b). The magnitudes of spectra decrease slightly for angles 30° and 45° in comparison to the case of 0°. Interestingly, the spectra of HTC are overlap completely for angles $\theta$ and 90°-$\theta$, due to the symmetric properties of tension matrix Eq. (1). For type II strain, the magnitudes of the spectra are almost unchanged as the strain modulus $\kappa$ increases from 0 to 0.2 as shown in Fig. 2(c). Nonetheless, the spectral HTC undergoes a red shift as the strain modulus increases. In a real situation, the strain directions for two graphene sheets can be controllable and may be different. Remarkably, the spectral HTC can be modified greatly by the directions of stretching as given in Fig. 2(d). The stretching angle for the upper sheet is fixed at 0° while the angle varies from 0° to 90° for the bottom one. As the angle increases, not only the resonant peaks are shifted, but also the magnitudes decrease.

Fig. 2. Modulation of spectral HTC for type I strain (a-b) and type II strain(c-d). In (a) and (c), the stretching direction is along the zigzag direction with $\theta =0^{\circ }$. In (b) and (d), the strain modulus is a constant with $\kappa =0.2$. The chemical potential of two graphene sheets are $E_{F1}=E_{F2}=0.2$ eV, and the electron mobility is $\mu _{1}=\mu _{2}=10^{4}$ cm$^{2}$/Vs. The temperature $T$=300 K and the separation distance $d$=100 nm.

Download Full Size | PDF

To understand the modulation of spectral HTC under mechanical strain, it is necessary to inspect the energy transmission coefficients $\xi (\omega , k_x, k_y)$ in $k$ space. To begin, we show the transmission coefficient of the unstrained situation in Fig. 3(a). Clearly, two concentric circles can be seen, representing the strong coupling of SPPs between two graphene sheets. For comparison, the isofrequency dispersion curve in single unstained graphene is illustrated by the dashed gray line. Figures 3(b)–3(d) show the transmission coefficient for type I strain. The transmission coefficient exhibits an ellipse-like diagram as the stretching angle at the zigzag direction, i.e., $\theta$=0°. The magnitude of $\xi (\omega , k_x, k_y)$ reduces greatly along the stretching direction(i.e., $k_x$ for $\theta$=0°), due to the mismatch of SPPs. Nevertheless, strong coupling of SPPs can also be found in some specific directions. For stretching angle 45°, and 90°, the shapes of transmission coefficient have little change compared with the case of 0°, except an additional rotation.

Fig. 3. Contours plots of energy transmission coefficient at 0.1 eV. (a)Unstrained situation. Type I strain for stretching angles (b)0°; (c)45° and (d)90°. Type II strain for stretching angles (e) (0°, 0°); (f)(0°, 45°) and (g)(0°, 90°). The strain modulus is $\kappa =0.2$ for both type I and type II. The gray dash lines represent the dispersion of SPPs of single graphene sheet without mechanical strain. The green lines in the styles of dot, dash and dash-dot represent the dispersion of SPPs in a single graphene sheet with strain angle 0°, 45° and 90°, respectively.

Download Full Size | PDF

Figures 3(d)–3(f) show the transmission coefficient $\xi (\omega , k_x, k_y)$ for type II strain. An ellipse-like diagram can be observed as well when the stretching direction for two graphene sheets is the same with (0°, 0°). The magnitude of the transmission coefficient in $k_x$ direction does not decrease, which differ considerably from those in Fig. 3(b). The corresponding SPPs supported by the upper and bottom sheets have strong coupling, due to the same stain modulus and direction. Figures 3(f) and 3(g) show the transmission coefficient for stretching angles (0°, 45°)and (0°, 90°), respectively. The transmission coefficient shows complicated diagrams and the coupling of SPPs is strong only in some specific directions, leading to the reduction of NFRHT.

The sensitivity of spectral HTC with respect to the stretching directions can be interpreted by the mismatch of SPPs between the upper and bottom sheets. The dispersions of SPPs in a single graphene sheet are shown in Figs. 3(a)–3(f) under different strain configurations. For the unstrained case, the isofrequency contour of dispersion indicates the isotropic property of plasmon. The dispersions become anisotropic as the strain is applied for stretching angle 0° and 45°. Moreover, the major(minor) axis of SPPs is parallel(perpendicular) to stretching direction. As a result, the mismatch of anisotropic SPPs for type II strain appears when the stretching directions are different. This finding is similar with those in anisotropic 2D materials [23–25] and one-dimensional grating systems [16], for which the modulations of NFRHT are realized by rotation of the principal axes. In our cases, the principal axes of anisotropic SPPs are generated automatically by strain, whereas the rotation of the sheets is not necessary.

The modulation of HTC controlled by strain engineering is given in Figs. 4(a)–4(d). The difference of HTC between strained and unstrained situation is defined as $\eta =H(T)/H_0(T)$, where $H(T)$ and $H_0(T)$ stand for HTC with and without mechanical strain, respectively. For the non-strain configuration, the mobility has a strong influence on the modulation as mentioned in [20]. The increase of mobility can reduce the line-widths of SPPs, which can improve the detuning of plasmonic resonances [20]. Indeed, ultrahigh electron mobility is possible for suspended graphene [45]. The strain modulation for type I strain is shown in Fig. 4(a) with different electron mobilities. As expected, the modulation contrast $\eta$ tends to decrease when the modulus $\kappa$ increases from 0 to 0.2. For mobilities 5000 cm$^2$/Vs, 10000 cm$^2$/Vs and 15000 cm$^2$/Vs, the HTC of unstrained configuration $H_0(T)$ is about 236, 158 and 121 times of the black-body limit($H_{bb}$=6.1 Wm$^{-2}$K$^{-1}$), and the corresponding reduction of HTC are respectively about 36 %, 46% and 51% when strain modulus $\kappa$=0.2. On the other hand, the stretching direction for type I strain has a rather weak influence on HTC as shown in Fig. 4(b). The three curves are almost flat as the $\theta$ increases from 0° to 90° and the dips at stretching angle 45° are smooth.

Fig. 4. Modulation of HTC under different values of electron mobilities. (a)Type I strain via strain modulus, and the stretching angle is 0°; (b)Type I strain via stretching angle; (c)Type II strain via modulus, and the stretching angle is (0°, 0°); (d)Type II strain via stretching angle ($\theta _1$=0°, $\theta _2$). In (b) and (d), the modulus $\kappa$=0.2 is fixed. The inset shows the the modulation contrast ($\mu$=10000 cm$^2$/Vs) for arbitrary stretching angle ($\theta _1$, $\theta _1+\Delta \theta$), and the white contour line represents $\eta =0.4$. Other parameters are kept the same as those of Fig. 2.

Download Full Size | PDF

The modulation of HTC for type II strain is shown in Fig. 4(c) with different strain modulus. Firstly, we consider the stretching angles of two graphene sheet are (0°, 0°). Unlike the case of type I strain, the HTC for type II strain can be enhanced slightly as the modulus $\kappa$ increases. However, the modulation is smaller than 6% even for strain modulus $\kappa =0.2$. On the other hand, the modulation contrast is sensitive to the differences of stretching directions as shown in Fig. 4(d). The stretching angle for the upper sheet is 0°, while the stretching angle for the bottom one varies from 0° to 90°. Remarkably, the HTC decreases quickly as the stretching angle of the bottom sheet increases. Again, the performance of the modulation is better for high mobilities. In a real situation, the stretching angle is arbitrary, as shown in the inset of Fig. 4(d) for mobility 10000 cm$^2$/Vs. By delicate choice of stretching angles, the modulation contrast $\eta$ can be tuned in a wide range, from 1.2 to 0.32, comparing to the non-strained case.

Finally, we consider the modulation of NFRHT as a function of separation distance in Figs. 5(a) and 5(b). The stretching angles for type I and type II strain are respectively 0° and (0°, 90°) and the strain modulus $\kappa$=0.2 is fixed for these two configurations. It is found that the modulation contrast $\eta$ tends to decrease as the separation distance $d$ increases from 10 nm to about 200 nm, and it increases as $d$ increases further. The trend of modulation as a function of separation distance can be explained by the bandwidth of spectral HTC. For small separation distance, the spectral HTC is broadband due to the contribution of high-frequency SPPs. As the separation distance increases further (e.g., $d$=200 nm), the spectral HTC becomes narrow, which can improve the thermal modulation. As $d$ increases to a large value (e.g., $d$=1000 nm), however, the component of far-field radiation becomes significant and the spectral HTC becomes broadband again. As a result, a moderate separation is preferred for optimal modulation. For $d$=200 nm, the reduction of HTC can be over 40% and 60% for type I and type II strain, respectively. Further reduction of HTC is expected if the mobility of graphene becomes larger. The inset in Fig. 5(a) shows the modulation as a function of chemical potential $E_F$ for type II strain. The ratio tends to decrease with the chemical potential increase from 0.1 eV, and there is a minimized value near 0.2 eV. As the chemical potential increases further, the modulation contrast increases. According to the work of Ghanekar et al [17], the strain sensitivity here can be given as $\alpha =|1-\eta |/\kappa$. It was reported that the strain sensitivity of NFRHT for meta-materials drops greatly as the separation distance $d$ increases [17] (e.g., $\alpha$ is smaller than 1 when $d>$300 nm). Here, the strain sensitivity of modulation is high within a wide range of separation distance for graphene sheets. The HTC for the unstrained configuration is shown in the dashed green curve. Clearly, the HTC decreases as $d$ increases, and it is still over 100 and 3 times of the black-body limit for $d$=100 nm and 1000 nm, respectively.

Fig. 5. (a)Modulation of HTC as a function of separation distance. The stretching angles for type I and type II strain are 0° and (0°, 90°), respectively. The inset shows the modulation as a function of chemical potential for type II strain. (b)Strain sensitivity as a function of separation distance. The green dashed line represents the HTC for non-strained graphene, normalized by the blackbody limit (6.1 Wm$^{-2}$K$^{-1}$). Other parameters are kept the same as those of Fig. 2.

Download Full Size | PDF

4. Conclusions

In summary, the strain-induced modulation of NFRHT between two parallel graphene sheets is investigated theoretically. Two types of strain are proposed and analyzed. For type I strain, the spectral HTC has a redshift and the magnitude decreases as the strain modulus increases. For type II strain, the stretching directions of two graphene sheets play a significant role in the modulation of NFRHT. As the stretching directions for two graphene sheet are different, the HTC can be reduced greatly, due to the mismatch of anisotropic SPPs. A large modulation with the reduction of HTC over 60% can be realized within the linear elastic regime. Our finding could be useful for thermal management in micro/nano-electromechanical devices. Although our work is focused on graphene, the modulation based on strain engineering is also expected in other promising 2D materials like silicene, phosphorene, dichalcogenide- and monochalcogenide-monolayers [27].

Funding

National Natural Science Foundation of China (NSFC) (11747100, 11804288); Innovation Scientists and Technicians Troop Construction Projects of Henan Province; Nanhu Scholars Program for Young Scholars of XYNU.

References

1. D. Polder and M. Van Hove, “Theory of radiative heat transfer between closely spaced bodies,” Phys. Rev. B 4(10), 3303–3314 (1971). [CrossRef]

2. A. I. Volokitin and B. N. J. Persson, “Near-field radiative heat transfer and noncontact friction,” Rev. Mod. Phys. 79(4), 1291–1329 (2007). [CrossRef]

3. B. Song, A. Fiorino, E. Meyhofer, and P. Reddy, “Near-field radiative thermal transport: From theory to experiment,” AIP Adv. 5(5), 053503 (2015). [CrossRef]

4. S. Boriskina, J. Tong, Y. Huang, J. Zhou, V. Chiloyan, and G. Chen, “Enhancement and tunability of near-field radiative heat transfer mediated by surface plasmon polaritons in thin plasmonic films,” Photonics 2(2), 659–683 (2015). [CrossRef]

5. K. Joulain, J.-P. Mulet, F. Marquier, R. Carminati, and J.-J. Greffet, “Surface electromagnetic waves thermally excited: Radiative heat transfer, coherence properties and casimir forces revisited in the near field,” Surf. Sci. Rep. 57(3-4), 59–112 (2005). [CrossRef]

6. S. Shen, A. Narayanaswamy, and G. Chen, “Surface phonon polaritons mediated energy transfer between nanoscale gaps,” Nano Lett. 9(8), 2909–2913 (2009). [CrossRef]

7. P. Ben-Abdallah and S.-A. Biehs, “Thermotronics: Towards nanocircuits to manage radiative heat flux,” Z. Naturforsch., A: Phys. Sci. 72(2), 151–162 (2017). [CrossRef]

8. N. Li, J. Ren, L. Wang, G. Zhang, P. Hänggi, and B. Li, “Colloquium: Phononics: Manipulating heat flow with electronic analogs and beyond,” Rev. Mod. Phys. 84(3), 1045–1066 (2012). [CrossRef]

9. C. R. Otey, W. T. Lau, and S. Fan, “Thermal rectification through vacuum,” Phys. Rev. Lett. 104(15), 154301 (2010). [CrossRef]

10. A. Ghanekar, J. Ji, and Y. Zheng, “High-rectification near-field thermal diode using phase change periodic nanostructure,” Appl. Phys. Lett. 109(12), 123106 (2016). [CrossRef]

11. V. Kubytskyi, S.-A. Biehs, and P. Ben-Abdallah, “Radiative bistability and thermal memory,” Phys. Rev. Lett. 113(7), 074301 (2014). [CrossRef]

12. K. Ito, K. Nishikawa, and H. Iizuka, “Multilevel radiative thermal memory realized by the hysteretic metal-insulator transition of vanadium dioxide,” Appl. Phys. Lett. 108(5), 053507 (2016). [CrossRef]

13. P. Ben-Abdallah and S.-A. Biehs, “Near-field thermal transistor,” Phys. Rev. Lett. 112(4), 044301 (2014). [CrossRef]

14. L. Cui, Y. Huang, J. Wang, and K.-Y. Zhu, “Ultrafast modulation of near-field heat transfer with tunable metamaterials,” Appl. Phys. Lett. 102(5), 053106 (2013). [CrossRef]

15. Y. Huang, S. V. Boriskina, and G. Chen, “Electrically tunable near-field radiative heat transfer via ferroelectric materials,” Appl. Phys. Lett. 105(24), 244102 (2014). [CrossRef]

16. S.-A. Biehs, F. S. Rosa, and P. Ben-Abdallah, “Modulation of near-field heat transfer between two gratings,” Appl. Phys. Lett. 98(24), 243102 (2011). [CrossRef]

17. A. Ghanekar, M. Ricci, Y. Tian, O. Gregory, and Y. Zheng, “Strain-induced modulation of near-field radiative transfer,” Appl. Phys. Lett. 112(24), 241104 (2018). [CrossRef]

18. O. Ilic, M. Jablan, J. D. Joannopoulos, I. Celanovic, H. Buljan, and M. Soljačić, “Near-field thermal radiation transfer controlled by plasmons in graphene,” Phys. Rev. B 85(15), 155422 (2012). [CrossRef]

19. Q. Zhao, T. Zhou, T. Wang, W. Liu, J. Liu, T. Yu, Q. Liao, and N. Liu, “Active control of near-field radiative heat transfer between graphene-covered metamaterials,” J. Phys. D: Appl. Phys. 50(14), 145101 (2017). [CrossRef]

20. O. Ilic, N. H. Thomas, T. Christensen, M. C. Sherrott, M. Soljačić, A. J. Minnich, O. D. Miller, and H. A. Atwater, “Active radiative thermal switching with graphene plasmon resonators,” ACS Nano 12(3), 2474–2481 (2018). [CrossRef]

21. R. Yu, A. Manjavacas, and F. J. G. de Abajo, “Ultrafast radiative heat transfer,” Nat. Commun. 8(1), 2 (2017). [CrossRef]

22. L. Ge, K. Gong, Y. Cang, Y. Luo, X. Shi, and Y. Wu, “Magnetically tunable multi-band near-field radiative heat transfer between two graphene sheets,” https://arxiv.org/pdf/1812.10648.

23. Y. Zhang, H.-L. Yi, and H.-P. Tan, “Near-field radiative heat transfer between black phosphorus sheets via anisotropic surface plasmon polaritons,” ACS Photonics 5(9), 3739–3747 (2018). [CrossRef]

24. L. Ge, Y. Cang, K. Gong, L. Zhou, D. Yu, and Y. Luo, “Control of near-field radiative heat transfer based on anisotropic 2d materials,” AIP Adv. 8(8), 085321 (2018). [CrossRef]

25. X. Liu, J. Shen, and Y. Xuan, “Pattern-free thermal modulator via thermal radiation between van der waals materials,” J. Quant. Spectrosc. Radiat. Transfer 200, 100–107 (2017). [CrossRef]

26. R. Roldán, A. Castellanos-Gomez, E. Cappelluti, and F. Guinea, “Strain engineering in semiconducting two-dimensional crystals,” J. Phys.: Condens. Matter 27(31), 313201 (2015). [CrossRef]

27. G. G. Naumis, S. Barraza-Lopez, M. Oliva-Leyva, and H. Terrones, “Electronic and optical properties of strained graphene and other strained 2d materials: a review,” Rep. Prog. Phys. 80(9), 096501 (2017). [CrossRef]

28. C. Si, Z. Sun, and F. Liu, “Strain engineering of graphene: a review,” Nanoscale 8(6), 3207–3217 (2016). [CrossRef]

29. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004). [CrossRef]

30. C. Lee, X. Wei, J. W. Kysar, and J. Hone, “Measurement of the elastic properties and intrinsic strength of monolayer graphene,” Science 321(5887), 385–388 (2008). [CrossRef]

31. F. Guinea, M. Katsnelson, and A. Geim, “Energy gaps and a zero-field quantum hall effect in graphene by strain engineering,” Nat. Phys. 6(1), 30–33 (2010). [CrossRef]

32. N. Levy, S. Burke, K. Meaker, M. Panlasigui, A. Zettl, F. Guinea, A. C. Neto, and M. Crommie, “Strain-induced pseudo–magnetic fields greater than 300 tesla in graphene nanobubbles,” Science 329(5991), 544–547 (2010). [CrossRef]

33. V. H. Nguyen, A. Lherbier, and J.-C. Charlier, “Optical hall effect in strained graphene,” 2D Mater. 4(2), 025041 (2017). [CrossRef]

34. M. Oliva-Leyva and G. G. Naumis, “Tunable dichroism and optical absorption of graphene by strain engineering,” 2D Mater. 2(2), 025001 (2015). [CrossRef]

35. A. D. Phan and T.-L. Phan, “Casimir interactions in strained graphene systems,” Phys. Status Solidi RRL 8(12), 1003–1006 (2014). [CrossRef]

36. D. Yoon, Y.-W. Son, and H. Cheong, “Strain-dependent splitting of the double-resonance raman scattering band in graphene,” Phys. Rev. Lett. 106(15), 155502 (2011). [CrossRef]

37. V. M. Pereira, A. C. Neto, and N. Peres, “Tight-binding approach to uniaxial strain in graphene,” Phys. Rev. B 80(4), 045401 (2009). [CrossRef]

38. M. Farjam and H. Rafii-Tabar, “Comment on “band structure engineering of graphene by strain: First-principles calculations”,” Phys. Rev. B 80(16), 167401 (2009). [CrossRef]

39. O. Blakslee, D. Proctor, E. Seldin, G. Spence, and T. Weng, “Elastic constants of compression-annealed pyrolytic graphite,” J. Appl. Phys. 41(8), 3373–3382 (1970). [CrossRef]

40. M. Oliva-Leyva and G. G. Naumis, “Anisotropic ac conductivity of strained graphene,” J. Phys.: Condens. Matter 26(12), 125302 (2014). [CrossRef]

41. B. Wunsch, T. Stauber, F. Sols, and F. Guinea, “Dynamical polarization of graphene at finite doping,” New J. Phys. 8(12), 318 (2006). [CrossRef]

42. L. Falkovsky, “Optical properties of graphene,” J. Phys.: Conf. Ser. 129, 012004 (2008). [CrossRef]

43. S.-A. Biehs, P. Ben-Abdallah, F. Rosa, K. Joulain, and J.-J. Greffet, “Nanoscale heat flux between nanoporous materials,” Opt. Express 19(S5), A1088–A1103 (2011). [CrossRef]

44. O. Kotov and Y. E. Lozovik, “Enhanced optical activity in hyperbolic metasurfaces,” Phys. Rev. B 96(23), 235403 (2017). [CrossRef]

45. K. I. Bolotin, K. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146(9-10), 351–355 (2008). [CrossRef]

Modulation of near-field radiative heat transfer between graphene sheets by strain engineering

Abstract

1. Introduction

2. Theoretical model

3. Results and discussions

4. Conclusions

Funding

References

Cited By

Figures (5)

Equations (10)

Optics Express