Contribution of the acoustic waves to near-field heat transfer

TL;DR

This paper investigates how electrostatic potential differences influence near-field heat transfer between metals, highlighting the significant role of acoustic waves and surface phonon polaritons in controlling nanoscale heat fluxes.

Contribution

It introduces a detailed analysis of the coupling between electromagnetic fields and acoustic waves in metals under potential differences, revealing new mechanisms for heat transfer control.

Findings

Potential difference greatly enhances heat flux via acoustic wave coupling.

Surface Rayleigh waves can generate surface phonon polaritons.

Heat transfer mechanisms depend on distance and potential difference.

Abstract

Calculations of the radiative and phonon heat transfer between metals in an extreme near field in presence of electrostatic potential difference are given. Potential difference leads to a coupling between the radiation field and acoustic waves in solid, as a result of which the heat flux between two gold plates associated with p -polarized electromagnetic waves increases by many orders of magnitude as the potential difference varies from 0 to 10V. The radiative heat transfer is compared with the phonon heat transfer associated with the electrostatic and van der Waals interactions between the surface displacements. For large potential difference and small distances the radiative heat transfer is reduced to the electrostatic phonon heat transfer. A particular case of surface acoustic waves - Rayleigh waves is studied in details. Conditions are obtained for the existence of surface phonon polaritons associated with the interaction of Rayleigh waves with an electromagnetic field. The surface Rayleigh and bulk acoustic waves can give contributions of the same order. The obtained results can be used to control heat fluxes at the nanoscale using the potential difference and to create coherent radiation sources based on the properties of the Rayleigh waves

Equations

$${\sigma }_0=\frac{\phi }{4\pi d}=\frac{E_0}{4\pi },$$

$$J^{rad}=\frac{1}{{\pi }^2}{\int }_0^{\infty }d\omega \left[{\Pi }_1(\omega )-{\Pi }_2(\omega )\right]{\int }_0^{\infty }{k}_{z}d{k}_z{e}^{-2k_{z}d}\left[\frac{\mathrm{Im}{R}_{1p}(\omega ,q)\mathrm{Im}{R}_{2p}(\omega ,q)}{\mid 1-e^{-2k_{z}d}{R}_{1p}(\omega ,q)R_{2p}(\omega ,q){\mid }^2}+(p\to s)\right],$$

$${\Pi }_i(\omega )=\frac{\hslash \omega }{e^{\hslash \omega /k_B{T}_i}-1},$$

$$E_z^{+}=\epsilon (\omega )E_z^{-},$$

$$E_q^{+}-E_q^{-}=-4\pi iq{p}_z=-4\pi iq{\sigma }_0^{2}M{E}_z^{+},$$

$$R_p=\frac{i\epsilon k_z-k_z^{\prime }+4\pi i{q}^2{\sigma }_0^{2}M\epsilon }{i\epsilon k_z+k_z^{\prime }-4\pi i{q}^2{\sigma }_0^{2}M\epsilon },$$

$${\sigma }_1=a{u}_1-b{u}_2,$$

$${\sigma }_2=a{u}_2-b{u}_1,$$

$$a=\frac{H}{2\pi d^4}+\frac{E_0^2}{4\pi }\frac{q\left(e^{qd}+e^{-qd}\right)}{e^{qd}-e^{-qd}},$$

$$b=\frac{H}{4\pi }\frac{q^2{K}_2(qd)}{d^2}+\frac{E_0^2}{2\pi }\frac{q}{e^{qd}-e^{-qd}},$$

$$u_1=u_1^f+M_1(a{u}_1-b{u}_2),$$

$$u_2=u_2^f+M_2(a{u}_2-b{u}_1),$$

$$⟨|u_i^f{|}^2⟩=\hslash \mathrm{Im}{M}_i(\omega ,q)\coth \frac{\hslash \omega }{2k_B{T}_i},$$

$$J^{ph}=\frac{1}{{\pi }^2}{\int }_0^{\infty }d\omega \left[{\Pi }_1(\omega )-{\Pi }_2(\omega )\right]{\int }_0^{\infty }dqq\frac{b^2\mathrm{Im}{M}_1\mathrm{Im}{M}_2}{\mid (1-a{M}_1)(1-a{M}_2)-b^2{M}_1{M}_2{\mid }^2}.$$

$$M=\frac{i}{\rho c_t^2}{\left(\frac{\omega }{c_t}\right)}^2\frac{p_l(q,\omega )}{S(q,\omega )},$$

$$S(q,\omega )={\left[{\left(\frac{\omega }{c_t}\right)}^2-2q^2\right]}^2+4q^2{p}_t{p}_l,$$

$$p_t={\left[{\left(\frac{\omega }{c_t}\right)}^2-q^2+i0\right]}^{1/2},p_l={\left[{\left(\frac{\omega }{c_l}\right)}^2-q^2+i0\right]}^{1/2},$$

$$\epsilon =1-\frac{{\omega }_p^2}{{\omega }^2+i\omega \nu },$$

$$\mathrm{Re}\left(\epsilon -i\frac{k_z^{\prime }}{q}-4\pi q{\sigma }_0^{2}M\epsilon \right)=0.$$

$$R_p=\frac{1+4\pi q{\sigma }_0^{2}M}{1-4\pi q{\sigma }_0^{2}M}.$$

$$J^{rad}=\frac{1}{{\pi }^2}{\int }_0^{\infty }d\omega \left[{\Pi }_1(\omega )-{\Pi }_2(\omega )\right]\left[\frac{E_0^4}{4{\pi }^2}{\int }_0^{\infty }dq{q}^3{\left(\frac{e^{-qd}}{1-e^{-2qd}}\right)}^2\mathrm{Im}{M}_1(\omega ,q)\mathrm{Im}{M}_2(\omega ,q)\right],$$

$$J^{ph}=\frac{1}{{\pi }^2}{\int }_0^{\infty }d\omega \left[{\Pi }_1(\omega )-{\Pi }_2(\omega )\right]{\int }_0^{\infty }dqq{\left(\frac{H}{4\pi }\frac{q^2{K}_2(qd)}{d^2}+\frac{E_0^2}{2\pi }\frac{q}{e^{qd}-e^{-qd}}\right)}^2\mathrm{Im}{M}_1\mathrm{Im}{M}_2.$$

$$J^{rad}=\frac{E_0^4{\omega }_t^3{k}_{B}T}{96{\pi }^2{\rho }^2{c}_t^6}{\int }_0^{1}dx{\left[\mathrm{Re}\frac{\sqrt{(c_t/c_l{)}^2-x^2}}{(1-2x^2{)}^2+4x^2\sqrt{1-x^2}\sqrt{(c_t/c_l{)}^2-x^2}}\right]}^2.$$

$$S(q,\omega )={\left[{\left(\frac{\omega }{c_t}\right)}^2-2q^2\right]}^2+4q^2{p}_t{p}_l=0.$$

$$M=-\frac{C}{\rho c_t(\omega -c_{s}q+i\gamma )},$$

$$C=\frac{c_{st}(1-c_{sl}^2)\sqrt{1-c_{sl}^2}}{4[\sqrt{1-c_{st}^2}\sqrt{1-c_{sl}^2}(c_{st}^2-2)+1+c_{tl}^2-2c_{sl}^2]},$$

$${\gamma }_0=0.17{\omega }_s=1.97\times 10^{2}q{s}^{-1}.$$

$$1-4\pi q{\sigma }_0^2\mathrm{Re}M=0.$$

$$1+\frac{c_{0}q(\omega -c_{s}q)}{(\omega -c_{s}q{)}^2+{\gamma }^2}=0,$$

$$c_0=\frac{4\pi {\sigma }_0^{2}C}{\rho c_t}=\frac{C}{4\pi \rho c_t}{\left(\frac{\phi }{d}\right)}^2.$$

Full text

PACS: 44.40.+a, 63.20.D-, 78.20.Ci

I Introduction

Radiative heat transfer at the nanoscale level is of fundamental importance Polder1971PRB ; Pendry1999JPCM ; Greffet2005SSR ; Volokitin2001PRB ; Volokitin2007RMP ; Volokitin2017Book and promises to find application in a variety of technologies ranging from thermophotovoltaic energy converters Jacob2014APL ; Greffet2006JAP ; BenAbdalakh2013SciRep ; Chen2003APL ; Park2008JQSRT ; Svetovoy2014PRAp , non-invasive thermal imaging Wilde2006Nature to thermomagnetic information recording and processing Challener2009NatPhot ; Biehs2014PRL ; Biehs2013APL ; Joulain2015APL and nanolithography Pendry1999JPCM . Based on the fluctuation electrodynamics developed by Rytov Rytov1953 ; Rytov1967 ; Rytov1987 , it was theoretically predicted Polder1971PRB ; Pendry1999JPCM ; Greffet2005SSR ; Volokitin2001PRB ; Volokitin2007RMP and experimentally confirmed Shen2009NanoLett ; Greffet2009NatPhot ; Reddy2015AIP ; Kittel2017NatCommun ; Reddy2017NatCommun ; Reddy2015Nature , that the radiative heat flux between two bodies with different temperatures in the near field (when the distance between the bodies $d<\lambda_{T}=c\hbar/k_{B}T$: at room temperature $\lambda_{T}\sim 10\mu$m) can be by many orders of magnitude larger than the limit, which is established by Planck’s law for blackbody radiation. With the development of new experimental techniques over the past decade, super-Planckian heat transfer has been observed for vacuum gaps between bodies in the interval from hundreds of nanometers to several Ångströms Reddy2015AIP ; Kittel2017NatCommun ; Reddy2017NatCommun ; Reddy2015Nature . Generally, the results of these measurements turned out to be in good agreement with the predictions based on the fluctuation electrodynamics for a wide range of materials and geometries.

Despite of the important progress achieved in understanding radiative heat transfer, there are still remain significant unresolved problems in understanding heat transfer between bodies in an extreme near field (gap size $<10$ nm)Kittel2017NatCommun ; Reddy2017NatCommun . In particular, the heat transfer observed in Ref.Kittel2017NatCommun for the separations from 1 to 10nm is orders of magnitude larger than the predictions of conventional Rytov theory and its distance dependence is not well understood. These discrepancies stimulate the search of the alternative channels of the heat transfer that can be activated in an extreme near-field. One of the obvious channel is related with phonon tunneling which stimulated active research of the phonon heat transfer in this region Persson2011JPCM ; Budaev2011APL ; Joulain2014PRB ; Pendry2016PRB ; Pendry2017Z.Nat ; Sellan2012PRB ; Esfarjani2015NCom ; KapitzRes2016PRB ; Roy2010PRL ; Meltaus2010PRL ; Henkel2019JOSA ; Volokitin2019JETPLett . Most papers consider phonon tunneling mediated by the van der Waals interaction. In Ref.Pendry2016PRB it was suggested another mechanism for phonon tunneling based on electrostatic potential differences between surfaces. For radiative heat transfer, phonon tunneling was considered for polar dielectrics whose surfaces can support surface phonon polaritonsVolokitin2017Book ; Volokitin2004PRB . Metal surfaces can support surface plasmon polaritons and they also can give important contribution to the radiative heat transferBoriskina2015Photonics ; Mahan2017PRB but for good conductors, like gold, the frequencies of the surface plasmon polaritons are too high and they do not contribute significantly to the radiative heat transfer. Giant enhancement of the radiative heat transfer due to resonant phonon tunneling between adsorbate vibrational modes occurs between metal surfaces covered by adsorbed layer with ionsVolokitin2017Book ; Volokitin2003JETPLett ; Volokitin2004PRB . The contribution from the acoustic waves are usually not considered in the conventional theory of the radiative heat transfer because they are optically nonactive. However, in Ref.Volokitin2019JETPLett it was shown that the acoustic waves can be activated by the electrostatic potential difference. The potential difference induces a surface charge density on the surface of the metals, as a result of which thermal fluctuations of the surface displacements give an additional contribution to the fluctuating electromagnetic field and, as a result, to the radiative heat transfer. For the charged surfaces all the acoustic waves in the frequency range $0<\omega<c_{s}/d<k_{B}T/\hbar$ can contribute to the heat transfer where $c_{s}$ is the sound velocity and $d$ is the separation between the surfaces. In an experiment electrostatic potential difference exists when STM is used for measurement of the heat transferKittel2017NatCommun ; Reddy2017NatCommun . Only the contribution from the bulk acoustic waves was considered in Ref. Volokitin2019JETPLett The surfaces can support Rayleigh acoustic waves that propagate near the surface of the medium and do not penetrate deep into it Landau1970ThElasticity . Rayleigh waves are one of the varieties of surface acoustic waves that are widely used in processing high-frequency signals, delay lines, sensors, and, more recently, for manipulating particles in microchannels. However, the contribution of the Rayleigh waves to the near-field radiative heat transfer was not considered so far because there is no coupling between the thermal radiation and acoustic waves. In this article the contribution from the Rayleigh waves is calculated and compared with bulk contribution. It is shown that these contributions can be of the same order. The radiative heat transfer is compared with the phonon heat transfer which is due to the electrostatic and van der Waals interactions between fluctuating surface displacements. For the electrostatic and van der Waals phonon heat transfer the contributions from the Rayleigh and bulk acoustic waves can be also of the same order what does not agree with the results of Ref.Pendry2016PRB .

II Theory

II.1 Radiative heat transfer

Consider two identical metal plates, which are separated from each other by a vacuum gap with a thickness $d$. Plate 1 has a surface that coincides with the $xy$ plane and is located for $z<0$, while parallel plate 2 is located for $z>d$. A potential difference $\varphi$ is applied between the surfaces, which induces a surface charge density

[TABLE]

where $E_{0}$ is the electric field in the gap between plates. The radiative heat transfer is associated with the fluctuating electromagnetic field created by thermal fluctuations of the charge and current densities inside the bodies. In the case of a charged surface, thermal fluctuations of the surface displacement will also contribute to the fluctuating electromagnetic field and radiative heat transfer. The heat flux between two surfaces separated by a vacuum gap $d$ due to evanescent (non-radiative) electromagnetic waves (for which $q>\omega/c$) is determined by the formula Volokitin2001PRB ; Volokitin2007RMP ; Volokitin2017Book

[TABLE]

where

[TABLE]

$R_{p}$ and $R_{s}$ are the reflection amplitudes for $p$- and $s$-polarized electromagnetic waves, $k_{z}=\sqrt{q^{2}-(\omega/c)^{2}}$, $\mathbf{q}$ is the component of the wave vector parallel to the surface. In the case of charged surfaces, the reflection amplitudes for them will no longer be determined by the Fresnel formulas, since the interaction of a charged surface with an electric field produces a mechanical stress, which will give rise to surface polarization due to the displacement of the surface. In the presence of a surface dipole moment, the boundary conditions for the electric field are determined by equations Volokitin2019JETPLett ; Langreth1989PRB

[TABLE]

[TABLE]

where $E_{z}^{\pm}=E_{z}(z=\pm 0)$, $E_{q}$ is the component of the electric field along $\hat{q}$-vector, $p_{z}=\sigma_{0}u$ is the normal component of the surface dipole moment, $u=M\sigma^{mech}$ is the surface displacement under the action of the mechanical stress $\sigma^{mech}=\sigma_{0}E_{z}^{+}$, $M$ is the mechanical susceptibility that determines the surface displacement under the action of mechanical stress. The reflection amplitude for p -polarized electromagnetic waves obtained using boundary conditions (3) and (4) is given by Volokitin2019JETPLett

[TABLE]

where $k^{\prime}_{z}=\sqrt{\varepsilon(\omega/c)^{2}-q^{2}}=\sqrt{(\varepsilon-1)(\omega/c)^{2}-k_{z}^{2}}$, $\varepsilon$ is the dielectric function of the medium.

II.2 Phonon heat transfer

For the electrostatic and van der Waals interactions between surfaces 1 and 2 the stresses that act on surfaces 1 and 2 due to their displacements are determined by equationsVolokitin2019JETPLett (see also Appendixes A and B for detailed derivation)

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where the first and second terms are due to the van der Waals and electrostatic interactions, respectively, $H$ is the Hamaker constant, $K_{2}(x)$ is the modified Bessel function of the second kind and second order. The surface displacements due to thermal and quantum fluctuations are determined by Volokitin2019JETPLett ; Persson2011JPCM

[TABLE]

[TABLE]

where according to the fluctuation-dissipation theorem, the spectral density of fluctuations of the surface displacements is determined by LandauStatisticalPhysics

[TABLE]

where $M_{i}$ is the mechanical susceptibility for surface $i$.

Due to the electrostatic and van der Waals interactions, the fluctuating displacements of the surfaces will create fluctuating stresses acting on the surfaces, resulting in a heat flux between the surfaces that is determined by Volokitin2019JETPLett

[TABLE]

III Numerical results

III.1 Contribution from bulk acoustic waves

In a elastic continuum model Persson2001JPCM

[TABLE]

where

[TABLE]

[TABLE]

where $\rho$, $c_{l}$, and $c_{t}$ is the density of the medium, the velocity of the longitudinal and transverse acoustic waves, respectively. For gold: $c_{l}=3240$ms*-1*, $c_{t}=1200$ms*-1*, $\rho=1.9280\times 10^{4}$kgm*-3*, $H=34.7\times 10^{-20}$J (see Ref.Hamaker2015 ). The dielectric function of gold Chapuis2008PRB

[TABLE]

where $\omega_{p}=1.71\times 10^{16}$s*-1*, $\nu=4.05\times 10^{13}$s*-1*.

The reflection coefficient (5) has poles, which are determined by the equation

[TABLE]

For high frequencies, when $\varepsilon\rightarrow 1-\omega_{p}^{2}/\omega^{2}$, $k_{z}^{\prime}\approx iq$ and $4\pi q\sigma_{0}^{2}M\ll 1$, Eq. (16) is reduced to $\varepsilon+1=0$ which determines the frequency of the surface plasmon polarotons $\omega_{sp}=\omega_{p}/\sqrt{2}$. In an extreme near field ($d\ll c/\omega|\varepsilon|$, $ik_{z}\approx k_{z}^{\prime}\approx iq$) for good conductors ($|\varepsilon|\gg$) and $4\pi q\sigma_{0}^{2}M\gg 1/|\varepsilon|$ (5) is reduced to

[TABLE]

For $1/|\varepsilon|\ll 4\pi q\sigma_{0}^{2}M\ll 1$, (2) and (13) are reduced to

[TABLE]

[TABLE]

For $qd\gg 1$: $K_{2}(qd)\approx\sqrt{\pi/2qd}e^{-2qd}$ (see Ref.Handbook ), thus the $q$-integration in (19) is restricted by $q<1/d$, from where follows that the electrostatic contribution to the phonon heat transfer dominates for $d>H/2\varphi^{2}$ when (19) is reduced to (18). The underlying physics of this result is that, with a large potential difference and small distances, the electrostatic field created by the fluctuating dipole moment of the charged surfaces mainly determines the fluctuating electromagnetic field. In this case, the radiative heat transfer is reduced to the electrostatic phonon heat transfer. For two identical bodies, introducing new variables $q=(\omega/c_{t})x$ and $\omega=\omega_{t}y$ where $\omega_{t}=c_{t}/d$, at $\hbar\omega_{t}/k_{B}T\ll 1$ Eq.(18) can be written in the form

[TABLE]

III.2 Contribution from Rayleigh waves

From formula (14) it follows that the mechanical susceptibility $M$ has poles, when

[TABLE]

These poles determine the frequency of Rayleigh surface waves that propagate near the surface of the medium and do not penetrate deep into it Landau1970ThElasticity . Eq. (21) has a solution for $\omega=\omega_{s}=c_{s}q=\xi c_{t}q$, where $\omega_{s}$ and $c_{s}$ are the frequency and the propagation velocity of the Rayleigh surface waves. The number $\xi$ depends only on $c_{t}/c_{l}$ which varies for different materials from $1/\sqrt{2}$ to 0 wherein $\xi$ varies from 0.874 to 0.955. For gold $c_{t}/c_{l}=0.37$ and $\xi=0.94$. Near the pole at $\omega\approx\omega_{s}$ the mechanical susceptibility can be written in the form

[TABLE]

where

[TABLE]

$c_{ij}=c_{i}/c_{j}$, for gold $C=0.36$, $\gamma$ is the damping constant for surface wave. If $\gamma=i0$, as it is assumed in Eq. (14), then the Rayleigh waves do not contribute to the heat transfer. As it was shown in Ref.Pendry2016PRB for the damping constant of the Rayleigh waves can be used estimate

[TABLE]

From (17) it follows that for a charged surface in the low-frequency region a resonance may occur, which is determined by the equation

[TABLE]

Equation (24) determines the frequencies of the surface phonon polaritons associated with the interaction of Rayleigh waves with electromagnetic field. Using (22) in (24) we get equation for the frequency of the surface phonon polaritons

[TABLE]

where

[TABLE]

Eq. (25) has solution at the condition $c_{0}q/\gamma=c_{0}/0.17\xi c_{t}>1$. However, for gold this condition can be fulfilled for non-realistic values of the potential difference and distances. In realistic case, when $c_{0}/c_{t}\ll 1$, the contribution of the Rayleigh waves to the heat transfer according to Eq. (18) is determined by

[TABLE]

At zero potential difference, i.e. for noncharged surfaces, the contribution of $p$-waves to the heat transfer is determined by

[TABLE]

At $d=0.5$nm: $J^{rad}_{R}(\varphi=10$V$)\approx 5\times 10^{7}Ish$W/m2 and $J^{rad}_{p}(\varphi=0$V$)\approx 2\times 10^{4}$W/m2. For the electrostatic and van der Waals interactions the contribution of the Rayleigh waves to the phonon heat transfer according to Eq. (19) is determined by

[TABLE]

The contribution from the electrostatic interaction dominates for $d>H/\varphi^{2}$ (see discussion after Eq.(19)), when $J^{ph}_{R}\approx J^{rad}_{R}$. The contribution from the van der Waals interaction dominates for $d<H/\varphi^{2}$ when

[TABLE]

where

[TABLE]

and it was used the value of the integral

[TABLE]

With distance, the contributions due to the van der Waals and electrostatic interactions decay as $d^{-9}$ and $d^{-7}$, respectively. Figs. 1, 2 and 3 compare the contributions from the Rayleigh and volume acoustic waves to the radiative and phonon heat transfer. It follows from the calculations that these contributions are of the same order and have almost the same distance dependence, which does not agree with the results of Ref.Pendry2016PRB . On Fig.3 the difference between the red and blue lines decreases when the potential difference increases what confirm theoretical prediction that at large potential difference and small separation the radiative heat transfer is reduced to the electrostatic phonon heat transfer. Fig. 4 shows the dependence of the contributions to the heat fluxes due to the Rayleigh waves on the damping constant of these waves.

IV Conclusion

The calculations of heat transfer between two plates of gold in an extreme near field are performed at the presence of the potential difference between the plates. The presence of a potential difference leads to a coupling between the radiation field and the acoustic waves, due to which the radiative heat transfer increases by many orders of magnitude when the potential difference varies from 0 to 10V. Radiative heat transfer was compared with the phonon heat transfer due to the electrostatic and van der Waals interactions between the surface displacements. For large potential difference and small distances the radiative heat transfer is reduced to the electrostatic phonon heat transfer. The presence of a potential difference leads to a contributions from Rayleigh surface waves, which are of the same order and have almost the same distance dependence as the contribution from bulk acoustic waves. Similar to the surface electromagnetic waves that are arising due to the interaction of the electromagnetic field with optical waves, Rayleigh waves on a charged surface can also create a thermal electromagnetic field with spatial and temporal coherence. The near-field properties of the thermal electromagnetic field in the presence of surface electromagnetic waves were reviewed in Ref.Joulain2005SSR . The heat flux between a gold coated near-field scanning thermal microscope tip and a planar gold sample in an extreme near field at nanometer distances of 0.2-7nm in the presence of the electrostatic potential difference of $\sim$1V between the tip and the sample was studied in Ref.Kittel2017NatCommun ; Reddy2017NatCommun . It was found that the the experimental results can not be explained by the conventional theory of the radiative heat transfer based on Rytov theory. In this article it was shown that the very large contribution, which is not included in the conventional Rytov theory, can be due to the fluctuating dipole moment induced on the surface by the potential difference. On the base of the presented theory it can be proposed that anomalously large nano-scale heat transfer between metals can be explained by the potential difference. In addition, based on the data obtained, it can be assumed that the fluctuating dipole moment associated with the surface double layer of the metal can also make a large contribution to heat transfer. Further research is needed for a detailed comparison of theory with experiment. In the extreme near field the electron tunnelling can also contribute to the heat transfer. However, this contribution is unlikely (see discussion in Ref. Kittel2017NatCommun ; Henkel2019JOSA ). According to the experimental dada Kittel2017NatCommun ; Reddy2017NatCommun for the distances above 1nm the tunnelling current is negligible. The tunnelling current grows exponentially for the distances below 0.5nm, while the heat flux has the power dependence in this distance range, thus the electron tunnelling can be excluded from the heat transfer mechanism even in the extreme near field. The obtained results can be used to control heat fluxes at the nanoscale using the potential difference.

Appendix A van der Waals interaction between surfaces

The van der Waals interaction energy between two semi-infinite media 1 and 2, whose surfaces have small displacements $u_{i}(\mathbf{x})$ with respect to the planes at $z=0$ and $z=d$, can be written in the form

[TABLE]

where $H$ is the Hamaker constant. Expanding $W$ in a series in powers of $u_{i}$ to terms that are quadratic in $u_{i}$ inclusive, we get

[TABLE]

[TABLE]

The stresses that act on the surfaces 1 and 2 when they are displaced are determined by the equations

[TABLE]

[TABLE]

where displacement-independent terms, which do not contribute to heat transfer, were omitted. Using representation of displacements as a Fourier integral

[TABLE]

we get Eqs. (6) and (7), in which Pendry2016PRB ; Pendry2017Z.Nat

[TABLE]

[TABLE]

where $K_{2}(x)$ is the modified Bessel function of the second kind and second order. In deriving (37), it was used the value of the integral Handbook

[TABLE]

where

[TABLE]

Appendix B Electrostatic interaction between surfaces

For metal surfaces, the potential difference $\varphi$ between them induces a surface charge density

[TABLE]

The interaction of closely spaced charged surfaces will be determined in the electrostatic limit, following the approach from Ref. Pendry2016PRB . We assume that surface 1 at $z=0$ is held at zero potential, and surface 2 at $z=d$ has a potential equal to $\varphi$. In the absence of displacements of the surfaces, the potential in the vacuum gap has the form

[TABLE]

Displacements of the surfaces

[TABLE]

will lead to a change of the electrostatic field in the vacuum gap at $0<z<d$, which can be described by potential

[TABLE]

In the electrostatic limit, the potential of metal surfaces should remain unchanged when the surfaces are displaced. From where follow the boundary conditions

[TABLE]

[TABLE]

where $E_{0}=\varphi/d=4\pi\sigma_{0}$. Eqs. (44) and (45) can be also obtained from the boundary condition (4). From (44) and (45) we get

[TABLE]

[TABLE]

The normal component of the electric field at $0<z<d$ has the form

[TABLE]

Using the Maxwell stress tensor, one can find the mechanical stresses that act on the surfaces 1 and 2 at their displacements

[TABLE]

[TABLE]

From Eqs. (49) and (50) follow Eqs. (6) and (7), where

[TABLE]

[TABLE]

From (51), (52), (36) and (37) follow, that $a\approx b$ for $qd\ll 1$. In this case, the spring model becomes valid, which was considered in Refs.Persson2011JPCM ; Joulain2014PRB ; Budaev2011APL . However, quantitatively the spring model is valid only for unrealistic short distances Pendry2016PRB .

The reported study was funded by RFBR according to the research project No 19-02-00453

∗alevolokitin@yandex.ru