Radiative Heat Transfer

Abstract

In this section, we consider the basic principles of radiative heat transfer. The general theory of the fluctuating electromagnetic field is applied for the calculation of the radiative heat transfer in the plate–plate and particle–plate configurations using the Green’s function and scattering matrix approaches. The Green’s function approach is used to calculate the radiative heat transfer between anisotropic materials and the scattering matrix approach is used to calculate the radiative heat transfer between plates in relative motion.

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this section, we consider the basic principles of radiative heat transfer. The general theory of the fluctuating electromagnetic field is applied for the calculation of the radiative heat transfer in the plate–plate and particle–plate configurations using the Green’s function and scattering matrix approaches. The Green’s function approach is used to calculate the radiative heat transfer between anisotropic materials, and the scattering matrix approach is used to calculate the radiative heat transfer between plates in relative motion. By considering one of the plates as sufficiently rarefied, we calculate the radiative heat transfer between a small neutral particle moving parallel to the plate and the plate in the different reference frames. Different limiting cases are considered. We consider the dependence of the heat transfer on the temperature $T\!$, the shape and the separation d, and discuss the role of non-local and retardation effects. We find that, for high-resistivity material, the heat transfer is dominated by retardation effects even for very short separations. The heat transfer at short separation between the plates may increase by many orders of magnitude when the surfaces are covered by adsorbates, or can support low-frequency surface plasmon polaritons (SPP) or surface phonon polaritons (SPhP). In this case, the heat transfer is determined by resonant photon tunneling between adsorbate vibrational modes, or SPP or SPhP modes. Using the nonlocal optic dielectric approach, we study the dependence of the heat flux between two metal surfaces on the electron concentration, and compare this with the predictions of the local optic approximation.

6.1 The Green’s Function Theory

The theory of electromagnetic fluctuations presented in Chap. 4 can be used to calculate the heat transfer between any two macroscopic bodies with different temperatures, $T_1$ and $T_2$, whose surfaces are separated by a distance d, much larger than the lattice constant of the solids. In this case, the problem can be treated macroscopically.

In order to calculate the radiative energy transfer between the bodies, we need the ensemble average of the Poynting’s vector

$$ \left\langle \mathbf {S}(\mathbf {r})\right\rangle _\omega =\left( c/{8\pi }\right) \left\langle \mathbf {E}(\mathbf {r})\times \mathbf {B}^{*}(\mathbf {r} )\right\rangle _\omega +c.c.= $$

$$\begin{aligned} =\frac{ic^2}{8\pi \omega }\left\{ \mathbf {\nabla }^{\prime }\left\langle \mathbf {E }(\mathbf {r})\cdot \mathbf {E}^{*}(\mathbf {r}^{\prime })\big \rangle -\left\langle ( \mathbf {E}( \mathbf {r})\cdot \mathbf {\nabla }^{\prime }\right) \mathbf {E}^{*}(\mathbf {r} ^{\prime })\right\rangle -c.c\right\} _{\mathbf {r}=\mathbf {r}^{\prime }}. \end{aligned}$$

(6.1)

Thus, the Poynting’s vector can be expressed through the average of the product of the components of the electric field. According to the theory of the fluctuating electromagnetic field, the spectral function of fluctuations of the electric field in the vacuum gap between the bodies 1 and 2 is given by (3.63) and (3.64) [13] (see also Appendix A)

$$ \left\langle \mathbf {E}(\mathbf {r})\mathbf {E}^*(\mathbf {r}^{\prime }\right\rangle _\omega = \frac{\big [\Pi _1(\omega )-\Pi _2(\omega )\big ]}{8\pi ^2\mathrm {i}k ^2\omega }\int \mathrm {d}\mathbf {S}_1^{{\prime \prime }}\Big \{ \hat{\mathbf {D}}(\mathbf {r},\mathbf {r} ^{\prime \prime })\mathbf {\nabla }^{\prime \prime } \hat{\mathbf {D}}^+(\mathbf {r}^{\prime },\mathbf {r} ^{\prime \prime })- $$

$$\begin{aligned} -\mathbf {\nabla }^{\prime \prime }\hat{\mathbf {D}}(\mathbf {r},\mathbf {r} ^{\prime \prime }) \hat{\mathbf {D}}^+(\mathbf {r}^{\prime },\mathbf {r} ^{\prime \prime })\Big \} +\frac{ \Pi _2(\omega )}{\pi \omega }\mathrm {Im}\hat{\mathbf {D}}(\mathbf {r},\mathbf {r} ^{\prime }), \end{aligned}$$

(6.2)

where the integration is over the surface of the body 1,

$$\begin{aligned} \Pi _i(\omega )=\hbar \omega \left( e^{\hbar \omega /k_BT_i}-1\right) ^{-1}. \end{aligned}$$

(6.3)

The Green’s function matrix of the electromagnetic field $\hat{\mathbf {D}}(\mathbf {r},\mathbf {r} ^{\prime })$ in the space between the bodies can be found by solving (3.33) and (3.34) with the appropriate boundary conditions [13, 183] (see also Appendix C). The second term in (6.1) does not give any contribution to the heat flux between two parallel surfaces because

$$\begin{aligned} \left\langle \left( \mathbf {E}(\mathbf {r})\cdot \mathbf {\nabla }^{\prime }\right) E_z^{*}(\mathbf {r} ^{\prime })\right\rangle =\left\langle \mathbf {\nabla } \cdot \mathbf {E}( \mathbf {r})E_z^{*}(\mathbf {r})\right\rangle =\nabla _z\left\langle E_z( \mathbf {r})E_z^{*}(\mathbf {r})\right\rangle . \end{aligned}$$

(6.4)

This term is purely real and does not give any contribution to the z-component of Poynting vector, which in this case is given by

$$\begin{aligned} \left\langle \mathbf {S}_{z}(\mathbf {r})\right\rangle _\omega =\left( c/8\pi \right) \left\langle \mathbf {E}(\mathbf {r})\times \mathbf {B}^{*}(\mathbf {r} )\right\rangle _\omega +c.c.=\frac{ic^2}{8\pi \omega }\left\{ \langle \mathbf {E }(\mathbf {r})\cdot \frac{d}{dz}\mathbf {E}^{*}(\mathbf {r})\rangle _{\omega } - c.c.\right\} . \end{aligned}$$

(6.5)

Using (6.2), in (6.5) we get

$$\begin{aligned} \left\langle S_z\right\rangle _\omega= & {} \frac{\big [ \Pi _1(\omega )-\Pi _2(\omega )\big ] }{64\pi ^3k ^4}\int \frac{\mathrm {d}^2\mathbf {q}}{ (2\pi )^2}\mathrm {Tr}\Bigg [ \hat{\mathbf {D}}\frac{\partial ^2}{\partial z\partial z^{\prime }}\hat{\mathbf {D}}^{+}(z, z^{\prime })- \nonumber \\&-\frac{\partial }{\partial z^{\prime }}\hat{\mathbf {D}}\frac{\partial }{\partial z}\hat{\mathbf {D}}^{+}(z, z^{\prime }) +c.c.\Bigg ]_{z\rightarrow z^{\prime }} \end{aligned}$$

(6.6)

For the plane geometry, the solution of (3.33) and (3.34) is most conveniently obtained by representing the Green’s function as Fourier integrals with respect to the transverse coordinates x, y (the z-axis being normal to the surfaces); this gives a system of linear inhomogeneous ordinary differential equations from which the Green’s functions can be obtained as functions of z [13, 183]. The solution of these equations is described in details in Appendix C. The Green’s function matrix for $z>z^{\prime }$ can be written in the form [195, 196]:

$$\begin{aligned} \hat{\mathbf {D}} =\frac{2\pi i\omega ^2}{k_z c^2}\Bigg [\hat{\mathbf {D}}_{12}\Bigg (\hat{\mathbf {I}}e^{\mathrm { i}k_z ( z-z^{\prime }) }+ \hat{\mathbf {R}}_1e^{\mathrm {i}k_z (z+z^{\prime })}\Bigg ) +\hat{\mathbf {D}}_{21}\Bigg ( \hat{\mathbf {R}}_2\hat{\mathbf {R}}_1e^{2\mathrm {i}k_z d} e^{\mathrm {i}k_z (z^{\prime }-z)} +\hat{\mathbf {R}}_2e^{2\mathrm {i} k_z d} e^{-\mathrm {i}k_z (z+z^{\prime })}\Bigg )\Bigg ], \nonumber \\ \end{aligned}$$

(6.7)

where

$$\begin{aligned}&\hat{\mathbf {R}}_1 = \hat{n}^+R_1\hat{n}^-,\,\hat{\mathbf {R}}_2 = \hat{n}^-R_2 \hat{n}^+,\,\hat{\mathbf {I}}=\hat{n}^+\mathrm {I}\hat{n}^+,\,\hat{\mathbf {D}}_{12}=\hat{n}^+D_{12}\hat{n}^+,\\&\hat{\mathbf {D}}_{21}=\hat{n}^-D_{21}\hat{n}^-,\, D_{ij}=\left[ \mathrm {I}-e^{2ik_z d} R_iR_j\right] ^{-1}, \end{aligned}$$

where $\mathrm {I}$ is the $2\times 2$ unit matrix, the $3\times 2$ matrix $\hat{n}^{\pm }=(\hat{n}_s^{\pm }, \hat{n}_p^{\pm })$, $k_z=((\omega /c)^2)-q^2)^{1/2},\,\lambda =(s, p),\,\,\hat{n}^{\pm }_s=[\hat{z}\times \hat{q}] = (-q_y, q_x, 0)/q,\,\hat{n}^{\pm }_p=[\hat{k}^{\pm }\times \hat{n}^{\pm }_s]=(\mp \mathbf {q}k_z, q^2/(kq),\, k=\omega /c ,\,\hat{k}^{\pm }=(\mathbf {q}\pm \hat{z}k_z)/k,\,\mathbf {q}=(q_x, q_y, 0)$. The $2\times 2$ reflection matrix R determines the reflection amplitudes for the waves with different polarization $\lambda =(s, p)$. This matrix is diagonal for isotropic materials. However, in the general cases of anisotropic materials, this matrix is not diagonal

$$ R_i=\left( \begin{array}{cc} R_{ss}^i &{} R_{sp}^i \\ R_{ps}^i&{} R_{ss}^i \end{array} \right) . $$

For the isotropic materials $R_{\lambda \lambda ^{\prime }} = R_{\lambda }\delta _{\lambda \lambda ^{\prime }}$ and the Green’s function matrix is simplified

$$ \hat{\mathbf {D}}^{isotropic} =\frac{2\pi i\omega ^2}{k_z c^2}\sum _{\lambda = (s, p)}\Bigg [\hat{\mathbf {n}}^+_{\lambda }\hat{\mathbf {n}}^+_{\lambda }e^{\mathrm { i}k_z ( z-z^{\prime }) }+ \hat{\mathbf {n}}^+_{\lambda }\hat{\mathbf {n}}^-_{\lambda }R_{1\lambda }e^{\mathrm {i}k_z (z+z^{\prime })} + \hat{\mathbf {n}}^-_{\lambda }\hat{\mathbf {n}}^-_{\lambda }R_{1\lambda }R_{2\lambda }e^{2\mathrm {i}k_z d} e^{\mathrm {i}k_z (z^{\prime }-z)} $$

$$\begin{aligned} +\hat{\mathbf {n}}^-_{\lambda }\hat{\mathbf {n}}^+_{\lambda }R_{2\lambda }e^{2\mathrm {i} k_z d} e^{-\mathrm {i}k_z (z+z^{\prime })}\Big ]\frac{1}{1-e^{2ik_zd}R_{1\lambda }R_{2\lambda }}, \end{aligned}$$

(6.8)

Substitution of (6.7) in (6.6) gives [195, 196]:

$$\begin{aligned} S_z= & {} \int _0^\infty \frac{\mathrm {d}\omega }{2\pi }\Big [\Pi _1(\omega )-\Pi _2(\omega )\Big ] \Big \{ \int _{q< \frac{\omega }{c}}\frac{\mathrm {d}^2q}{(2\pi )^2} \mathrm {Tr}\left[ \left( \mathrm {I}- R_{2}^+R_2\right) D_{12}\left( \mathrm {I}- R_{1}R_{1}^+\right) D_{12}^+\right] \nonumber \\&+\int _{q>\frac{\omega }{c}}\frac{\mathrm {d}^2q}{(2\pi )^2}\mathrm {e}^{-2\mid k_z \mid d} \mathrm {Tr}\left[ \left( R_{2}^+-R_2\right) D_{12}\left( R_{1}-R_1^+\right) D_{12}^+\right] \Big \}. \end{aligned}$$

(6.9)

6.2 The Scattering Matrix Theory

We introduce two coordinate systems, K and $K^{\prime }$ with coordinate axes xyz and $x^{\prime }y^{\prime }z^{\prime }$. In the K system, body 1 is at rest while body 2 is moving with the velocity V, along the x-axis (the xy and $x^{\prime }y^{\prime }$ planes are in the surface of body 1, x and $x^{\prime }$-axes have the same direction, and the z and $z^{\prime }$-axes point toward body 2). In the $K^{\prime }$ system, body 2 is at rest while body 1 is moving with velocity $-V$ along the x-axis. Since the system is translational invariant in the $\mathbf {x}=(x, y)$ plane, the electromagnetic field can be represented by the Fourier integrals (see (5.22) and (5.23)). In the vacuum region, the electromagnetic field is determined by (5.22). Decomposing the electromagnetic field into s- and p-electromagnetic waves and substituting (5.22) in (6.5), we obtain the heat flux, transferred through the surface 1:

$$\begin{aligned} S_1 =\frac{1}{4\pi }\int _0^{\infty }d\omega \int \frac{d^2q}{(2\pi )^2}\frac{\omega }{k^2} \left[ (k_z+k_z^{*})\left( \left\langle \mid w_p\mid ^2\right\rangle \right. +\left\langle \mid w_s\mid ^2\right\rangle -\right. \nonumber \\ \left. -\left\langle \mid v_p\mid ^2\right\rangle - \left. \left\langle \mid v_s\mid ^2\right\rangle \right) +(k_z-k_z^{*})\left\langle (w_pv_p^* + w_sv_s^* - c.c\right\rangle \right] . \end{aligned}$$

(6.10)

In (6.10), the integration is performed only over positive values of $\omega$, which introduce an additional factor of 2. After averaging in (6.10) over the fluctuating electromagnetic field with the use of (2.15)–(2.19) from Appendix B, we obtain the heat flux through surface 1, separated from the surface 2 by vacuum gap (thickness d) [128]:

$$ S_1 =\frac{\hbar }{8\pi ^3}\int _0^\infty d\omega \int _{q< \frac{\omega }{c}}d^2q\frac{\omega }{|\Delta |^2}\Big [\left( q^2 - \beta kq_x \right) ^2 + \beta ^2k_z^2q_y^2\Big ]\times $$

$$ \times \, \Big [\left( q^2 - \beta kq_x \right) ^2 \left( 1-\mid R_{1p}\mid ^2 \right) \left( 1-\mid R_{2p}^{\prime }\mid ^2 \right) |D_{ss}|^2+ $$

$$ +\,\beta ^2k_z^2q_y^2 \left( 1-\mid R_{1p}\mid ^2 \right) \left( 1-\mid R_{2s}^{\prime }\mid ^2 \right) |D_{sp}|^2 + \big (p\leftrightarrow s \big )\Big ] \left( n_2(\omega ^{\prime })-n_1(\omega )\right) + $$

$$ +\,\frac{\hbar }{2\pi ^3}\int _0^\infty d\omega \int _{q> \frac{\omega }{c}}d^2q\frac{\omega }{|\Delta |^2}\Big [\left( q^2 - \beta kq_x \right) ^2 + \beta ^2k_z^2q_y^2 \Big ] e^{-2\mid k_z\mid d}\times $$

$$ \times \,\Big [\left( q^2 - \beta kq_x \right) ^2 \mathrm {Im}R_{1p}\mathrm {Im}R_{2p}^{\prime }|D_{ss}|^2- \beta ^2k_z^2q_y^2 \mathrm {Im}R_{1p}\mathrm {Im} R_{2s}^{\prime }|D_{sp}|^2+ $$

$$\begin{aligned} + \big (p\leftrightarrow s \big )\Big ]\left( n_2(\omega ^{\prime })-n_1(\omega )\right) . \end{aligned}$$

(6.11)

The quantities entering in (6.11) have the same meaning as in (5.30). There is also a heat flux $S_2$ from body $\mathbf 2$ in the $K^{\prime }$-reference frame. Actually, $S_1$ and $S_2$ are the same quantities, looked at from different coordinate systems. These quantities are related by the equation:

$$\begin{aligned} F_xV = S_1 + {S_2}/{\gamma }, \end{aligned}$$

(6.12)

where $F_x$ is the friction force, due to the relative motion of the two bodies. Equation (6.12) has a simple meaning: the power of friction force is equal to heat flux through the surfaces of both bodies. At $V=0$, (6.11) is reduced to the formula obtained in [13]

$$\begin{aligned} S_z= & {} \int _0^\infty \frac{\mathrm {d}\omega }{2\pi }\big [\Pi _1(\omega )-\Pi _2(\omega )\big ] \Big \{ \int _{q<\frac{\omega }{c}}\frac{\mathrm {d}^2q}{(2\pi )^2}\times \nonumber \\&\times \, \frac{\left( 1-\mid R_{1p}(\mathbf {q},\omega )\mid ^2 \right) \left( 1-\mid R_{2p}(\mathbf {q},\omega )\mid ^2 \right) }{\mid 1-\mathrm {e}^{2\mathrm {i}k_z d}R_{1p}(\mathbf {q},\omega )R_{2p}(\mathbf {q}\omega )\mid ^2}+ \nonumber \\&+\, 4\int _{q>\frac{\omega }{c}}\frac{\mathrm {d}^2q}{(2\pi )^2}\mathrm {e}^{-2\mid k_z \mid d}\times \nonumber \\&\times \, \frac{\mathrm {Im}R_{1p}(\mathbf {q},\omega )\mathrm {Im}R_{2p}( \mathbf {q},\omega )}{\mid 1-\mathrm {e}^{-2\mid k_z\mid d}R_{1p}(\mathbf {q} ,\omega )R_{2p}(\mathbf {q},\omega )\mid ^2} +\nonumber \\&\left. +\,\left[ p\rightarrow s\right] \Big \}, \right. \end{aligned}$$

(6.13)

where the symbol $\left[ p\rightarrow s\right]$ stands for the terms that are obtained from the first two terms by replacing the reflection amplitude $R_p$ for the p-polarized electromagnetic waves with the reflection amplitude $R_s$ for the s-polarized electromagnetic waves, and where $k_z =((\omega /c)^2-q^2)^{1/2}$. The contributions to the heat transfer from the propagating ($q<\omega /c$) and evanescent ($q>\omega /c$) electromagnetic waves are determined by the first and the second terms in (6.13), respectively. Because of the presence of the exponential factor in the integrals in (6.13), the q-integration is effectively limited by to $q<\lambda _T^{-1}$ for the propagating waves, and $q<d^{-1}$ for the evanescent waves. Thus from phase space arguments, it follows that the number of the available channels for the heat transfer for the evanescent waves will be by a factor of $(\lambda _T/d)^2$ larger than the number of the available channels for the propagating waves. For $d=1$ nm and $T=300$ K, this ratio is of the order of ${\sim }{10^{8}}$.

In the local optic approximation, and in the non-retarded limit, the formula (6.13) reduces to the results first obtained in [93] and [115], respectively.

Equation (6.13) can be understood qualitatively as follows. The heat flux per unit frequency is (thermal energy) $\times$ (transmission coefficient). The transmission coefficient can be written in the form

$$ \mid D\mid ^2 \quad =\quad \mid D_1D_2e^{ik_z d}\big (1+R_1R_2e^{2ik_z d}+ \left( R_1R_2e^{2ik_z d}\right) ^2 + \ldots \big )\mid ^2= $$

$$\begin{aligned} = \frac{\mid D_1\mid ^2\mid \mid D_2\mid ^2e^{-2\mathrm {Im}(k_z) d}}{\mid 1-R_1R_2e^{2ik_z d}\mid ^2}, \end{aligned}$$

(6.14)

where $D_1$ and $D_2$ are the transmission amplitude for the surfaces 1 and 2, respectively. The transmission amplitude D, for two interfaces with reflection amplitude $R_1$ and $R_2$ is obtained by geometrical progression of the subsequent reflections of the electromagnetic waves between two surfaces, while keeping proper track of the accumulated phase. Finally, one integrates over $\omega$ and the allowed q, and subtracts the opposite fluxes.

6.3 General Formulas and Limiting Cases

Let us first consider some general consequences of (6.13). In the case of heat transfer through the propagating photons ($q\le \omega /c$), the heat transfer is maximal for black bodies with zero reflection amplitude, $R=R_r+iR_i=0$. Now, what is the photon-tunneling equivalent of a black body? For $q>\omega /c$ there are no constraints on the reflection amplitude, $R(q,\omega )$, other than that $\mathrm {Im}R(q,\omega )$ is positive, and $R_r$ and $R_i$ are connected by the Kramers–Kronig relation. Therefore, assuming identical surfaces, we are free to maximize the transmission coefficient corresponding to the photon tunneling

$$\begin{aligned} T =|D|^2=\frac{R_i^2\mathrm {e}^{-2kd}}{\left| 1-\mathrm {e}^{-2kd}R^2\right| ^2} \end{aligned}$$

(6.15)

(where $k=|k_z|$) with respect to $R_i$ (or $R_r$). This function is maximal when [115]:

$$\begin{aligned} R_r^2+R_i^2=\mathrm {e}^{2kd}, \end{aligned}$$

(6.16)

so that $T=1/4$. Substituting this result in (6.13) gives the maximal contribution from the evanescent waves:

$$\begin{aligned} (S_z)_{max}^{evan}=\frac{k_B^2T^2q_c^2}{24\hbar }, \end{aligned}$$

(6.17)

where $q_c$ is a cut-off in q, determined by the properties of the material. It is clear that the largest possible $q_c\sim 1/b$, where b is an inter-atomic distance. Thus, from (1.2) and (6.17) we get the ratio of the maximal heat flux connected with evanescent waves to the heat flux due to black body radiation $(S_z)_{max}/S_{BB}\approx 0.25(\lambda _T/b)^2$, where $\lambda _T=c\hbar /k_BT$. At room temperature, the contribution to the heat flux from the evanescent waves will be approximately eight orders of magnitude larger than the contribution from black body radiation, and the upper boundary for the radiative heat transfer at room temperature $(S_z)_{max}\sim 10^{11}$ Wm$^{-2}$. The result that there is a maximum heat flow in a given channel links with more profound ideas of entropy flow. It was shown [96] from very general arguments that the maximum of the information flow in a single channel is linked to the flow of energy. Briefly, the arguments were that the flow of information in a channel is limited by

$$\begin{aligned} \dot{E}\ge \frac{3\hbar \mathrm {ln}^22}{\pi }\dot{I}^2, \end{aligned}$$

(6.18)

where $\dot{E}$ is the energy flow and $\dot{I}$ the information flow. Identifying the energy flow with heat flow, S,

$$\begin{aligned} \dot{E}=S \qquad \dot{I}=\frac{S}{k_BT\mathrm {ln}2} \end{aligned}$$

(6.19)

we have

$$\begin{aligned} S\le \frac{\pi k_B^2T^2}{3\hbar }, \end{aligned}$$

(6.20)

hence,

$$\begin{aligned} S\le S_{\mathrm {max}}=2\sum _{\mathbf {q}}\frac{\pi k_B^2T^2}{3\hbar }, \end{aligned}$$

(6.21)

where factor 2 takes into account two possible polarizations. Therefore, the maximum in the heat flux per channel is interpreted as conducting the maximum allowed amount of entropy per channel.

Let us apply the general theory to concrete materials. For the local optic case, the reflection amplitudes are determined by the Fresnel formulas (3.17). For metals, the dielectric function can be written in the form

$$\begin{aligned} \varepsilon =1+4\pi i\sigma /\omega , \end{aligned}$$

(6.22)

where $\sigma$ is the conductivity that can be considered as constant in the mid- and far-infrared region. For good conductors, when $k_BT/4\pi \sigma \ll 1$ and $\lambda _T|\varepsilon (\omega _T)|^{-3/2}<d<\lambda _T|\varepsilon (\omega _T)|^{1/2}$, where $\omega _T=c/\lambda _T=k_BT/\hbar$, the contribution to the heat transfer from p-polarized waves is determined by

$$\begin{aligned} S_p\approx 0.2\frac{(k_BT)^2}{\hbar \lambda _Td}\left( \frac{k_BT}{4\pi \hbar \sigma }\right) ^{1/2}, \end{aligned}$$

(6.23)

while the s-wave contribution for $d<\lambda _T|\varepsilon (\omega _T)|^{-1/2}$ is distance independent:

$$\begin{aligned} S_s\approx 0.02\frac{4\pi \sigma k_BT}{\lambda _T^2}. \end{aligned}$$

(6.24)

For good conductors, the heat flux associated with p-polarized electromagnetic waves decreases with the separation as ${\sim }{d^{-1}}$, and increases with decreasing conductivity as $\sigma ^{-1/2}$. When $k_BT/4\pi \hbar \sigma \ge 1$ the heat flux decreases with separation as $d^{-2}$. Figure 6.1a shows the heat transfer between two semi-infinite silver bodies separated by the distance d, and at the temperatures $T_1=273$ K and $T_2=0$ K. The s- and p-wave contributions are shown separately, and the p-wave contribution has been calculated using non-local optics, i.e. spatial dispersion of the dielectric function was taken into account (the dashed line shows the result using local optics). It is remarkable how important the s-contribution is even for short distances. The nonlocal optics contribution to $(S_z)_p$, which is important only for $d<l$ (where l is the electron mean free path in the bulk), is easy to calculate for free electron like metals. The non-local surface contribution to $\mathrm {Im}R_p$ is given by [117]

Fig. 6.1

a The heat transfer flux between two semi-infinite silver bodies, one at temperature $T_1=273\ \mathrm {K}$ and another at $T_2=0\ \mathrm {K}$, as a function of the separation d. b The same as (a) except that we have reduced the Drude electron relaxation time $\tau$ for solid 1 from a value corresponding to a mean free path $v_F\tau =l=560$ to 20 Å. c The same as (a) except that we have reduced l to 3.4 Å. The dashed lines correspond to the results obtained within local optic approximation. (The base of the logarithm is 10.)

$$\begin{aligned} \left( \mathrm {Im}R_p\right) _{\mathrm {surf}}=2\xi {\frac{\omega }{\omega _p}} {\frac{q}{k_F}}, \end{aligned}$$

(6.25)

where $\xi (q)$ depends on the electron-density parameter $r_s$ but typically $\xi (0)\sim 1$. Using this expression for $\mathrm {Im}R_p$ in (6.13) gives the (surface) contribution:

$$\begin{aligned} S_{\mathrm {surf}}\approx \frac{\xi ^2k_B^4}{\omega ^2k_F^2d^4\hbar ^3} \left( T_1^4-T_2^4 \right) . \end{aligned}$$

(6.26)

Note from Fig. 6.1a that the local optic contribution to $(S_z)_p$ depends nearly linearly on 1 / d in the studied distance interval, and that this contribution is much smaller than the s-wave contribution. Both of these observations are in agreement with the analytical formulas presented above. However, for very high-resistivity materials, the p-wave contribution becomes much more important, and a crossover to a $1/d^2$-dependence of $(S_z)_p$ is observed at very short separations d. This is illustrated in Fig. 6.1b and c, which have been calculated with the same parameters as in Fig. 6.1a, except that the electron mean free path has been reduced from $l=560$ Å (the electron mean free path for silver at room temperature) to 20 Å (approximately the electron mean free path in lead at room temperature) (Fig. 6.1b) and 3.4 Å (of the order of the lattice constant, representing the minimal possible mean free path) (Fig. 6.1c). Note that when l decreases, the p-wave contribution to the heat transfer increases while the s-wave contribution decreases. Since the mean free path cannot be much smaller than the lattice constant, the result in Fig. 6.1c represents the largest possible p-wave contribution for normal metals. However, the p-wave contribution may be even larger for other materials, such as semimetals, with lower carrier concentration than in normal metals. For high resistivity material, when $k_BT/4\pi \hbar \sigma >1$ the heat flux is proportional to the conductivity:

$$\begin{aligned} S_p\approx 0.2\frac{k_BT\sigma }{d^2}. \end{aligned}$$

(6.27)

By tuning the resistivity of the material we can optimize the photon transmission coefficient across the vacuum gap and hence the potential for heat transport by tunneling. The transmission coefficient $|T(\omega , q)|^2$ is proportional to the energy density of the electromagnetic field associated with evanescent waves:

$$\begin{aligned} T(\omega , q)\sim \mathrm {Im}R_p(\omega , q)e^{-qd}. \end{aligned}$$

(6.28)

For $q \gg |\varepsilon (\omega )|^{1/2}\omega /c$

$$\begin{aligned} \mathrm {Im}R_p(\omega , q)\approx \mathrm {Im}\frac{\varepsilon -1}{ \varepsilon +1}\approx \frac{8\pi \sigma /\omega }{4+(4\pi \sigma /\omega )^2 }. \end{aligned}$$

(6.29)

Assuming that the conductivity, $\sigma$, is independent of $\omega$ and q the energy density is maximal when

$$\begin{aligned} \sigma _{\max }=\frac{\omega }{2\pi }\approx \frac{k_BT}{2\pi \hbar }=2.3T(\Omega \cdot \mathrm {m})^{-1}, \end{aligned}$$

(6.30)

where we have replaced $\hbar \omega$ with the typical thermal energy $k_BT$. At room temperature the optimum electrical conductivity is 690 ($\Omega \cdot$m$)^{-1}$.

Fig. 6.2

The thermal flux as a function of the conductivity of the solids. The surfaces are separated by $d=10$ Å. The heat flux for other separations can be obtained using scaling ${\sim }{1/d^{2}}$, which holds for high-resistivity materials. (The base of the logarithm is 10.)

To illustrate this case, Fig. 6.2 shows the thermal flux as a function of the conductivity of the solids. Again, we assume that one body is at zero temperature and the other at $T=273$ K. The solid surfaces are separated by $d=10$ Å. The heat flux for other separations can be obtained using scaling ${\sim }{1/d^{2}}$, which holds for high-resistivity materials. The heat flux is maximal when $\sigma \approx 920$ ($\Omega$ m)$^{-1}$.

For good conductors, the most important contribution to the radiative heat transfer comes from the non-local optic effects in the surface region. However, it was shown above that the radiative heat transfer becomes much larger for high-resistivity material, for which the volume contribution from non-local effects is also important. Non-local optic refers to the fact that the current at point $\mathbf {r}$ depends on the electric field not only at point $\mathbf {r}$, as it is assumed in the local optic approximation, but also at points $\mathbf {r}^{\prime }\ne \mathbf {r}$ in a finite region around the point $\mathbf {r}$. In the case when both points are located outside of the surface region, the dielectric response function can be expressed through the dielectric function appropriate for the semi-infinite electron gas. However, if one of the points $\mathbf {r}$ or $\mathbf {r}^{\prime }$ is located in the surface region, the dielectric response function will be different from its volume value, and this gives the surface contribution from non-locality. In order to verify the accuracy of the local optic approximation, we study the dependence of the radiative heat transfer on the dielectric properties of the materials within the non-local dielectric approach, which was proposed some years ago for the investigation of the anomalous skin effects [198] (see Appendix D).

Figure 6.3 shows the thermal flux between two clean metal surfaces as a function of the electron density n. In the calculations, we have assumed that one body is at zero temperature and the other at $T=273$ K, and the Drude relaxation time $\tau =4 \times 10^{-14}$ s. When the electron density decreases, there is transition from a degenerate electron gas ($k_BT\ll \varepsilon _F$, where $\varepsilon _F$ is the Fermi energy) to a non-degenerate electron gas ($k_BT \gg \varepsilon _F$) at the density $n_F\sim (K_BTm)^{3/2}/\pi ^2\hbar ^3$, where m is the electron mass. At $T=273$ K, the transition density $n_F \sim$ 10$^{25}$ m$^{-3}$. The full line was obtained by interpolation between the two dashed curves, calculated in the non-local dielectric function formalism for the non-degenerate electron gas (valid for $n<n_F\approx$ $10^{25}$ m$^{-3})$, and for the degenerate electron gas (for $n>n_F)$ [198]. The thermal flux reaches the maximum $S_{\max }\approx 5\times 10^8$ W$\cdot$m$^{-2}$ at $n_{\max }\approx 10^{25}$ m$^{-3}$, which corresponds to the DC conductivity $\sigma \approx 3 \times 10^3$ (${\Omega }{\cdot }$m)$^{-1}$. Within the local optic approximation, the radiative heat transfer is maximal at $n_{L\max }\approx 10^{24}$ m$^{-3}$ where $S_{L\max }\approx 10^9$ W$\cdot$m$^{-2}$. The thermal flux due to traveling electromagnetic waves is determined by formula (1.2) which gives $S_{BB}=308$ W$\cdot$m$^{-2}$ for $\, T=273$ K.

Fig. 6.3

The heat flux between two metal surfaces as a function of the free electron concentration n. One body is at zero temperature and the other at $T=273$ K. The full line was obtained by interpolation between curves (dashed lines) calculated in the non-local dielectric formalism for a non-degenerate electron gas for $n<n_F\sim$ $10^{25}\,\mathrm{m}^{-3}$, and for a degenerate electron gas for $n>n_F$. Also shown are results (dashed lines) obtained within the local optic approximation. The calculations were performed with the damping constant $\tau ^{-1}=2.5 \times 10^{13}\,\mathrm{s}^{-1}$, separation $d=10$ Å and $n_0=8.6 \times 10^{28}\,\mathrm{m}^{-3}$. (The log-function is with basis 10.)

Fig. 6.4

The heat flux between two semi-infinite silver bodies coated with a 10 Å thick layer of a high resistivity ($\rho =0.14\ \mathrm {\Omega cm}$) material. Also shown is the heat flux between two silver bodies, and two high-resistivity bodies. One body is at zero temperature and the other at $T=273$ K. a, b show the p- and s-wave contributions, respectively. (The base of the logarithm is 10.)

Finally, we note that a thin high-resistivity coatings can drastically increase the heat transfer between two solids. Figure 6.4 shows the heat flux when thin films (${\sim }{10}$ Å) of a high resistivity material $\rho =0.14\ \Omega$ cm, are deposited on silver surfaces. One body is at zero temperature and the other at $T=273$ K. (a) and (b) show the p and s-contributions, respectively. Also shown are the heat fluxes when the two bodies are made from silver, and from a high resistivity material. It is interesting to note that while the p-wave contribution to the heat flux for the coated surfaces is strongly influenced by the coating, the s-contribution is nearly unaffected.

6.4 Resonant Photon Tunneling Enhancement of the Radiative Heat Transfer

Another case where the transmission coefficient can be close to unity is connected with resonant photon tunneling between surface states localized on the different surfaces. The resonant condition corresponds to the case when the denominators in (6.13) are small. For two identical surfaces and $R_i\ll 1\le R_r$, where $R_i$ and $R_r$ are the imaginary and real parts of the reflection amplitude, this corresponds to the resonant condition $R_r^2\exp (-2qd)=1$. The resonance condition can be fulfilled even for the case when $\exp (-2qd)\ll 1$, since, for the evanescent electromagnetic waves, there is no restriction on the magnitude of real part or the modulus of R. This opens up the possibility of resonant denominators for $R_r^2 \gg 1$. Close to the resonance we can use the approximation

$$\begin{aligned} R=\frac{\omega _a}{\omega -\omega _0-i\eta }, \end{aligned}$$

(6.31)

where $\omega _a$ is a constant. Then from the resonant condition ($R_r=\pm e^{qd}$) we get the positions of the resonance

$$\begin{aligned} \omega _{\pm }=\omega _0\pm \omega _ae^{-qd}. \end{aligned}$$

(6.32)

For the resonance condition to be valid, the separation $\Delta \omega =|\omega _{+}-\omega _{-}|$ between two resonances in the transmission coefficient must be greater than the width $\eta$ of the resonance. This condition is valid only for $q\le q_c\approx \ln (2\omega _a/\eta )/d$. For $\omega _0>\omega _a$ and $q_cd>1$, we get

$$\begin{aligned} S_{p\pm }=\frac{\eta q_c^2}{8\pi }\big [\Pi _1(\omega _0)-\Pi _2(\omega _0)\big ]. \end{aligned}$$

(6.33)

Note, that the explicit d dependence has dropped out of (6.33). However, S may still be d-dependent, through the d-dependence of $q_c$. For small distances, one can expect that $q_c$ is determined by the dielectric properties of the material, and thus does not depend on d. In this case, the heat transfer will also be distance independent.

Resonant photon tunneling enhancement of the heat transfer is possible for two semiconductor surfaces that can support low-frequency surface plasmon modes in the mid-infrared frequency region. The reflection amplitude $R_p$ for a clean semiconductor surface at $d<\lambda _T\left| \varepsilon (\omega _T)\right| ^{-1/2}$ is given by Fresnel’s formula (see Appendix O.2). As an example, consider two clean surfaces of silicon carbide (SiC). The optical properties of this material can be described using an oscillator model [180]:

$$\begin{aligned} \varepsilon (\omega )=\epsilon _\infty \left( 1+\frac{\omega _L^2-\omega _T^2}{ \omega _T^2-\omega ^2-i\Gamma \omega }\right) , \end{aligned}$$

(6.34)

with $\varepsilon _\infty =6.7,\,\omega _L=1.8 \times 10^{14}$ s$^{-1},\,\omega _T=1.49 \times 10^{14}$ s$^{-1}$, and $\Gamma =8.9 \times 10^{11}$ s$^{-1}$. The frequency of surface plasmons is determined by condition $\varepsilon ^{\prime }(\omega _0)=-1$ and from (6.34) we get $\omega _0=1.78 \times 10^{14}$ s$^{-1}$. The resonance parameters are

Using the above parameters in (6.33), and assuming that one surface is at temperature $T=300$ K and the other at $T=0$ K, we get the heat flux S(d) between two clean surfaces of SiC:

$$\begin{aligned} S\approx \frac{8.4 \times 10^9}{d^2}\text{ W }\cdot \text{ m }^{-2}, \end{aligned}$$

(6.35)

where the distance d is in Å. Note that this heat flux is several orders of magnitude larger than between two clean good conductor surfaces (see Fig. 6.1).

6.5 Adsorbate Vibrational Mode Enhancement of the Radiative Heat Transfer

Another mechanism for resonant photon tunneling enhancement of the heat transfer is possible between adsorbate vibrational modes localized on different surfaces. If the distance between adsorbates $d \gg R$, where R stands for effective radius of the adsorbate, each adsorbate is equivalent to a point dipole. As shown in [101], the dipole approximation is valid for distances larger than a few adsorbate diameters. Let us consider two particles with dipole polarizabilities $\alpha _1(\omega )$ and $\alpha _2(\omega )$ and with the fluctuating dipole moments $p_1^f$ and $p_2^f$ normal to the surfaces. According to the fluctuation-dissipation theorem [8, 184], the spectral function of fluctuations for the dipole moment is given by

$$\begin{aligned} \left\langle p_i^fp_j^f\right\rangle _\omega =\frac{\hbar }{\pi }\left( \frac{1}{2}+n_i(\omega )\right) \mathrm {Im}\alpha _i(\omega )\delta _{ij}. \end{aligned}$$

(6.36)

Assume that the particles are situated opposite to each other on two different surfaces, at the temperatures $T_1$ and $T_2$, respectively, and separated by the distance d. The fluctuating electric field of a particle $\mathbf {1}$ does work on a particle $\mathbf {2}$. The rate of work is determined by

$$\begin{aligned} P_{12}=2\int _0^\infty d\omega \,\omega \mathrm {Im}\alpha _2(\omega )\langle E_{12}E_{12}\rangle _\omega , \end{aligned}$$

(6.37)

where $E_{12}$ is the electric field created by particle $\mathbf {1}$ at the position of particle $\mathbf {2}$:

$$\begin{aligned} E_{12}=\frac{8p_1^f/d^3}{1-\alpha _1\alpha _2{\left( 8/d^3\right) }^2}. \end{aligned}$$

(6.38)

From (6.36)–(6.38) we get $P_{12}$, and the rate of cooling of a particle $\mathbf {2}$ can be obtained using the same formula by reciprocity. Thus, the total heat power exchange between the particles is given by

$$\begin{aligned} P=P_{12}-P_{21}=\frac{2\hbar }{\pi }\int _0^\infty d\omega \,\omega \frac{ \mathrm {Im}\alpha _1\mathrm {Im}\alpha _2\left( 8/d^3 \right) ^2}{\left| 1-\left( 8/d^3 \right) ^2\alpha _1\alpha _2\right| ^2}\big (n_1(\omega )-n_2(\omega )\big ). \end{aligned}$$

(6.39)

Let us first consider some general consequences of (6.39). There are no constraints on the particle polarizability $\alpha (\omega )=\alpha ^{\prime }+i\alpha ^{\prime \prime }$ other than that $\alpha ^{\prime \prime }$ is positive, and $\alpha ^{\prime }$ and $\alpha ^{\prime \prime }$ are connected by the Kramers–Kronig relation. Therefore, assuming identical surfaces, we are free to maximize the photon-tunneling transmission coefficient (for comparison see also (6.15))

$$\begin{aligned} t=\frac{\left( 8\alpha ^{\prime \prime }/d^3 \right) ^2}{\left| 1-\left( 8\alpha /d^3 \right) ^2\right| ^2}. \end{aligned}$$

(6.40)

This function has a maximum when $\alpha ^{\prime 2}+\alpha ^{\prime \prime 2}=\left( d^3/8 \right) ^2$ so that $t=1/4$. Substituting this result in (6.39) gives the upper bound for the heat transfer between two particles:

$$\begin{aligned} P_{max}=\frac{\pi k_B^2}{3\hbar }\left( T_1^2-T_2^2 \right) . \end{aligned}$$

(6.41)

For adsorbed molecules at the concentration $n_a=10^{19}$ m$^{-2}$, when one surface is at zero temperatures and the other is at the room temperature, the maximal heat flux due to the adsorbates $S_{max}=n_aP_{max}=10^{12}$ Wm$^{-2}$, which is nearly 10 orders of magnitude larger than the heat flux due to the black body radiation, $S_{BB}=\sigma _BT=4 \times 10^2$ Wm$^{-2}$, where $\sigma _B$ is the Boltzmann constant.

The conditions for resonant photon tunneling are determined by

$$\begin{aligned} \alpha ^{\prime }(\omega _{\pm })=\pm d^3/8. \end{aligned}$$

(6.42)

Close to resonance, we can use the approximation

$$\begin{aligned} \alpha \approx \frac{C}{\omega -\omega _0-i\eta }, \end{aligned}$$

(6.43)

where $C=e^{*2}/2M\omega _0$, and where $e^{*}$ and M are the dynamical charge and mass of the adsorbate, respectively.

For $\eta \ll 8C/d^3$, from (6.39) we get

$$\begin{aligned} P=\frac{\hbar \eta }{2} \left[ \omega _{+}\big (n_1(\omega _{+})-n_2(\omega _{+})\big )+(+\rightarrow -)\right] , \end{aligned}$$

(6.44)

where $\omega _{\pm }=\omega _0\pm 8C/d^3$. Using (6.44), we can estimate the heat flux between identical surfaces covered by adsorbates with concentration $n_a$: $J\approx n_aP$. For $8C/d^3<\eta$ we can neglect multiple scattering of the photon between the particles, so that the denominator in the integrand in (6.39) is equal to unity. For $d \gg b$, where b is the interparticle spacing, the heat flux between two surfaces covered by adsorbates with concentration $n_{a1}$ and $n_{a2}$ can be obtained after integration of the heat flux between two separated particles. We get

$$\begin{aligned} S=\frac{24n_{a1}n_{a2}}{d^4}\int _0^\infty d\omega \,\mathrm {Im}\alpha _1 \mathrm {Im}\alpha _2 \big [\Pi _1(\omega )-\Pi _2(\omega )\big ]. \end{aligned}$$

(6.45)

Assuming that $\alpha$ can be approximated by (6.43), for $\omega _0\ll \eta$ (6.45) gives the heat flux between two identical surfaces:

$$\begin{aligned} S=\frac{12\pi C^2n_a^2}{d^4\eta }\big [\Pi _1(\omega _0)-\Pi _2(\omega _0) \big ]. \end{aligned}$$

(6.46)

We note that (6.46) can be obtained directly from the heat flux between two semi-infinite solids (determined by (6.13)), since in the limit $d>b$ we can use a macroscopic approach, where all information about optical properties of the surface is included in reflection amplitude.

The reflection amplitude for p-polarized electromagnetic waves which takes into account the contribution from a adsorbate layer, can be obtained using the approach proposed in [99]. Using this approach we get [170, 171] (see also Appendix E):

$$\begin{aligned} R_p=\frac{1-s/(q\varepsilon ) +4\pi n_a[s\alpha _{\parallel }/\varepsilon +q\alpha _{\perp }]-qa\left( 1-4\pi n_aq\alpha _{\parallel }\right) }{1+s/(q\varepsilon ) +4\pi n_a[s\alpha _{\parallel }/\varepsilon -q\alpha _{\perp }]+qa\left( 1+4\pi n_aq\alpha _{\parallel }\right) }, \end{aligned}$$

(6.47)

where $s=\sqrt{q^2-(\omega /c)^2\epsilon }$. The polarizability for ion vibrations (with dynamical charge $e^{*}$) normal and parallel to the surface, is given by

$$\begin{aligned} \alpha _{\perp (\Vert )}=\frac{e^{*2}}{M\big (\omega _{\perp (\Vert )}^2-\omega ^2-i\omega \eta _{\perp (\Vert )}\big )}, \end{aligned}$$

(6.48)

where $\omega _{\perp (\Vert )}$ is the frequency of the normal (parallel) adsorbate vibration, $\eta _{\perp (\Vert )}$ the corresponding damping constant, and M is the adsorbate mass. In comparison with the expression obtained in [199], (6.47) takes into account the fact that the centers of the adsorbates are located at distance a away from the image plane of the metal. Although this gives corrections of the order of $qa\ll 1$ to the reflection amplitude, for parallel adsorbate vibrations on the good conductors ($|\epsilon | \gg 1$), this correction becomes important (see Sect. 7.8.2). As an illustration, in Fig. 6.5 we compare the heat flux between two Cu(100) surfaces covered by low concentration of potassium atoms ($n_a=10^{18}$ m$^{-2}$), with the heat flux between two clean Cu(100) surfaces. At separation $d=1$ nm, the heat flux between the adsorbate-covered surfaces is enhanced by five orders of magnitude in comparison with the heat flux between the clean surfaces, and by seven orders of magnitude in comparison with the black body radiation. However, this enhancement of the heat flux disappears if only one of the surfaces is covered by adsorbates. For $d \gg b$ the numerical data can be approximated by the formula

$$\begin{aligned} S\approx 5.6 \times 10^{-24}\frac{n_a^2}{d^4}\,\text{ W } \text{ m }^{-2}, \end{aligned}$$

(6.49)

where d is in Å.

Fig. 6.5

The heat flux between two surfaces covered by adsorbates and between two clean surfaces, as a function of the separation d. One body is at zero temperature and the other at $T=273$ K. For parameters corresponding to K/Cu(001) and Cu(001) [200] ($\omega _{\perp }=1.9 \times 10^{13}$ s$^{-1}, \omega _{\parallel }= 4.5 \times 10^{12}$ s$^{-1}, \eta _{\parallel }=2.8 \times 10^{10}$ s$^{-1}, \eta _{\perp }=1.6 \times 10^{12}$ s$^{-1}, e^{*}=0.88e$). (The base of the logarithm is 10.)

For $d<b$, the macroscopic approach is not valid and we must sum the heat flux between each pair of adatoms. For $\eta =10^{12}$ s$^{-1}$ and $d<10$ Å, when one surface has $T=300$ K and the other $T=0$ K, from (6.44) we get a distance independent $P\approx 10^{-9}$ W. In this case, for $n_a=10^{18}$ m$^{-2}$ the heat flux $S\approx Pn_a\approx 10^9$ Wm$^{-2}$. Under the same conditions, the s-wave contribution to the heat flux between two clean surfaces is $S_{clean}\approx 10^6$ Wm$^{-2}$. Thus, the photon tunneling between the adsorbate vibrational states can strongly enhance the radiative heat transfer between the surfaces.

It is interesting to note that in the strong coupling case ($8c/d^3 \gg \eta$), the heat flux between two molecules does not depend on the dynamical dipole moments of the molecules (see (6.44)). However, in the opposite case of weak coupling ($8C/d^3\ll \eta$), the heat flux is proportional to the product of the squares of the dynamical dipole moments (see (6.45)).

6.6 Vibrational Heating by Localized Photon Tunneling

The radiative heat transfer due to the evanescent electromagnetic waves (photon tunneling) may be used for surface modification. Thus, if a hot tip is brought ${\sim }{1}$ nm from a surface with a thin layer of heat sensitive polymer, one may induce local polymerization and this may be used for nanoscale lithography. This non-contact mode of surface modification may have several advantages compared with the contact mode: for example, no wear or contamination of the tip will occur.

Let us consider the radiative heat transfer between an adsorbed molecule on a tip and another molecule adsorbed on a substrate. The temperature increase at the adsorbed molecule may be very large, which may induce local chemical reactions, such as diffusion or desorption. Heat transfer to some adsorbate vibrational mode, i.e., vibrational heating, will be particularly important when the energy relaxation time $\tau _\mathrm{b}$ of the adsorbate mode is long compared with a relaxation time characterizing the photon tunneling from the tip to the substrate adsorbate. High-frequency adsorbate vibrations on metals typically have very short energy relaxation times (in the pico-second range) owing to the continuum of low-energy electronic excitations [201, 202]. However, low-frequency adsorbate vibrations, e.g., frustrated translations, may have rather long relaxation times (typically in the order of nanoseconds for inert adsorbates on noble metals) [203], and in these cases photon tunneling heat transfer may be important. Adsorbate vibrational modes on insulators may have very long relaxation times if the resonance frequency is above the top of the bulk phonon band. In these cases, energy relaxation is caused by multi-phonon processes, which are often very slow. One extreme example is CO adsorbed on NaCl crystals [204], where $\tau _\mathrm{b} \approx 10^{-3} \ \mathrm{s}$. For this case, even a very weak coupling to a hot tip may result in heating of the C–O stretch vibration.

The photon tunneling energy transfer per unit time from a vibrational mode of the tip adsorbate (frequency $\omega _\mathrm{a}$ and vibrational relaxation time $\tau ^*_\mathrm{a}=\eta _\mathrm{a}^{-1}$) to a vibrational mode $(\omega _\mathrm{b},\tau ^*_\mathrm{b})$ of the substrate adsorbate (see Fig. 6.6) is given by (6.39). The molecular polarizability is given by (6.48). The energy transfer rate from the tip to the tip adsorbate is given by [205]

Fig. 6.6

The heat transfer (photon tunneling) between a tip atom (or molecule) and a substrate atom (or molecule)

$$\begin{aligned} J_\mathrm{a} = {\hbar \omega _\mathrm{a} \over \tau _\mathrm{a}} \big [n(\omega _\mathrm{a} /T_0)-n(\omega _\mathrm{a} /T_\mathrm{a})\big ], \end{aligned}$$

(6.50)

and the energy transfer rate from the substrate adsorbate to the substrate

$$\begin{aligned} J_\mathrm{b} = {\hbar \omega _\mathrm{b} \over \tau _\mathrm{b}} \big [n(\omega _\mathrm{b} /T_\mathrm{b})-n(\omega _\mathrm{b} /T_1)\big ], \end{aligned}$$

(6.51)

where $\tau _a$ and $\tau _b$ are the vibrational energy relaxation times. Note that, in general $1/\tau _a^* > 1/\tau _a$ and $1/\tau ^*_b > 1/\tau _b$ since the vibrational relaxation rate $1/\tau ^*$ which enters in the polarizability has contributions from both energy relaxation and pure dephasing. In general, the integral in (6.39) must be performed numerically, but as an illustration, let us consider the case where the relaxation time $\tau ^*_\mathrm{b} \gg \tau ^*_\mathrm{a}$. In this case, (6.39) reduces to

$$\begin{aligned} P = r {\hbar \omega _\mathrm{b} \over \tau _\mathrm{b}} \big [n(\omega _\mathrm{b} /T_\mathrm{a})-n(\omega _\mathrm{b} /T_\mathrm{b})\big ], \end{aligned}$$

(6.52)

where

$$\begin{aligned} r= {(\tau _b/\tau ^*_b) s \over 1+\big [2(\omega _\mathrm{a}-\omega _\mathrm{b})\tau ^*_\mathrm{a}\big ]^2 +4s}, \end{aligned}$$

(6.53)

$$\begin{aligned} s= 64 \omega _\mathrm{a} \tau ^*_\mathrm{a} \omega _\mathrm{b} \tau ^*_\mathrm{b}\alpha _\mathrm{va} \alpha _\mathrm{vb}/d^6, \end{aligned}$$

(6.54)

where $\alpha _\mathrm{va}=e^{*2}_\mathrm{a}/M_\mathrm{a}\omega _\mathrm{a}^2$ and similar for mode b.

Note that the energy transfer rate $J_\mathrm{a}$ (and similar for $J_\mathrm{b}$) depends only on the energy relaxation rate $1/\tau _\mathrm{a}$ and not on the relaxation rate $1/\tau _\mathrm{a}^*$ which determines the width of the vibrational resonance state. The latter is the sum of $1/\tau _\mathrm{a}$ and a pure dephasing contribution, which reflects the fluctuation $\omega _\mathrm{a} (t)$ in the vibrational level position due to the irregular thermal motion of the atoms in the system, and which depends on the anharmonic coupling between the different vibrational modes. This level-fluctuation contributes to the vibrational linewidth as observed using, for example, infrared spectroscopy, but not to the energy transfer between the adsorbate and the solid on which it is adsorbed. On the other hand, the energy transfer rate P between the two adsorbates is determined by the overlap in the vibrational resonance states and it is therefore determined by $\tau _\mathrm{a}^*$ (and $\tau _\mathrm{b}^*$), and does not depend on $\tau _\mathrm{a}$ (or $\tau _\mathrm{b}$). (Note: the $\tau _\mathrm{b}^{-1}$ factor which appears in (6.52) cancels out against the factor $\tau _\mathrm{b}$ in the expression for r.)

In the high temperature limit $n(\omega /T) \approx k_\mathrm{B} T / \hbar \omega$, and assuming that this relation holds for all modes and temperatures relevant here, we get

$$\begin{aligned} P = r \tau _\mathrm{b}^{-1} k_\mathrm{B} (T_\mathrm{a}-T_\mathrm{b}). \end{aligned}$$

(6.55)

We also get

$$\begin{aligned} J_\mathrm{a} = \tau _\mathrm{a}^{-1} k_\mathrm{B} (T_0-T_\mathrm{a}), \end{aligned}$$

(6.56)

$$\begin{aligned} J_\mathrm{b} = \tau _\mathrm{b}^{-1} k_\mathrm{B} (T_\mathrm{b}-T_1). \end{aligned}$$

(6.57)

Assuming first a steady state situation so that $P=J_\mathrm{a}=J_\mathrm{b}$ we get from (6.55)–(6.57) $T_\mathrm{a} \approx T_0$ and

$$\begin{aligned} T_\mathrm{b} \approx T_1 + {r \over 1+r} (T_0-T_1), \end{aligned}$$

(6.58)

where we have assumed that $\tau _\mathrm{b} \gg \tau _\mathrm{a}$.

The theory above can also be used to estimate the time it takes to reach the steady state where the (ensemble averaged) adsorbate temperature equals to (6.58). In general we have

$$ \hbar \omega _\mathrm{b} {d \over dt} n(\omega _\mathrm{b}/T_\mathrm{b}(t)) = J(t) - J_\mathrm{b}(t). $$

In the classical limit, this gives

$$ {d T_\mathrm{b} \over dt} = -{1\over \tau _\mathrm{b}} (1+r) T_\mathrm{b}+{1\over \tau _\mathrm{b}} (T_1+rT_0). $$

If we assume $T_\mathrm{b} (0) = T_1$, this gives

$$T_\mathrm{b} (t) = T_1 + {r \over 1+r} (T_0-T_1) \left( 1-e^{-(1+r)t/\tau _\mathrm{b}}\right) .$$

Thus for $t \gg \tau$, where $\tau = \tau _\mathrm{b} /(1+r)$, the steady state temperature has been reached.

For adsorbates on insulating substrates $\tau _\mathrm{b}$ will, in general, be very large if the resonance frequency $\omega _\mathrm{b}$ is well above the highest substrate phonon frequency. We now consider this case, which is equivalent to low temperature. Assume for simplicity that the temperature of the substrate vanishes ($T_1=0$) and assume that $\hbar \omega _\mathrm{a} \gg k_\mathrm{B}T_0$ and $\hbar \omega _\mathrm{b} \gg k_\mathrm{B}T_0$. In this case, is easy to show from (6.50)–(6.52) that $T_\mathrm{a} \approx T_0$ and

$$\begin{aligned} T_\mathrm{b} \approx {\omega _\mathrm{b} T_0 \over \omega _\mathrm{b} +T_0 \mathrm{ln}\!\left[ (1+r)/r\right] }, \end{aligned}$$

(6.59)

where we have measured frequency in units of $k_\mathrm{B}/\hbar$.

Let us now assume arbitrary tip and substrate temperatures but still assume $\tau _\mathrm{b} \gg \tau _\mathrm{a}$. Using (6.50)–(6.52) we get $T_\mathrm{a} \approx T_0$ and

$$\begin{aligned} {\omega _\mathrm{b} \over T_\mathrm{b}} = \mathrm{ln} \left[ 1+{1+r \over r \left( e^{\omega _\mathrm{b}/T_0} - 1\right) ^{-1}+ \left( e^{\omega _\mathrm{b}/T_1} - 1 \right) ^{-1}}\right] . \end{aligned}$$

(6.60)

This expression reduces to (6.58) for high temperatures and to (6.59) for low temperatures. In Fig. 6.7, we show the effective temperature $T_\mathrm{b}$ as a function of the tip temperature $T_0$ when $T_1=0.1 \hbar \omega _\mathrm{b}/k_\mathrm{B}$ and $r=0.25$.

Fig. 6.7

The adsorbate temperatures $T_a$ and $T_b$ as a function of the tip temperature $T_0$ (all in units of $\hbar \omega _b/k_B$). For $r=9.25$ and $T_1=0.1 \hbar \omega _b/k_B$

The effective adsorbate temperature $T_\mathrm{b}$, calculated above may be used to calculate (or estimate) the rate w of an activated process: $w=w_0 \mathrm{exp} (-E/k_\mathrm{B} T_\mathrm{b})$. When the barrier height E is large, even a very small temperature increase will result in a large increase in the reaction rate. It is also important to note that the excitation of a high frequency mode, such as the C–O stretch mode, can result in reactions involving other reaction coordinates; for example, diffusion, rotation or desorption. This is possible because of anharmonic coupling between the high frequency mode and the reaction coordinate mode. This has already been observed in STM studies of several different adsorption systems [207, 208]. Let us give an example of an adsorption system where photon tunneling may give rise to a strong temperature increase. We focus on $^{13}$C$^{18}$O on NaCl(100) at $T_1=30 \ \mathrm{K}$ which has been studied in detail by Chang and Ewing [204]. In this case, $\omega _\mathrm{b} \approx 2040 \ \mathrm{cm}^{-1}$, $\tau _\mathrm{b} \approx 10^{-3} \ \mathrm{s}$ (due mainly to decay via multi-phonon emission [204]), and the (pure dephasing dominated) relaxation time [209] $\tau ^*_\mathrm{b} \approx 10^{-10} \ \mathrm{s}$. We assume a Pt-tip at room temperature with an adsorbed $^{12}$C$^{18}$O with $\omega _\mathrm{a} \approx 2064 \ \mathrm{cm}^{-1}$ and $\tau ^*_\mathrm{a}\approx 3 \times 10^{-12} \ \mathrm{s}$ [210] mainly due to decay by excitation of electron-hole pairs. Using the experimental measured vibrational polarizability $\alpha _\mathrm{av} \approx 0.2$ Å$^3$ and $\alpha _\mathrm{bv} \approx 0.04$ Å$^3$ and assuming the tip–substrate separation of $d=1 \ \mathrm{nm}$, we get $s \approx 20$ and $2(\omega _\mathrm{a}-\omega _\mathrm{b})\tau ^*_\mathrm{a} \approx 16$. Thus, from (6.53) we get $r \approx 10^6$ and from (6.60) we get $T_\mathrm{b} \approx 300 \ \mathrm{K}$ where we have assumed the tip temperature $T_0=300 \ \mathrm{K}$. The CO/NaCl case is an extreme case because of the exceptional long vibrational energy relaxation time. However, the analysis presented above remains unchanged for any $\tau _\mathrm{b}$ larger than $10^{-8} \ \mathrm{s}$, so the conclusions are very general. Thus, we expect strong heating effects due to photon tunneling for high-frequency modes in adsorbed layers or films on insulating substrates.

The temperature increase for the C–O stretch vibration found above is similar to the temperature increase observed (or calculated) for CO on Pd(110) during STM experiments [207]. In this the case the excitation of the C–O stretch vibration is caused by inelastic tunneling. The temperature increase in the C–O stretch mode resulted in CO diffusion as a result of energy transfer to the parallel frustrated translation because of anharmonic coupling. This has been observed for CO molecule on Pd(110) [207], and on Ag(110) [211]. We expect similar decay processes for vibrational excited CO on NaCl resulting, in, for example, diffusion or desorption of the CO molecule.

6.7 Radiative Heat Transfer Between a Small Particle and a Plane Surface

The problem of radiative heat transfer between a small particle—considered to be a point-like dipole—and a plane surface without motion of the particle relative to the surface has been studied by several authors [16, 115, 139]. The more general case of the radiative heat transfer at relativistic motion of the particle parallel to the surface was studied in [150–152, 155, 156]. This case will be considered in Sect. 9.2. The particle could be a single molecule, or a dust particle, and it is modeled by a sphere with radius $R\ll d$, where d is the separation between the particle and the plane surface. The heat flux between the particle and substrate can be obtained from for the heat flux between two semi-infinite bodies (see (6.11)), considering one of them (say the body 2) as sufficiently “rarefied”. As in Sect. 5.3, from (6.11) at $V=0$ we get contribution to heat flux through the surface of a semi-infinite body. We assume $d\ll \lambda _T$ so that the heat flow result from the evanescent waves. In this case, in K-reference frame the heat flux is given by

$$\begin{aligned} S =\frac{4\hbar }{ \pi } \int _0^\infty \mathrm {d}\omega \omega \big ( n(\omega , T_1)-n(\omega , T_2)\big ) \int _0^\infty \mathrm {d}qq^2\mathrm {e}^{-2qd}\times \nonumber \\ \times \left\{ \mathrm {Im}R_p(\omega )\mathrm {Im}\alpha _E (\omega )+\mathrm {Im}R_s(\omega )\mathrm {Im}\alpha _H (\omega )\right\} . \end{aligned}$$

(6.61)

Due to presence of factor $e^{-2qd}$ in the q-integral, the most important contribution comes from $q\approx 1/d$, and in the $\omega$-integration due to presence of the factor $n(\omega )$ the most important contribution comes from $\omega \approx \omega _T=k_BT/\hbar$. For $d\ll \lambda _0\ll \lambda _T$, where $\lambda _0=\lambda _T|\varepsilon (\omega _T)|^{-1/2},\, \lambda _T=c/\omega _T$, Fresnel’s formulas for reflection amplitudes can be written in the form

$$\begin{aligned} R_p = \frac{\varepsilon k_z - k_{z1}}{\varepsilon k_z + k_{z1}}\approx \frac{\varepsilon -1}{\varepsilon +1}, \end{aligned}$$

(6.62)

$$\begin{aligned} R_s = \frac{k_z-k_{z1}}{k_z-k_{z1}}\approx \frac{1}{4}\left( \frac{\omega }{cq}\right) ^2(\varepsilon -1), \end{aligned}$$

(6.63)

where $k_z = \sqrt{(\omega /c)^2 - q^2}$, $k_{z1} = \sqrt{(\omega /c)^2\varepsilon - q^2}$. The dielectric and magnetic susceptibility of a spherical particle with radius R are determined by (5.58) and (5.59). For metals in the low frequency range $\omega \ll \nu$ the dielectric permittivity $\varepsilon = 4\pi i\sigma /\omega$, where $\nu ^{-1}$ is the electron relaxation time and $\sigma$ is the conductivity. Using this expression for dielectric permittivity in (5.58), (6.62) and (6.63) and assuming $\omega \ll 4\pi \sigma$, we get

$$ \mathrm {Im}R_p \approx \frac{2\omega }{4\pi \sigma },\qquad \mathrm {Im}R_s \approx \frac{1}{4}\left( \frac{\omega }{cq}\right) ^2\frac{4\pi \sigma }{\omega }, $$

$$\begin{aligned} \mathrm {Im}\alpha _E(\omega ) = R^3\frac{3\omega }{4\pi \sigma }. \end{aligned}$$

(6.64)

From (5.59) and (5.60) at $R\ll \delta$, where $\delta = c/\sqrt{2\pi \omega \sigma }$, we get

$$\begin{aligned} \mathrm {Im}\alpha _H=\frac{1}{30}\left( \frac{R}{\delta }\right) ^2R^3=\frac{2\pi \sigma \omega R^5}{15c^2}. \end{aligned}$$

(6.65)

Using (6.64) and (6.65) in (6.61), we get $S=S_E +S_H$, where the contributions $S_E$ and $S_H$ due to the electric and magnetic dipole moment S are given by

$$\begin{aligned} S_E\approx \frac{2\pi ^3}{5}\left( \frac{R}{d}\right) ^3\left( \frac{k_B^4}{16\pi ^2\hbar ^3\sigma _1\sigma _2}\right) \left( T_1^4-T_2^4\right) , \end{aligned}$$

(6.66)

$$\begin{aligned} S_H\approx \frac{\pi ^3}{225}\left( \frac{R}{d}\right) \left( \frac{16\pi ^2k_B^4\sigma _1\sigma _2R^4}{\hbar ^3c^4}\right) \left( T_1^4-T_2^4\right) , \end{aligned}$$

(6.67)

where $\sigma _{1(2)}$ is the conductivity of substrate (sphere). For a example, for $T_2=300$ K, $d=2R=10$ nm, $\sigma _1=\sigma _2=4 \times 10^{17}$ s (which corresponds to gold), $S_E \approx$ $\approx 10^{-17}$ W and $S_H \approx 10^{-11}$ W.

As pointed out in [115], large heat transfer is expected for high-resistivity materials. The heat flux (6.66) is maximized when $k_BT/4\pi \hbar \sigma \approx 1$. In this case, for a particle at room temperature, and a distance $d=2R=10$ nm above a cold ($T=0$ K) sample, we get $P_p/d^2\approx 10^7$ Wm$^{-2}$. This should be compared with the heating from black body radiation. When the sample surface at temperature $T_2$ is illuminated with black body radiation at temperature $T_1$, taking into account the surface reflectivity, the heat flux to the sample from black body radiation is approximately [15, 93, 94]:

$$\begin{aligned} S_{BB}= & {} \frac{1}{8\pi ^3}\int _0^\infty \mathrm {d}\omega \big [\Pi _1(\omega )-\Pi _2(\omega )\big ]\int _{\frac{q<\omega }{c}}\mathrm {d}^2q\left( 1-\mid R_p(\omega )\mid ^2\right) +\left[ p\rightarrow s\right] = \nonumber \\= & {} 0.4\frac{k_B^4T_1^4}{\hbar ^3c^2}\left( \frac{k_BT_1}{4\pi \hbar \sigma }\right) ^{1/2}-[T_1\rightarrow T_2], \end{aligned}$$

(6.68)

where $[T_1\rightarrow T_2]$ stands for the term obtained from the first term by replacing $T_1$ with $T_2$. For $T_2\approx 300$ K and $T_1\approx 0$ K, and for $k_BT/\hbar \approx 4\pi \sigma$, (6.68) gives $S_{BB}\approx 100$ Wm$^{-2}$. Thus, a particle may give rise to a large local enhancement of the heating of the surface, compared with the uniform black body radiation. When the substrate and the particle are made from the same materials, which can support surface plasmons with frequencies $\omega _s$ and $\omega _p$, respectively, in the two poles, an approximation the rate of the heat transfer between them is given by

$$\begin{aligned} P=\left( \frac{3\Gamma R}{d}\right) \left\{ \left[ \Pi _1(\omega _{sp})-\Pi _2(\omega _{sp})\right] +[\omega _{sp}\rightarrow \omega _{ss}]\right\} . \end{aligned}$$

(6.69)

For SiC with $\omega _s=1.79 \times 10^{14}$ s$^{-1}$ and $\omega _p=1.76 \times 10^{14}$ s$^{-1}$, $d=10$ nm, $R=5$ nm, $T_1=300$ K and $T_2=0$ K, (6.62) gives $P\approx 1.6 \times 10^{-10}$ W. We note that a much larger heat transfer can be achieved if the surfaces are covered with adsorbates with matched frequencies [14, 15] (see Sect. 6.5).

6.8 Near-Field Radiative Heating in Ion Traps

Electric-field noise near surfaces is a common problem in diverse areas of physics and a limiting factor for many precision measurements. There are multiple mechanisms by which such noise is generated, many of which are poorly understood. Laser-cooled, trapped ions provide one of the most sensitive systems to probe electric-field noise at MHz frequencies and over a distance range 30–3000 $\upmu$m from a surface [212]. This experimental setup represents one of the most promising systems for the implementation of large-scale quantum information processing. Most of the basic requirements for building a quantum computer have been demonstrated in the laboratory and the generation of entangled states of up to 14 ions has been achieved. Many experimental efforts are now focused on the development of miniaturization and microfabrication techniques for ion traps to realize more efficient and also fully scalable quantum computing architectures. However, when devices are miniaturized, physics at the short distance becomes a challenge. This is evident in measurements of the Casimir force or of noncontact friction; in the case of trapped ions, it manifests itself in the appearance of an excess (‘anomalous’) heating rate and electric-field noise near surfaces, as the trap-surface distance is decreased. Therefore, a detailed understanding of the origin of this noise will be essential for the future progress of trapped-ion quantum computing, as well as the development of several hybrid quantum computing approaches where, for example, ions, Rydberg atoms, polar molecules, or charged nanomechanical resonators are operated in the vicinity of solid-state systems.

The heat transfer between an electrode and an ion trap, due to the fluctuating electric field near the electrode, can for a spherically symmetric trap be written in the form

$$\begin{aligned} P=\int _{-\infty }^{\infty }\frac{d\omega }{2\pi }\omega \mathrm {Im}\alpha _t(\omega )S_E(\omega ), \end{aligned}$$

(6.70)

where the spectral density of the electric field fluctuations

$$ S_E(\omega )=\big \langle |\mathbf {E(r_0)}|^2\big \rangle _{\omega }=\int _{-\infty }^{\infty }dt e^{i\omega t}\big \langle \mathbf {E}(\mathbf {r}_0, t+\tau )\cdot \mathbf {E}(\mathbf {r}_0,\tau )\big \rangle , $$

where $\mathbf {r}_0$ is the position of the trap. The polarizability of the trap is given by

$$\begin{aligned} \alpha _t(\omega )=\frac{Q^2}{M}\frac{1}{\omega _t^2-\omega ^2-i\omega \eta _t} \end{aligned}$$

(6.71)

where Q and M are the charge and mass of the ion in the trap, respectively; $\omega _t$ and $\eta _t$ are the frequency and damping constant of the ion vibrations in the trap, respectively. Using (6.71) in (6.70) and taking into account that for $\eta \ll \omega _t$

$$ \mathrm {Im}\alpha _t(\omega )=\frac{\pi Q^2}{2M}\big [\delta (\omega -\omega _t) - \delta (\omega +\omega _t)\big ], $$

we get

$$\begin{aligned} P=\frac{Q^2}{2M} S_E(\omega _t). \end{aligned}$$

(6.72)

When the ion is laser cooled to the vibrational ground state, the fluctuating electric fields couple to the motion of the ion and lead to an increase of the average vibrational occupation number $\bar{n}$ with a characteristic rate

$$\begin{aligned} \dot{\bar{n}}=\frac{P}{\hbar \omega _t}=\frac{Q^2}{2M\hbar \omega _t} S_E(\omega _t) \end{aligned}$$

(6.73)

Using (4.2) in (6.72) for $q\sim 1/d \gg \omega _t/c$ we get

$$\begin{aligned} S_E(\omega _t)=4\hbar n(\omega _t, T)\int _0^{\infty }dqq^2e^{-2qd}\mathrm {Im}R_p(\omega _t, q)\approx 4\frac{k_BT}{\omega _t} \int _0^{\infty }dqq^2e^{-2qd}\mathrm {Im}R_p(\omega _t, q), \end{aligned}$$

(6.74)

where d is a distance between trapped ion and a metal surface. $R_p$ is the reflection amplitude for p-polarized waves. Equation (6.74) can be obtained directly from (6.61). For a clean surface and $c/\omega _t\sqrt{|\varepsilon |}\ll d \ll c/\omega _t$ the reflection amplitude can be written in the form

$$\begin{aligned} R_p = \frac{\varepsilon k_z - k_{z1}}{\varepsilon k_z + k_{z1}}\approx \frac{q-\frac{\omega }{c\sqrt{\varepsilon }}}{q+\frac{\omega }{c\sqrt{\varepsilon }}}\approx 1-2\frac{\omega }{cq\sqrt{\varepsilon }} \end{aligned}$$

(6.75)

Using (6.75) in (6.74) with $\varepsilon = 4\pi \sigma /\omega$ we get

$$\begin{aligned} S_E(\omega _t)\approx \frac{2k_BT}{cd^2}\sqrt{\frac{\omega _t}{2\pi \sigma }} \end{aligned}$$

(6.76)

For typical parameters of the ion trap ($Q=10^{-19}$ K, $M=10^{-26}$ kg, $\omega _t=10^8$ s$^{-1}$, $d=10^{-4}$ m, $T=300$ K, $\sigma = 10^{18}$ s$^{-1}$) we get $\dot{\bar{n}}\approx 0.1$ s$^{-1}$, and $S_E(\omega _t)\approx 10^{-16}$ (V/m)$^2$s$^{-1}$ which is much smaller than the experimentally observed values.

Now we consider a surface covered by adsorbates. In this case, the reflection amplitude is given by (6.47). For typical parameters of the ion trap, $\varepsilon \sim 10^{11}$ and $qa\sim 10^{-6}$. In this case, the main contribution is provided by adsorbate vibrations that are normal to the surface, and the reflection amplitude can be written in the form

$$\begin{aligned} R_p\approx 1+8\pi n_a q\alpha _{\perp } \end{aligned}$$

(6.77)

where $n_a$ is the concentration of adsorbates, and $\alpha _{\perp }$ is the adsorbate polarizability l. Using (6.78) in (6.74) we get

$$\begin{aligned} S_E(\omega _t)=\frac{12\pi k_BTn_a\mathrm {Im}\alpha _{\perp }(\omega _t)}{\omega _td^4}, \end{aligned}$$

(6.78)

where, according to (6.48)

$$\begin{aligned} \alpha _{\perp }(\omega _t)=\frac{e^{*2}}{M_a}\frac{\omega _t\eta _a}{{\left( \omega _a^2-\omega _t^2\right) }^2+\omega _t^2\eta _a^2}, \end{aligned}$$

(6.79)

where $e^*$ and $M_a$ are the adsorbate charge and mass, $\omega _a$ and $\eta _a$ are the adsorbate frequency and damping constant. Using (6.79) in (6.77), we get

$$\begin{aligned} S_E(\omega _t)=\frac{12\pi k_BTn_ae^{*2}}{M_ad^4}\frac{\eta _a}{{\left( \omega _a^2-\omega _t^2\right) }^2+\omega _t^2\eta _a^2} \end{aligned}$$

(6.80)

The expression (6.80) can be very large in the case of resonance. In particular at $\omega _a=10\eta _a$ and $\eta _a\sim \omega _t\sim 10^8$ s$^{-1}$, we get $S_E(\omega _t)\sim 10^{-4}$ (V/m)$^2$s and $\dot{\bar{n}}\sim 10^{10}$ s$^{-1}$. For the frequencies out of resonance we can neglect displacement of adsorbates relative substrate. In this case displacement of the substrate surface under the action of the stress applied to adsorbed layer is given by

$$\begin{aligned} u_z(\mathbf {x})=M_{zz}n_ae^{*}E_ze^{i\mathbf {q}\cdot \mathbf {x}-i\omega t} \end{aligned}$$

(6.81)

The explicit form of the stress tensor $M_{ij}$ in the elastic continuum model is given in [213] (see also Appendix T). The displacement (6.81) will give rise to the dipole moment per unit area

$$\begin{aligned} p_z(\mathbf {x})=n_ae^{*}u_z(\mathbf {x})=M_{zz}(n_ae^{*})^2E_ze^{i\mathbf {q}\cdot \mathbf {x}-i\omega t}. \end{aligned}$$

(6.82)

Thus, for this mechanism, $n_a\alpha$ in (6.78) should be replaced by $n_ae^{*2}M_{zz}$. For $d \gg c_l/\omega _t$, where $c_l$ is the longitudinal sound velocity and $\rho$ is the density of substrate, the stress tensor is given by

$$ M_{zz}=\frac{i}{\rho \omega c_l} $$

and instead of (6.80), we get

$$\begin{aligned} S_E(\omega _t)=\frac{12\pi k_BT(n_ae^{*})^2}{\rho d^4\omega _t^2 c_l}. \end{aligned}$$

(6.83)

For the same (typical) typical parameters as above, we get $S_E(\omega _t)\sim 10^{-7}$ (V/m)$^2$s and $\dot{\bar{n}}\sim 10^{7}$ s$^{-1}$. Thus, this mechanism also gives an important contribution to the heating rate.

6.9 Radiative Heat Transfer Between Two Dipole Inside a N-Dipole System

Let us consider a discrete set of N objects located at positions $\mathbf {r}_i$ and maintained at different temperatures $T_i$ with $i = 1, \ldots , N$. Suppose that the size of these objects is small enough compared with the smallest thermal wavelength $\lambda _{T_i}= c\hbar /(k_BT)$ so that all individual objects can be modeled as simple radiating electrical dipoles. For metals, one has also to include the magnetic dipole moments due to the induction of eddy currents. Such an extension is straightforward and as such, for convenience, we will consider electric dipoles only. The Fourier component of the electric field at the frequency $\omega$ (with the convention $\hat{f}(t)=\int \frac{d\omega }{2\pi }f(\omega )e^{-i\omega t}$) generated at the position $\mathbf {r}_{i}$ by the fluctuating part $p^{f}_j$ of electric dipole moment of the particle j which is located at $\mathbf {r}_j$ reads

$$\begin{aligned} \mathbf {E}_{ij}=k^2\mathbf {G}^{ij}\mathbf {p}_j^{f}, \end{aligned}$$

(6.84)

where $\mathbf {G}^{ij}=\mathbf {G}(\mathbf {r}_i, \mathbf {r}_j, \omega )$ is the dyadic Green’s tensor (i.e. the propagator) between the particles i and j inside the set of N particles and $k=\omega /c$. On the other hand, by summing the contribution of fields radiated by each particle, the dipolar moment induced by the total field on the i-th particle is given by

$$\begin{aligned} \mathbf {p}_i^{ind}=\alpha _i\sum _{j\ne i}\mathbf {E}_{ij}, \end{aligned}$$

(6.85)

where $\alpha _i$ is the particle’s polarizability. Then, the power dissipated inside the particle i at a given frequency $\omega$ due to the fluctuating field $\mathbf {E}_{ij}$ generated by the particle j can be calculated from the work of the fluctuating electromagnetic field on the charge carriers as

$$\begin{aligned} P_{j\rightarrow i}= 2\mathrm {Re}\left\langle -i\omega \mathbf {p}_i^{ind}\cdot \mathbf {E}_{ij}^*\right\rangle , \end{aligned}$$

(6.86)

where the brackets represent the ensemble average. Using the relations (6.84) and (6.85) between the dipole moments, and the fluctuation dissipation theorem, i.e.

$$ \left\langle p_{j, \alpha }^f p_{i,\beta }^f\right\rangle =2\hbar n_i(\omega )\delta _{\alpha \beta }\delta _{ij}, $$

we find after a straightforward calculation that

$$\begin{aligned} P_{j\rightarrow i}= 3\int _0^{\infty }\frac{d\omega }{2\pi }\Pi (\omega , T_j)T_{i, j}(\omega ), \end{aligned}$$

(6.87)

where the transmission coefficient:

$$\begin{aligned} T_{i, j}(\omega ) =\frac{3}{4}k^4\mathrm {Im}(\alpha _i)\mathrm {Im}(\alpha _j)\mathrm {Sp} \left[ \mathbf {G}^{ij}\mathbf {G}^{ij+}\right] . \end{aligned}$$

(6.88)

In order to present the heat flux in an obvious Landauer-like manner, we rewrite the heat flux in terms of the conductance $G_{i, j} = \partial P_{j\rightarrow i}/\partial T_{j}$ so that $P_{j\rightarrow i} = G_{i, j}\Delta T$. Then we find

$$\begin{aligned} P_{j\rightarrow i}=3\left( \frac{\pi ^2k_B^2T}{3h}\right) \overline{T}_{i, j}\Delta T, \end{aligned}$$

(6.89)

where $\overline{T}_{i, j}=\int dx f(x)T_{i, j}(x)/(\pi ^2/3)$ is the mean transmission coefficient with $f(x)= x^2\exp {(-x)}/(\exp {(x)}-1)^2$. In the case of two particles (N $=$ 2), one can show (see Sect. 6.5) that $T_{i, j}(\omega , d) \in [0, 1]$ and therefore $\overline{T}_{i, j}(\omega , d) \in [0, 1]$ as well. Hence, the conductance between two dipoles is limited by three times the quantum of thermal conductance $\pi ^2k_B^2T/(3h)$. In other words, only three channels contribute to the heat flow between two dipoles, namely the channels due to the coupling of the three components $p_{j,\alpha }$ with the same three components $p_{j,\alpha }$ (i.e. same polarization). Of course, by adding further particles, this limit cannot be exceeded, whereas the heat flux can be increased or decreased with respect to the case of two particles. Nevertheless, the number of channels increases if electric multipoles as well as the magnetic moments come into play.

Now, for calculating the Green’s function for a system of N particles, we use the set of 3N self-consistent equations

$$\begin{aligned} \mathbf {E}_{ij}=k^2\mathbf {G}_0^{ij}\mathbf {p}_{j\ne i}+k^2\sum _{k\ne i}\mathbf {G}_0^{ik}\alpha _k/\mathbf {E}_{kj} \end{aligned}$$

(6.90)

for $i=1,\ldots , N$ with the free space Green’s function

$$ \mathbf {G}_0^{ij}= \frac{k^2e^{ikr_{ij}}}{r_{ij}}\left[ \left( 1+\frac{ikr_{ij}-1}{k^2r_{ij}^2}\right) \mathbf {1}+\frac{3-3ikr_{ij}-k^2r_{ij}^2}{k^2r^2_{ij}}\hat{\mathbf {r}}_{ij}\otimes \hat{\mathbf {r}}_{ij}\right] , $$

where the unit vector $\hat{\mathbf {r}}_{ij}\equiv \mathbf {r}_{ij}/r_{ij},\,\mathbf {r}_{ij}=\mathbf {r}_{i}-\mathbf {r}_{j}$ and $\mathbf {1}$ stands for the unit dyadic tensor. Comparing (6.84) and (6.90) we get the Green’s tensor

$$\begin{aligned} \left( \begin{array}{c} \mathbf {G}^{1k}\\ \vdots \\ \mathbf {G}^{Nk} \end{array} \right) =[\mathbf {1}-\mathbf {A}_0]^{-1} \left( \begin{array}{c} \mathbf {G}^{1k}_0\\ \vdots \\ \mathbf {G}^{(k-1)k}_0\\ 0\\ \mathbf {G}^{(k+1)k}_0\\ \vdots \\ \mathbf {G}^{Nk}_0 \end{array} \right) \end{aligned}$$

(6.91)

for $k=1,\ldots , N$ with

$$\begin{aligned} \mathbf {A}_0=\frac{\omega ^2}{c^2}\left( \begin{array}{cccc} 0&{}\mathbf {G}^{12}_0\alpha _2&{}\ldots &{}\mathbf {G}^{1N}_0\alpha _N\\ \mathbf {G}^{21}_0\alpha _1&{}\ddots &{}\ddots &{}\vdots \\ \vdots &{}\ddots &{}\ddots &{}\mathbf {G}^{(N-1)N}_0\alpha _N\\ \mathbf {G}^{N1}_0\alpha _1&{}\ldots &{}\mathbf {G}^{N(N-1)}_0\alpha _{N-1}&{}0 \end{array} \right) \end{aligned}$$

(6.92)

With this relation and (6.87), it is possible to determine the interparticle heat flux in a system of N particles out of equilibrium.

Let us now apply this theoretical formalism to describe some emerging many-particle effects. We consider the simplest possible configuration where such effects occur that is a triplet of particles. We consider only the interparticle heat flux between particle 1 and 2 separated by a distance 2l in the presence of the third particle. Here, we assume that $T_1 = 300$ K and $T_2 = T_3 = 0$. The interparticle heat flux is then given by

$$\begin{aligned} \phi _{1\rightarrow 2}=P_{1\rightarrow 2}-P_{2\rightarrow 1}. \end{aligned}$$

(6.93)

In this case, the dyadic Green’s function has the form

$$\begin{aligned} \mathbf {G}^{21}=\mathbf {D}^{-1}_{213}\left[ \mathbf {G}^{21}_0+\frac{\omega ^2}{c^2}\mathbf {B}^{213}\mathbf {D}^{-1}_{31}\mathbf {G}^{31}_0\right] \end{aligned}$$

(6.94)

where

$$ \mathbf {D}_{213}=\mathbf {D}_{21} - \frac{\omega ^4}{c^4}\mathbf {B}^{213}\mathbf {D}^{-1}_{31}\mathbf {B}^{312}, $$

$$ \mathbf {D}_{21}= \mathbf {1}-\frac{\omega ^4}{c^4}\mathbf {G}^{12}_0\alpha _1\mathbf {G}^{12}_0\alpha _2, $$

and

$$ \mathbf {B}^{213}=\mathbf {G}^{23}_0\alpha _3 + \frac{\omega ^2}{c^2}\mathbf {G}^{21}_0\alpha _1\mathbf {G}^{13}_0\alpha _3. $$

Figure 6.8 shows the resulting interparticle flux between particle 1 and 2 in the presence of body 3 normalized to the flux for two isolated dipoles. The position of the particles, for which the interparticle flux is calculated, is fixed, but the position of the third particle is changed. It can be seen that for some geometric configurations, the heat flux mediated by the presence of the third particle can be larger than the value for two isolated dipoles. In particular, we observe an exaltation of heat flux of approximately one order of magnitude when the third particle is located between the two other particles, i.e., when all three particles are aligned. Hence, the heat flux between two dipoles can dramatically be increased when inserting a third particle in between.

Fig. 6.8

Normalized heat flux between two spherical particles with the same radius. A third particle is located equidistant to particles 1 and 2 ($T_1 = 300$ K, $T_2 = T_3 = 0$ K). From [214]

The heat flux enhancement can be attributed to a three-body effect that is a resonant surface mode coupling mediated by the third particle. This effect could be used to improve, for example, the performance of near-field thermophotovoltaic devices, by placing nanoparticles on the surface of photovoltaic cells.

6.10 Local Heating of the Surface by an Atomic Force Microscope Tip

An atomic force microscope tip, at a distance d above a flat sample surface with the radius of curvature $R \gg d$, can be approximated by a sphere with radius R. In this case, the heat transfer flux between the tip and the surface can be estimated using the approximate method of Derjaguin [215], later called the proximity force approximation (PFA) [216]. According to this method, the radiative flux in the gap between two smooth curved surfaces at short separations can be approximately calculated as a sum of fluxes between pairs of small parallel plates corresponding to the curved geometry of the gap. Specifically, the sphere–plane heat flux is given by

$$\begin{aligned} P=2\pi \int _0^R d\rho \rho S(z(\rho )), \end{aligned}$$

(6.95)

where R is the radius of the sphere, $z(\rho )=d+R-\sqrt{R^2-\rho ^2}$ denotes the tip–surface distance as a function of the distance $\rho$ from the tip symmetry axis, and the heat flux per unit area $S(z(\rho ))$ is determined for flat surfaces. This scheme was proposed in [215, 217] for the calculation of the conservative van der Waals interaction; in this case, the error is not larger than 5–10% in an atomic force application, and 25% in the worst case situation [218]. We assume that the same scheme is also valid for the calculation of heat transfer. We assume that the tip has a paraboloid shape given [in cylindrical coordinates ($z,\rho$)] by the formula: $z=d+\rho ^2/2R$, where d is the distance between the tip and the flat surface. If

$$\begin{aligned} S=\frac{C}{\left( d+\rho ^2/2R\right) ^n}, \end{aligned}$$

(6.96)

we get

$$\begin{aligned} P=\frac{2\pi R}{n-1}\frac{C}{d^{n-1}}=\frac{2\pi Rd}{n-1}S(d)\equiv A_{ \mathrm {eff}}S(d), \end{aligned}$$

(6.97)

where $A_{\mathrm {eff}}=2\pi Rd/(n-1)$ is the effective surface area. In a more general case, one must use numerical integration to obtain the heat transfer.

As an illustration, consider the heat flux between a SiC tip and a flat SiC surface. From (6.35) and (6.97), we get that the heat transfer power between SiC tip at $T_1=300$ K and a cold SiC surface ($T_2\,{=}\, 0$ K) at $d \ll R$: $P(d)=5.2 \times 10^{-10}(R/d)$ W.

Fig. 6.9

Comparison of the numerically exact solution for NFRHT between a sphere and a plane with approximation schemes for various sphere diameters and gap sizes. From [228]

The most general method available for calculating both the Casimir force and the radiative heat transfer between many bodies of arbitrary shapes, materials, temperatures and separations, expressed the Casimir force and radiative heat transfer in terms of the scattering matrices of individual bodies [89, 219–227]. Specifically, a numerically exact solution for the near-field radiative heat transfer between a sphere and an infinite plane was first obtained using the scattering matrix approach (Fig. 6.9).

6.11 A Nanoscale ‘Heat Stamp’

It has been proposed [229] that-near field optics could be exploited to write extremely fine details for integrated circuits. The basic idea of this proposal is that the components of the electromagnetic field with short wavelengths (and therefore the potential for high resolution) are naturally evanescent, and do not contribute to the far field. Hence, fine details in any patterned mask will rapidly disappear with distance from the mask. However, if it is possible to position the wafer close to the mask, then fine details can be resolved. Roughly speaking, the separation between mask and wafer must be of the same order as the lateral details to be resolved.

An extension of this idea was suggested [115] in the form of the ‘heat stamp’. It is possible to imagine a mask consisting of a surface patterned alternatively in highly reflecting (and therefore poorly emitting) material and a second material chosen to maximize emission of heat into the evanescent modes. For adsorbate-covered structures, it is possible to have a ‘heat stamp’ with atomic resolution.