Near-field radiative heat transfer and noncontact friction

All material bodies are surrounded by a fluctuating electromagnetic field because of the thermal and quantum fluctuations of the current density inside them. Close to the surface of planar sources (when the distance $d⪡{\lambda }_T=c\hslash ∕k_{B}T$), thermal radiation can be spatially and temporally coherent if the surface can support surface modes like surface plasmon polaritons, surface phonon polaritons, or adsorbate vibrational modes. The fluctuating field is responsible for important phenomena such as radiative heat transfer, the van der Waals interaction, and the van der Waals friction between bodies. A general formalism for the calculation of the power spectral density for the fluctuating electromagnetic field is presented and applied to the radiative heat transfer and the van der Waals friction using both the semiclassical theory of the fluctuating electromagnetic field and quantum field theory. The radiative heat transfer and the van der Waals friction are greatly enhanced at short separations $(d⪡{\lambda }_T)$ between the bodies due to the evanescent electromagnetic waves. Particularly strong enhancement occurs if the surface of the body can support localized surface modes like surface plasmons, surface polaritons, or adsorbate vibrational modes. An electromagnetic field outside a moving body can also be created by static charges which are always present on the surface of the body due to inhomogeneities, or due to a bias voltage. This electromagnetic field produces electrostatic friction which can be greatly enhanced if on the surface of the body there is a two-dimensional electron or hole system, or an incommensurate adsorbed layer of ions exhibiting acoustic vibrations. Applications of radiative heat transfer and noncontact friction to scanning probe spectroscopy are discussed. The theory gives a tentative explanation for the experimental noncontact friction data.

Figure 1

(Color online) There are two modes for exchange of heat between two surfaces separated by vacuum: (a) conventional radiative heat transfer via propagating electromagnetic waves and (b) photon tunneling via evanescent waves.

Figure 2

(Color online) Evanescent waves play no role in thermal radiation from a hot dielectric surface to vacuum (left), but evanescent waves can carry heat from a hot to a cold dielectric surface (right).

Figure 3

(Color online) At short distances, evanescent states dominate in phase space: propagating photon modes carry heat flux within the inner circle, evanescent modes within the outer circle.

Figure 4

(Color online) The electromagnetic waves emitted in the opposite direction from the body at the bottom will experience an opposite Doppler shift in the reference frame in which the body at the top is at rest. Due to the frequency dispersion of the reflection amplitude these electromagnetic waves will reflect differently from the surface of the body at the top, which gives rise to momentum transfer between the bodies. This momentum transfer is the origin of van der Waals friction.

Figure 5

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

Figure 6

The thermal flux as a function of the conductivity of the solids. The surfaces are separated by $d=10\mathrm{\AA }$. 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.)

Figure 7

The heat flux between two semi-infinite silver bodies coated with $10\mathrm{\AA }$ of high resistivity $(\rho =0.14\Omega \mathrm{cm})$ 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 $T=273\mathrm{K}$. (a), (b) The $p$- and $s$-wave contributions, respectively. (The base of the logarithm is 10.)

Figure 8

(Color online) 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 $T=273\mathrm{K}$. For parameters corresponding to $\mathrm{K}∕\mathrm{Cu}$ (001) and Cu (001) (116) ${\omega }_{\perp }=1.9\times {10}^{13}{\mathrm{s}}^{-1}$, ${\omega }_{\Vert }=4.5\times {10}^{12}{\mathrm{s}}^{-1}$, ${\eta }_{\Vert }=2.8\times {10}^{10}{\mathrm{s}}^{-1}$, ${\eta }_{\perp }=1.6\times {10}^{12}{\mathrm{s}}^{-1}$, and $e^{*}=0.88e$. (The base of the logarithm is 10.)

Figure 9

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

Figure 10

(Color online) The adsorbate temperature $T_a$ and $T_b$ as a function of the tip temperature $T_0$ (all in units of $\hslash {\omega }_b∕k_B$). For $r=9.25$ and $T_1=0.1\hslash {\omega }_b∕k_B$.

Figure 11

(Color online) The friction coefficient for two flat surfaces in parallel relative motion as a function of separation $d$ at $T=273\mathrm{K}$ with parameters chosen to correspond to copper (${\tau }^{-1}=2.5\times {10}^{13}{\mathrm{s}}^{-1}$, ${\omega }_p=1.6\times {10}^{16}{\mathrm{s}}^{-1}$). The contributions from the $s$- and $p$-polarized electromagnetic fields are shown separately. The full and dashed curves have been obtained using the nonlocal optical dielectric formalism and local optical approximation, respectively. (The base of the logarithm is 10.)

Figure 12

(Color online) The same as Fig. 11 but for normal relative motion.

Figure 13

The friction coefficient between the copper tip and copper substrate of surfaces which are covered by low concentration of cesium atoms, as a function of the separation $d$. The cylindrical tip is characterized by the radius of curvature $R=1\mu \mathrm{m}$ and the width $w=7\mu \mathrm{m}$. The other parameters correspond to Cs adsorbed on the Cu (100) surface at coverage $\theta \approx 0.1$ (116; 140): $e^{*}=0.28e$, $\eta =3\times {10}^9{\mathrm{s}}^{-1}$, $a=2.94\mathrm{\AA }$, and $T=293\mathrm{K}$. (The base of the logarithm is 10.)

Figure 14

Two ways for studying the van der Waals friction. Left: A metallic block sliding relative to the metallic substrate with the velocity $v$. An electronic frictional stress will act on the block (and on the substrate). Right: The shear stress $\sigma$ can be measured if instead of sliding the upper block, a voltage $U_2$ is applied to the block resulting in a drift motion of the conduction electrons (velocity $v$). The resulting frictional stress $\sigma$ on the substrate electrons will generate a voltage difference $U_1$ (proportional to $\sigma$) as indicated.

Figure 15

(Color online) The frictional drag coefficient for two quantum wells at $T=3\mathrm{K}$ as a function of separation $d$. The $s$- and $p$-wave contributions are shown separately. The calculations were performed with surface electron density $n_s=1.5\times {10}^{15}{\mathrm{m}}^{-2}$, damping constant $\eta =1.3\times {10}^{10}{\mathrm{s}}^{-1}$, effective electron mass $m^{*}=0.067m_e$, and dielectric constant $\epsilon =10$. (The base of the logarithm is 10.)

Figure 16

(Color online) The frictional drag coefficient for two quantum wells at $T=273\mathrm{K}$ as a function of separation $d$. The $s$- and $p$-wave contributions are shown separately. The calculations were performed using the surface electron density $n_s=1.05\times {10}^{19}{\mathrm{m}}^{-2}$, damping constant $\eta =2.5\times {10}^{13}{\mathrm{s}}^{-1}$, effective electron mass $m^{*}=m_e$, and dielectric constant $\epsilon =1$. (The base of the logarithm is 10.)

Figure 17

The frictional drag coefficient for two quantum wells at $T=3\mathrm{T}$ as a function of electron concentration $n_s$. The full curve was obtained by interpolation between the curves (dashed lines) obtained within the nonlocal optical dielectric approach, with the dielectric functions corresponding to a degenerate electron gas $(n_s>n_F\sim {10}^{14}{\mathrm{m}}^{-2})$, and to the nondegenerate electron gas $(n_s<n_F)$. The electron density parameter $n_{s0}=1.5\times {10}^{15}{\mathrm{m}}^{-2}$, damping constant $\eta =1.3\times {10}^{10}{\mathrm{s}}^{-1}$, effective electron mass $m^{*}=0.067m_e$, separation $d=175\mathrm{\AA }$, and the dielectric constant $\epsilon =10$. (The base of the logarithm is 10.)

Figure 18

Scheme of the tip-sample system. The tip shape is characterized by its length $L$ and the tip radius of curvature $R$.