# Extreme Near-Field Heat Transfer Between Gold Surfaces

Takuro Tokunaga  Department of Mechanical Engineering, University of Utah,
Salt Lake City, Utah 84112, United States.  Amun Jarzembski  Sandia National
Laboratories, Albuquerque, NM 87185, United States.  Takuma Shiga  Department
of Mechanical Engineering, The University of Tokyo, Bunkyo, Tokyo 113-8656,
Japan.  Keunhan Park<EMAIL_ADDRESS>Department of Mechanical
Engineering, University of Utah, Salt Lake City, Utah 84112, United States.
Mathieu Francoeur<EMAIL_ADDRESS>Department of Mechanical
Engineering, University of Utah, Salt Lake City, Utah 84112, United States.

###### Abstract

Extreme near-field heat transfer between metallic surfaces is a subject of
debate as the state-of-the-art theory and experiments are in disagreement on
the energy carriers driving heat transport. In an effort to elucidate the
physics of extreme near-field heat transfer between metallic surfaces, this
Letter presents a comprehensive model combining radiation, acoustic phonon and
electron transport across sub-10-nm vacuum gaps. The results obtained for gold
surfaces show that in the absence of bias voltage, acoustic phonon transport
is dominant for vacuum gaps smaller than $\sim$2 nm. The application of a bias
voltage significantly affects the dominant energy carriers as it increases the
phonon contribution mediated by the long-range Coulomb force and the electron
contribution due to a lower potential barrier. For a bias voltage of 0.6 V,
acoustic phonon transport becomes dominant at a vacuum gap of 5 nm, whereas
electron tunneling dominates at sub-1-nm vacuum gaps. The comparison of the
theory against experimental data from the literature suggests that well-
controlled measurements between metallic surfaces are needed to quantify the
contributions of acoustic phonon and electron as a function of the bias
voltage.

Radiative heat transfer between two surfaces separated by a sub-wavelength
vacuum gap can exceed the far-field blackbody limit by a few orders of
magnitude owing to tunneling of evanescent electromagnetic (EM) waves Polder
and Van Hove (1971). Theoretical predictions of near-field radiative heat
transfer based on fluctuational electrodynamics Rytov et al. (1989) are well-
established, and have been experimentally validated in various configurations
for nanosized vacuum gaps Rousseau et al. (2009); Shen et al. (2009); Song et
al. (2015, 2016); St-Gelais et al. (2016); Bernardi et al. (2016); Watjen et
al. (2016); Ito et al. (2017); Fiorino et al. (2018); Lim et al. (2018);
Ghashami et al. (2018); DeSutter et al. (2019); Shi et al. (2019); Tang et al.
(2020); Ying et al. (2020). However, fluctuational electrodynamics may not be
able to accurately describe heat transfer in the extreme near-field regime,
defined here as sub-10-nm vacuum gap distances, due to its macroscopic nature
involving local field averaging (i.e., local dielectric function) Chiloyan et
al. (2015). In addition, fluctuational electrodynamics solely accounts for EM
waves (i.e., radiation), whereas acoustic phonons and electrons may also
contribute to heat transport between surfaces separated by single-digit
nanometer vacuum gaps prior to contact. A few theoretical works have
investigated acoustic phonon transport across vacuum gaps Prunnila and Meltaus
(2010); Ezzahri and Joulain (2014); Xiong et al. (2014); Chiloyan et al.
(2015); Pendry et al. (2016); Sasihithlu et al. (2017); Zhang et al. (2018);
Messina et al. ; Alkurdi et al. (2020), and some have explicitly shown the
inadequacy of fluctuational electrodynamics for modeling heat transfer in the
extreme near field Ezzahri and Joulain (2014); Chiloyan et al. (2015); Pendry
et al. (2016); Sasihithlu et al. (2017); Zhang et al. (2018); Messina et al. ;
Alkurdi et al. (2020).

Experimental measurements of extreme near-field heat transfer are scarce
Kittel et al. (2005); Altfeder et al. (2010); Kim et al. (2015); Kloppstech et
al. (2017); Cui et al. (2017); Jarzembski et al. . Jarzembski et al.
Jarzembski et al.  measured a thermal conductance exceeding fluctuational
electrodynamics predictions by three orders of magnitude between a silicon tip
and a platinum nanoheater separated by sub-10-nm vacuum gaps down to contact.
By considering all energy carriers, it was shown quantitatively that heat
transfer across vacuum gaps was dominated by acoustic phonon transport
mediated by van der Waals and Coulomb forces. Extreme near-field heat transfer
between metallic surfaces have also been measured, but drastically different
results have been reported. Kim et al. Kim et al. (2015) measured heat
transfer between a thermocouple-embedded tip fabricated at the free-end of a
stiff cantilever and a suspended resistive microheater at 305 K. Their
measurements between a gold (Au)-coated tip and an Au surface are in good
agreement with fluctuational electrodynamics down to a vacuum gap of $\sim$2
nm. On the other hand, Kloppstech et al. Kloppstech et al. (2017) reported the
measurement of heat transfer between an Au thermocouple-embedded scanning
tunneling microscope tip and a cryogenically-cooled Au surface, observing heat
transfer largely exceeding fluctuational electrodynamics for vacuum gaps from
7 nm down to 0.2 nm. However, no comprehensive modeling was performed to
quantitatively support their observation.

Subsequently, Cui et al. Cui et al. (2017) measured heat transfer between an
Au-coated tip and an Au surface, both subjected to various surface-cleaning
procedures, for vacuum gaps from 5 nm down to a few Å. They hypothesized that
the large heat transfer reported by Kloppstech et al. Kloppstech et al. (2017)
may be due to a low apparent potential barrier for electron tunneling mediated
by surface contaminants bridging the tip and the surface prior to bulk
contact.

A few theoretical works have analyzed extreme near-field heat transfer between
Au surfaces Pendry et al. (2016); Zhang et al. (2018); Messina et al. ;
Alkurdi et al. (2020). Using a surface perturbation approach, Pendry et al.
Pendry et al. (2016) predicted that the heat transfer coefficient due to
acoustic phonon exceeds that obtained with fluctuational electrodynamics for
vacuum gaps smaller than 0.4 nm. Alkurdi et al. Alkurdi et al. (2020) compared
the contributions of radiation, phonons and electrons to heat transfer between
Au surfaces separated by vacuum gaps varying from 1.5 nm down to 0.2 nm. Using
a three-dimensional (3D) lattice dynamics framework, it was found that
acoustic phonon transport exceeds radiation for all vacuum gaps considered,
whereas electron tunneling slightly surpasses the phonon contribution for
vacuum gaps smaller than 0.4 nm. In stark contrast, Messina et al. Messina et
al.  predicted that acoustic phonon transport does not play any role in
extreme near-field heat transfer between Au surfaces. Their results suggested
that for vacuum gaps smaller than $\sim$1 nm, electron tunneling largely
dominates heat transport, whether or not a bias voltage is applied between the
two surfaces.

Clearly, the mechanisms driving heat transfer between metallic surfaces
separated by sub-10-nm vacuum gaps are not well understood. The objective of
this work is to elucidate the physics underlying extreme near-field heat
transfer between Au surfaces, and to determine vacuum gap ranges in which
radiation, phonon, and electron are the dominant carriers. This is achieved by
implementing a comprehensive model that accounts for all energy carriers, and
by comparing theoretical predictions against the experimental data from Refs.
Kim et al. (2015); Kloppstech et al. (2017); Messina et al. . It is shown that
in the absence of bias voltage, thermal transport is mediated by acoustic
phonon transport for vacuum gaps smaller than $\sim$2 nm. In addition, the
results demonstrate that the application of a bias voltage enhances the
contributions of acoustic phonon and electron tunneling owing to the long-
range Coulomb force and the lower potential barrier, respectively.

To compare the theory against experimental results, the thermal conductance
$G$ between a tip and a surface separated by a vacuum gap of thickness $d$ is
calculated from local heat transfer coefficients $h$ between two parallel
surfaces, modeled as semi-infinite layers, using the Derjaguin approximation
Rousseau et al. (2009); Derjaguin (1934):

$G\ =\ \int_{0}^{r_{\mathrm{tip}}}drh(\tilde{d}){2\pi}r$ (1)

where $r$ is the radial direction, $r_{\mathrm{tip}}$ is the tip radius, and
$\tilde{d}=d+r_{\mathrm{tip}}-\sqrt{{r_{\mathrm{tip}}}^{2}-r^{2}}$ is the
vacuum gap distance between the surfaces (see Fig. 1). The heat transfer
coefficient in Eq. (1) is the sum of contributions from radiation, phonon, and
electron, as detailed hereafter.

Figure 1: The thermal conductance between a tip and a surface separated by a
vacuum gap $d$ is calculated from the heat transfer coefficients between two
parallel surfaces, modeled as semi-infinite layers, separated by a vacuum gap
$\tilde{d}$ using the Derjaguin approximation. Top-most atoms are located at
the Au-vacuum interfaces. The interatomic vacuum distance $d^{\prime}$ used
for predicting acoustic phonon transport is the same as the vacuum gap
thickness $\tilde{d}$ used for radiation and electron tunneling calculations.
The left (L) and right (R) semi-infinite layers are assumed to be at constant
and uniform temperatures $T_{\mathrm{L}}$ and $T_{\mathrm{R}}$.

The heat transfer coefficient due to radiation between two parallel surfaces
(media L and R) separated by a vacuum gap of thickness $\tilde{d}$ (medium 0)
is calculated using fluctuational electrodynamics to account for the near-
field effects Polder and Van Hove (1971); Rytov et al. (1989):

$h_{\mathrm{rad}}\ =\
\frac{1}{\pi^{\mathrm{2}}(T_{\mathrm{L}}-T_{\mathrm{R}})}\int_{0}^{\infty}d{\omega}\left[\Theta\left(\omega,T_{\mathrm{L}}\right)-\Theta\left(\omega,T_{\mathrm{R}}\right)\right]\\\
\\\
\int_{0}^{\infty}dk_{\mathrm{\rho}}k_{\mathrm{\rho}}\sum_{\gamma=\mathrm{TE,TM}}{\cal
T_{\mathrm{rad}}^{\gamma}}\left(\omega,k_{\mathrm{\rho}}\right)$ (2)

where $\omega$ is the angular frequency, $k_{\mathrm{\rho}}$ is the component
of the wavevector parallel to an interface, and $\Theta(\omega,T)$ is the mean
energy of an EM state calculated as
$\hbar\omega/[\mathrm{exp}({\hbar\omega/k_{\mathrm{b}}T})-1]$. The
transmission functions in polarization state $\gamma$ for propagating
($k_{\mathrm{\rho}}$ $\leq$ $k_{\mathrm{0}}$) and evanescent
($k_{\mathrm{\rho}}$ $>$ $k_{\mathrm{0}}$) EM waves in vacuum are respectively
given by:

${\cal T_{\mathrm{rad,prop}}^{\gamma}}\left(\omega,k_{\mathrm{\rho}}\right)\
=\frac{\left(1-\left|r_{\mathrm{0L}}^{\gamma}\right|^{2}\right)\left(1-\left|r_{\mathrm{0R}}^{\gamma}\right|^{2}\right)}{4\left|1-r_{\mathrm{0L}}^{\gamma}r_{\mathrm{0R}}^{\gamma}e^{2i\mathrm{Re}\left(k_{\mathrm{z0}}\right)\tilde{d}}\right|^{2}}$
(3) ${\cal
T_{\mathrm{rad,evan}}^{\gamma}}\left(\omega,k_{\mathrm{\rho}}\right)\
=e^{-2\mathrm{Im}\left(k_{\mathrm{z0}}\right)\tilde{d}}\frac{\mathrm{Im}\left(r_{\mathrm{0L}}^{\gamma}\right)\mathrm{Im\left(r_{\mathrm{0R}}^{\gamma}\right)}}{\left|1-r_{\mathrm{0L}}^{\gamma}r_{\mathrm{0R}}^{\gamma}e^{-2\mathrm{Im}\left(k_{\mathrm{z0}}\right)\tilde{d}}\right|^{2}}$
(4)

where $k_{\mathrm{z0}}$ is the component of the vacuum wavevector
perpendicular to an interface, and $r_{0j}^{\gamma}$
($j=\mathrm{L},\mathrm{R}$) is the Fresnel reflection coefficient Yeh (1988).
The frequency-dependent dielectric function of Au is calculated using the
Drude model provided in Ref. Chapuis et al. (2008).

Acoustic phonons can tunnel across vacuum gaps via force interactions.
Acoustic phonon transport is modeled using the one-dimensional (1D) atomistic
Green’s function (AGF) method Zhang et al. (2007); Sadasivam et al. (2014).
The heat transfer coefficient due to acoustic phonon transport across an
interatomic vacuum distance $d^{\prime}$ for a 1D atomic chain is written as:

$h_{\mathrm{ph}}\ =\
\frac{1}{A(T_{\mathrm{L}}-T_{\mathrm{R}})}\int_{0}^{\infty}d{\omega}\frac{\hbar\omega}{2\pi}{\cal
T_{\mathrm{ph}}}({\omega})[N(\omega,T_{\mathrm{L}})-N(\omega,T_{\mathrm{R}})]$
(5)

where $N=1/[\mathrm{exp}({\hbar\omega/k_{\mathrm{b}}T})-1]$ is the Bose-
Einstein distribution function and $A$ is the cross-sectional area of an atom.
The atomic radius of Au is taken as 1.740 Å Clementi et al. (1967). The top-
most atoms are located at the Au-vacuum interfaces Nakamura et al. (1997),
such that the interatomic vacuum distance $d^{\prime}$ is the same as the
vacuum gap thickness $\tilde{d}$ used for radiation and electron tunneling
calculations (see Fig. 1). Note that the lattice constant of Au (4.065 Å) is
used as the criterion for bulk contact. The minimum distance at which the heat
transfer coefficient is calculated is therefore 4.065 Å. The phonon
transmission function is given by the Caroli formula Caroli et al. (1971):

${\cal T_{\mathrm{ph}}}({\omega})\ =\
\mathrm{Trace}[\Gamma_{\mathrm{L}}G_{\mathrm{d}}\Gamma_{\mathrm{R}}G_{\mathrm{d}}^{\mathrm{\dagger}}]$
(6)

where the superscript $\dagger$ denotes conjugate transpose,
$\Gamma_{\mathrm{L,R}}$ is the escape rate of phonons from the device region
to the semi-infinite layers, and $G_{\mathrm{d}}$ is the Green’s function of
the device region. In the present study, the device region encompasses the
vacuum gap and five atoms in each of the semi-infinite layers. Increasing the
number of atoms in the device region does not affect the phonon transmission
function. Details about the 1D AGF method have been provided in Refs. Zhang et
al. (2007); Sadasivam et al. (2014); Jarzembski et al. .

For the case of two Au surfaces, the short- and long-range interactions across
the vacuum gap are respectively described by the Lennard-Jones and Coulomb
force models Chiloyan et al. (2015); Alkurdi et al. (2020). The Lennard-Jones
model accounts for the van der Waals force and overlapping electron cloud
repulsive force, whereas the Coulomb force is effective only when there is a
bias voltage Horn and Smith (1992); Heinze et al. (1999); Lang and Kohn
(1973). The rationale behind the selection of these forces and the force
models are provided in Sec. S1 of the Supplemental Material sup .

Electrons can contribute to heat transfer across vacuum gaps via tunneling and
thermionic emission. Owing to the low temperatures considered ($\sim$300 K),
the contribution of thermionic emission is negligible. The heat transfer
coefficient due to electron tunneling across a vacuum gap can be written as
Hishinuma et al. (2001); Westover and Fisher (2008); Wang et al. (2016):

$\begin{split}h_{\mathrm{el}}\ =\
\frac{1}{(T_{\mathrm{L}}-T_{\mathrm{R}})}\int_{-\infty}^{W_{\mathrm{max}}}dE_{z}[&\left(E_{z}+{k_{\mathrm{b}}}{T_{\mathrm{L}}}\right)N_{\mathrm{L}}\left(E_{z},T_{\mathrm{L}}\right)\\\
-&\left(E_{z}+k_{\mathrm{b}}{T_{\mathrm{R}}}\right)N_{\mathrm{R}}\left(E_{z}-eV_{\mathrm{bias}},T_{\mathrm{R}}\right)]{\cal
T_{\mathrm{el}}}\left(E_{z}\right)\end{split}$ (7)

where $e=1.602{\times}10^{-19}{\,}\mathrm{C}$ is the electron charge,
$V_{\mathrm{bias}}$ is the bias voltage, $E_{z}$ is the electron energy
perpendicular to the surfaces, and $W_{\mathrm{max}}$ is the maximum potential
barrier. The term $N_{j}(E_{z},T_{j})$, denoting the number of electrons at
energy level $E_{z}$ per unit area and per unit time, is calculated as
Hishinuma et al. (2001):

$N_{j}\left(E_{z},T_{j}\right)\ =\
\frac{m_{\mathrm{e}}k_{\mathrm{b}}T_{j}}{2{\pi}^{2}\hbar^{3}}\mathrm{ln}\left[1+\mathrm{exp}\left(-\frac{E_{z}-E_{F,j}}{k_{\mathrm{b}}T_{j}}\right)\right]$
(8)

where $m_{\mathrm{e}}=\mathrm{9.109\times 10^{-31}{\,}kg}$ is the electron
mass, and $E_{F,j}$ is the Fermi level used as a reference for computing the
electron energy Hishinuma et al. (2001). The electron transmission function is
calculated using the Wentzel-Kramers-Brillouin (WKB) approximation Berry and
Mount (1972):

${\cal T_{\mathrm{el}}}\left(E_{z}\right)\ =\
\mathrm{exp}\left[-\frac{\sqrt{8m_{\mathrm{e}}}}{\hbar}\int_{z_{1}}^{z_{2}}dz\sqrt{W\left(z\right)-E_{z}}\right]$
(9)

where $W(z)$ is the potential barrier profile in the vacuum gap, while $z_{1}$
and $z_{2}$ are the roots of $W(z)-E_{z}=0$ delimiting the width of the
electron tunneling barrier $E_{z}$. The potential barrier profile can be
expressed as Hishinuma et al. (2001):

$W(z)=W_{id}(z)+W_{ic}(z)$ (10)

Note that the space-charge effect is assumed to be fully suppressed for
sub-10-nm vacuum gaps Wang et al. (2016). The ideal barrier profile Wang et
al. (2016) and image-charge perturbation Bâldea and Köppel (2012) are
respectively calculated as:

$W_{id}(z)=\Phi_{\mathrm{L}}-(\Phi_{\mathrm{L}}-\Phi_{\mathrm{R}}-eV_{\mathrm{bias}})\left(\frac{z}{\tilde{d}}\right)$
(11)

$W_{ic}(z)=\frac{e^{2}}{16{\pi}{\epsilon}_{0}d}\left[-2{\Psi}\left(1\right)+{\Psi}\left(\frac{z}{\tilde{d}}\right)+{\Psi}\left(1-\frac{z}{\tilde{d}}\right)\right]$
(12)

where $\Phi_{\mathrm{L},\mathrm{R}}=5.10{~{}}\mathrm{eV}$ is the work function
of Au Zhou et al. (2012) and $\Psi$ is the digamma function.

The WKB approximation may lead to an overestimation of the electron heat
transfer coefficient below vacuum gaps of $\sim$0.5 nm Olesen et al. (1996);
Westover and Fisher (2008). As such, for small vacuum gaps, the electron heat
transfer coefficient is also calculated via the 1D non-equilibrium Green’s
function (NEGF) approach described in Ref. Westover and Fisher (2008). In the
NEGF framework, the electron transmission function is given by:

${\cal T_{\mathrm{el}}}(E_{z})\ =\
\mathrm{Trace}[\Gamma_{\mathrm{L}}G^{\mathrm{R}}\Gamma_{\mathrm{R}}\left(G^{\mathrm{R}}\right)^{\mathrm{\dagger}}],$
(13)

where $\Gamma_{\mathrm{L,R}}$ is the energy broadening matrix, whereas
$G^{\mathrm{R}}$ is the retarded Green’s function.

Figure 2: (a) Heat transfer coefficients due to radiation, phonon, and
electron between two Au surfaces separated by a vacuum gap $\tilde{d}$
($T_{\mathrm{{L}}}$ = 305 K, $T_{\mathrm{{R}}}$ = 300 K, $V_{\mathrm{bias}}$ =
0 V). The electron heat transfer coefficient is calculated via the WKB
approximation and the NEGF. The phonon heat transfer coefficient predicted via
the 1D AGF method is compared against the 3D lattice dynamics results of
Alkurdi et al. Alkurdi et al. (2020). (b) Thermal conductance between a
450-nm-radius tip (300 K) and a surface (305 K), both made of Au, separated by
a vacuum gap $d$. The total conductance is calculated using the electron heat
transfer coefficient from the WKB approximation. The predicted conductance is
compared against the experimental results of Kim et al. Kim et al. (2015).

The heat transfer coefficients between two Au surfaces due to radiation,
acoustic phonon transport, and electron tunneling are presented in Fig. 2(a)
for the experimental conditions taken from Kim et al. Kim et al. (2015), where
$T_{\mathrm{{L}}}$ = 305 K, $T_{\mathrm{{R}}}$ = 300 K, and
$V_{\mathrm{bias}}$ = 0 V. The Au-vacuum interfaces support surface plasmon
polaritons at a high frequency ($\sim\mathrm{1.2\times 10^{16}}$ rad/s) that
cannot be thermally excited at room temperature. As such, the radiation heat
transfer coefficient saturates as the vacuum gap decreases Chapuis et al.
(2008). It should be noted that while fluctuational electrodynamics is not
expected to be valid for sub-2-nm vacuum gaps owing to non-local effects
Chapuis et al. (2008), radiation predictions below 2 nm vacuum gaps are shown
throughout the paper for reference. The accuracy of the 1D AGF approach for
predicting acoustic phonon transport between Au surfaces is verified in Fig.
2(a) by comparison against the results obtained from a 3D lattice dynamics
framework Alkurdi et al. (2020). In the absence of bias voltage, only the van
der Waals and overlapping electron cloud forces, represented by the Lennard-
Jones model, contribute to acoustic phonon transport. The phonon and electron
heat transfer coefficients exceed that of radiation for vacuum gaps smaller
than $\sim$2 nm and $\sim$1 nm, respectively. Provided that the NEGF
calculates the electron transmission function more accurately than the WKB
approximation for vacuum gaps smaller than $\sim$0.5 nm, the results indicate
that the phonon contribution always exceed that of the electron for sub-2-nm
vacuum gaps near room temperature.

Figure 2(b) shows the predicted thermal conductance between a 450-nm-radius Au
tip at 300 K and an Au surface at 305 K by applying the Derjaguin
approximation with the heat transfer coefficients shown in Fig. 2(a). These
predictions are compared against the conductance measured in Ref. Kim et al.
(2015) for vacuum gaps down to $\sim$2 nm. Clearly, for vacuum gaps larger
than $\sim$1.5 nm, heat transfer is mediated by radiation. The calculated
total conductance is in good agreement with the experimental data of Ref. Kim
et al. (2015). The results from the comprehensive model presented in this work
therefore support the conclusions of Kim et al. Kim et al. (2015). Vacuum gaps
approximately equal to or smaller than 1 nm would have been required to
observe a significant enhancement of the conductance due to acoustic phonon
transport.

Figure 3(a) presents the heat transfer coefficients between two Au surfaces
due to radiation, acoustic phonon transport, and electron tunneling for the
experimental conditions taken from Kloppstech et al. Kloppstech et al. (2017),
where $T_{\mathrm{L}}$ = 280 K, $T_{\mathrm{R}}$ = 120 K, and
$V_{\mathrm{bias}}$ = 0.6 V. It should be noted that an electric field
generated by a bias voltage not only enhances electron tunneling but also
induces surface charges Lang and Kohn (1973); Heinze et al. (1999), which are
at the origin of the Coulomb force Horn and Smith (1992). Since the long-range
Coulomb force can drive acoustic phonon transport across vacuum gaps Chiloyan
et al. (2015), the surface charge density under a bias voltage should be
determined and taken into account for the calculations. Jarzembski et al.
Jarzembski et al.  estimated the surface charge density of a biased platinum
surface based on their experimental conditions. A surface charge density of
$\mathrm{8\times 10^{-4}~{}C/{m^{2}}}$ was estimated for a bias voltage of 0.8
V. In the present work, a value of $\mathrm{6\times 10^{-4}~{}C/{m^{2}}}$ is
used under the assumption that the surface charge density varies linearly with
the bias voltage Chen et al. (1999); Liu et al. (2020). Note that the phonon
heat transfer coefficient is calculated with and without bias voltage, and the
shaded area in Fig. 3(a) shows its possible values.

Figure 3: (a) Heat transfer coefficients due to radiation, phonon, and
electron between two Au surfaces separated by a vacuum gap $\tilde{d}$
($T_{\mathrm{{L}}}$ = 280 K, $T_{\mathrm{{R}}}$ = 120 K, $V_{\mathrm{bias}}$ =
0.6 V). The electron heat transfer coefficient is calculated via the WKB
approximation and the NEGF. The phonon heat transfer coefficient is calculated
both with and without bias voltage. (b) Thermal conductance between a 30-nm-
radius tip (280 K) and a surface (120 K), both made of Au, separated by a
vacuum gap $d$. The total conductance is calculated using the electron heat
transfer coefficient from the WKB approximation. The predicted conductance is
compared against the experimental results of Kloppstech et al. Kloppstech et
al. (2017) and Messina et al. Messina et al. .

Owing to the long-range Coulomb force, the phonon heat transfer coefficient
exceeds that of radiation for vacuum gaps smaller than $\sim$5 nm. The Coulomb
force, however, has a low impact on the phonon heat transfer coefficient for
vacuum gaps smaller than 1 nm (see Sec. S1 of the Supplemental Material for
more details sup ). The electron heat transfer coefficient is enhanced due to
the bias voltage that lowers the potential barrier for tunneling. The WKB
approximation suggests that electron-mediated heat transfer exceeds acoustic
phonon transport for vacuum gaps smaller than 1 nm. The NEGF also suggests
that the electron heat transfer coefficient is slightly larger than that of
phonon, but for sub-0.5-nm vacuum gaps.

The thermal conductance calculated between a 30-nm-radius Au tip and an Au
surface is shown in Fig. 3(b) under the same experimental conditions as in
Fig. 3(a) (temperatures of 280 and 120 K, $V_{\mathrm{bias}}$ = 0.6 V). The
results are compared against the experimental data of Kloppstech et al.
Kloppstech et al. (2017), in addition to a more recent set of data from the
same group (Messina et al. Messina et al. ). Note that the most recent
experimental data were obtained for surface and tip temperatures of
respectively 195 K and 295 K. These slight temperature differences have no
noticeable impact on the heat transfer coefficients (see Sec. S2 of the
Supplemental Material sup ). The total conductance, which include the
radiation, phonon, and electron contributions, display three distinct regimes.
For vacuum gaps larger than 5 nm, radiation transport via evanescent EM waves
drives heat transfer. For vacuum gaps from 5 nm down to $\sim$1.5 nm,
radiation transitions to acoustic phonon-mediated heat transfer. Acoustic
phonon transport becomes dominant between vacuum gaps of 1.5 nm and 0.9 nm.
For vacuum gaps smaller than 0.9 nm, electron tunneling drives heat transfer
owing to the applied bias voltage. The magnitude of the experimental and
theoretical data are quite different. Nevertheless, a power law analysis may
enable determining the energy carriers driving heat transport in the
experiments of Refs. Kloppstech et al. (2017); Messina et al. . In the vacuum
gap range of 1 nm to 1.5 nm where acoustic phonon transport is dominant, the
predicted total conductance follows a $d^{-3.2}$ power law. Within that range,
the data of Messina et al. Messina et al.  vary as $d^{-2.9}$, which is close
to the theory. This regime arises due to the Coulomb force induced by the bias
voltage. For vacuum gaps smaller than 0.9 nm where electron tunneling
dominates heat transport, the total conductance follows a $d^{-12.6}$ power
law. Interestingly, a similar trend is observed in the experimental data of
Kloppstech et al. Kloppstech et al. (2017) but at much larger vacuum gaps
($d\gtrsim$ 5 nm).

Figure 4: Thermal conductance between a 30-nm-radius tip (280 K) and a surface
(120 K), both made of Au, separated by a vacuum gap $d$. The electron heat
transfer coefficient is calculated with the WKB approximation using an
apparent potential barrier of 1 eV. The phonon heat transfer coefficient is
calculated with a bias voltage of 0.6 V. The predicted conductance is compared
against the experimental results of Messina et al. Messina et al. .

It has been hypothesized that contaminants bridging the tip and the surface
prior to bulk contact may lower the potential barrier for electron tunneling
Cui et al. (2017). Using their measured tunneling current, Messina et al.
Messina et al.  estimated an apparent potential barrier of 1 eV, which induces
an enhancement of the electron heat transfer coefficient. Fig. 4 shows the
thermal conductance predicted by considering a simple potential barrier of 1
eV for electron tunneling. As expected, electron tunneling for that case
becomes dominant at a larger vacuum gap ($\sim$1.5 nm). The magnitude of the
total conductance is closer to the experimental data of Messina et al. Messina
et al. , but the trends are different. Electron-mediated heat transport
follows a $d^{-10.2}$ power law, which is not experimentally observed. In the
gap range between 1.5 nm and 2 nm where all energy carriers contribute to the
conductance, a $d^{-2.7}$ power law is predicted. This is again close to the
$d^{-2.9}$ power law of Messina et al. Messina et al.  which however arises in
a slightly different vacuum gap range (1 nm to 1.5 nm). It is concluded that
the measured conductance reported by Messina et al. Messina et al.  is not
solely due to electron tunneling. Heat transport for that case is likely to be
a combination of acoustic phonon transport, enhanced via the bias voltage, and
electron tunneling possibly enhanced by the low apparent potential barrier
mediated by contaminants. It has been pointed out that the data of Kloppstech
et al. Kloppstech et al. (2017) were obtained at different apparent potential
barrier heights Messina et al. , which make their interpretation challenging.
Yet, the variation of the conductance with respect to the vacuum gap suggests
electron-mediated heat transfer.

In summary, a comprehensive model combining fluctuational electrodynamics for
radiation, the AGF method for phonon, and the WKB approximation in addition to
the NEGF approach for electron has been implemented to analyze extreme near-
field heat transfer between Au surfaces. In the absence of bias voltage and
near room temperature, heat transfer is mediated by radiation through
evanescent EM waves down to a vacuum gap of $\sim$2 nm, below which acoustic
phonon transport is the dominant energy carrier. It was shown that the
application of a bias voltage increases not only electron tunneling, but also
acoustic phonon transport mediated by the long-range Coulomb force. As a
result, for a bias voltage of 0.6 V, acoustic phonon transport can affect heat
transfer for vacuum gaps from 5 nm down to $\sim$1 nm, and electron tunneling
should be a responsible heat transfer mechanism below 1 nm. However,
comparison of the comprehensive model against state-of-the-art measurements
Kim et al. (2015); Kloppstech et al. (2017); Messina et al.  reveals that
additional data are needed to experimentally quantify the contributions of
acoustic phonon and electron as a function of the bias voltage in the heat
transfer between Au surfaces separated by sub-10-nm vacuum gaps.

This work was supported by the National Science Foundation (grants
CBET-1605584 and CBET-1952210). T.T. also appreciate the financial support by
the Yamada Science Foundation and the Fujikura Foundation.

## References

  * Polder and Van Hove (1971) D. Polder and M. Van Hove, Phys. Rev. B 4, 3303 (1971).
  * Rytov et al. (1989) S. M. Rytov, Y. A. Kravtsov, and V. I. Tatarskii, _Principles of Statistical Radiophysics. 3 Elements of Random Fields_ (Springer, New York, 1989).
  * Rousseau et al. (2009) E. Rousseau, A. Siria, G. Jourdan, S. Volz, F. Comin, J. Chevrier, and J. J. Greffet, Nat. Photonics 3, 514 (2009).
  * Shen et al. (2009) S. Shen, A. Narayanaswamy, and G. Chen, Nano Lett. 9, 2909 (2009).
  * Song et al. (2015) B. Song, Y. Ganjeh, S. Sadat, D. Thompson, A. Fiorino, V. Fernández-Hurtado, J. Feist, F. J. Garcia-Vidal, J. C. Cuevas, P. Reddy, et al., Nat. Nanotechnol. 10, 253 (2015).
  * Song et al. (2016) B. Song, D. Thompson, A. Fiorino, Y. Ganjeh, P. Reddy, and E. Meyhofer, Nat. Nanotechnol. 11, 509 (2016).
  * St-Gelais et al. (2016) R. St-Gelais, L. Zhu, S. Fan, and M. Lipson, Nat. Nanotechnol. 11, 515 (2016).
  * Bernardi et al. (2016) M. P. Bernardi, D. Milovich, and M. Francoeur, Nat. Commun. 7, 12900 (2016).
  * Watjen et al. (2016) J. I. Watjen, B. Zhao, and Z. M. Zhang, Appl. Phys. Lett. 109, 203112 (2016).
  * Ito et al. (2017) K. Ito, K. Nishikawa, A. Miura, H. Toshiyoshi, and H. Iizuka, Nano Lett. 17, 4347 (2017).
  * Fiorino et al. (2018) A. Fiorino, D. Thompson, L. Zhu, B. Song, P. Reddy, and E. Meyhofer, Nano Lett. 18, 3711 (2018).
  * Lim et al. (2018) M. Lim, J. Song, S. S. Lee, and B. J. Lee, Nat. Commun. 9, 4302 (2018).
  * Ghashami et al. (2018) M. Ghashami, H. Geng, T. Kim, N. Iacopino, S. K. Cho, and K. Park, Phys. Rev. Lett. 120, 175901 (2018).
  * DeSutter et al. (2019) J. DeSutter, L. Tang, and M. Francoeur, Nat. Nanotechnol. 14, 751 (2019).
  * Shi et al. (2019) K. Shi, Y. Sun, Z. Chen, N. He, F. Bao, J. Evans, and S. He, Nano Lett. 19, 8082 (2019).
  * Tang et al. (2020) L. Tang, J. DeSutter, and M. Francoeur, ACS Photonics 7, 1304 (2020).
  * Ying et al. (2020) X. Ying, P. Sabbaghi, N. Sluder, and L. Wang, ACS Photonics 7, 190 (2020).
  * Chiloyan et al. (2015) V. Chiloyan, J. Garg, K. Esfarjani, and G. Chen, Nat. Commun. 6, 6755 (2015).
  * Prunnila and Meltaus (2010) M. Prunnila and J. Meltaus, Phys. Rev. Lett. 105, 125501 (2010).
  * Ezzahri and Joulain (2014) Y. Ezzahri and K. Joulain, Phys. Rev. B 90, 115433 (2014).
  * Xiong et al. (2014) S. Xiong, K. Yang, Y. A. Kosevich, Y. Chalopin, R. D’Agosta, P. Cortona, and S. Volz, Phys. Rev. Lett. 112, 114301 (2014).
  * Pendry et al. (2016) J. B. Pendry, K. Sasihithlu, and R. V. Craster, Phys. Rev. B 94, 075414 (2016).
  * Sasihithlu et al. (2017) K. Sasihithlu, J. B. Pendry, and R. V. Craster, Z. Naturforsch. A 72, 181 (2017).
  * Zhang et al. (2018) Z. Q. Zhang, J. T. Lü, and J. S. Wang, Phys. Rev. B 97, 195450 (2018).
  * (25) R. Messina, S.-A. Biehs, T. Ziehm, A. Kittel, and P. Ben-Abdallah, eprint arXiv:1810.02628v1.
  * Alkurdi et al. (2020) A. Alkurdi, C. Adessi, S. Li, K. Termentzidis, and S. Merabia, Int. J. Heat Mass Transfer 158, 119963 (2020).
  * Kittel et al. (2005) A. Kittel, W. Muller-Hirsch, J. Parisi, S. A. Biehs, D. Reddig, and M. Holthaus, Phys. Rev. Lett. 95, 224301 (2005).
  * Altfeder et al. (2010) I. Altfeder, A. A. Voevodin, and A. K. Roy, Phys. Rev. Lett. 105, 166101 (2010).
  * Kim et al. (2015) K. Kim, B. Song, V. Fernández-Hurtado, W. Lee, W. Jeong, L. Cui, D. Thompson, J. Feist, M. T. Reid, F. J. García-Vidal, et al., Nature 528, 387 (2015).
  * Kloppstech et al. (2017) K. Kloppstech, N. Könne, S. A. Biehs, A. W. Rodriguez, L. Worbes, D. Hellmann, and A. Kittel, Nat. Commun. 8, 14475 (2017).
  * Cui et al. (2017) L. Cui, W. Jeong, V. Fernández-Hurtado, J. Feist, F. J. García-Vidal, J. C. Cuevas, E. Meyhofer, and P. Reddy, Nat. Commun. 8, 14479 (2017).
  * (32) A. Jarzembski, T. Tokunaga, J. Crossley, J. Yun, C. Shaskey, R. A. Murdick, I. Park, M. Francoeur, and K. Park, eprint arXiv:1904.09383.
  * Derjaguin (1934) B. Derjaguin, Colloid Polym. Sci. 69, 155 (1934).
  * Yeh (1988) P. Yeh, _Optical Waves in Layered Media_ (Wiley-VCH, 1988).
  * Chapuis et al. (2008) P.-O. Chapuis, S. Volz, C. Henkel, K. Joulain, and J.-J. Greffet, Phys. Rev. B 77, 035431 (2008).
  * Zhang et al. (2007) W. Zhang, T. S. Fisher, and N. Mingo, ASME J. Heat Transfer 129, 483 (2007).
  * Sadasivam et al. (2014) S. Sadasivam, Y. Che, Z. Huang, L. Chen, S. Kumar, and T. S. Fisher, Annu. Rev. Heat Transf. 17, 89 (2014).
  * Clementi et al. (1967) E. Clementi, D. L. Raimondi, and W. P. Reinhardt, J. Chem. Phys. 47, 1300 (1967).
  * Nakamura et al. (1997) K. Nakamura, H. Hasegawa, T. Oguchi, K. Sueoka, K. Hayakawa, and K. Mukasa, Phys. Rev. B 56, 3218 (1997).
  * Caroli et al. (1971) C. Caroli, R. Combescot, P. Nozieres, and D. Saint-James, J. Phys. C 4, 916 (1971).
  * Horn and Smith (1992) R. G. Horn and D. T. Smith, Science 256, 362 (1992).
  * Heinze et al. (1999) S. Heinze, X. Nie, S. Blügel, and M. Weinert, Chem. Phys. Lett 315, 167 (1999).
  * Lang and Kohn (1973) N. D. Lang and W. Kohn, Phys. Rev. B 7, 3541 (1973).
  * (44) See Supplemental Material at [URL will be inserted by publisher] for discussions on the Lennard-Jones and Coulomb force models and the impact of temperature on the heat transfer coefficient.
  * Hishinuma et al. (2001) Y. Hishinuma, T. H. Geballe, B. Y. Moyzhes, and T. W. Kenny, Appl. Phys. Lett. 78, 2572 (2001).
  * Westover and Fisher (2008) T. L. Westover and T. S. Fisher, Phys. Rev. B 77, 115426 (2008).
  * Wang et al. (2016) Y. Wang, T. Liao, Y. Zhang, X. Chen, S. Su, and J. Chen, J. Appl. Phys. 119, 045106 (2016).
  * Berry and Mount (1972) M. V. Berry and K. E. Mount, Rep. Prog. Phys. 35, 315 (1972).
  * Bâldea and Köppel (2012) I. Bâldea and H. Köppel, Phys. Status Solidi B Basic Res. 249, 1791 (2012).
  * Zhou et al. (2012) Y. Zhou, C. Fuentes-hernandez, J. Shim, J. Meyer, A. J. Giordano, H. Li, P. Winget, T. Papadopoulos, H. Cheun, J. Kim, et al., Science 336, 327 (2012).
  * Olesen et al. (1996) L. Olesen, M. Brandbyge, M. R. Sørensen, K. W. Jacobsen, E. Lægsgaard, I. Stensgaard, and F. Besenbacher, Phys. Rev. Lett. 76, 1485 (1996).
  * Chen et al. (1999) X. Q. Chen, H. Yamada, H. T., and M. K., J. Vac. Sci. Technol. B 17, 1930 (1999).
  * Liu et al. (2020) Y. Liu, W. Liu, Z. Wang, W. He, Q. Tang, Y. Xi, X. Wang, H. Guo, and C. Hu, Nat. Commun. 11, 1599 (2020).