# Heat transfer across a vacuum gap induced by piezoelectrically mediated
acoustic phonon tunneling

Zhuoran Geng<EMAIL_ADDRESS>Ilari J. Maasilta<EMAIL_ADDRESS>Nanoscience
Center, Department of Physics, University of Jyväskylä, FI-40014 Jyväskylä,
Finland

###### Abstract

In contradictin to the common concept that acoustic phonons can only travel
inside a material medium, they can in fact ”tunnel” across a vacuum gap with
the help of piezoelectricity, transmitting a significantly stronger heat flux
than that of blackbody radiation. Here, we present a theoretical formulation
for the heat flux of such piezoelectrically mediated heat transfer, applicable
to any anisotropic piezoelectric crystals with an arbitrary orientation. A few
numerical results are demonstrated and compared to heat transfer driven by
other close-range mechanisms, including near-field radiative heat transfer and
other acoustic phonon tunneling mechanisms. We find that piezoelectrically
mediated heat transfer has a significant effect when the vacuum gap size is
smaller than the phonon characteristic thermal wavelength, and its heat flux
can dominate heat transfer between piezoelectric solids over all other known
heat transfer mechanisms at temperatures below 50 K.

It is known that heat can be transfered between macroscopic bodies via three
different channels, namely, conduction, convection and radiation. Out of
those, radiation is the only possible channel between two materials separated
by a vacuum gap, and the corresponding heat flux is well understood based on
Planck’s law of radiation. However, when the separation between two material
bodies decreases, heat flux exceeding that of Planck’s law by several orders
of magnitude has been observed in experiments[1, 2, 3]. Among the mechanisms
that can lead to such super-Planckian radiation, near field radiative heat
transfer (NFRHT) mediated by photon tunneling[4, 5, 6, 7] is the most studied
one. Analogous to quantum mechanical tunneling, this photon tunneling becomes
relevant when the vacuum gap distance is below the photon thermal wavelength,
which is about $\sim 10\,\mu$m at room temperature.

With advances in nanotechnology, vacuum gaps with sizes in the nanometer-to
sub-nanometer range can be achieved in experiments[8, 9, 10, 11]. This has
stimulated active research in recent years also on the heat transfer driven by
the tunneling of acoustic phonons[12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22,
23, 11] in addition to photons, as their thermal wavelengths are at that
length scale. However, the concept of acoustic phonon tunneling is far from
obvious, as a phonon, being a vibration of the atomic lattice, requires the
presence of a medium to propagate. In the past decade, a few mechanisms that
can mediate acoustic phonon tunneling have been suggested, the van der Waals
force[16, 17, 18, 19] the electrostatic force[19, 20, 21], or the non-local
contribution of acoustic phonons to NFRHT in polar crystals [24].
Nevertheless, these studies show that the heat flux due to these mechanisms
decay rapidly with the gap width ($d^{-7}$ and $d^{-9}$ for van der Waals and
electrostatic mechanisms, respectively), and hence provide non-trivial
contributions to the heat transfer only when a vacuum gap $d<1$ nm at room
temperature[19].

There exists yet another, much less studied acoustic phonon tunneling
mechanism, which utilizes piezoelectricity. A thermally excited acoustic
phonon impinging on a free surface of a piezoelectric solid can create a
decaying, evanescent electric field leaking into the vacuum. Such field
couples to the lattice deformations of a second piezoelectric solid placed
within the phonon wavelength, leading to a transmission of heat across the
vacuum. Such piezoelectrically mediated heat transfer (PEMHT) was previously
suggested and studied in Ref. [12], however the theoretical estimation in that
study was based on a highly simplified model, in which the isotropic material
parameters were assumed and only two phonon modes were considered. Moreover,
their results are not in agreement with those derived from the more general
acoustic wave tunneling formalism for piezoelectric materials developed in
Refs.[25, 26], and hence the PEMHT phenomenon needs to be re-examined.

In this work, we present a general formulation for the piezoelectrically
mediated heat transfer via acoustic phonon tunneling, which can be applied to
arbitrarily anisotropic and oriented piezoelectric crystals. Numerical
examples are investigated and compared to other close-range heat transfer
mechanisms, including non-piezoelectric tunneling of acoustic phonons and
near-field radiative transfer (photon tunneling). We find that PEMHT can
dominate heat transfer at temperatures below 50 K. At the end, numerical
examples of PEMHT using different piezoelectric materials with varying crystal
orientations are discussed.

Figure 1: Two piezoelectric solids 1, 2, at temperatures $T_{1}$ and $T_{2}$,
are separated by a vacuum gap of width $d$. An incoming acoustic phonon (red)
with an incident zenith angle $\theta$ and azimuth angle $\varphi$ tunnels
across the vacuum from solid 1 and transfers heat to the adjacent solid 2. In
this study, we demonstrate results for hexagonal crystals, (ZnO,AlN), whose
orientation can be described by a rotation angle $\vartheta$ between the
crystal $c$-axis and the laboratory $z$-axis.

We consider two piezoelectric, semi-infinite solids which are placed parallel
to each other and separated by a vacuum gap of width $d$, as shown in Fig.1.
Both solids can be rotated to arbitrary orientations described by a set of
Euler angles denoting the relation between the intrinsic crystal coordinates
and the laboratory coordinates $xyz$ (see [27] for more details on crystal
orientation). In our model, these solids are assumed to be continuous and
anisotropic, satisfying linear elastic constitutive equations[28, 25].

We assume that the thermal acoustic phonons in solid 1 impinging on the gap
consist of three bulk acoustic wave modes $\alpha=1,2,3$[25], where each wave
mode can be described by a time-harmonic displacement field

$\pmb{u}_{\alpha}=b_{\alpha}\pmb{A}_{\alpha}\exp(i\omega_{\alpha}t-i\pmb{k}\cdot\pmb{r}),$
(1)

where $\pmb{k}$ is the wave vector, $\pmb{r}=(x,y,z)$ is the position vector
in the laboratory coordinates, $\omega_{\alpha}$ is the angular frequency of
the phonon mode, $\pmb{A}_{\alpha}$ is the Stroh-normalized[25] polarization
vector and $b_{\alpha}$ is the dimensionless amplitude of the mode[25].

In PEMHT, the emitted heat flux $J^{\textrm{PE}}$ from the solid $\gamma=1,2$
across the gap can then be expressed as

$\displaystyle J^{\textrm{PE}}_{\gamma}(T_{\gamma},d)=$
$\displaystyle\sum_{\alpha}\int\frac{\mathrm{d}^{3}k}{(2\pi)^{3}}\hbar\omega_{\alpha}(\pmb{k})n(\omega_{\alpha},T_{\gamma})$
(2)
$\displaystyle\times\bigg{(}\pmb{\hat{n}}_{\gamma}\cdot\frac{\partial\omega_{\alpha}}{\partial\pmb{k}}\bigg{)}\Theta\bigg{(}\pmb{\hat{n}}_{\gamma}\cdot\frac{\partial\omega_{\alpha}}{\partial\pmb{k}}\bigg{)}\mathcal{T}_{\alpha}(\theta,\varphi,k,d),$

where
$n(\omega_{\alpha},T_{\gamma})=[\exp(\hbar\omega_{\alpha}/k_{B}T_{\gamma})-1]^{-1}$
is the Bose-Einstein distribution describing the thermal occupation of the
phonon mode of energy $\hbar\omega_{\alpha}(\pmb{k})$,
$\pmb{\hat{n}}_{\gamma}$ is the _outward_ unit normal of the vacuum-solid
interface, and $\mathcal{T}_{\alpha}$ is the power transmittance of mode
$\alpha$ [26].

The term $\pmb{\hat{n}}_{\gamma}\cdot\partial\omega_{\alpha}/\partial\pmb{k}$
in Eq.(2) describes the group velocity of the phonon wave mode $\alpha$ in the
direction of the outward normal of the vacuum-interface, and can be expressed
as[27]:

$\pmb{\hat{n}}_{\gamma}\cdot\frac{\partial\omega_{\alpha}}{\partial\pmb{k}}=\frac{1}{2}\frac{\sin\theta}{\rho
v_{\alpha}(\theta,\varphi)|\pmb{A}_{\alpha}(\theta,\varphi)|^{2}}\hat{\xi}_{\gamma}(\theta,\varphi),$
(3)

where $\rho$ is the density of the solid. The phase velocity term
$v_{\alpha}=\omega_{a}/k$ depends on the material, crystal orientation and
incident angles $(\theta,\varphi)$, and can be solved from the
piezoelectrically stiffened Christoffel equation [Eq.(8.147) in Ref.[28]].
Furthermore, the polarization vector $\pmb{A}_{\alpha}$ is obtained from the
normalized eigenvector of the extended Stroh matrix [Eq.(3) in Ref.[25]],
whereas for the bulk waves, $\hat{\xi}_{\gamma}=\pm 1$ (J/m) is obtained from
the Stroh-normalization[26], whose sign determines the energy flow direction
of the phonon along the unit vector $\pmb{\hat{n}}_{\gamma}$.

In addition, a Heaviside step function $\Theta(f)$, which equals to unity
(zero) when $f>0$ ($f<0$), is used to correctly select the phonons whose group
velocities point from the solid towards the vacuum. It is important to note
that in anisotropic crystals, the direction of phonon propagation, signified
by the direction of the group velocity
$\partial\omega_{\alpha}/\partial\pmb{k}$, is generally different from the
wave front direction given by the wave vector $\pmb{k}$. It is possible for a
phonon that tunnels outward from solid 1 to 2 to have an inward wave vector in
the normal direction, i.e.
$(\pmb{\hat{n}}_{\gamma}\cdot\partial\omega_{\alpha}/\partial\pmb{k})<0$ but
$(\hat{\pmb{n}}_{1}\cdot\pmb{k})>0$. Therefore, to fully account for all the
possible incident acoustic phonons impinging on the surface towards the
vacuum, we have to integrate over the complete k-space in Eq.(2) including the
inward half-hemisphere, but choosing the outward traveling phonons with the
help of the Heaviside step function
$\Theta[\pmb{\hat{n}}_{\gamma}\cdot(\partial\omega_{\alpha}/\partial\pmb{k})]$.

By following the methods presented in Ref.[25], the total tunneled power
transmittance of an incoming wave mode $\alpha$, coupling into all possible
bulk modes in the second solid, takes the form[26]:

$\mathcal{T}_{\alpha}(\theta,\varphi,k,d)=\frac{2\mathrm{Re}\big{[}r_{V}^{(2)}\big{]}\big{|}t_{\alpha\rightarrow
V}^{(1)}\big{|}^{2}e^{-2kd\sin\theta}}{\big{|}1-r_{V}^{(1)}r_{V}^{(2)}e^{-2kd\sin\theta}\big{|}^{2}},$
(4)

where $t^{(i)}$ and $r^{(i)}$ are the single surface transmission and
reflection coefficients, which describe the scattering of the acoustic wave at
the surface of solid $i=1,2$ as if there is no adjacent second solid. To be
more specific, we denote with $t_{\alpha\rightarrow V}^{(1)}$ the coefficient
of an incoming $\alpha$ mode wave transmitted into an evanescent wave
(electrical potential) in vacuum from solid 1, and with $r_{V}^{(i)}$ the
coefficient of an evanescent wave coming from vacuum reflected on the surface
of solid $i$. For given material parameters, crystal orientations and incident
wave mode $\alpha$, these coefficients are functions of the incident angles
$\theta$ and $\varphi$, independent of $k$ and $d$, and can be obtained
numerically using the boundary conditions at the solid-vacuum interface,
following the formalism presented in Ref.[25].

The net heat flux between solid 1 at temperature $T_{1}$ and solid 2 at
temperature $T_{2}$ is the difference between their corresponding emitted heat
fluxes, and reads as $\Delta
J^{\textrm{PE}}=J_{1}^{\textrm{PE}}(T_{1},d)-J_{2}^{\textrm{PE}}(T_{2},d)$.
Furthermore, in the limit where the power transmittance [Eq.(4)] is set to
unity and isotropy is assumed for the group velocity, given then by
$(\pmb{\hat{n}}_{\gamma}\cdot\partial\omega_{\alpha}/\partial\pmb{k})=v_{\alpha}\cos\theta$,
the heat flux Eq.(2) simplifies to
$J_{\gamma}=\sum_{\alpha}\pi^{2}k_{B}^{4}T_{\gamma}^{4}/120v_{\alpha}^{2}\hbar^{3}$,
recovering the expression for phonon blackbody radiation for isotropic
matter[29].

To demonstrate PEMHT numerically, we consider the case of two ZnO crystals
with a hexagonal 6mm symmetry, with their material constants taken from
Ref.[28]. These solids are separated by a vacuum gap of width $d$, and are
rotated identically such that their crystal $c$-axes are aligned with the
laboratory $y$-axis (hence $\vartheta=\pi/2$, see Fig.1). In Fig.2 we plot the
emitted heat flux $J^{\mathrm{PE}}(T,d)$ between the two solids as a function
of the emitter temperature $T$ and gap width $d$. By introducing the
characteristic thermal wavelength $\lambda_{T}$ [12, 19, 21], defined as
$\lambda_{T}=2\pi v_{\alpha}\hbar/k_{B}T$, we find that the plot can be
divided into two regions roughly separated by the black dotted line signifying
the condition $d=\lambda_{T}\approx(0.2\textrm{ $\mu$m})/T$, where we used an
average phonon phase velocity [$v=(\sum_{\alpha}v_{\alpha})/3$] of $3900$ m/s
for ZnO.

Figure 2: Contour plot of the total emitted heat flux $J^{\mathrm{PE}}$ of all
phonon wave modes for ZnO as function of temperature $T$ and gap width $d$.
The dotted line is $T=2\pi\hbar v/k_{B}d$, denoting the condition
$d=\lambda_{T}$ where $v=3900$ m/s. The dashed line is
$J^{\mathrm{PE}}=\pi^{2}k_{B}^{4}T^{4}/60c^{2}\hbar^{3}$, signifying the
blackbody radiation limit.

Towards the lower-left region, the heat flux decreases strongly with
decreasing temperature, but saturates with the gap width. This saturation
comes about because only the power transmittance $\mathcal{T}_{\alpha}$ is a
function of the gap width $d$ in Eq.(2), and this dependency is only expressed
via the exponential term $\sim\exp(-kd)$ in Eq.(4)[25]. As a result, for a gap
width $d\ll\lambda_{T}\sim 1/k$, $\mathcal{T}$ becomes constant, and the heat
flux $J^{\textrm{PE}}$ is hence determined by the thermal distribution
function $n(\omega,T)$, which is strongly modified by the temperature. In
contrast, the exponential decay of $\mathcal{T}$ dominates the heat flux
towards the upper-right section. In the large-gap limit ($d\gg\lambda_{T}$),
the heat flux is practically ”switched-off” and becomes insensitive to the
change of temperature. In addition, a black dashed line marks where the heat
flux from PEMHT equals to that of the blackbody radiation at the same
temperature [$J^{\mathrm{PE}}(T,d)=\pi^{2}k_{B}^{4}T^{4}/60c^{2}\hbar^{3}$
where $c$ the speed of light]. The comparison between the dashed and dotted
lines shows that PEMHT, if ”switched-on” ($d\leq\lambda_{T}$), generally
contributes at least three orders of magnitude stronger heat flux than
blackbody radiation at a given temperature, making it a non-trivial source in
the context of near-field heat transfer.

In addition, at room temperature, PEMHT is stronger than blackbody radiation
even at a gap width close to $10$ nm. This is very different to the other
acoustic phonon tunneling mechanisms described in literature [17, 19, 14, 20,
21, 24], for which non-trivial heat flux occurs in the sub-nanometer length
scale at room temperature. In Fig.3, the heat flux carried by various relevant
close-range mechanisms are compared, including the near-field radiative heat
transfer (NFRHT, [5, 7]) for ZnO [30, 31] and Au [32], phonon tunneling for Au
mediated by van der Waals force [19] and electrostatic force (for $1$ V bias
across the vacuum) [20, 21], and the blackbody radiation (maximal far-field
radiative heat transfer).

Figure 3: Comparison of the emitted heat fluxes driven by different close-
range mechanisms. (a) The heat fluxes as a function of the gap width at $300$
K. Purple square marker is PEMHT of ZnO, the dotted lines are NFRHT for Au
(blue) and ZnO (orange). The dashed lines are the acoustic phonon tunneling in
Au mediated by van der Waals force (green) and $1$ V electrostatic potential
difference (red). (b) The heat fluxes as function of temperature with a fix
gap width of $1$ nm. (c) The heat fluxes as a function of gap width at $4$ K.

In panel (a) of Fig.3, the heat fluxes of all the above mentioned mechanisms
at $300$ K are plotted as a function of the gap width $d$. It is clear that at
room temperature, NFRHT for Au and ZnO, denoted by the blue and orange dotted
lines, respectively, is significantly stronger than the heat transfer mediated
by the acoustic phonon mechanisms. Moreover, the heat fluxes driven by the van
der Waals force and the electrostatic force scale as $d^{-9}$ and $d^{-7}$,
respectively [21], hence they only have non-trivial contributions below $d<1$
nm at room temperature, and will fall off rapidly with the increase of the gap
width. In contrast, PEMHT (square symbols) scales as $d^{-3}$, similar to that
for NFRHT (photon tunneling). Consequently, PEMHT quickly dominates the other
phonon tunneling mechanisms at larger-than-nanometer scales, being a
relatively ”long-range” phenomenon.

One interesting observation in Fig.3 (a) is that the NFRHT of ZnO is
particularly strong. This happens[7] because the surface-phonon polaritons of
ZnO can be excited at infrared frequencies [30, 31], matching the spectrum of
the room temperature thermal photons, and hence enhancing the heat flux. But
this also infers that one should expect a strong attenuation of the NFRHT once
the excitations are stopped, _i.e._ when the temperature is lowered. This is
confirmed in Fig.3(b), in which the heat fluxes are plotted as a function of
the temperature for a fixed gap width of $1$ nm. There is a clear cut-off of
the NFRHT flux at about $100$ K for ZnO.

More interestingly, PEMHT becomes stronger than the fluxes from all other
mechanisms, including NFRHTs, between 0.1 K to 50 K. At this temperature
range, the power transmittance of the PEMHT increases exponentially as the
thermal wavelengths of the acoustic phonons increase, eventually saturating at
sub-Kelvin range (see Fig.2). With the temperature lowered even further, the
PEMHT flux is determined by the phonon state energy term (phonon thermal
spectrum), and therefore has a $T^{4}$ dependence on the temperature, similar
to the NFRHT of ZnO in the low temperature limit. Meanwhile, the other
acoustic phonon tunneling mechanisms, scaling more slowly than the PEMHT,
begin to dominate the heat transfer at the temperatures below 0.1 K.

From the above numerical analyses, we believe that PEMHT can be experimentally
observed using modern nanofabrication techniques at cryogenic temperatures. As
an example,in Fig.3 (c) we demonstrate the heat fluxes from all the above
mentioned mechanisms, as a function of the gap width at 4 K. One finds that
the PEMHT of ZnO (purple squares) dominates the heat flux up to a gap width as
high as $d\sim 300$ nm, compared to the NFRHT of ZnO (orange dotted line) and
blackbody radiation (black line). With further lowering of the temperature or
reduction of the gap width, PEMHT will become more prominent and easier to
observe.

Next, we plot in Fig.4 the emitted PEMHT for two different piezoelectric
materials, ZnO and AlN[33], as a function of the orientation angle
$\vartheta$, with a fixed gap width of $1$ nm and a temperature of $0.1$ K, to
illustrate the influence of the material choice and orientations.

Figure 4: Comparison of emitted PEMHT as a function of the crystal rotation
angle $\vartheta$ for ZnO (solid line) and AlN (dotted line). The heat fluxes
are calculated for a gap width of $1$ nm at a temperature of $0.1$ K.

Obvious similarities are shared by the heat fluxes of ZnO (solid line) and AlN
(dotted line). For example, for both materials the flux is generally stronger
around the orientation $\vartheta=\pi/2$, contrasting to those around $0$ and
$\pi$, and sharp slopes appear around $20^{\circ}$ and $40^{\circ}$ leading to
more than five-fold, step-like variations of the heat flux. These similarities
come about because both materials have the $6mm$-wurtzite crystal symmetry.
The underlying physics of these features is that as $\vartheta$ is the angle
between the normal of the solid-vacuum interface and the crystal’s
piezoelectric axis (the $c$-axis), and when $\vartheta$ is close to
$90^{\circ}$, the $c$-axis is more aligned with the surface. As a result, the
reflected evanescent modes, which propagate only on the surface, can excite a
stronger piezoelectric response, leading to an enhanced electrostatic coupling
across the vacuum. It has been shown that even complete tunneling of certain
acoustic modes can be achieved for ZnO, when $\vartheta$ is between
$60^{\circ}$ and $120^{\circ}$[26].

In addition, PEMHT for ZnO is about one order of magnitude stronger than that
of AlN. This is mostly due to the differences in their phase velocities. When
the characteristic wavelength $\lambda_{T}$ is much larger than the gap width,
the heat flux scales roughly with $v_{\alpha}^{-3}$. As a result, a ten-fold
difference in the heat flux is expected since AlN has an average phase
velocity $v\approx 7500$ m/s, whereas it is about $v\approx 3900$ m/s for ZnO.

To summarize, we presented a general formulation for piezoelectrically
mediated heat transfer (PEMHT) via acoustic phonon tunneling. Such a
formulation can be used to compute the heat flux across a vacuum gap, between
arbitrarily anisotropic and oriented piezoelectric crystals. Our analytical
and numerical studies reveal that PEMHT provides a significant heat flux,
generally more than three orders of magnitude stronger than that from the
blackbody radiation when the gap width $d$ is smaller than the phonon
characteristic thermal wavelength $\lambda_{T}$. By comparing to other near-
field mechanisms, our study shows that PEMHT can dominate the heat flux at
temperatures below 50 K.

Finally we remark that with the currently available nano-fabrication and
cryogenic measurement techniques, PEMHT could possibly be investigated
experimentally. We believe that further understanding and engineering of
acoustic phonon tunneling can be crucial for many application areas such as
nano-electronics, low temperature detectors, quantum information devices and
others, for which increasing demand for heat manipulation and management
exists.

This study was supported by the Academy of Finland project number 341823.

## References

  * Domoto  _et al._ [1970] G. Domoto, R. Boehm, and C. Tien, Experimental investigation of radiative transfer between metallic surfaces at cryogenic temperatures, J. Heat Transfer 92, 412 (1970).
  * Song  _et al._ [2015] B. Song, A. Fiorino, E. Meyhofer, and P. Reddy, Near-field radiative thermal transport: From theory to experiment, AIP Adv. 5, 053503 (2015).
  * Lucchesi  _et al._ [2021] C. Lucchesi, R. Vaillon, and P. O. Chapuis, Radiative heat transfer at the nanoscale: Experimental trends and challenges, Nanoscale Horiz. 6, 201 (2021).
  * Polder and Van Hove [1971] D. Polder and M. Van Hove, Theory of Radiative Heat Transfer between Closely Spaced Bodies, Phys. Rev. B 4, 3303 (1971).
  * Pendry [1999] J. B. Pendry, Radiative exchange of heat between nanostructures, J. Phys.: Condens. Matter 11, 6621 (1999).
  * Volokitin and Persson [2001] A. I. Volokitin and B. N. Persson, Radiative heat transfer between nanostructures, Phys. Rev. B 63, 205404 (2001).
  * Joulain  _et al._ [2005] K. Joulain, J.-P. Mulet, F. Marquier, R. Carminati, and J.-J. Greffet, Surface electromagnetic waves thermally excited: Radiative heat transfer, coherence properties and Casimir forces revisited in the near field, Surf. Sci. Rep. 57, 59 (2005).
  * 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, J. C. Cuevas, E. Meyhofer, and P. Reddy, Radiative heat transfer in the extreme near field, 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, Giant heat transfer in the crossover regime between conduction and radiation, Nat. Commun. 8, 14475 (2017).
  * Cui  _et al._ [2017] J. Cui, J. Zhang, T. Barayavuga, X. Wang, X. He, L. Yang, H. Xie, X. Mei, and W. Wang, Nanofabrication with the thermal AFM metallic tip irradiated by continuous laser, Integr. Ferroelectr. 179, 140 (2017).
  * Jarzembski  _et al._ [2022] A. Jarzembski, T. Tokunaga, J. Crossley, J. Yun, C. Shaskey, R. A. Murdick, I. Park, M. Francoeur, and K. Park, Role of Acoustic Phonon Transport in Near- to Asperity-Contact Heat Transfer, Phys. Rev. B 106, 205418 (2022).
  * Prunnila and Meltaus [2010] M. Prunnila and J. Meltaus, Acoustic phonon tunneling and heat transport due to evanescent electric fields, Phys. Rev. Lett. 105, 125501 (2010).
  * Sellan  _et al._ [2012] D. P. Sellan, E. S. Landry, K. Sasihithlu, A. Narayanaswamy, A. J. McGaughey, and C. H. Amon, Phonon transport across a vacuum gap, Phys. Rev. B. 85, 024118 (2012).
  * Persson  _et al._ [2011] B. N. Persson, A. I. Volokitin, and H. Ueba, Phononic heat transfer across an interface: Thermal boundary resistance, J. Phys.: Condens. Matter 23, 045009 (2011).
  * Chiloyan  _et al._ [2015] V. Chiloyan, J. Garg, K. Esfarjani, and G. Chen, Transition from near-field thermal radiation to phonon heat conduction at sub-nanometre gaps, Nat. Commun. 6, 6755 (2015).
  * Budaev and Bogy [2011] B. V. Budaev and D. B. Bogy, On the role of acoustic waves (phonons) in equilibrium heat exchange across a vacuum gap, Appl. Phys. Lett. 99, 053109 (2011).
  * Ezzahri and Joulain [2014] Y. Ezzahri and K. Joulain, Vacuum-induced phonon transfer between two solid dielectric materials: Illustrating the case of Casimir force coupling, Phys. Rev. B 90, 115433 (2014).
  * Sasihithlu  _et al._ [2017] K. Sasihithlu, J. B. Pendry, and R. V. Craster, Van der Waals Force Assisted Heat Transfer, Z. Naturforsch. 72, 181 (2017).
  * Pendry  _et al._ [2016] J. B. Pendry, K. Sasihithlu, and R. V. Craster, Phonon-assisted heat transfer between vacuum-separated surfaces, Phys. Rev. B 94, 075414 (2016).
  * Volokitin [2019] A. I. Volokitin, Effect of an Electric Field in the Heat Transfer between Metals in the Extreme Near Field, JETP Lett. 109, 749 (2019).
  * Volokitin [2020] A. I. Volokitin, Contribution of the acoustic waves to near-field heat transfer, J. Phys.: Condens. Matter 32, 215001 (2020).
  * Fong  _et al._ [2019] K. Y. Fong, H. K. Li, R. Zhao, S. Yang, Y. Wang, and X. Zhang, Phonon heat transfer across a vacuum through quantum fluctuations, Nature 576, 243 (2019).
  * Biehs  _et al._ [2020] S. A. Biehs, A. Kittel, and P. Ben-Abdallah, Fundamental limitations of the mode temperature concept in strongly coupled systems, ZNA 75, 803 (2020).
  * Viloria  _et al._ [2023] M. G. Viloria, Y. Guo, S. Merabia, R. Messina, and P. Ben-Abdallah, Radiative heat exchange driven by the acoustic vibration modes between two solids at the atomic scale (2023), arXiv:2302.00520.
  * Geng and Maasilta [2022] Z. Geng and I. J. Maasilta, Acoustic wave tunneling across a vacuum gap between two piezoelectric crystals with arbitrary symmetry and orientation, Phys. Rev. Res. 4, 033073 (2022).
  * Geng and Maasilta [2023] Z. Geng and I. J. Maasilta, Complete tunneling of acoustic waves between closely spaced piezoelectric crystals, arXiv:2209.08287 (2023).
  * [27] See Supplemental Material at [URL will be inserted by publisher].
  * Auld [1990] B. Auld,  _Acoustic fields and waves in solids_ , 2nd ed. (Krieger, Malabar, Florida, 1990).
  * Swartz and Pohl [1989] E. T. Swartz and R. O. Pohl, Thermal boundary resistance, Rev. Mod. Phys. 61, 605 (1989).
  * Ashkenov  _et al._ [2003] N. Ashkenov, B. N. Mbenkum, C. Bundesmann, V. Riede, M. Lorenz, D. Spemann, E. M. Kaidashev, A. Kasic, M. Schubert, M. Grundmann, G. Wagner, H. Neumann, V. Darakchieva, H. Arwin, and B. Monemar, Infrared dielectric functions and phonon modes of high-quality ZnO films, J. Appl. Phys. 93, 126 (2003).
  * Ooi  _et al._ [2011] P. Ooi, S. Lee, S. Ng, Z. Hassan, and H. A. Hassan, Far Infrared Optical Properties of Bulk Wurtzite Zinc Oxide Semiconductor, J. Mater. Sci. Technol. 27, 465 (2011).
  * Chapuis  _et al._ [2008] P.-O. Chapuis, S. Volz, C. Henkel, K. Joulain, and J.-J. Greffet, Effects of spatial dispersion in near-field radiative heat transfer between two parallel metallic surfaces, Phys. Rev. B 77, 035431 (2008), arXiv:0802.1899v1 .
  * Tsubouch and Mikoshiba [1985] K. Tsubouch and N. Mikoshiba, Zero-Temperature-Coefficient SAW Devices on AlN Epitaxial Films, IEEE Trans. Sonics Ultrason. 32, 634 (1985).
  * Laude  _et al._ [2005] V. Laude, A. Reinhardt, and A. Khelif, Equality of the energy and group velocities of bulk acoustic waves in piezoelectric media, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 52, 1869 (2005).
  * Every and Neiman [1992] A. G. Every and V. I. Neiman, Reflection of electroacoustic waves in piezoelectric solids: Mode conversion into four bulk waves, J. Appl. Phys. 71, 6018 (1992).