Complete tunneling of acoustic waves between piezoelectric crystals

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ Complete tunneling of acoustic waves between piezoelectric crystals © 2023 The Author(s). Published version Geng, Zhuoran; Maasilta, Ilari J. Geng, Z., & Maasilta, I. J. (2023). Complete tunneling of acoustic waves between piezoelectric crystals. Communications Physics, 6, Article 178. https://doi.org/10.1038/s42005-023-01293-y 2023 ARTICLE Complete tunneling of acoustic waves between piezoelectric crystals Zhuoran Geng 1✉& Ilari J. Maasilta 1✉ The mechanical displacements in piezoelectric materials carry along macroscopic electric fields, allowing tunneling of acoustic waves across a vacuum gap beyond the charge-charge interaction distance. However, no rigorous proof of complete acoustic wave tunneling has been presented, and the conditions to achieve complete tunneling have not been identified. Here, we demonstrate analytically the condition for such phenomenon for arbitrary anisotropic crystal symmetries and orientations, and that complete transmission of the incoming wave occurs at the excitation frequency of leaky surface waves. We also show that the complete transmission condition can be related to the surface electric impedance and the effective surface permittivity of the piezoelectric material, relevant to realize the complete tunneling experimentally. We support our findings with numerical results for the maximum power transmittance of a slow transverse wave tunneling between identical ZnO crystals. The results show that complete tunneling can be achieved for a large range of orientations. https://doi.org/10.1038/s42005-023-01293-y OPEN 1Nanoscience Center, Department of Physics, University of Jyvaskyla, P. O. Box 35, FI-40014 Jyväskylä, Finland. ✉email: zhgeng@jyu.fi;maasilta@jyu.fi COMMUNICATIONS PHYSICS | (2023) 6:178 | https://doi.org/10.1038/s42005-023-01293-y | www.nature.com/commsphys 1 1234567890():,; Acoustic waves (acoustic phonons) are deformations or vibrations propagating through a material medium. As such, they do not exist in vacuum, leading to the initial conclusion that it is impossible for the vacuum to transmit the energy of an acoustic wave between two separated media. However, at the atomic scale the vibrations of the nuclei can propagate via their electrical interactions through vacuum. Thus, a question can be raised, whether acoustic phonons can also be transmitted across larger than atomic scale vacuum gaps through some electromagnetic mechanism. This is a relevant question, as with the advances in experimental techniques, nanometer to subnanometer scale vacuum gaps can be achieved1–4. The possibility of such acoustic phonon tunneling, as it is often called in the literature, has attracted a considerable amount of theoretical work in recent years to investigate possible mechanisms of the effect such as Casimir and van der Waals forces, particularly in the context of near-field heat transfer5–18. One possible mechanism for acoustic wave tunneling is piezoelectricity, as in piezoelectric materials mechanical displacements carry along macroscopic electric fields. When an acoustic wave in a piezoelectric solid impinges on a free surface, it extends a decaying, evanescent electric field into the vacuum19. The length scale of this decay is determined by the wavelength of the acoustic wave, so by bringing another piezoelectric solid within a wavelength, acoustic power can be transmitted into the second piezoelectric solid across the vacuum gap. What makes this piezoelectrically mediated acoustic wave tunneling particularly attractive is its length scale: it is not fixed to be in the nanoscale, but operates on the typically much larger wavelength scale defined by the frequency (1 GHz would correspond to ~5 μm). The effect was introduced20,21 and observed22 long ago (for more detailed background, see Geng and Maasilta23), but developed further more recently5,23,24. In particular, a general formalism was introduced23 that is applicable to any incident bulk wave mode for any anisotropic crystallographic orientation. One of the most interesting suggestions5,21,24 is the possibility of unity transmission for some particular conditions, meaning that the incident wave could perhaps be completely transmitted into the adjacent solid. However, the discussions in previous literature5,21,24 are limited either by the simplified models used, or only show numerical results for the highest symmetry crystal orientations. Until now, no rigorous proof of complete acoustic wave tunneling has been presented, nor have generally valid complete tunneling conditions been put forward. In this work, we focus on the power transmittance of acoustic wave tunneling. We use the general formalism developed for piezoelectric acoustic wave tunneling23 to analytically prove the existence of the complete tunneling phenomenon between two vacuum separated identical solids. In addition, a resonant tunneling condition is also derived, corresponding to the excitation of leaky surface waves. We also propose that this condition could be checked experimentally. Further discussion of the results are presented with a few numerical examples for ZnO crystals. In particular, we find our results differ from those obtained before5. Results and discussion Tunneling of acoustic waves. We study a system of two anisotropic, semi-infinite piezoelectric solids separated by a vacuum gap of width d, as shown in Fig. 1. Two coordinate systems describe the relation between the crystal intrinsic orientation, denoted by XYZ, and the external laboratory space, denoted by xyz. The surfaces of the solids are assumed to be mechanically and electrically free23, with surface normals aligned with the z-axis. We consider an incoming homogeneous acoustic plane (bulk) wave expðikrþiωtÞ, where kand ωare the wave vector and angular frequency, propagating inside the xz-plane (sagittal plane) from the positive z-axis direction toward the surface at z=0, with a positive x-component of wave vector (k x > 0). In addition, we only consider low frequency acoustic waves with linear dispersion and assume the usual quasistatic approximation for piezoelectric acoustic waves19 satisfying E=−∇Φ, where Eand Φare the electric field and the electric potential, respectively. An incident bulk wave scatters into a linear combination of partial waves at an interface. These partial waves are either reflected or transmitted, and can either be homogeneous (bulk) waves or inhomogeneous (evanescent) waves bound on the surface of the solid23. The single surface reflection and transmission coefficients, which describe the amplitudes of these scattered waves, can be calculated following the multiple reflection method presented in Section III.B in Geng and Maasilta23. We denote these coefficients with an overhead bar, as follows:  tð1Þ in!Vis the coefficient of an incoming wave from solid 1 transmitted into a vacuum electric wave,  tð2Þ V!αis the coefficient of an vacuum wave transmitted into mode αin solid 2, and  rðiÞ Vis the coefficient of an vacuum wave reflected on the vacuum side of the interface of solid i=1, 2. In these coefficients, α=1, ..., 4 correspond to the four physically allowed electroacoustic partial wave modes in the corresponding solids 1 or 2. It should be noted that these coefficients are not the direct analogs of the Fresnel coefficients25 from optics. A total transmission coefficient t α , which describes the amplitude ratio of a transmitted partial wave αin solid 2 to an incoming bulk wave from solid 1, takes a form23 (for details, see “Methods”section): tα¼  tð1Þ in!V tð2Þ V!α ekxd rð1Þ V rð2Þ Vekxd¼ tð1Þ in!V tð2Þ V!αfmðdÞ;ð1Þ with k x the wave vector component along the surfaces, which is conserved in the tunneling process. This expression can be interpreted as two single surface transmission Fig. 1 Schematic of the system under study. Two piezoelectric solids 1, 2 are separated by a vacuum gap of width d. An incoming acoustic wave from solid 1 (positive z-axis of a laboratory coordinates xyz) with an incident angle θ i tunnels across the vacuum gap into solid 2 inside the xz-plane. XYZ describe the intrinsic crystal coordinates, which can be rotated w.r.t. the xyz coordinates. ARTICLE COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-023-01293-y 2COMMUNICATIONS PHYSICS | (2023) 6:178 | https://d oi.org/10.1038/s42005-023-01293-y | www.nature.com/commsphys coefficients  tð1Þ in!Vand  tð2Þ V!αcoupled by a geometrical multiple reflection factor for evanescent electrical waves in the gap fmðdÞ¼½expðkxdÞ rð1Þ V rð2Þ VexpðkxdÞ123. It implicitly depends on the incident angle θ i not only via kx¼ksin θi, but also via the coefficients  tand  r, which are functions of vx¼ω=ðksin θiÞ(“Methods”). Derivation of the condition for complete tunneling. To fully describe the tunneling of the acoustic wave, we also look at the energy transfer between the solids. The time-averaged power flow density (energy flux, units [Wm−2]) of a transmitted partial wave in the direction normal to the surfaces (denoted as P α ) can be obtained from the real part of the normal component of piezoelectric Poynting vector (“Methods”). For the tunneled bulk partial waves, the transmitted power relates to the normal component of the incident power by P α =∣t α ∣2P in , in which the input power P in can be from a coherent bulk wave or from a thermal phonon, whereas the reflected or transmitted evanescent partial waves in solids 1,2 are bound onto the surface and carry no power in the normal direction (P α =0ifαis an evanescent mode). As there is no dissipation inside the vacuum gap, the normal direction power flow density inside the vacuum (denoted by P V ) is equal to the total normal direction transmitted power density (denoted by P Σ ). It is clear that P Σ is the sum of P α over all the transmitted bulk waves in solid 2, and we can write it using Eq. (1)asPΣ¼∑αj tð1Þ in!V tð2Þ V!αfmðdÞj2Pin, where αruns only over the bulk modes. The number of transmitted bulk modes can be from zero to three (in some cases four26), and if there is no bulk mode available, the power flow in both the vacuum and solid 2 are zero. On the other hand, the normal power flow inside the vacuum gap can be expressed using the Poynting’s theorem under the quasistatic approximation as PV¼2j tð1Þ in!VfmðdÞj2Re½ rð2Þ VPin (see Supplementary Note 1 for the derivation). As a result, from P V =P Σ we find a relation: 2Re  rð2Þ V hi ¼∑ α¼bulk j tð2Þ V!αj2:ð2Þ Furthermore, if we assume that the two solids consist of the same material with identical crystal orientations, two additional relations that link the single surface coefficients of the two solids can be found by exploiting the completeness of the eigensolutions of the scattering problem (see Supplementary Note 2 for the derivations). The first one relates the reflection coefficients  rðiÞ Vof the two solids as:  rV rð2Þ V¼  rð1Þ V:ð3Þ The second one states that if the transmitted bulk wave mode γin the solid 2 is the same mode as the incident wave in solid 1, there exists a relation:  tð1Þ γ!V¼ tð2Þ V!γ:ð4Þ In addition, by comparing the relation (4) with Eq. (2), we find the condition: 2Reð rVÞ≥j tð1Þ in!Vj2;ð5Þ where the equality is satisfied when there exists only one transmitted bulk wave mode in solid 2 and the mode is the same as the incident wave in solid 1. By applying the relations (2) and (3), P Σ can then be simplified to: PΣ Pin ¼2Reð rVÞj tð1Þ in!Vj2 4Reð rVÞ2þe2kxdj  rVj2  2e2kxd;ð6Þ which explicitly depends only on two single surface coefficients:  tð1Þ in!Vand  rV. Equation (6) shows that the total transmitted power P Σ is always less than the incident power P in if more than one transmitted bulk wave modes exist, since in that case the inequality Eq. (5) takes the greater-than sign. This result has the implication that complete tunneling, i.e., the full transmission of the incident power, can’t be achieved if the transmitted wave consists of multiple partial bulk waves, in contradiction to previous work5. In contrast, if there is only one transmitted homogeneous bulk mode and it is the same mode as the incident wave, then the equal sign of Eq. (5) is valid, and Eq. (6) simplifies to: PΣ Pin ¼4Reð rVÞ2 4Reð rVÞ2þe2kxdj  rVj2  2e2kxd;ð7Þ which very much resembles the Fabry-Perot-like form of transmission coefficients for the near-field radiative heat transfer27. From Eq. (7), it is clear that the maximum transmitted power is exactly equal to the incident power (P Σ =P in ) when the resonance condition: j rVj¼ekxd;ð8Þ is satisfied, similar to the corresponding condition for perfect photon tunneling in near-field heat transfer28,29. This proves that (1) unity transmission (complete tunneling) of an acoustic wave across a vacuum gap is possible, and (2) the condition for it depends explicitly only on the single surface reflection coefficient  rV, the wave vector component k x and the gap width d. The physical explanation of such complete tunneling is the excitation of resonant coupled leaky surface waves on both interfaces (more details below and in Supplementary Note 4), which is fundamentally different from the principle of antireflection in optics25. In particular, with a given material and crystal orientation,  rV is only a function of the incident angle and is independent of the gap width or the existence of the adjacent solid. We propose that, as a material parameter,  rVcould be determined experimentally by measuring the effective surface permittivity ϵ eff (v x )24,30,31 or the TM-wave surface impedance Z p (ω,v x )32,33 of the piezoelectric solid. They are found to be related by expressions (see Supplementary Note 3):  rV¼iϵeff ϵ0 ϵeff þϵ0 ; rV¼i1þivxϵ0Zp 1ivxϵ0Zp ;ð9Þ where ϵ 0 is the vacuum permittivity. The effective surface permittivity concept is useful in the study of piezoelectric materials, for example for the generation and detection of acoustic waves by transducers31 or for determining the gap wave modes between piezoelectric solids24.Wefind the symmetric and antisymmetric gap wave conditions can be simply expressed by  rV¼±iexpðkxdÞ, which are the poles of the transmission coefficient of Eq. (1) (Supplementary Note 3). Numerical examples and physical interpretation of complete tunneling between identical ZnO crystals. We now turn to demonstrate the complete tunneling effect with numerical examples for two identical ZnO crystals, using the formalism developed before23. The first example is shown in Fig. 2, where the two crystals are separated with a scaled gap width of kd = 0.01, and are both rotated first with respect to the x-axis by ϑ=46.89∘and then to the z-axis by φ=88° (see Geng and Maasilta23 for details on the crystal rotation procedure). The mode of the incident wave in this example is chosen to be the slowest quasi-transversal wave (ST), so that there exists a critical COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-023-01293-y ARTICLE COMMUNICATIONS PHYSICS | (2023) 6:178 | https://doi.org/10.1038/s42005-023-01293-y | www.nature.com/commsphys 3 incident angle beyond which only one bulk transmitted wave can be found, thus satisfying the general condition for complete tunneling. In Fig. 2a, we plot the transmittance into each bulk mode P α /P in as a function of the incident angle θ i , where αcan be the quasi-longitudinal (L), the fast quasi-transversal (FT) or the slow quasi-transversal (ST) mode, categorized based on their phase velocities. We see that for most angles, transmittance is low, except for the two sharp transmission peaks for the ST mode giving exactly unity transmission at angles between 75° and 80°. Abrupt cut-offs are visible for the transmitted L and FT modes, corresponding to the critical incident angles θLc28and θFTc63:5. Beyond these critical angles, the corresponding modes become evanescent, bound on the surface of the solid with no direct energy transmission into the bulk. Figure 2b provides a zoomed view on the resonant transmission peaks, now with two different scaled gap values kd =1 (blue solid line) and kd =0.01 (orange solid line), with an overlay of the j rVjcurve (black dashed line), helping us also to understand the doublet structure. The two additional horizontal dashed lines represent the values of the RHS of Eq. (8) for the two kd values, whereas the dashed black curve represents the LHS of Eq. (8). It is clear that the unity transmission occurs where the resonance condition is valid, proving consistency between the analytical theory and the numerical approach. In addition, we see that with the increase of the scaled gap width from 0.01 to 1, the separation of the peaks is reduced, and with a further increase the two solutions would merge into one at the maximum of j rVj. With this particular ZnO crystal orientation, this maximum is about 4 as shown in the plot, which leads to a maximum gap width of kd ≈1.4 to observe complete tunneling (merged unity transmission peak). For ZnO (ST wave velocity v=2780 m/s), and a 2 GHz frequency relevant for device applications, this corresponds to a quite long physical distance of d=300 nm with the parameters and the orientation used in the example. In addition, it is also useful to briefly discuss how sensitive the power transmittance is to deviations from the resonance condition, based on the above numerical example. For a fixed frequency of 2 GHz, the transmittance as a function of both the incident angle θ i and the gap width dis presented in Fig. 2c. We see that when the vacuum gap is small, e.g., d< 100 nm, the two resonances are well separated and are sensitive to deviations of both dand θ i . For example, a 50% drop in transmittance (white dashed lines) occurs within a 10 nm change in dor a 0.1 degree change in θ i for the right branch. However, the merging of the two resonances leads to a higher deviation tolerance. θ i =76. 4° is an example, where the transmittance remains higher than 50% for a wide range 150 nm < d< 500 nm, significantly relaxing the constraint for the gap width control in measuring the tunneling. Similar discussion can also be applied for a fixed gap width, for which the transmittance becomes a function of the angular frequency ωand the in-plane wave vector k x , as illustrated in the inset of panel (c). Fig. 2 Angular and gap width dependence of the power transmittance of an incoming ST wave. a Power transmittance P α /P in of the longitudinal α=L (green), the fast transverse α=FT (orange) and the slow transverse α=ST (blue) waves, for an incoming ST wave as function of the incident angle θ i , for two identical ZnO crystals separated by a scaled gap kd =0.01andorientedwithazenithangleϑ=46.89°andanazimuthangleφ=88° (inset). We used the anisotropic crystal parameters c 11 =20.97 × 1010 Nm−2,c 33 =21.09 × 1010 Nm−2,c 44 =4.247 × 1010 Nm−2,c 12 =12.11 × 1010 Nm−2, c 13 =10.51 × 1010 Nm−2,c 66 =(c 11 −c 12 )/2, ϵ xx =8.55ϵ 0 ,ϵ zz =10.2ϵ 0 ,e x5 =−0.48 Cm−2,e z1 =−0.573 Cm−2,e z3 =1.32 Cm−2, and the density ρ=5680 kgm−3,takenfromAuld 19.bZoomed view on the peaks of the transmittance (left axis) with two values kd =1 (blue solid line) and kd =0.01 (orange solid line). The single surface reflection coefficient j rVjcurve (black dashed line) is overlayed (right axis) together with expðkxdÞ¼expðsin θiÞ (blue horizontal dashed line) and expðkxdÞ¼expð0:01 sin θiÞ(orange horizontal dashed line) to demonstrate the resonance condition, Eq. (8), for the twoscaledgaps,respectively.cPower transmittance (color scale) as a function of incident angle θ i and gap width dat a fixed frequency of 2 GHz (main panel), and as a function of ω/k x and k x ,withafixed d=300 nm (inset panel). The black solid line represents the resonance condition (Eq. (8)), and the white dashed lines indicate 50% power transmittance. dFrequencies of the symmetric (blue) and antisymmetric (orange) resonances as functions of gap width dfor a fixed k x of 2π/kx =1μm, where the symmetry refers to the shape of the electrical potential function in the gap. The green dashed line shows the frequency difference between the two resonances. ARTICLE COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-023-01293-y 4COMMUNICATIONS PHYSICS | (2023) 6:178 | https://d oi.org/10.1038/s42005-023-01293-y | www.nature.com/commsphys Furthermore, it is possible to take advantage of the resonances in experiments and potential applications, such as the precise control of a gap distance. By exciting a bulk wave with a known in-plane wave vector k x (for example with an interdigital transducer of finger spacing 2π/k x 31), the frequencies of the two tunneling resonances become functions of the gap distance, as demonstrated for our numerical example case in Fig. 2d. We see that while the higher-frequency resonance f sym depends weakly on d, the lower-frequency resonance f anti is highly sensitive to d, with ∂f/∂d≈0.3 MHz/nm at d< 20 nm. In addition, the frequency difference Δf=f sym −f anti (green dashed line, right axis scale) is also sensitive to d, reaching a responsivity ~0.5 MHz/nm at d< 20 nm. Such relations can be envisioned to be used not only to experimentally demonstrate acoustic wave tunneling, but also to control a buried nanoscale gap distance with nanometer accuracy. In general, the complete resonant tunneling can take place for a range of crystal orientations. In Fig. 3, we show the numerically calculated maximal power transmittance P ST /P in (over all θ i )ofan incident ST mode to a transmitted ST mode, as a function of all possible crystal rotations (ZnO has a crystallographic 6 mm system with uniaxial symmetry19, hence all the unique orientations of the crystal can be represented by the direction of the crystal c-axis (Z-axis) using a zenith angle ϑ∈(0°, 180°) and an azimuth angle φ∈(−90°, 90°)), using again parameters for anisotropic ZnO19 and a fixed scaled gap k x d=0.01. We find a significant parameter space for orientations, with multiple separate regions, where complete tunneling is possible (dark red regions). To validate the consistency of the numerics with the analytical condition, Eq. (8), we also plot a set of dotted contour lines in Fig. 3a to encircle the orientations satisfying j rVj>1, where unity transmission is possible, finding excellent agreement. Another observation is that the incident angle θ i satisfying complete tunneling varies for different crystal orientations (Fig. 3b, c), reaching as low values as 60° in some cases (Fig. 3c). To understand the physics, we first consider the three ellipsoidal unity transmission areas around ϑ=90° (Fig. 3). Inside these areas, the incident ST waves are not pure shear waves and therefore couple to the other partial waves (L, FT) at the surface. As a result, when the incoming ST wave has an incident angle beyond the critical angle of the FT mode, the reflected FT wave becomes evanescent, with its energy concentrated on the surface. For those orientations the FT-mode waves are predominately polarized in the direction of the c-axis, the direction of the piezoelectric dipole, creating a strong piezoelectric response. That excites large electric potential differences on the surface and hence gives rise to a strong electric coupling across the gap, which finally enables the resonant transmission. On the other hand, when the azimuth rotations approach φ=±90° with ϑ=90°, the c-axis aligns with the x-axis and the ST mode becomes a pure shear mode, polarized perpendicular to the sagittal plane. Then the incident ST waves are very weakly piezoelectric, and also decouple from all other partial modes. Other features can also be observed in Fig. 3a. Nodes having low transmission at around φ=±25° and ϑ=90° appear. This is because the electric potential excited by the reflected FT wave mode change polarity around these nodes, leading to minimized potential differences and weak coupling between the two surfaces. In addition, unity transmission is also observed in four small areas around φ=±90°, where the single surface reflection coefficient  rð1Þ ST!FT of the reflected FT partial waves increases significantly. This indicates an enhanced mode conversion between the ST and FT partial wave modes at these orientations, providing large electric potential differences on the solid-vacuum interface again via the evanescent FT wave, leading to strong tunneling signal. A more detailed discussion of the physical interpretation of the resonance can be found in Supplementary Note 4. Our numerical formalism can also be applied to the particular case studied with a simplified model before5, the details of which can be found in Supplementary Note 5. We do not find complete tunneling for the incoming modes and the crystal orientation in question, in contradiction to previous work5. Conclusions In conclusion, we have analytically and numerically proven it is possible for acoustic waves to completely tunnel across a vacuum gap between two piezoelectric solids, up to gap sizes of about a wavelength. We showed that such complete tunneling, with unity power transmittance, is possible only if one transmitted partial bulk mode is excited, it being the same mode as the incident wave. We derived a simple resonance tunneling condition for the complete tunneling effect, Eq. (8), and proved its validity and range of applicability with numerical examples for arbitrarily rotated ZnO crystals. As this is a strong and not a rare effect, it could have an impact in future acoustic wave devices, as well as in other application areas concerning phonons, such as controlling heat transport, optomechanics and quantum information science. Methods Extended Stroh formalism and multiple reflection approach. In piezoelectric solids, the dynamics of a propagating plane (bulk) wave expðikrþiωtÞare governed by the elastic equation of motion ∇⋅σ=ρ∂2u/∂t2and Gauss’s law ∇⋅ D=0, together with the piezoelectric constitutive relations19: σ¼cE:SeE D¼e:SþϵSEð10Þ where S,σ,u,Dare the elastic strain, elastic stress, mechanical displacement and electric displacement fields, ρ,cE,e,ϵSare the mass density, elastic stiffness tensor at constant electric field, piezoelectric stress tensor and electric permittivity tensor at constant strain, respectively. The double dot product indicates summation over paired indices between second-rank and higher-rank tensors, and the straindisplacement relation reads as S ij =(∂u i /∂r j +∂u j /∂r i )/2. An incident plane wave is scattered into a linear combination of partial waves at an interface, which are either reflected or transmitted. The general solutions of such Fig. 3 Crystal rotation map for complete tunneling between ZnO crystals. aColor scale of the ST-to-ST mode maximum power transmittance P ST /P in over all incident angles, plotted as function of crystal rotation angles ϑand φfor anisotropic ZnO. The dotted lines encircle the regions where j rVj>1. b,cshow the range of θ i where j rVj>1 for two fixed φ(b), or ϑ(c). Here, we fixkxd¼kd sin θiinstead of kd, as complete tunneling can be achieved by tuning k x either by changing the incident angle θ i or by the angular frequency ω. COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-023-01293-y ARTICLE COMMUNICATIONS PHYSICS | (2023) 6:178 | https://doi.org/10.1038/s42005-023-01293-y | www.nature.com/commsphys 5 partial waves that satisfy the governing equations take the expressions34–36: u¼∑ αbαAαeiðkxxþkyyþpαkxzωtÞ Φ¼∑ αbαϕαeiðkxxþkyyþpαkxzωtÞ nσ¼ikx∑ αbαLαeiðkxxþkyyþpαkxzωtÞ nD¼ikx∑ αbαDαeiðkxxþkyyþpαkxzωtÞ; ð11Þ in which nis the unit vector of the z-axis. A α ,ϕ α ,L α ,D α are the normalized constants describing the polarization vector, the electric potential, the traction force and the normal projection of the electric displacement of a partial wave mode α, respectively. b α are dimensionless amplitudes of the partial waves, and p≡k z /k x .To avoid redundant writing in the following expressions, we omit the common phase factor expðikxxikyyþiωtÞshared by all solutions. In this study, we solved these governing equations under the framework of extended Stroh formalism23, in which Eq. (10) is combined and rearranged into an eight-dimensional eigenvalue problem37,38 in the form of: NðvxÞξα¼pαξα;ð12Þ where Nis 8 × 8 real matrix and v x ≡ω/k x is the x-component of the phase velocity. Generally, eight linearly independent eigenvectors ξα¼½Aα;ϕα;Lα;DαTand corresponding eigenvalues p α can be obtained for partial wave modes α=1, ..., 8. These eigenvectors follow the orthonormalization and completeness conditions: ξT α ^ Tξβ¼δαβ ð13Þ ∑ξα^ Tξα¼^ I8´8;ð14Þ where the operator ⊗denotes the outer product of two matrices, δ αβ is the Kronecker delta, ^ I8´8is 8 × 8 unit matrix, and ^ Ttakes the form: ^ T¼O4´4^ I4´4 ^ I4´4O4´4 "# ð15Þ where O 4×4 and ^ I4´4are 4 × 4 zero and unity matrices. The continuity of the electric potential (Φ(i)=Φ V , where the subscript i=1, 2 indicates the medium index and the subscript V indicates the vacuum) and the normal component of electric displacement (n⋅D(i)=n⋅D V ), as well as the condition of a mechanically free surface (n⋅σ(i)=0) enforce the boundary conditions of the two solid-vacuum interfaces: bð1Þ in Uð1Þ in þ∑ 4 α¼1bð1Þ αUð1Þ α¼bVþUVþþbVUV; ∑ 4 α¼1 ~ bð2Þ αUð2Þ α¼bVþUVþekxdþbVUVekxd; ð16Þ in which we introduce 5 × 1 column vectors UðiÞ γ¼½ϕðiÞ γ;DðiÞ γ;LðiÞ γTfor wave modes γ=in, α, where the subscript in indicates the incident wave mode, α=1, ..., 4 corresponds to four physically allowed wave modes in their corresponding medium i¼1;2;UV±¼½ϕV±;DV±;0;0;0T, and ~ bð2Þ αbð2Þ αexpðipð2Þ αkxdÞfor simplicity. In the vacuum region, the electric potential and displacement fields take the form: ΦVðzÞ¼bVþϕVþekxzþbVϕVekxz nDVðzÞ¼ϵ0kxbVþϕVþekxzþϵ0kxbVϕVekxz;ð17Þ where ϕV±¼1=ffiffiffiffiffiffiffiffiffiffiffiffi ±2iϵ0 p, noting the normalization condition for the vacuum mode 2ϕV±DV±¼1. The amplitude factors bðiÞ αcan be solved from the boundary conditions of Eq. (16), following the multiple reflection method introduced in Geng and Maasilta23: the single surface reflection ( rðiÞ in!α; rðiÞ V) and transmission ( tðiÞ V!α; tðiÞ in!V) coefficients are calculated first via scattering matrices SðiÞfor solid i=1, 2, and the total transmission coefficient t α for a partial mode αis then obtained by coupling the single surface coefficients with a multiple reflection factor f m (d), which explicitly depends on the gap distance d. We note here that the overlined single surface coefficients describe the scattering of the electroacoustic wave as if there is no second adjacent solid. The 5 × 2 scattering matrices Sð1Þand Sð2Þtake the following form: Sð1Þ¼ rð1Þ tð1Þ  tð1Þ in!V rð1Þ V "# ¼Uð1Þ 1; :::; Uð1Þ 4;UVþ hi 1Uð1Þ in ;UV hi ; Sð2Þ¼ rð2Þ tð2Þ  tð2Þ in!V rð2Þ V "# ¼Uð2Þ 1; :::; Uð2Þ 4;UV hi 1Uð2Þ in ;UVþ hi ð18Þ where the expression  rðiÞ¼½  rðiÞ in!1; :::; rðiÞ in!4Tand  tðiÞ¼½  tðiÞ V!1; :::; tðiÞ V!4Tare the single surface reflection and transmission coefficients of modes α=1, ..., 4. The total transmission coefficient t α from an incoming bulk wave in solid 1 into a partial wave of mode αin solid 2 can be obtained as: tα ~ bð2Þ α bð1Þ in ¼ tð1Þ in!V tð2Þ V!αfmðdÞð19Þ where the multiple reflection factor is fmðdÞ¼½expðkxdÞ rð1Þ V rð2Þ VexpðkxdÞÞ1. Equation (19) is identical to Eq. (1) in the main text. The time-averaged transmitted power flow density in the direction normal to the surfaces from solid 1 to 2 can be expressed by the real part of the piezoelectric Poynting vector in the normal direction34: Pα¼ωkx 4jbαj2ξT α ^ Tξ α:ð20Þ For transmitted homogeneous (bulk) waves, ξT α ^ Tξ α¼±ξT α ^ Tξα¼±1, due to the Stroh-normalization condition (Eq. (13)). Therefore ∣t α ∣2can be interpreted as the power flow ratio (the transmittance) of the transmitted bulk partial wave over the incident wave in the normal direction: Pα¼jtαj2Pin:ð21Þ Data availability All relevant data are available from the authors upon request. Code availability All relevant code for simulations are available from the authors upon request. Received: 26 January 2023; Accepted: 30 June 2023; References 1. Kim, K. et al. Radiative heat transfer in the extreme near field. Nature 528, 387 (2015). 2. Kloppstech, K. et al. Giant heat transfer in the crossover regime between conduction and radiation. Nat. 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Existence considerations. J. Appl. Phys. 47, 1799 (1976). Acknowledgements This study was supported by the Academy of Finland project number 341823 and by the European Union’s Horizon 2020 research and innovation program under the grant agreement number 800923 (SUPERTED). Author contributions Z.G. and I.J.M. conceived the idea, carried out the analytical derivations and wrote the manuscript. Z.G. carried out all the numerical calculations. Competing interests The authors declare no competing interests. Additional information Supplementary information The online version contains supplementary material available at https://doi.org/10.1038/s42005-023-01293-y. Correspondence and requests for materials should be addressed to Zhuoran Geng or Ilari J. Maasilta. Peer review information Communications Physics thanks Takuma Shiga, Svend Age Biehs and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. A peer review file is available. 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