Introduction

Electromagnetic fluctuations between objects separated by a gap and the exchange of spatially mismatch of modes between them give rise to a ubiquitous interaction. This type of force was originally considered for perfect metals by H. Casimir1. Since then, many authors have shown that the Casimir interaction can serve as a probing platform for the basic physics of the materials making up the objects and their geometry2,3. Experimental advances have demonstrated that Casimir forces play an important role in quantum levitation and trapping, mechanical actuators and parametric amplifiers, and self-assembly processes among others4,5,6,7,8.

Although the Casimir force is universally present between objects, its scaling law, magnitude, and even sign are strongly dependent on the properties of the interacting materials. Particularly, the optical response, which, of course, is closely related to the electronic structure, plays a crucial role in specific features of the Casimir force. For example, topologically nontrivial properties in Dirac materials, Weyl semimetals, Chern insulators, and twisted bilayer graphenes at magic angles result in unique fingerprints in Casimir phenomena9,10,11,12. Also, objects with reduced dimensionality experience much enhanced thermal fluctuation effects in the interaction, while strong anisotropy can even result in Casimir torque13,14.

New materials have stimulated research in the field of Casimir interactions, and much of the work has been always associated with in-depth studies of the optical response properties as dictated by the underlying electronic structure of the systems. Phonons, however, are rarely discussed in the context of Casimir physics. This is typically justified as one expects negligible phonon participation in the electromagnetic exchange at separations exceeding the very short phonon mean free path due to the weak phonon-photon coupling. Nevertheless, in polar materials, the hybridization between transverse optical phonon modes and photon excitations leads to tunable phonon polariton resonances in the dielectric function, which can affect the magnitude of the Casimir energy15,16.

A subclass of polar materials composed of piezoelectrics is characterized by strong coupling between the macroscopic electric field E and the mechanical deformation vector u induced by phonons17,18. The rising surface phonon polaritons (PhPs) create additional channels of fluctuation-induced interactions between piezoelectric materials. Depending on the surface of the material, as is the case for different SiC polytypes, hybrid longitudinal-transverse modes due to additional Bragg peaks are also possible. These hybrid modes are especially pronounced in the electromagnetic coupling between substrates with finite width potentially leading to changes in the scaling law, magnitude, and even repulsion in the force. Covering the substrates with two-dimensional layers, such as graphene for example, may lead to further modifications in the Casimir interaction.

In this paper, we investigate piezoelectric materials as a materials platform that can support surface PhP modes to study phonon-modulated Casimir interactions. Phonon polariton modes are captured explicitly in the coupled elastic and electromagnetic boundary conditions, which are then taken in the Casimir force calculations within a real frequency formalism. Hybrid longitudinal-transverse excitations result in unexpected functionalities, especially for substrates with finite thickness due to the unusual interplay between quantum and thermal effects. Graphene monolayers covering the surfaces of the piezoelectric objects result in much enhanced thermal fluctuations which ultimately diminish the manifestation of piezoelectricity in the Casimir force.

Results and discussion

Electromagnetic fields and phonon polaritons in piezoelectrics

The distinct feature of a piezoelectric material is that an acoustic wave impinging on its surface extends outside of the material as a decaying evanescent electric field giving rise to PhP modes18,19,20. Bringing another piezoelectric nearby enables the exchange of fluctuating modes in the separating gap, creating conditions for phonon-modulated interactions. The system considered here is schematically given in Fig. 1 showing two identical planar piezoelectric objects with finite thickness L and separated by a distance D. Two-dimensional layers, such as graphene, may also cover the piezoelectric surfaces, forming the gap region between them.

Fig. 1: Schematics of the considered system and phonon dispersion properties of the piezoelectric material.
figure 1

a The system of two graphene-coated SiC films with thickness L separated by a distance d. b Phonon dispersions in bulk SiC with PhP branches together with LO and TO modes. The light cone in vacuum (black dotted) and in a dielectric with ε∞ (blue dotted), εst (red dotted) are also given as references.

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Let is first consider the phonon and electromagnetic modes and their hybridization in a single material. For this purpose, we utilize the generalized Born-Huang hydrodynamic model19,20,21,22,23, in which the time evolution of the ions relative mechanical displacement u is given as a driven harmonic oscillator governed by the electric field E,

$$\begin{array}{l}{\rho }_{{{{\rm{i}}}}}\left[\frac{{\partial }^{2}}{\partial {t}^{2}}+\gamma \frac{\partial }{\partial t}+{\omega }_{{{{\rm{TO}}}}}^{2}\right]{{{\bf{u}}}}={e}_{{{{\rm{i}}}}}{{{\bf{E}}}}+{\rho }_{{{{\rm{i}}}}}\left[{\beta }_{{{{\rm{TO}}}}}^{2}{{{\boldsymbol{\nabla }}}}\times \left({{{\boldsymbol{\nabla }}}}\times {{{\bf{u}}}}\right)-{\beta }_{{{{\rm{LO}}}}}^{2}{{{\boldsymbol{\nabla }}}}({{{\boldsymbol{\nabla }}}}\cdot {{{\bf{u}}}})\right],\end{array}$$
(1)

where ρi and ei are the ionic mass and charge densities, respectively, and γ is a damping constant. The speeds of the transverse optical (TO) and longitudinal optical (LO) phonon are denoted as βTO and βLO, respectively, and ωTO is the TO phonon frequency at the center of the Brillouin zone of the material. The relation \({\omega }_{{{{\rm{LO}}}}}^{2}-{\omega }_{{{{\rm{TO}}}}}^{2}={e}_{{{{\rm{i}}}}}^{2}/{\varepsilon }_{\infty }{\varepsilon }_{0}{\rho }_{{{{\rm{i}}}}}\) with ε∞ being the high-frequency relative permitivity of the material (ε0—vacuum permittivity) establishes an inter-dependence between the characteristic LO and TO phonon frequencies19,20,23.

The coupling between the electric field and mechanical wave propagation is further expressed in the electrical polarization \({{{\bf{P}}}}=\left({\varepsilon }_{\infty }-1\right){\varepsilon }_{0}{{{\bf{E}}}}+{e}_{{{{\rm{i}}}}}{{{\bf{u}}}}\) of the piezoelectric. As a result, the Gauss’ law ∇ ⋅ D= 0 is also modified since the displacement electric field now not only depends on electric field, but also on mechanical deformation D = ε0E + P =ε∞ε0E + eiu.

The mechanical wave propagation in P together with the electromagnetic fields from the Maxwell’s equations must be supplemented by appropriate boundary conditions at the piezoelectric surface. The generalized Born-Huang model distinguishes three types of phonon modes for the ionic vibrations24,25. One type is the LO phonons characterized by longitudinal displacement that satisfies ∇ × uLO = 0. Another type is the TO phonons obeying ∇ ⋅ uTO= 0 and they are a purely mechanical wave that does not induce any electromagnetic field in the material. The third type is the PhP modes associated with vibrations confined at the surface of the material whose displacement vector satisfies ∇ ⋅ uPhP = 0. The characteristic properties of the LO, TO, and PhP modes in terms of their dispersion relations, effective dielectric functions, and corresponding electromagnetic fields according to the Born-Huang model are summarized in Table 1.

Table 1 Dispersion relations, effective dielectric functions, and electromagnetic fields of the LO, TO, and PhP modes in the generalized Born-Huang model
Full size table

To better understand the different excitations in a bulk piezoelectric, we consider 2H-SiC as a representative material. In Fig. 1b we show the dispersion relations of LO, TO and PhP excitations using the following parameters22: ωTO = 795 cm−1, ε∞=6.5, εst=10.0, βTO = 9.15× 105 cm. s−1, βLO= 15.39 × 105 cm.s−1, γ =4 cm−1 while the LO phonon frequency is determined from Lyddane–Sachs–Teller relation \({\omega }_{{{{\rm{LO}}}}}={\omega }_{{{{\rm{TO}}}}}\sqrt{{\varepsilon }_{{{{\rm{st}}}}}/{\varepsilon }_{\infty }}\). There are two PhP branches and they both are of transverse polarization due to hybridized TO phonons and photons. The propagating PhP+ branch is optically active for wave vectors q < 0.65 × 107 cm−1, while the decaying evanescent PhP− branch is optically active for all q.

We now continue with examining the boundary conditions at the surface of the piezoelectric material. The electromagnetic boundary conditions are consistent with Maxwell’s equations, and they include continuity of Dz, Ex, Ey, Bz, Hx, Hy components at the surface z = 0. Additionally, the mechanical ionic displacement satisfies the elastic boundary conditions of u = (ux, uy, uz) being continuous at z = 0. Since the vacuum has no ionic displacement, then uLO + uTO+ uPhP = 0 at z =  0. The resolution of both types of conditions requires all three types of phonons, LO, TO, and PhP excitations, which leads to complex inter-relations for the reflection coefficients. For the system with isotropic materials in Fig. 1a, the Fresnel scattering matrix2,3 has a diagonal form \({{{\bf{diag}}}}\left\{{r}_{ss}({{{{\bf{q}}}}}_{\parallel },\omega ),{r}_{pp}({{{{\bf{q}}}}}_{\parallel },\omega )\right\}\), with coefficients for α= (s, p) polarization found as

$${r}_{\alpha \alpha }({{{{\bf{q}}}}}_{\parallel },\omega )={r}_{\alpha \alpha }^{12}({{{{\bf{q}}}}}_{\parallel },\omega )+\frac{{t}_{\alpha \alpha }^{12}({{{{\bf{q}}}}}_{\parallel },\omega ){r}_{\alpha \alpha }^{23}({{{{\bf{q}}}}}_{\parallel },\omega ){t}_{\alpha \alpha }^{21}({{{{\bf{q}}}}}_{\parallel },\omega ){e}^{2i{q}_{{{{\rm{PhP}}}},z}L}}{1-{r}_{\alpha \alpha }^{21}({{{{\bf{q}}}}}_{\parallel },\omega ){r}_{\alpha \alpha }^{23}({{{{\bf{q}}}}}_{\parallel },\omega ){e}^{2i{q}_{{{{\rm{PhP}}}},z}L}},$$
(2)
$${\lim }_{L\to \infty }{r}_{ss}\left({{{{\bf{q}}}}}_{\parallel },\omega \right)=\frac{{q}_{z}-{q}_{{{{\rm{PhP}}}},z}-{\mu }_{0}\omega {\sigma }_{{{{\rm{s}}}}}({{{{\bf{q}}}}}_{\parallel },\omega )}{{q}_{z}+{q}_{{{{\rm{PhP}}}},z}+{\mu }_{0}\omega {\sigma }_{{{{\rm{s}}}}}({{{{\bf{q}}}}}_{\parallel },\omega )}$$
(3)
$${\lim }_{L\to \infty }{r}_{pp}\left({{{{\bf{q}}}}}_{\parallel },\omega \right)=\frac{{\varepsilon }_{{{{\rm{PhP}}}}}(\omega ){q}_{z}-\left({q}_{{{{\rm{PhP}}}},z}+\Omega ({{{{\bf{q}}}}}_{\parallel },\omega )\right)+{\mu }_{0}c{\sigma }_{{{{\rm{s}}}}}({{{{\bf{q}}}}}_{\parallel },\omega )\left(\frac{{q}_{z}c}{\omega }\right)\left({q}_{{{{\rm{PhP}}}},z}+\Omega ({{{{\bf{q}}}}}_{\parallel },\omega )\right)}{{\varepsilon }_{{{{\rm{PhP}}}}}(\omega ){q}_{z}+\left({q}_{{{{\rm{PhP}}}},z}+\Omega ({{{{\bf{q}}}}}_{\parallel },\omega )\right)+{\mu }_{0}c{\sigma }_{{{{\rm{s}}}}}({{{{\bf{q}}}}}_{\parallel },\omega )\left(\frac{{q}_{z}c}{\omega }\right)\left({q}_{{{{\rm{PhP}}}},z}+\Omega ({{{{\bf{q}}}}}_{\parallel },\omega )\right)},$$
(4)

where μ0 is the vacuum magnetic permeability, q∥ = (qx, qy) is the wave vector along the surface, \({q}_{z}=\sqrt{{\omega }^{2}/{c}^{2}-{q}_{\parallel }^{2}}\), and \({q}_{{{{\rm{PhP}}}},z}=\sqrt{{\varepsilon }_{{{{\rm{PhP}}}}}\left(\omega ,q\right){\omega }^{2}/{c}^{2}-{q}_{\parallel }^{2}}\). The indices i, j=1, 2, 3 denote the regions shown in Fig. 1a, while rij, tij correspond to the reflection and transmission coefficients at the interface between the two media. For semi-infinite plates for which L→ ∞, Eq. (2) becomes Eqs. (3), (4). Explicit expressions for \({r}_{\alpha \alpha }^{ij}\) and \({t}_{\alpha \alpha }^{ij}\) and details of their derivations are given in Sections I and II in the Supplementary Information. The atomic layers in the gap region are taken into account via their surface conductivity, which in the case of graphene is taken to be its universal value σs(q∥, ω)= e2/4ℏ26.

A distinct feature arising from the boundary conditions is the surface PhP (SPhP) mode. Such excitations are a direct consequence of the coupled electromagnetic-elastic boundary conditions, such that

$$\Omega ({{{{\bf{q}}}}}_{\parallel },\omega )=\frac{{q}_{\parallel }^{2}\left({q}_{{{{\rm{PhP}}}},z}-{q}_{{{{\rm{TO}}}},z}\right)}{{q}_{\parallel }^{2}+{q}_{{{{\rm{TO}}}},z}{q}_{{{{\rm{LO}}}},z}}\left({\varepsilon }_{{{{\rm{PhP}}}}}(\omega )-{\varepsilon }_{\infty }\right),$$
(5)

in which \({q}_{{{{\rm{TO}}}},z}=\sqrt{{\omega }_{{{{\rm{TO}}}}}^{2}-{\omega }^{2}}/{\beta }_{{{{\rm{TO}}}}}\) and \({q}_{{{{\rm{LO}}}},z}=\sqrt{{\omega }_{{{{\rm{LO}}}}}^{2}-{\omega }^{2}}/{\beta }_{{{{\rm{TO}}}}}\) are the local out-of-plane components of the wave vectors of the TO and LO phonons. Eqs. (2)-(4) show that, in fact, the elastic boundary conditions affect only the p-polarized modes, while the s-polarization remains the same as from standard electromagnetic boundary conditions. The factor \(\left({\varepsilon }_{{{{\rm{PhP}}}}}(\omega )-{\varepsilon }_{\infty }\right)\) indicates that the electromagnetic-elastic boundary conditions term Ω(q∥, ω) is significant, especially at a small frequency. Consequently, the dielectric function for the piezoelectric material is determined by solving the PhP excitations dispersion ω2 = q2c2/εPhP(ω, q), which takes into account the nonlocal dielectric function of PhP mode in the Born-Huang model given in Table 1. Since βTO, βLO ≪ c, the PhP dielectric function is found as

$$\frac{{\varepsilon }_{{{{\rm{PhP}}}}}(\omega )}{{\varepsilon }_{\infty }}\approx \frac{{\omega }_{{{{\rm{LO}}}}}^{2}-\omega (\omega +i\gamma )-\frac{{\varepsilon }_{\infty }{\omega }^{2}{\beta }_{{{{\rm{TO}}}}}^{2}}{{c}^{2}}\frac{{\omega }_{{{{\rm{LO}}}}}^{2}-\omega (\omega +i\gamma )}{{\omega }_{{{{\rm{TO}}}}}^{2}-\omega (\omega +i\gamma )}}{{\omega }_{{{{\rm{TO}}}}}^{2}-\omega (\omega +i\gamma )-\frac{{\varepsilon }_{\infty }{\omega }^{2}{\beta }_{{{{\rm{TO}}}}}^{2}}{{c}^{2}}\frac{{\omega }_{{{{\rm{LO}}}}}^{2}-\omega (\omega +i\gamma )}{{\omega }_{{{{\rm{TO}}}}}^{2}-\omega (\omega +i\gamma )}}.$$
(6)

The Born-Huang hydrodynamics model is based on a continuum approximation and it has been successfully applied in a variery of piezoelectric materials22,27,28,29,30. SiC, the material of interest here, has several polytypes with distinct electromagnetic spectra. Specifically, for polytypes, such as 4H-, 6H-, 8H-, 10H-SiC, an elongated c-axis in the lattice results in additional Bragg peaks in the spectrum causing zone folding of the LO phonon dispersion ω(q) in the center of the Brillouin zone, such that q∥→ q∥ + qm,n where qm,n = 2mπ/nc2 and m= 1, 2, …, n − 1 is the number of Bragg peak scattering. Signatures of the zone-folding lying in the Reststrahlen band between TO and LO phonon frequencies have been observed experimentally in the Raman spectra of various SiC polytypes22,31,32,33. For a 4H-SiC polytype, considered here, zone-folding is accounted by taking \({q}_{\parallel }^{2}\to {q}_{\parallel }({q}_{\parallel }+{q}_{m,n})\) in Eq. (5), where m = 1, n=  2 and q1,2 =π/c2 with c2= 5 Å22 (Details in Section IA in the Supplementary Information).

In Fig. 2a, b, the density plots of the real and imaginary parts of rpp for a semi-infinite 4H-SiC plate are given. We find that in addition to the PhP+ and PhP− branches (broad dark red regions in Fig. 2a), rather localized modes close to the photon line and between ωTO, ωLO (blue regions) appear. These modes are a direct consequence of the combined effect from the zone-folded phonons and SPhPs, and they are associated with εPhP(ω)qz + qPhP,z+ Ω(q∥, ω) = 0. One obtains that the regions near the surface modes SPhP+ and SPhP− branches are characterized by \({{{\bf{Re}}}}({r}_{pp})\to \pm \infty\). On the other hand, \({{{\bf{Im}}}}({r}_{pp})\to \infty\) is mostly localized along SPhP− and \({{{\bf{Im}}}}({r}_{pp})\to -\infty\) along SPhP+ near LO line. Anti-crossing due to the strong interaction between zone-folded LO phonons and SPhPs is observed in Fig. 2b. In fact, the resultant quasiparticle, longitudinal-transverse SPhP (LT-SPhP), contains an admixture of longitudinal and transverse polarization types. LT-SPhPs are accessible for a wide range of wave vectors and they have recently been observed in SiC systems22.

Fig. 2: Density plots of the Fresnel reflection coefficients of a semi-infinite SiC substrate.
figure 2

Density plots in the (ℏω, q) domain showing different PhP, LO, and TO branches and their dispersions in a \({{{\bf{Re}}}}({r}_{pp})\) and b \({{{\bf{Im}}}}({r}_{pp})\). Density plots in the (iℏξ, q) domain of a semi-infinite SiC substrate showing \(| {{{\bf{Im}}}}({r}_{pp})|\) of a SiC substrate supporting: c SPhP and d LT-SPhP.

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Casimir force for planar objects

Lifshitz formalism34 is widely used for obtaining interaction energies and forces between objects. Calculations are typically done using summation over the imaginary Matsubara frequency domain ω = iξ, where dielectric functions are monotonically decreasing functions. However, the reflection coefficient \({r}_{pp}({{{{\bf{q}}}}}_{\parallel },i\xi )\) is a complex quantity, as can be seen for plates supporting SPhP modes (Fig. 2c) or LT-SPhP excitations (Fig. 2d), especially in regions for larger wave vectors. This behavior is tightly associated with the different dispersion relations of the photon excitations and the optical phonons inside the piezoelectric: the photon frequency is proportional to the wave vector, while the optical phonon starts from a characteristic frequency and decreases as a function of the wave vector (Table 1). Thus to calculate the force, we first consider Rytov’s theory of fluctuating electrodynamics in real frequencies35, as originally proposed by Lifshitz34. Particularly, the pressure between two identical planar objects34,36,37,38 is found as

$$P(D,T)= {\sum}_{\alpha =s,p}\left[{P}_{{{{\rm{prop}}}}}^{\alpha }(D,T)+{P}_{{{{\rm{evan}}}}}^{\alpha }(D,T)\right],$$
(7)
$${P}_{{{{\rm{prop}}}}}^{\alpha }(D,T)=\frac{\hslash }{4{\pi }^{3}}{{{\bf{Re}}}}\int\limits_{0}^{+\infty }d\omega \,\eta (\omega ,T) {\iint}_{q < \omega /c}{d}^{2}{{{{\bf{q}}}}}_{\parallel }\frac{{q}_{z}{r}_{\alpha \alpha }^{(1)}({{{{\bf{q}}}}}_{\parallel },\omega ){r}_{\alpha \alpha }^{(2)}({{{{\bf{q}}}}}_{\parallel },\omega ){e}^{2i{q}_{z}D}}{1-{r}_{\alpha \alpha }^{(1)}({{{{\bf{q}}}}}_{\parallel },\omega ){r}_{\alpha \alpha }^{(2)}({{{{\bf{q}}}}}_{\parallel },\omega ){e}^{2i{q}_{z}D}},$$
(8)
$${P}_{{{{\rm{evan}}}}}^{\alpha }(D,T)=\frac{\hslash }{4{\pi }^{3}}{{{\bf{Re}}}}\int_{0}^{+\infty }d\omega \,\eta (\omega ,T){\iint}_{q > \omega /c}{d}^{2}{{{{\bf{q}}}}}_{\parallel }\frac{{q}_{z}{r}_{\alpha \alpha }^{(1)}({{{{\bf{q}}}}}_{\parallel },\omega ){r}_{\alpha \alpha }^{(2)}({{{{\bf{q}}}}}_{\parallel },\omega ){e}^{2i{q}_{z}D}}{1-{r}_{\alpha \alpha }^{(1)}({{{{\bf{q}}}}}_{\parallel },\omega ){r}_{\alpha \alpha }^{(2)}({{{{\bf{q}}}}}_{\parallel },\omega ){e}^{2i{q}_{z}D}},$$
(9)

where \({r}_{\alpha \alpha }^{1,2}\) are the Fresnel reflection coefficients for objects (1,2) for polarization α= s, p and \(\eta (\omega ,T)=\coth (\frac{\hslash \omega }{2{k}_{{{{\rm{B}}}}}T})\) is the Planck distribution function at temperature T. The quantum mechanical limit is found by taking η(ω, 0) → 1, while the thermal limit corresponds to η(ω, T)→2kBT/ℏω.

Applying Wick rotation ω → iξ to \({P}^{\alpha }(D,T)={P}_{{{{\rm{prop}}}}}^{\alpha }(D,T)+{P}_{{{{\rm{evan}}}}}^{\alpha }(D,T)\) and combining with the fact that the Matsubara frequencies ξn =2πnkBT/ℏ appear as poles of η(iξ, T), the Casimir pressure is found as

$${P}^{\alpha }(D,T) = {P}_{{{{\rm{prop}}}}}^{\alpha }(D,T)+{P}_{{{{\rm{evan}}}}}^{\alpha }(D,T)\\ = -\frac{{k}_{{{{\rm{B}}}}}T}{2{\pi }^{2}}{{{\bf{Re}}}}{\mathop{\sum}_{n = 0}^{\infty }}^{{\prime} }\int\limits_{0}^{+\infty }{d}^{2}{{{{\bf{q}}}}}_{\parallel }\left\{{\kappa }_{z}\frac{{r}_{\alpha \alpha }^{(1)}({{{{\bf{q}}}}}_{\parallel },i{\xi }_{n}){r}_{\alpha \alpha }^{(2)}({{{{\bf{q}}}}}_{\parallel },i{\xi }_{n}){e}^{-2{\kappa }_{z}D}}{1-{r}_{\alpha \alpha }^{(1)}({{{{\bf{q}}}}}_{\parallel },i{\xi }_{n}){r}_{\alpha \alpha }^{(2)}({{{{\bf{q}}}}}_{\parallel },i{\xi }_{n}){e}^{-2{\kappa }_{z}D}}\right\}.$$
(10)

where \({\kappa }_{z}=\sqrt{{\xi }_{n}^{2}/{c}^{2}+{q}_{\parallel }^{2}}\). The prime in the summation sign indicates that the n= 0 Matsubara term is weighted by 1/2. The above equation can now be applied to obtain the interaction pressure between the piezoelectric materials by taking into account the complex nature of their reflection coefficients.

Casimir force between piezoelectric materials

Semi-infinite plates

We first consider the case of semi-infinite plates in Fig. 1a) with corresponding Fresnel reflection coefficients given in Eqs. (3), (4). Results for the interaction between plates without graphene at T = 0 K are shown in Fig. 3a. One finds that Ps and Pp follow similar scaling laws, however, the contribution from Pp to the tolal pressure is dominant. Both Ps and Pp are bound by their respective limits associated with the ε∞ and εst dielectric constants. It appears that Ps, Pp ~ D−4 everywhere except in the 0.5–10 microns region, in which the Casimir interaction transitions between the limiting coupling found with εst and ε∞. This transition region is comparable to the phonon wavelengths λLO = c/ωLO =1.6 and λTO=c/ωTO =2 microns. The calculations show that the Casimir interaction for substrates supporting SPhPs or LT-SPhPs excitations is identical at small separations, while in the sub-micron and larger region (consistent with the long-wavelength limit), P(LT-SPhS) is slightly larger than P(SPhS) (see Section III in the Supplementary Information for details).

Fig. 3: Casimir pressure between two semi-infinite SiC plates.
figure 3

The Casimir pressure P normalized by the perfect metal limit Pm = − π2ℏc/240D4 a at T= 0 K; b at T = 100 K; c for substrates covered by graphene sheets at T=100 K. The characteristic wavelengths are denoted as λTO = c/ωTO, λLO = c/ωLO, λth=  ℏc/kBT, λgr = ℏvF/kBT where ωTO = 795 cm−1, ωLO=986 cm−1, vF=105 m/s. The Casimir pressures for semi-infinite plates with ε∞= 6.5 and εst= 10.0 (characteristic dielectric constants for SiC) are also shown.

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As an example representation for the role of temperature, the Casimir pressure at T =100 K is given in Fig. 3b, where the characteristic thermal wavelength λth = ℏc/kBT are also shown. One finds that in the (λLO, λth) region, Ps, Pp experience a scaling law transition from D−4 to D−3 signaling the onset of thermal fluctuations at larger separations. More precisely, one can find the thermal transition distance Dth from P(T =0) =Pn=0(T), which yields Dth ≈ 8 microns at T =100 K. (see Section V in the Supplementary Information). Similar to the quantum mechanical limit, the Casimir pressure is dominated by the p-polarized modes with little distinction between SPhPs and LT-SPhPs modes at sub-micron and larger separations.

The inclusion of graphene layers on top of the surfaces of the semi-infinite piezolectrics does not change significantly the Casimir interaction at T= 0 K. This is evident in Fig. 3c, which also shows the greatly diminished Ps role in the total pressure. We further find that the distance marking the quantum-to-thermal transition is reduced by a factor of 1.5 due to the graphene presence. This is not surprising given the much reduced characteristic thermal wavelength of graphene λgr=  ℏvF/kBT, where vF =105 m.s−1 is the Fermi velocity13,39,40. As a result, the D−4→ D−3 transition happens in a broader region between λgr and λth. From P(T = 0) = Pn=0(T) a transition distance Dth≈5 microns is found for T =100 K (see Section V in the Supplementary Information). As T increases, however, the λgr <Dth < λth shrinks, indicating a sharper quantum-to-thermal transition in the Casimir interaction as shown for T  = 300 K in the Supplementary Information.

Finite thickness plates

Let us now consider the Casimir interaction between two SiC plates with a finite thickness distinguishing SPhPs vs. LT-SPhPs and the role of graphene. Numerical calculations for L = 20, 82, and 500 nm at T =0 K are given in Fig. 4a, c, e. We find that for SiC supporting both types of modes, the pressure decreases non-monotonically as D increases. It appears that for D≲L, P exhibits similar behavior as the one found for semi-infinite plates (Fig. 3a). As D becomes larger, however, the P/Pm experiences non-monotonical decrease with a sharper descent for thinner plates. Particularly, our calculations show that for plates with SPhPs modes P/Pm ~ D−2, in other words, the scaling law has changed from D−4 for semi-infinite plates to D−6 upon increasing the separation distance41,42. This type of scaling law is indicative that the SiC plates behave as two dielectrics.

Fig. 4: Casimir pressure between two SiC plates with finite thickness.
figure 4

Casimir pressure P normalized by the perfect metal limit Pm = − π2ℏc/240D4 for plates with a thickness L =20 nm, T = 0 K; b thickness L =20 nm, T =100 K; c thickness L = 82 nm, T= 0 K; d thickness L = 82 nm, T = 100 K; e thickness L =  500 nm, T = 0 K; f thickness L=500 nm, T = 100 K. Results are shown for plates supporting SPhPs (solid blue), LT-SPhPs (solid red), plates supporting SPhPs and covered with graphene (dashed blue), plates supporting LT-SPhPs and covered with graphene (dashed red); Density plots showing the D vs. L dependence of the Casimir pressure P/Pm between two plates supporting LT-SPhPs: g at T = 0 K and h at T = 100 K.

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In fact, since \(\Omega ({{{{\bf{q}}}}}_{\parallel },\omega ) \sim {q}_{\parallel }^{2}\) (Eq. (5)) the elastic boundary terms have no contribution to the long-wavelength limit of the interaction, thus the reflection and transmission coefficients \({r}_{pp}^{ij},{t}_{pp}^{ij}\) of p-polarized waves obeys the same constrains as dielectrics described with εst. Consequently, the Fresnel reflection coefficient from Eq. (2) is found to have the following asymptotic behavior rαα(q∥, iξ) ∝ ξL/c for both type of polarizations at long-wavelength limit q∥→ 0. It is the rαα~ ξ relation that gives the additional powers in the separation scaling law of the Casimir pressure, such that (details in Section IV.A in the Supplementary Information)

$$P(D\, \gg \, L)\approx \left\{\begin{array}{ll}-\frac{{({\varepsilon }_{{{{\rm{st}}}}}-1)}^{2}\left(9{\varepsilon }_{{{{\rm{st}}}}}^{2}+10{\varepsilon }_{{{{\rm{st}}}}}+4\right){L}^{2}\hslash c}{32{\pi }^{2}{\varepsilon }_{{{{\rm{st}}}}}^{2}{D}^{6}}\quad &T=0\\ -\frac{3{({\varepsilon }_{{{{\rm{st}}}}}-1)}^{2}{L}^{2}{k}_{{{{\rm{B}}}}}T}{32\pi {D}^{5}}\quad &T\,\, \gg \, \frac{\hslash c}{D{k}_{{{{\rm{B}}}}}}\end{array}\right.$$
(11)

The inclusion of graphene has a minor effect as long as D ≲ L indicating that the quantum regime is dominated by the properties of the piezoelectric plate itself. At sub-micron and larger separations, however, the interaction is almost completely dominated by graphene P  ≈ Pgr ≈ 5.38 × 10−2Pm recovering the expected D−4 scaling law.

When the SiC plates support LT-SPhPs, however, P may have significant deviations from the one obtained for plates with SPhPs modes, which are especially pronounced at larger D. For example, the interaction becomes negative with D−4 scaling law at D > 104 nm for L = 20 nm plates and D > 105 nm for L=500 nm plates. This repulsive effect is reduced upon the inclusion of graphene layers. On the other hand, for plates with L = 82 nm, the interaction becomes strongly attractive, approaching the perfect metal limit for D > 104 nm. Graphene layers reduce by close to half this enhancement. Thus, unlike SiC plates supporting SPhPs, the LT-SPhPs modes in the long-range limit cannot be completely dominated by graphene partly due to the same D−4 scaling law.

Numerical calculations for L =20, 82, 500 nm at T  = 100 K are given in Fig. 4b, d, f. The results show that temperature alters the scaling law from D−6 to D−5 for plates supporting only SPhPs. LT-SPhP modes, however, have a much-pronounced effect as Fig. 4b, f shows a stronger repulsive effect arising at smaller separations (about an order smaller as compared to the T = 0 K case) with the typical D−3 scaling law of thermal fluctuation. The transition to D−3 behavior is also enhanced when compared to plates with SPhPs, as Fig. 4d shows a steeper scaling law change with a larger magnitude of attraction. The inclusion of graphene accentuates the role of thermal fluctuations. As mentioned earlier, since graphene is responsible for the almost complete reflection of the low-frequency p-modes, the Casimir pressure becomes dominated by thermal effects around D ≈ 5 microns consistent with the thermal wavelength of graphene. The repulsion from LT-SPhPs is washed away rendering a very similar behavior of finite thickness piezoelectric plates with SPhPs.

Figure 4a–f shows that there is an unusual interplay between D and L in the interaction between plates supporting LT-SPhPs. To further examine this relation, Fig. 4g, h shows a D vs L density map of the Casimir pressure at T = 0 and 100 K. The broad yellow region corresponds to D ≤ L showing that the interaction is similar to the one of semi-infinite plates (discussed earlier). An unusual feature is the oscillatory-like behavior outside of that limit marked by positive and negative peaks. This is unique to the LT-SPhPs excitations, and it is directly linked to Ω± ~ q1,2/qLO at the long-wavelength limit with qLO=  ωLO/βL being the LO phonon wave vector. Thus, unlike SPhPs for which \(\Omega \sim {q}_{\parallel }^{2}\), the characteristics frequencies for LT-SPhPs entering the Fresnel coefficients are constant associated with the phonon zone-folding parameter q1,2 = π/c2. We find that

$$P(D\gg L)\approx \left\{\begin{array}{ll}-\frac{{\pi }^{2}\hslash c}{240{D}^{4}}{{{\mathcal{R}}}}(L)\quad &T=0\\ -\frac{\zeta (3){k}_{{{{\rm{B}}}}}T}{8\pi {D}^{3}}{{{\mathcal{R}}}}(L)\quad &T\gg \frac{\hslash c}{{k}_{{{{\rm{B}}}}}D}\end{array}\right.,$$
(12)
$${{{\mathcal{R}}}}(L)=\frac{1-{\left(\frac{\pi {\varepsilon }_{{{{\rm{st}}}}}{q}_{{{{\rm{LO}}}}}}{2\left({\varepsilon }_{{{{\rm{st}}}}}-{\varepsilon }_{\infty }\right){q}_{1,2}}\right)}^{2}{\cot }^{2}\left(\frac{{q}_{{{{\rm{LO}}}}}L}{2}\right)}{{\left(1+{\left(\frac{\pi {\varepsilon }_{{{{\rm{st}}}}}{q}_{{{{\rm{LO}}}}}}{2\left.({\varepsilon }_{{{{\rm{st}}}}}-{\varepsilon }_{\infty })\right){q}_{1,2}}\right)}^{2}{\cot }^{2}\left(\frac{{q}_{{{{\rm{LO}}}}}L}{2}\right)\right)}^{2}}$$
(13)

Apparently, confining the LO phonon into the cavity results in resonant-like trapped LT-SPhPs excitations controlled by the thickness and associated with the periodic \(\cot \left(\frac{{q}_{{{{\rm{LO}}}}}L}{2}\right)\). The attractive “resonances” occur when \(\cot \left(\frac{{q}_{{{{\rm{LO}}}}}L}{2}\right)=0\), while repulsive “resonances” happen for \(\cot \left(\frac{{q}_{{{{\rm{LO}}}}}L}{2}\right)=\pm \frac{2\sqrt{3}\left({\varepsilon }_{{{{\rm{st}}}}}-{\varepsilon }_{\infty }\right){q}_{1,2}}{\pi {\varepsilon }_{{{{\rm{st}}}}}{q}_{{{{\rm{LO}}}}}}\) (see Section IV.B in the Supplementary Information). These resonant-like features are controlled by the piezoelectric properties and the finite width of the plate through the factor \({{{\mathcal{R}}}}(L)\), and they appear broadened as temperature is increased. At the same time, the characteristic scaling laws at the quantum and thermal limit are preserved. The numerical simulations shown in Fig. 4g, h are consistent with the analytical expressions at the long-wavelength limit in the above equations.

Conclusions

The ubiquitous Casimir force is typically considered a macroscopic manifestation of quantum vacuum excitations, where phonons are not relevant. However, in piezoelectric materials, the phonon excitations can lead to surface hybrid modes, which can significantly modify the force. These developments are made possible by employing a generalized Born-Huang model. Taking the SiC plate as a representative example, the model describes SPhPs, decaying quasiparticles with longitudinal polarization. The Born-Huang model can also accommodate LO phonon zone-folding in SiC polytypes, whose strong interaction with SPhPs results in a hybrid quasiparticle, LT-SPhPs, which is of mixed longitudinal and transverse polarization.

A similar type of polarization hybridization also occurs in plasmonic systems due to the finite thickness of the substrate and the electronic pressure43,44,45. In such systems, the hybrid modes are possible at a finite wave vector range. In plasmonic materials, one has to resort to ultrathin films (on the order of a few nanometers) where the strong localization of the electric field leads to hybrid plasmon polarizations characterized by a vanishing permittivity at a small frequency range43,44. For such systems, the Casimir force exhibits unusual scaling as a function of the film thickness and distance separation.

In the case of piezoelectrics, however, the hybrid phonon modes are not restricted to a specific wave vector range. One does not have to utilize ultrathin plates to observe SPhPs signatures in the Casimir interaction. Because of the SPhP excitations, the effective dielectric constants of piezoelectrics are different for low and high-frequency modes. Consequently, the Casimir pressure exhibits a transition at short and long-distance limits with associated transition distance around the phonon wavelengths λTO,LO. However, since SPhPs and LT-SPhPs modes behave differently in the long-wavelength limit, the Casimir force acquires specific to each type of excitation features.

Taking into account the finite thickness of the piezoelectric plates, this distinction is even more significant. Indeed, when the separation distance is larger than the thickness of the piezoelectric plates D ≫ L, their Casimir pressure assisted by SPhPs decays as D−6 at larger separations, similar to substrates made of typical dielectrics41,42. This behavior is primarily associated with the SiC dielectric function approaching the εst limit and the small long-wavelength contributions of SPhS modes. The force between plates with LT-SPhPs decays as D−4 in the long-range limit. It appears that in this case the hybrid modes, however, the confined LO phonons in the cavity can result in resonant-like conditions associated with the periodic \(\tan \left({q}_{{{{\rm{LO}}}}}L\right)\). Positive or negative “resonances” in the Casimir pressure are controlled by the thickness and the optical properties of the piezoelectric material, as discussed previously.

Thermal fluctuation at nonzero temperatures dominates the quantum fluctuations as long as the separation distance is larger than thermal wavelength D ≫ λth. Thermal effects change the scaling law of the quantum Casimir pressure between finite thickness plates supporting SPhP modes from D−6 to D−5 for SPhP modes, while D−4 to D−3 transition is found for finite thickness plates supporting LT-SPhP modes. Similar to zero temperature, confining LO phonon into cavity modes affects LT-SPhP modes and makes the force repulsive at appropriate thickness.

Coating the piezoelectric plates by graphene monolayers also has interesting consequences in the Casimir interaction. Due to its constant optical conductivity at small frequencies, there is an almost complete reflection of the low-frequency p-modes. As a result, the effects from SPhPs and LT-SPhPs excitations at finite temperature are significantly reduced now in the Casimir force for separations D > λth. Graphene, however, is almost inconsequential for smaller separations regardless of temperature.

Our study suggests that phonons do have strong effects on the Casimir force, whose “typical” source is electromagnetic fluctuations between objects. Piezoelectric materials appear to be a suitable platform for observing the consequences of surface plasmon polaritons through the exchange of quantum vacuum excitations. Depending on the material polytype, as is the case for SiC, plasmon polaritons with longitudinal or hybrid longitudinal-transverse polarizations are possible, and they can be resolved simply by tracking the scaling law and/or sign of the Casimir interaction. From our study, we find that for separations less than 10 microns, the magnitude of Casimir pressure is around 10−8– 10−3 mPa. This range is accessible experimentally, as demonstrated recently by the Casimir And Non-Newtonian force EXperiment (CANNEX) reporting measure Casimir pressures in the 10−8–10−3 mPa range46,47. In fact, signatures of the interaction in terms of scaling law, magnitude, and/or sign can be used to probe the different plasmon polariton modes in piezoelectric materials and their structural differences. Our study further expands the materials perspective of the Casimir phenomena by demonstrating the unique role of phonons and their coupling to electromagnetic excitations.