Phononic Casimir Effect in Planar Materials

Abstract

The Phononic Casimir effect between planar objects is investigated by deriving a formalism from the quantum partition function of the system following multiscattering approach. This fluctuation-induced coupling is mediated by phonons modeled as an effective elastic medium. We find that excitations with three types of polarizations arise from the resolved boundary conditions, however the coupling is dominated by only one of these degrees of freedom due to exponential suppression effects in the other two. The obtained scaling laws and dependence on materials properties and temperature suggest effective pathways of interaction control. Scenarios of materials combinations are envisioned where the Phononic Casimir effect is of similar order as the standard Casimir interaction mediated by electromagnetic fluctuations.

Figures (5)

• Figure 1: Schematics of the considered system where two substrates made of solid Material 1 interact via phonon modes exchange through a gap filled with solid Material 0.
• Figure 2: (a) Phononic Casimir energy per unit area between Ge plates separated by Si at $T=10^{-15}$ K; (b) Phononic Casimir energy normalized by the electromagnetic Casimir energy for several materials combinations at $T=300$ K. The ratio $\Gamma=\large (\frac{(\mu_0-\mu_1)/(\epsilon_0-\epsilon_2)}{(\mu_0+\mu_1)/(\epsilon_0+\epsilon_2)}\large )^2$ is also shown. The materials parameters are summarized in Table S1 in the Supplementary Information supp.
• Figure 3: Schematics of the incident, reflected, and transmitted elastic waves in two isotropic media with a shared planar boundary. Blue arrows represent transversal waves with $\bm{M}$ corresponding to an SH wave and $\bm{N}$ corresponding to an SV wave. Red arrows represent longitudinal waves labeled as P waves, given by the $\bm{L}$ functions. The angles that the incident, reflected, and transmitted elastic waves with the transverse and longitudinal polarizations make with the vertical are also denoted. The reflected and transmitted planar waves have the same frequency $\omega$ and parallel momentum $\bm{k}_{\parallel} = (k_{x},k_{y})$, but they have different sound velocities, thus $k_{z,\ell} \neq k_{z,t}$).
• Figure 4: Phononic and electromagnetic Casimir interaction energies for (a) Ge/Si/Ge, (b) Diamond/Si/Diamond, and (c) Diamond/BN$_{\text{hex}}$/Diamond configurations. The contributions from the quantum mechanical $\mathcal{E}_M^{qm}$ and the thermal $\mathcal{E}_M^{T}$ and $\mathcal{E}_{L,N}^{T}$ limits in Eqs. (17), (18), and (16) from the main text, respectively, are shown. Similarly, contributions from the quantum mechanical $E_{em}^{qm}$ and thermal $E_{em}^T$ limits are also shown.
• Figure 5: Typical values of the magnitude of the unitless constant $B$ in Eqs (15, 16) in the main text.