Boundary condition for phonon distribution functions at a smooth crystal interface and interfacial angular momentum transfer

TL;DR

This work develops two-tier boundary conditions for phonon distribution functions at smooth crystal interfaces: coarse-grained conditions that capture energy and SAM transfer via acoustic-mismatch-inspired reflectance/transmittance, and detailed conditions that preserve center-of-mass information to enforce TAM conservation and predict transverse (Imbert-Fedorov–like) shifts. By linking elastic-plane-wave scattering to lattice vibrations, the authors show that interfacial SAM flux discontinuities are compensated by interfacial OAM generation, yielding a circulating edge flow of phonons and a TAM flux continuous across the interface. The theory provides analytic expressions for OAM density and flux, and demonstrates OAM generation in junctions with temperature gradients, with OAM magnitude comparable to SAM in some cases. These results establish a rigorous framework for polarization-dependent phonon transport and have potential applications in heat transport, phonon-dynamical spin-orbit effects, and interfacial boson transport more broadly.

Abstract

We theoretically elucidate the boundary conditions for phonon distribution functions of long-wavelength acoustic phonons at smooth crystal interfaces. We first derive boundary conditions that fully incorporate reflection, transmission, and mode conversion. We obtain these conditions for phonons from those for classical lattice vibrations, using the correspondence between the quantum and classical descriptions. This formulation provides a theoretical foundation for the acoustic mismatch model, widely used to analyze Kapitza resistance. We then refine the boundary conditions to include spatial dependence parallel to the interface. The refined form captures transverse shifts of elastic wave packets, analogous to the optical Imbert--Fedorov shift, and ensures conservation of total angular momentum. Consequently, circularly polarized phonons carrying spin angular momentum (SAM) generate phonon orbital angular momentum (OAM) at the interface. We analytically determine the spatial profile of this OAM and demonstrate that SAM and OAM are both involved in the interfacial diffusion of chiral phonons. Our theory provides concise boundary conditions for phonons, with applications ranging from heat transport to phonon angular momentum transport.

Figures (10)

• Figure 1: Schematic illustration of the reflection and transmission of a phonon or elastic wave packet at the interface $z = 0$ between two crystals, and the associated generation of orbital angular momentum (OAM). (a) When the incident wave packet (black line) is circularly polarized, the reflected and transmitted wave packets (colored lines) emerge from positions laterally displaced from the incident plane by $\Delta r_{\text{IF}}$. The wavevector in the region $z < 0$ is denoted by $\bm{k}$, and in $z > 0$ by $\bm{q}$, with equal in-plane components: $\bm{k}_{\parallel} = \bm{q}_{\parallel}$. (b) Definition of right-handed (RH) and left-handed (LH) circular polarizations as used in this study. They are defined from the point of view of the source. (c) Orthogonal coordinates $x'y'$ introduced along the in-plane wavevector component $\bm{k}_{\parallel} = \bm{q}_{\parallel}$. (d) Spin angular momentum (SAM) $S_z$ associated with circular polarization accumulates in the region $z < 0$ and diffuses across the interface, which in turn induces a finite OAM $L_z$ at the interface. The phonon wave packet is depicted as an orange bell shape with a black ring indicating circular polarization. The OAM $L_z$ arises from the transverse shifts $\Delta r_{\text{IF}}$, which yield an extrinsic angular momenta (AM) of the form $\Delta r_{\text{IF}} \times \hbar \left|\bm{k}_{\parallel}\right|$.
• Figure 2: Flow of the discussion in this paper.
• Figure 3: Schematics of the adjacent two crystals, and the reflection and transmission of plane waves at a smooth interface $z = 0$. The polarization vectors of each wave mode are indicated by orange arrows. The interface corresponds to the $x'y'$ plane, and the incident plane is the $zx'$ plane. Note that $x'$ and $y'$ axes are in the direction of $\bm{k}_{\parallel}$ and $\hat{\bm{z}}\times \bm{k}_{\parallel}$, respectively. We also use a laboratory coordinate $(x, y, z)$ without primes that is independent of the direction of wavevectors, as shown in Fig. \ref{['fig: intro schematics']}(c).
• Figure 4: Spatial profile of phonon angular momentum (AM), including spin (SAM), orbital (OAM), and total (TAM) components, at four different interfaces. The interface separates a chiral quartz crystal at $z < 0$ under a thermal gradient $\partial T/\partial z$ from an achiral crystal at $z > 0$ without external driving. For reference, the rightmost panels illustrate the configuration in which two quartz crystals are joined. (a) AM densities: $S_z$ (SAM), $L_z$ (OAM), and $J_z = S_z + L_z$ (TAM). (b) AM fluxes: $j^{\text{S}}_z$ (SAM), $j^{\text{L}}_z$ (OAM), and $j^{\text{J}}_z = j^{\text{S}}_z + j^{\text{L}}_z$ (TAM). AM densities and fluxes are normalized by the bulk SAM density $S_0$ and transverse-wave velocity $v_{\text{T}}$ in the quartz. We set parameters as $v_{\text{L}}/v_{\text{T}} = 1.59$ and [Quartz--Vacuum] $\widetilde{\rho} c_{\text{T}}/\rho v_{\text{T}} = 0$, [Quartz--Platinum] $c_{\text{L}}/c_{\text{T}} = 2.25$, $c_{\text{T}}/v_{\text{T}} = 0.493$, and $\widetilde{\rho} c_{\text{T}}/\rho v_{\text{T}} = 3.99$, [Quartz--Lead] $c_{\text{L}}/c_{\text{T}} = 2.84$, $c_{\text{T}}/v_{\text{T}} = 0.183$, and $\widetilde{\rho} c_{\text{T}}/\rho v_{\text{T}} = 0.947$, [Quartz--Quartz] $c_{\text{L}}/v_{\text{T}} = 1.59$ and $c_{\text{T}}/v_{\text{T}} = \widetilde{\rho} /\rho = 1$Rikanempyo2024. The results of the SAM in (a) and (b) are identical to Fig. 2(a) in our companion paper SuzukiSumitaKato2024a.
• Figure 5: Schematic illustration of the OAM distribution carried by phonon wave packets (orange bell shapes) with central wavevector $\bm{k}_{\parallel}$ for cross sections with square [(a),(d)], circular [(b),(e)], and arbitrary [(c),(f)] shapes. The origin $O$ is located at the centroid of each cross section. The OAM of a wave packet, $L^{z}=(\bm{R}_{\parallel}\times\hbar\bm{k}_{\parallel})_{z}$, varies with its center-of-mass coordinate $\bm{R}_{\parallel}$. Within the narrow strip parallel to $\bm{k}_{\parallel}$, shown as the dark gray regions in (d)–(f), $L^{z}$ remains constant. The ratio of the strip area to the total cross-sectional area $\varSigma$ defines the OAM density function $W_{\bm{k}_{\parallel}}(L^{z})$.