Spin Phononics: Coupling Spin and Lattice Dynamics

• Spin phononics is the study of coupling between spin degrees of freedom and lattice vibrations, enabling the conversion and hybridization of angular momentum in solids.
• It employs techniques such as Brillouin scattering and optomechanical arrays to probe chiral and topological phononic states with practical implications in quantum control.
• Recent advances allow ultrafast manipulation of spin and lattice dynamics, paving the way for innovative applications in spintronics and phononic devices.

Spin phononics is the field that investigates the interplay, coupling, and mutual conversion between spin degrees of freedom (of electrons, magnons, spinons, or orbital angular momentum) and lattice vibrations (phonons) in solids. This interdisciplinary area encompasses microscopic mechanisms of spin-phonon interaction, the transport and manipulation of phonon angular momentum, the emergence of chiral and topological phononic states, and practical methods for probing and harnessing spin–lattice effects in quantum materials, metasurfaces, and devices. Advances in spin phononics provide new means to control magnetic, thermal, and quantum information at the femtosecond-to-nanosecond timescale, both in bulk and at mesoscopic scales.

1. Phonon Angular Momentum and the Concept of Phonon Spin

The angular momentum in elastic media comprises the sum of orbital (OAM) and spin (SAM) components. In classical elasticity, Noether’s theorem applied to the elastic Lagrangian density,

$$\mathcal{L}\left(u_i, \partial_\mu u_i, x^\mu\right) = \frac{1}{2} \left[ \rho (\dot{u}_i)^2 - C_{ijkl} (\partial_j u_i)(\partial_l u_k) \right]$$

reveals that

• OAM: $L = \rho\,\dot{u}\cdot(-\mathbf{r}\times\nabla)\,u$
• SAM: $S = \rho\,u\times\dot{u}$

For monochromatic waves, the time-averaged spin density reduces to $S = (\rho\omega/2) \operatorname{Im}(u^* \times u)$ (Ren, 2022). Upon quantization, this “phonon spin” survives in the operator sense and is analytically connected to the polarization state of the phonon modes. Each circularly polarized phonon, indexed by wavevector $k$ and helicity $\pm$, carries angular momentum $\pm\hbar$ along $k$ (Garanin et al., 2015). Importantly, phonon spin is not restricted to transverse (shear) phonons; local rotation of the polarization can also arise from:

• Interference between longitudinal phonon modes of differing wavevectors,
• Hybridization between longitudinal and transverse, or surface, phonons,
• Elliptically polarized surface (e.g., Rayleigh or Lamb) modes (Ren, 2022, Yang et al., 2023).

This establishes a direct analogy between phonon spin and photonic spin, and imparts a “spin degree of freedom” to phonons, even though phonon modes themselves are quantized atomic displacements.

2. Microscopic Spin–Phonon Coupling Mechanisms

Spin–phonon interactions emerge in various forms depending on the context:

• In quantum spin liquids, charge-neutral spinon quasiparticles couple to phonons identically to electrons in the long-wavelength limit, via deformation potentials that distort the Fermi surface (Zhou et al., 2010):

$$H_{\mathrm{sp-ph}} = \sum_{k,q,\lambda,\sigma} M_{k,\lambda}(q)\, f^\dagger_{k+q,\sigma}\, f_{k,\sigma}\, (a_{q,\lambda} + a^\dagger_{-q,\lambda})$$

with $$M_{k,\lambda}(q) = (k\cdot\hat{\varepsilon}_{q,\lambda}) [q\cdot v(k)]/\sqrt{2\rho_{\text{ion}} \omega_{q,\lambda}}$$.

• In lattice-magnetic systems, the effective Hamiltonian can be written as

$$H = H_\text{spin}(J, K, ...) + H_\text{lattice}(u, \pi) + H_\text{int},$$

where magnetoelastic coupling arises from the dependence of the exchange $J$ and anisotropy $K$ on lattice distortions and strains (Rückriegel et al., 2019).

• Near resonance, coupling between magnon and phonon branches hybridizes their polarizations, producing elliptically polarized magnon-polaron modes (MPs) that carry angular momentum at the Brillouin zone center without symmetry breaking (Ning et al., 14 Oct 2024). The mixing Hamiltonian matrix is:

$$H = \begin{pmatrix} E_{m1} & 0 & g_1 & 0 \ 0 & E_{m2} & 0 & g_2 \ g_1 & 0 & E_{p1} & 0 \ 0 & g_2 & 0 & E_{p2} \end{pmatrix}$$

Diagonalizing yields nondegenerate elliptically polarized eigenmodes whose trajectories and angular momentum are precisely measurable via phase-resolved polarimetry (Ning et al., 14 Oct 2024).

• For spin waves in ferromagnets, application of a spin-polarized current introduces spin-transfer torque, modifying the allowed polarizations (RPSW, LPSW) via Doppler shifts and stabilizing both branches within certain current windows (Zhou et al., 2018).
• All these mechanisms are underpinned by conservation of total angular momentum, typically partitioned among the electronic/magnetic OAM and SAM, the phonon OAM and spin, and any rigid-body lattice motion (Garanin et al., 2015, Rückriegel et al., 2019).

3. Quantum and Topological Effects in Spin Phononics

Spin phononics research has demonstrated that:

• Coupling of phonons to internal degrees of freedom (orbital, spin, valley) can yield phonon band structures with nontrivial topology, including nonzero Berry curvature and topologically protected edge states (Pai et al., 2022, Sanavio et al., 2020).
• In optomechanical arrays, the interplay between engineered photonic spin–orbit coupling and linearized optomechanical interactions can induce chiral one-way edge modes for phonons with robust backscattering protection. The effective model is

$$\hat{H} = \sum_i [-\Delta\,\hat{a}^\dagger_i \hat{a}_i + \Omega\,\hat{b}^\dagger_i \hat{b}_i + (G\,\hat{a}^\dagger_i \hat{b}_i + \mathrm{h.c.}) ] + \sum_{\langle i, j\rangle} J e^{i\phi_{ij}}\hat{a}^\dagger_i \hat{a}_j$$

leading to hybridized photon–phonon bands with well-defined Chern numbers (Sanavio et al., 2020).

• Experience in acoustic metamaterials with artificial “micropolar” elasticity demonstrates that longitudinal-to-transverse conversion and spin–orbit coupling effects such as acoustic spin-Hall and vortex generation are achievable, expanding functionality beyond conventional scalar acoustics (Wang et al., 2021).
• Rolling waves with non-paraxial spins—i.e., waves where the spin is not aligned along the propagation vector—emerge in engineered anisotropic media satisfying $c_L = c_T$, yielding broadband spin textures not possible in isotropic solids (Zhang et al., 2020).

4. Spin–Phonon Transport, Manipulation, and Detection

Spin currents mediated by phonons can be both generated and probed:

• Acoustic spin pumping experiments demonstrate that surface acoustic wave (SAW) pulses tuned to the ferromagnetic resonance condition induce magnetization precession via magnetoelastic coupling, leading to spin current injection in an adjacent metal detected by the inverse spin Hall effect (Weiler et al., 2011).
• Spin currents of phononic origin may also be generated by dynamic, spatially inhomogeneous (AC) magnetic fields that couple to chiral phonon modes via Zeeman interactions. Parametric pumping efficiently rectifies the drive into a DC spin current:

$$\frac{\partial s}{\partial t}\bigg|_{\mathrm{DC}} = \omega \frac{(m^*\mu_0\gamma_e h_0)^2}{2\pi\hbar}\,\hat{z}$$

where $h_0$ is the field amplitude, as demonstrated in van der Waals semiconductors (Cai et al., 8 Apr 2024).

• Brillouin–Mandelstam spectroscopy has become an indispensable tool for mapping phonon and magnon dispersions, measuring nonreciprocal and topological effects, and probing spin–lattice coupling at submicron scales (Kargar et al., 2020).
• In nano- and micromechanical devices, phonon-mediated coupling between distant spins (such as NV centers in diamond cantilevers) has been achieved, enabling phonon "quantum buses" for distributing entanglement in quantum networks (Oeckinghaus et al., 2019).

5. Nonlinear Phononics, Symmetry Breaking, and Spin Control

Nonlinear phononics introduces the ability to induce or enhance spin–lattice phenomena using light-driven, symmetry-lowering lattice distortions:

• Resonant excitation of IR-active phonons by intense mid-infrared lasers, with cubic or quartic nonlinear coupling to Raman phonons, generates static or dynamic lattice displacements $Q_R$ that modify exchange interactions, and can induce or switch spin/orbital order (Subedi, 2021).
• In bilayer van der Waals systems, such nonlinear drive can lower the symmetry, enabling interlayer Dzyaloshinskii–Moriya interactions (IL-DMI) that are forbidden in equilibrium (Lin et al., 9 Aug 2024). The resulting magnetic Hamiltonian is

$$H = \sum_{\langle i, j \rangle, \eta} [ J_1 \mathbf{S}_{i,\eta}\cdot\mathbf{S}_{j,\eta} - \Delta S^z_{i,\eta} ] + \sum_i [J_2\, \mathbf{S}_{i,\mathrm{top}}\cdot\mathbf{S}_{i,\mathrm{bottom}} + \mathbf{d}_i\cdot(\mathbf{S}_{i,\mathrm{top}} \times \mathbf{S}_{i,\mathrm{bottom}}) ] + ...$$

The light-induced IL-DMI drives a transition from collinear to canted magnetic order, induces magnon band gaps, and can effect topological transitions of magnon bands characterized by nonzero Chern numbers. This process is reversible on ultrafast timescales and can be externally controlled (Lin et al., 9 Aug 2024).

• The nonlinear induction of chiral phonons and the corresponding topological properties of phonon bands (Berry curvature, phonon Hall effect) enable routes for robust phononic transport immune to disorder (Pai et al., 2022).

6. Experimental Observations and Applications

• Observation of acoustic spin in fluids, including direct torque measurement on meta-atoms and mapping of spin-momentum locking in acoustic metamaterials demonstrates control over longitudinal and transverse spin degrees of freedom in sound (Shi et al., 2018).
• Inelastic neutron, Raman, and Brillouin scattering experiments have confirmed the existence of spin-lattice hybrid excitations, phonon-assisted spin modes in frustrated magnets, and nontrivial phonon angular momentum distributions (Sugimoto et al., 2011, Yang et al., 2023).
• Selectively excited phonon–magnon hybrid modes (elliptical magnon polarons) in van der Waals antiferromagnets have been characterized in phase-resolved THz polarimetry, revealing that nondegenerate phonons with intrinsic angular momentum can occur even without external symmetry breaking (Ning et al., 14 Oct 2024).
• Advances in device engineering and MEMS/NEMS integration now allow for spin-controlled routing, nonreciprocal phononic logic, and phononic memories, leveraging the topological and spin textures of sound and vibrations (Sanavio et al., 2020, Zhou et al., 2018).

7. Theoretical Frameworks and Extensions

• Angular momentum conservation in spin–phonon processes is rigorously maintained by partitioning the total into contributions from atomic spins, phonon OAM, phonon spin, and macroscopic body rotation:

$$\boldsymbol{\Omega} = \mathbf{I}^{-1}\cdot ( \mathbf{J} - \mathbf{L} - \mathbf{S} )$$

where $\Omega$ is the angular velocity, $I$ is the moment-of-inertia tensor (Rückriegel et al., 2019).

• The notion of "spin–phonon dispersion” extends beyond traditional single-parameter coupling constants: the full spatial and mode-resolved effect of vibrational motion on spin amplitudes is captured by the eigenvalue structure of the spin force matrix,

$$S_{ij}^{\alpha\beta}(L) = \frac{\partial^2 M_s}{\partial R_{i,\alpha}\partial R_{j,\beta}}$$

with negative/positive eigenvalues corresponding to phonon-induced demagnetization/enhancement, respectively (Gu et al., 2022).

• The Elliott–Yafet theory of spin relaxation, originally confined to nonmagnetic metals, is generalized by these frameworks to account for ultrafast demagnetization and complex relaxation in ferromagnets, antiferromagnets, and spin liquids where spin amplitudes are modified by lattice motion throughout the Brillouin zone (Gu et al., 2022).

Summary Table: Major Spin Phononics Phenomena and Approaches

Phenomenon / Method	Physical System / Mechanism	Key Reference(s)
Phonon spin and OAM	Classical/quantum lattices, metamaterials	(Ren, 2022, Garanin et al., 2015)
Spinon–phonon interaction	Quantum spin liquids	(Zhou et al., 2010)
Nonlinear phononics, IL-DMI	Driven van der Waals magnets	(Subedi, 2021, Lin et al., 9 Aug 2024)
Magnon–phonon hybridization	van der Waals antiferromagnets	(Ning et al., 14 Oct 2024)
Chirality via vibronic bound states	QSLs, rare-earth magnets	(Pai et al., 2022)
Spin currents via phonons	Ferromagnetic thin films, MEMS, vdW	(Weiler et al., 2011, Cai et al., 8 Apr 2024)
Topological phononics	Optomechanical arrays, metamaterials	(Sanavio et al., 2020, Yang et al., 2023)

The field of spin phononics unifies the paper of angular momentum transfer, hybridization, and topology among spins, magnons, and phonons on both the theoretical and experimental levels. By leveraging symmetry, nonlinear effects, and nanofabricated structures, it facilitates ultrafast and dissipationless control of spin, lattice, and information flows in solids and devices.