Phonon Edelstein Effect

• Phonon Edelstein effect is the generation of net phonon angular momentum in noncentrosymmetric crystals through external perturbations like electrical currents or temperature gradients.
• It relies on chiral phonons and strong electron–phonon as well as spin–orbit coupling, modeled by a minimal effective Hamiltonian and linear-response theory.
• The phenomenon offers practical routes for spintronics and phononics by enabling hybrid devices that convert spin and charge signals through lattice excitations.

The Phonon Edelstein Effect (PEE) is the phenomenon by which a net phonon angular momentum density is generated in a crystal under nonequilibrium conditions, typically via an applied electrical current, temperature gradient, or lattice distortion. It arises in systems with broken spatial inversion symmetry combined with spin–orbit or related electron–phonon coupling mechanisms; the effect is directly analogous to the electronic Edelstein effect, where a current induces a spin accumulation in the presence of spin–orbit interactions. The PEE expands the concept of Edelstein-type conversion phenomena beyond electronic spin degrees of freedom to the field of lattice excitations, with consequences for both fundamental symmetry analysis and applied device design in spintronics and phononics.

1. Physical Principles and Symmetry Requirements

The phonon Edelstein effect requires the following symmetry conditions: inversion symmetry must be broken, but time-reversal symmetry can be present or absent, depending on the specific form (thermal, electrical, or current-driven). In chiral (gyrotropic) crystals with spin–orbit coupling, lattice vibrations (phonons) may carry intrinsic angular momentum when their atomic displacements are circularly polarized, leading to the formation of so-called chiral phonons. Applying an electric current or other perturbation can induce a net phonon angular momentum, denoted as $L_{\mathrm{ph}}^\alpha$, with the linear response form:

$$L_{\mathrm{ph}}^\alpha = \chi_{\mathrm{ph}}^{\alpha\beta} j^\beta$$

Here, $\chi_{\mathrm{ph}}^{\alpha\beta}$ is the "phonon Edelstein susceptibility tensor," and $j^\beta$ is the electric current density. In analogy to the electronic Edelstein effect ($$\langle S^\alpha \rangle = \chi_e^{\alpha\beta} j^\beta$$), inversion symmetry breaking and nonvanishing coupling between the electronic and lattice degrees of freedom are essential (Yokoyama, 15 May 2025).

2. Microscopic Hamiltonians and Theoretical Formalism

The minimal effective Hamiltonian describing the PEE in a chiral metal includes the following components (Yokoyama, 15 May 2025):

• An electronic Hamiltonian featuring strong spin–orbit coupling and inversion symmetry breaking (e.g., a Weyl-type term),
• A phonon sector supporting circularly polarized modes,
• An electron–phonon coupling term of Zeeman (spin–angular-momentum) type,
• Coupling of the system to external electric or current fields.

Explicitly, the Hamiltonian reads:

$H = H_e + H_p + H_{ep} + H_A$

With:

• $$H_e = \frac{\mathbf{k}^2}{2m} + \alpha \mathbf{k} \cdot \boldsymbol{\sigma}$$
• $$H_p = \sum_{\mathbf{q},\nu=\pm} \omega_{\mathbf{q}\nu} a_{\mathbf{q}\nu}^\dagger a_{\mathbf{q}\nu}$$
• $H_{ep} = \lambda L^z \sigma_z$
• $H_A = -j_z A_z$, where $j_z = -e(k_z/m + \alpha \sigma_z)$ and $A_z$ encodes the applied electric field.

The induced phonon angular momentum is computed using linear-response theory (Kubo–Green’s-function approach), yielding for the chiral metal:

$$\langle L^z \rangle = \chi E_z, \qquad \chi = -\frac{5}{12} e \alpha \lambda \nu_e \tau_e \tau_p$$

Here, $\alpha$ is the spin–orbit coupling strength, $\lambda$ is the spin–phonon coupling, $\nu_e$ the electronic density of states at the Fermi level, and $\tau_e$, $\tau_p$ are the electron and phonon scattering times, respectively (Yokoyama, 15 May 2025).

3. Related Effects: Thermal and Rotoelectric Phonon Edelstein Effects

Beyond current-induced PEE, related effects include:

• Phonon Thermal Edelstein Effect: Under a temperature gradient in non-magnetic (time-reversal invariant), non-centrosymmetric crystals, a net out-of-equilibrium phonon angular momentum arises:

$$J_{\mathrm{ph},i} = \sum_j \alpha_{ij}^{(\nabla T)} \partial_j T$$

Here, $\alpha_{ij}^{(\nabla T)}$ is the thermal phonon Edelstein tensor, generally a rank-2 axial tensor, allowed in magnetic point groups supporting such tensors. The effect vanishes if inversion symmetry is restored or if both $T$ and inversion are broken but their product ($PT$ symmetry) is preserved (Hamada et al., 2020).

• Phonon Rotoelectric Effect: In magnetic crystals with both time-reversal and inversion symmetry broken but $PT$ symmetry, a net phonon angular momentum can be generated via lattice distortions induced by an applied electric field:

$J_{\mathrm{ph},i} = \sum_j \beta_{ij} E_j$

The tensor $\beta_{ij}$ is again rank-2 axial, in analogy to linear magnetoelectric response (Hamada et al., 2020). The physical origin is piezoelectric or Born-charge-induced distortion of lattice equilibrium positions, breaking the $PT$ symmetry of the phonon Hamiltonian.

4. Phenomenology, Magnitude, and Materials Realizations

The magnitude of the current-induced PEE in gyrotropic metals is governed by the strengths of spin–orbit and spin–phonon couplings, and by the product of electron and phonon lifetimes. For a typical chiral metal such as Te (tellurium) with experimental parameter values ($\tau_p\sim 10^{-10}$ s, $\tau_e\sim 10^{-12}$ s, $\alpha\sim 1$ eV$\cdot$\AA, $\lambda \sim 1$ meV, $\nu_e\sim 1$/eV$\cdot$unit cell, $E_z\sim 10^4$ V/m), the estimated angular momentum density is $\langle L^z \rangle \sim 0.1\,\hbar$ per unit cell, with a susceptibility $$\chi^{zz}_{\rm ph}\sim 10^{-12} \hbar\text{ m}^2/\text{A}$$ (Yokoyama, 15 May 2025). The effect scales linearly with $\alpha$, $\lambda$, and $\tau_p$.

Materials maximizing the response are those with strong chiral phonons (large $\lambda$) and long phonon lifetimes ($\tau_p\gg \tau_e$), such as tellurium and selenium.

5. Connections to Spin and Charge Conversion Phenomena

In hybrid devices, phonons can act as mediators or converters in spin–charge interconversion regimes:

• In Ni/Cu(Ag)/Bi$_2$O$_3$ devices, surface acoustic waves (SAWs) pump spin current via magnon–phonon coupling in Ni, which is then converted to a charge voltage by the inverse Edelstein effect (IEE) at the Rashba interface. SAW-induced spin current densities on the order of $10^8$ A/m$^2$ and measurable voltages $\sim 1$ μV have been achieved (Xu et al., 2018).
• The angular dependencies of the induced voltages reveal symmetry-based decompositions into symmetric and asymmetric components, attributed to the interference of longitudinal and shear strains and the spatial distribution of electrodes with respect to the SAW field (Xu et al., 2018).

A plausible implication is that the PEE, through such hybrid mechanisms, enables the design of phononic or acoustic spintronic devices that manipulate spin and charge signals via lattice degrees of freedom.

6. Experimental Detection and Device Applications

Direct probing of phonon angular momentum presents significant challenges. Primary experimental strategies include:

• Detection via circular dichroism in Raman or Brillouin scattering under applied current;
• Use of magneto-optical Kerr or Faraday effects to probe effective magnetization associated with lattice rotations;
• NV center magnetometry or SQUID detection of the minute magnetic fields from the Barnett effect of rotating ions;
• Pump–probe spectroscopies (e.g., ultrafast x-ray diffraction, time-resolved circularly polarized SHG) to monitor phonon angular momentum dynamics post electronic excitation (Yokoyama, 15 May 2025).

For device applications, tunability of the electron–phonon coupling via substrate selection, strain engineering, or electrostatic gating is an essential asset, enabling reversible switching between spin-polarized and depolarized regimes. In 2D materials, achievable values of phonon energy ($\omega_0 \sim$ 10–100 meV) and coupling ($\lambda_c \lesssim 1$) enable practical implementation of PEE-based on-demand control over spin polarization, with direct relevance to next-generation spin logic and low-dissipation device architectures (Yarmohammadi et al., 2 Oct 2025).

7. Outlook and Open Problems

The phonon Edelstein effect expands the scope of multiphysics interconversion phenomena in solids, bridging electronic, spin, and lattice angular momentum. Experimental realization hinges on advances in chiral material synthesis, control of spin–phonon coupling, and improved angular momentum detection techniques. The symmetry analysis underpinning PEE also suggests avenues for discovering new classes of magnetoelectric and acousto-spin transport responses, especially in low-dimensional and topological materials.

A common misconception is to interpret the PEE as confined to purely thermal (temperature-gradient-driven) or piezoelectric contexts; in reality, current-driven and electrically driven scenarios are all symmetry-allowed variants, provided the requisite tensor properties and noncentrosymmetry are present (Hamada et al., 2020). Emerging results on substrate-tunable EPC in altermagnets further point toward device-level manipulation of phonon-mediated spin physics (Yarmohammadi et al., 2 Oct 2025), reinforcing the centrality of the PEE in contemporary materials research.