---
title: Phonon Rotoelectric Effect in Noncentrosymmetric Crystals
url: https://www.emergentmind.com/topics/phonon-rotoelectric-effect
type: topic
---

# Phonon Rotoelectric Effect in Noncentrosymmetric Crystals

The phonon rotoelectric effect is the generation of net phonon angular momentum by an applied electric field in magnetic, noncentrosymmetric crystals possessing combined time-reversal and inversion (TR×I) symmetry. Unlike the phonon thermal Edelstein effect, in which a temperature gradient produces a nonequilibrium phonon angular momentum under broken inversion symmetry and preserved time-reversal symmetry, the rotoelectric effect arises only when both time-reversal and inversion are individually broken but their product is conserved. This phenomenon is symmetry-distinct from thermo-phononic analogues and is characterized by a linear coupling between the electric field and the phonon angular momentum, mediated by a specific response tensor with magnetoelectric symmetry properties [2006.00017].

## 1. Definition and Phenomenological Framework

Microscopically, each normal-mode phonon in a crystal carries a "microscopic" angular momentum given by
\[
l_{\sigma,\alpha}(\mathbf k)
=\hbar\,\epsilon_{\sigma}^\dagger(\mathbf k)\,M_\alpha\,\epsilon_{\sigma}(\mathbf k)\,,
\]
where $\epsilon_{\sigma}(\mathbf k)$ is the polarization vector for the phonon mode $\sigma$ at wavevector $\mathbf k$, and the operator $M_\alpha$ encodes angular momentum components. The equilibrium angular momentum density is
\[
J^{\rm ph}_\alpha
=\frac1V\sum_{\mathbf k,\sigma} l_{\sigma,\alpha}(\mathbf k)\left[f_0(\omega_{\sigma}(\mathbf k))+\tfrac12\right],
\]
with $f_0(\omega)$ the Bose occupation factor.

While the phonon thermal Edelstein effect features a linear response of $J^{\rm ph}_\alpha$ to temperature gradient $\nabla T$ in noncentrosymmetric, time-reversal symmetric crystals, the rotoelectric effect occurs in a different symmetry class. In crystals with both TR and inversion ($I$) broken but TR×I preserved, the total equilibrium phonon angular momentum and the phonon thermal Edelstein tensor vanish identically:
\[
J^{\rm ph}_\alpha = 0,\qquad \alpha_{\alpha\beta} = 0.
\]
However, a coupling to an external electric field is permitted:
\[
J^{\rm ph}_\alpha = \beta_{\alpha\beta} E_\beta,
\]
where $\beta_{\alpha\beta}$ is an axial response tensor analogous to the magnetoelectric tensor in multiferroics.

## 2. Symmetry Considerations

The emergence and selection rules of the phonon rotoelectric effect are determined by distinct transformation properties of involved vectors under inversion and time-reversal:
- Under inversion ($I$): The electric field $\mathbf E$ is a polar vector and flips sign, while the phonon angular momentum $J^{\rm ph}$ is an axial vector and does not transform.
- Under time-reversal (TR): $J^{\rm ph}$ flips sign, while $\mathbf E$ remains unchanged.

If TR is preserved and $I$ is broken, then $l_\sigma(\mathbf k) = -l_\sigma(-\mathbf k)$, leading to vanishing equilibrium sum but permitting a response to $\nabla T$. If both $I$ and TR are broken but TR$\times$I is preserved, then $l_\sigma(\mathbf k) \equiv 0$ for all modes and wavevectors—precluding any thermal Edelstein effect. In this symmetry class, both $\mathbf E$ (polar) and $J^{\rm ph}$ (axial) transform identically under TR$\times$I, allowing the linear rotoelectric coupling $J^{\rm ph}_\alpha = \beta_{\alpha\beta} E_\beta$ but forbidding any $\alpha_{\alpha\beta} \nabla T$ term [2006.00017].

## 3. Linear Response and Kubo Formalism

The rotoelectric effect can be formulated within the zero-frequency, zero-wavevector Kubo response theory. The relevant operators are:
\[
\hat J^{\rm ph}_\alpha = \sum_{\mathbf k, \sigma} \hbar\,\epsilon_\sigma^\dagger(\mathbf k)\,M_\alpha\,\epsilon_\sigma(\mathbf k)\,\left(a_{\mathbf k\sigma}^\dagger a_{\mathbf k\sigma}+\tfrac12\right)
\]
for the phonon angular momentum and
\[
\hat P_\beta = \sum_{\ell\kappa}q_\kappa\,\hat u_{\ell\kappa,\beta}
\]
for the ionic polarization. The response tensor is then
\[
\beta_{\alpha\beta}
= \lim_{\omega\to0}\frac{1}{\hbar\omega}
\Im\,\langle\langle \hat J^{\rm ph}_\alpha;\,\hat P_\beta\rangle\rangle_{\omega}
= \frac1{\hbar}\int_0^\infty dt\, \langle [\,\hat J^{\rm ph}_\alpha(t),\,\hat P_\beta(0)\,]\rangle_0,
\]
so that $J^{\rm ph}_\alpha = \beta_{\alpha\beta} E_\beta$ in the limit of weak fields. While a full microscopic Kubo evaluation is not performed in the cited work, toy-model calculations serve as concrete realizations of this formalism [2006.00017].

## 4. Temperature Dependence of $\alpha_{\alpha\beta}$ and $\beta_{\alpha\beta}$

The temperature dependencies of the phonon thermal Edelstein tensor $\alpha_{\alpha\beta}$ and the rotoelectric tensor $\beta_{\alpha\beta}$, as derived under the Debye approximation for acoustic modes in three-dimensional crystals, are summarized as follows:

| Response              | High-$T$ limit                           | Low-$T$ limit                            |
|-----------------------|------------------------------------------|------------------------------------------|
| $\alpha_{\alpha\beta}$| $\propto T^0$ (constant)                 | $\propto T^3$                            |
| $\beta_{\alpha\beta}$ | $\propto T^{-1}$                         | Constant $+ \mathcal{O}(T^5)$            |

At high temperatures ($\hbar\omega \ll k_B T$), $\beta_{\alpha\beta}$ decays as $T^{-1}$ due to the vanishing of the leading $T$-dependent part by symmetry, with only a $1/T$ piece surviving. In the low-temperature regime, the equilibrium (zero-point) angular momentum—activated by $\mathbf E$—produces a constant contribution to $\beta_{\alpha\beta}$, with corrections scaling as $T^5$ [2006.00017].

## 5. Toy Model Realization and Quantitative Estimates

The effect is elucidated using a two-dimensional spring-mass model featuring three ions per unit cell and antiferromagnetic ordering, with spin-phonon coupling mediated via a Raman interaction:
\[
H_{\rm SPI} = -g\sum_{l\kappa} \mathbf S_\kappa\cdot\left(\mathbf u_{l\kappa}\times m_\kappa\dot{\mathbf u}_{l\kappa}\right).
\]
Application of an electric field $E_x$ perturbs the equilibrium positions, endowing each phonon band with nonzero $l_{\sigma,z}(\mathbf k)$ and yielding $J^{\rm ph}_z = \beta_{zx} E_x \neq 0$, linearly proportional to both the shift $\tilde d \propto E_x$ and $E_x$ itself. No such angular momentum arises for $E\parallel\hat y$, consistent with model symmetries.

A numerical parameter set typical of real magnets such as $\mathrm{Cr_2O_3}$ yields the order-of-magnitude estimate:
\[
J^{\rm ph} \sim \frac{(\varepsilon-\varepsilon_0)E\,g\hbar^2}{ea\,\omega_D}
\sim 10^{-8}\hbar\ \text{per unit cell at } T\sim300~\mathrm{K}
\]
for $\omega_D \sim 10\,\mathrm{THz}$ and $g\hbar\sim1\,\mathrm{cm}^{-1}$, assuming ionic displacements induced by applied fields of order $10$ V/mm [2006.00017].

## 6. Relation to Magnetoelectric and Edelstein Effects

The phonon rotoelectric effect is the phononic analogue of the magnetoelectric effect in multiferroics, with $\beta_{\alpha\beta}$ possessing identical symmetry transformation properties to the magnetoelectric tensor $\alpha^{\rm ME}$. By contrast, the phonon thermal Edelstein effect is the direct phonon counterpart of the electronic Edelstein effect, appearing only under broken $I$ and preserved TR symmetry. In the combined TR$\times$I symmetry scenario, the rotoelectric effect is the exclusive allowed phonon angular momentum response, reflecting a different fundamental origin based on symmetry-enforced zero-point lattice rotations rather than nonequilibrium distribution shifts [2006.00017].

Source: https://www.emergentmind.com/topics/phonon-rotoelectric-effect