Dynamical breaking of inversion symmetry, strong second harmonic generation, and ferroelectricity with nonlinear phonons

TL;DR

The paper demonstrates that inversion symmetry can be dynamically broken in crystals by nonlinear chiral phonons driven near half their resonance, leading to strong second harmonic generation and ferroelectric rectification. A minimal model with two degenerate phonon coordinates and Kerr-like nonlinearity shows a Duffing-type response and a Mathieu-type parametric instability for the even-harmonic component, yielding a stable, SHG-rich steady state with a DC lattice displacement. The authors extend the framework to resonant enhancement via an auxiliary phonon and to collective instabilities involving both x and y modes, showing that ellipticity of the drive is essential for instability and that nondegenerate chiral modes can be addressed separately. These results offer a route to on-demand SHG, rectification, and driven ferroelectricity, with potential implications for coupling to electronic and magnetic degrees of freedom in out-of-equilibrium materials. Key concepts include nonlinear chiral phonons, parametric instability, Duffing dynamics, and Mathieu-type resonance as mechanisms for dynamical symmetry breaking.

Abstract

We show how crystalline inversion symmetry can be dynamically broken by optical phonons with generic, hardening Kerr-like non-linearities. The symmetry-broken state is reached through a parametric instability that can be accessed by driving close to half the phonon resonance. After the onset of the instability, the system settles to a steady state with inversion-symmetry breaking phonon trajectories and strong second harmonic generation. The time averaged positions of the atoms are displaced relative to equilibrium, resulting in a ferroelectric rectification of the driving signal.

Figures (3)

• Figure 1: a) Phonons [Eq. (\ref{['eq:Phonon_H']})] driven with a linearly polarized electric field $E_{x}\cos\left(\omega t\right)$ oscillating at frequency $\omega=0.6\Omega_{0}$. The effective phonon frequency $\tilde{\Omega}_{0}\left(E_{x}\right)$, given in Eq. (\ref{['eq:shifting_frequency']}), exhibits a blue shift as $E_{x}$ is increased. At $\tilde{\Omega}_{0}\left(E_{x}\right)=2\omega$, the system enters the symmetry breaking state with strong second harmonic generation. In the symmetry-breaking regime, the resonance curve $\tilde{\Omega}_{0}\left(E_{x}\right)$ is interrupted. b) Spectrum of $Q_{x}\left(t\right)$ in the symmetry breaking state. c) Amplitude ratio of second and first harmonics across the symmetry breaking transition for different dampings $\gamma$. The white dashed line shows the result of Eq. (\ref{['eq:threshold_field']}). d) The Lissajous trajectory of phonon coordinates $Q_{x}\left(t\right)$ and $Q_{y}\left(t\right)$ when driven into the symmetry-breaking state using elliptically polarized light with $\mathbf{E}=E_{x}\left[\cos\left(\omega t\right),0.25\sin\left(\omega t\right)\right]$. The inversion symmetry of Eq. (\ref{['eq:Phonon_H']}) is broken dynamically. e) A higher order symmetry-breaking steady state with $\omega=0.3\Omega_{0}$ and $\tilde{\Omega}_{0}\left(E_{x}\right)=4\omega$. The forth harmonic dominates the response of $Q_{x}\left(t\right)$. We used $\beta=\Omega_{0}^{2}/\left(\text{Å}u\right)$ for all simulations.
• Figure 2: The instability window $\Delta_{\mathrm{max}}-\Delta_{\mathrm{min}}$ [see the discussion above Eq. (\ref{['eq:instability_Fx_Fy_expand']})] as a function of $F_{y,1}/F_{x,1}$ is plotted for $F_{x,1}=0.1$. The full solution for the ansatz of Eq. (\ref{['eq:4_freq_ansatz']}) is plotted as a blue solid line in the main figure and in the inset. The dashed red line indicates the approximation of Eq. (\ref{['eq:instability_Fx_Fy_expand']}). The dashed green line in the inset shows the approximation of Eq. (\ref{['eq:instability_Fx=00003DFy_expand']}), which is valid close to $F_{y,1}=F_{x,1}$. Stable regions, where $\Delta_{\mathrm{max}}\leq\Delta_{\mathrm{min}}$ are marked yellow [see discussion below Eq. (\ref{['eq:instability_Fx=00003DFy_expand']})]. We conclude that the system is stable at $F_{x,1}=F_{y,1}$ which holds for driving with perfectly circularly polarized light, i.e. $E_{x}=E_{y}$. Some amount of ellipticity of the driving electromagnetic field is necessary to access the instability.
• Figure 3: Symmetry breaking with non-degenerate chiral phonons [see Eq. (\ref{['eq:chiral_ham']})]. The phonon frequencies are split according to Eq. (\ref{['eq:_split_spectra']}): $\Omega_{+}$ corresponds to right-handed motion, while $\Omega_{-}$ corresponds to a left-handed rotation. The two modes are accessed with light of opposite polarizations, fitting their respective sense of motion. To excite the $\Omega_{+}$ mode, we use $\mathbf{E}=E_{0}\left[-\left(1-\delta\right)\cos\omega t,\sin\omega t,0\right]$, and for the $\Omega_{-}$ mode, $\mathbf{E}=E_{0}\left[\cos\omega t,\left(1-\delta\right)\sin\omega t,0\right]$, with $\delta=0.25$ and $\omega=0.62\Omega$. As for degenerate chiral phonons, a slight detuning from circularity $\delta$ is necessary, in order to trigger the instability at half the resonance frequency [see Eq. (\ref{['eq:instability_Fx=00003DFy_expand']})]. The figure combines the results of two runs, in which the two chiralities were simulated separately. The resonance frequencies exhibit a driving amplitude dependent blue-shift, such that the instability occurs at different powers, for the two chiralities.