Electrical Switching of the Nonreciprocal Directional Microwave Response in a Triplon Bose-Einstein Condensate

We present a microwave electron spin resonance study of the quantum spin dimer system ${\mathrm{TlCuCl}}_3$, which shows the magnetic-field-induced ordering with both antiferromagnetic spin order and ferroelectricity by the Bose-Einstein condensation (BEC) of triplon quasiparticles. Our main achievement is an electrical switching of the nonreciprocal directional microwave response in the triplon BEC phase. High-speed directional control of microwave absorption by applying an electric field has been accomplished in this Letter. The strength of the observed nonreciprocal microwave response well agrees with the calculation based on Kubo theory with the parameters, evaluated from the static electric polarization in this material.

Figure 1

(a) Magnetic-field dependence of the magnetic excitation energies at the magnetic zone center in ${\mathrm{TlCuCl}}_3$ for $\mathit{H}\parallel [201]$, calculated based on the bond operator formulation. The dotted line represents the critical field $H_c$. Open circles represent the ESR points at 50 GHz, where the NDD measurements are performed in this study. Closed circles are the data, obtained from the previous neutron scattering [28]. Closed squares below 80 GHz are the data obtained from the microwave ESR measurements [32], and those above 160 GHz are the data obtained in this study. Three branches of the triplon excitation, which are triply degenerated at zero magnetic field in the case of an isotropic system, split by a rhombic anisotropy. Owing to the triplon BEC, the field dependence of the two higher energy modes is renormalized, whereas the lowest mode changes to the Nambu-Goldstone mode for $H>H_c$ and the anisotropy causes reopening of the excitation gap in the Nambu-Goldstone mode. As shown by a dashed curve, the spontaneous electric polarization $\mathit{P}$ perpendicular to the ($10\overline{2}$) plane appears for $H>H_c$. Thus, the triplon BEC brings about a finite $\mathit{P}\times \mathit{M}$ parallel to [010] in ${\mathrm{TlCuCl}}_3$ for $\mathit{H}\parallel [201]$. (b) Precession motion of the two-sublattice-ordered spin moments, driven by the excitation of the Nambu-Goldstone magnon for $H>H_c$.

Figure 2

Transition matrix elements (a) $⟨0|P_i|1⟩$ and (b) $⟨0|M_i|1⟩$, ($i=x$, $y$, $z$) between the ground state $|0⟩$ and the excited state $|1⟩$ for the excitation of the lowest energy triplon or the Nambu-Goldstone magnon, calculated for $\mathit{H}\parallel [201]$. (Inset) Both longitudinal oscillation of $\mathit{M}$ along [201] and transverse oscillation of $\mathit{P}$ perpendicular to ($10\overline{2}$) plane are induced by the excitation of the Nambu-Goldstone magnon for $H>H_c$. In the ESR experiment, the linearly polarized microwave with the oscillating magnetic field $\mathit{h}\parallel [201]$ and the electric field $\mathit{e}\perp (10\overline{2})$ is illuminated to a single crystal of ${\mathrm{TlCuCl}}_3$ (see inset).

Figure 3

(a) Microwave absorption due to the excitation of the Nambu-Goldstone mode at 50 GHz at 2.0 K. Reversal of $\mathit{P}$ causes large change in the absorption coefficient. (Inset) Change in the absorption coefficient $\mathrm{\Delta }\alpha ={\alpha }_{+}-{\alpha }_{-}$ by the reversal of $\mathit{P}$. Solid and dashed curves represent the experimental and theoretical results, respectively. (b) Microwave absorption spectra observed in reversed magnetic fields. The black arrow indicates the absorption due to the excitation of the triplon below $H_c$. Change in the absorption due to the excitation of the Nambu-Goldstone mode by the reversal of $\mathit{M}$, as well as by the reversal of $\mathit{P}$ for $H>H_c$, is observed, whereas the absorption due to the excitation of the triplon shows almost no change.

Figure 4

(a) Change in the absorption coefficient by polarity reversal of the applied static electric field. The measurement was performed at 12.2 T, where the absorption peak due to the excitation of the Nambu-Goldstone mode is observed. (b) Switching of the nonreciprocal directional microwave response by $\mathit{E}$. The propagation direction $\mathit{k}$ with stronger microwave absorption is reversed by the reversal of $\mathit{P}$.