Fractionalized excitations probed by ultrasound

In this work, we study magnetoelastic interactions by means of ultrasound experiments in $\alpha \text{−}{\mathrm{RuCl}}_3$—a prototypical material for the Kitaev spin model on the honeycomb lattice, with a possible spin-liquid state featuring Majorana fermions and ${\mathbb{Z}}_2$-flux excitations. We present results of the temperature and in-plane magnetic-field dependence of the sound velocity and sound attenuation for several longitudinal and transverse phonon modes propagating along high-symmetry crystallographic directions. A comprehensive data analysis above the magnetically ordered state provides strong evidence of phonon scattering by Majorana fermions. This scattering depends sensitively on the value of the phonon velocities relative to the characteristic velocity of the low-energy fermionic excitations describing the spin dynamics of the underlying Kitaev magnet. Moreover, our data displays a distinct reduction of anisotropy of the sound attenuation, consistent with the presence of thermally excited ${\mathbb{Z}}_2$ visons. We demonstrate the potential of phonon dynamics as a promising probe for uncovering fractionalized excitations in $\alpha \text{−}{\mathrm{RuCl}}_3$ and provide new insights into the $H\text{−}T$ phase diagram of this material.

Figure 1

Temperature dependence of the relative sound velocity (blue, left scales) and sound attenuation (black and red, right scales) of selected acoustic modes in (a)–(c) zero and in (d)–(f) magnetic field along the $\mathbf{a}$ direction with ${\mu }_{0}H=8$ T. The modes shown are (a), (d) $c_{11}$, (b), (e) $c_{44}$, and (c), (f) $(c_{11}-c_{12})/2$ with $\mathbf{q}\parallel \mathbf{b}$. Geometries and measurement frequencies are given in Table 1. The inset in panel (a) shows the sound velocity change for the $c_{33}$ acoustic mode up to 180 K ($f=60\mathrm{MHz}$). Here, up and down sweeps are marked by arrows. Note the different attenuation scales for the data obtained in (a)–(c) zero field and (d)–(f) at ${\mu }_{0}H=8$ T.

Figure 2

(a) Change of the sound attenuation versus temperature in $\alpha \text{−}{\mathrm{RuCl}}_3$ beyond the magnetically ordered state. We show results for the longitudinal $c_{11}$ and transverse $(c_{11}-c_{12})/2$ acoustic modes for two high-symmetry in-plane propagation directions, $\mathbf{q}\parallel \mathbf{a}$ and $\mathbf{q}\parallel \mathbf{b}$, in zero (black) and magnetic field applied along a (red, orange). Results for the transverse $c_{44}$ acoustic mode ($\mathbf{q}\parallel \mathbf{b},\mathbf{u}\parallel \mathbf{c}$) are also shown. The zero-field data were removed close to and below $T_N$ (below the vertical dashed blue line) to mask the dominant attenuation anomaly [shown in Figs. 1–1] at the ordering temperature $T_N$. Panels (b) and (c) show the data from panel (a) for the acoustic mode $(c_{11}-c_{12})/2$ and $c_{11}$, respectively, on an enlarged scale. Measurement frequencies were in the range 20–40 MHz . Note that, for different experimental geometries (different acoustic modes and different propagation directions) the related attenuation is arbitrarily shifted for clarity. The zero- and in-field curves belonging to the same acoustic mode for a chosen propagation direction [and with the same $A^{\prime }(T_0,H_0)$ defined in Sec. 2] are not shifted with respect to each other. The straight dashed gray lines are guides to the eye. Panels (d) and (e) illustrate different scattering channels near the bottom of the Dirac cone with acoustic-phonon velocities $v_s<v_F$ and $v_s>v_F$, respectively. See text for details.

Figure 3

(a) Temperature and (b) magnetic-field dependence of the relative sound velocity of the longitudinal acoustic mode $c_{11}$ ($\mathbf{q}\parallel \mathbf{u}\parallel \mathbf{a}$) at selected magnetic fields ($\mathbf{H}\parallel \mathbf{a}$) and temperatures, respectively. The curves are arbitrarily shifted for clarity. The measurement frequency was $f=22\mathrm{MHz}$. The vertical arrows indicate positions of the sound-velocity anomalies. $T_N, H_{c1}$, and $H_{c2}$ are also shown. The gray dashed line in panel (a) corresponds to the Néel temperature $T_N$ at zero field. (c) $H\text{−}T$ ($\mathbf{H}\parallel \mathbf{a}$) phase diagram of $\alpha \text{−}{\mathrm{RuCl}}_3$. Characteristic temperatures (red symbols) and magnetic fields (black symbols) are extracted from the ultrasound data shown in panels (a) and (b). The lines are guides to the eye. The inset shows crystallographic directions in a honeycomb plane with the notations used in this work.

Figure 4

Schematic illustration of the ultrasound setup based on the homodyne method for the simultaneous measurement of the sound velocity and sound attenuation.

Figure 5

Echo train for the transverse $(c_{11}-c_{12})/2$ mode ($q\parallel a$) in $\alpha \text{−}{\mathrm{RuCl}}_3$ at 200 K. The $Q$ (blue) and $I$ signal (purple) are shown versus time together with the ranges averaged by means of gated integrator (green and red gates). Vertical, dashed lines show the onset of crosstalk (CT, gray) and onset of echoes (black) $n=0$ (first transmitting signal), 1, and 2. In the present work all measurements were performed at zero echo. Note, that for different echoes, $n$, the effective sample length should be used $L^{\prime }=L(2n+1)$. $\tau =L/v$ is the time required by the sound to travel through the sample length $L$ with the velocity $v$. Hence, the absolute sound velocity can be obtained.

Figure 6

Ultrasound-attenuation change for the elastic modes $c_{44}$ ($\mathbf{k}\parallel \mathbf{b}, \mathbf{u}\parallel \mathbf{c}$), $(c_{11}-c_{12})/2$ ($\mathbf{k}\parallel \mathbf{b}, \mathbf{u}\parallel \mathbf{a}$), and $c_{33}$ ($\mathbf{k}\parallel \mathbf{u}\parallel \mathbf{c}$) versus temperature in $\alpha \text{−}{\mathrm{RuCl}}_3$ up to 140 K in zero field.