Field-induced transitions in the Kitaev material $\alpha -{\mathrm{RuCl}}_3$ probed by thermal expansion and magnetostriction

High-resolution thermal expansion and magnetostriction measurements were performed on single crystals of $\alpha -{\mathrm{RuCl}}_3$ in magnetic fields applied parallel to the Ru-Ru bonds. The length changes were measured in the direction perpendicular to the honeycomb planes. Our data show clear thermodynamic characteristics for the field-induced phase transition at the critical field ${\mu }_0{H}_{\mathrm{c}1}=7.8\left(2\right)\mathrm{T}$ where the antiferromagnetic zigzag order is suppressed. At higher fields, a kink in the magnetostriction coefficient signals an additional transition or crossover around ${\mu }_0{H}_{\mathrm{c}2}\approx 11\mathrm{T}$. The extracted Grüneisen ratio shows typical hallmarks for quantum criticality near $H_{\mathrm{c}1}$ but also displays anomalous behavior above $H_{\mathrm{c}1}$. We compare our experimental data with spin-wave calculations employing a minimal Kitaev-Heisenberg model in the semiclassical limit. Most of the salient features are in agreement with each other, however, the peculiar features in the region above $H_{\mathrm{c}1}$ cannot be accounted for in our semiclassical modeling and hence suggest a genuine quantum nature. We construct a phase diagram for $\alpha -{\mathrm{RuCl}}_3$ in a magnetic field along the Ru-Ru bonds, displaying a zigzag ordered state below $H_{\mathrm{c}1}$, a quantum paramagnetic regime between $H_{\mathrm{c}1}$ and $H_{\mathrm{c}2}$, and a semiclassical partially polarized state above $H_{\mathrm{c}2}$.

Figure 1

Normalized linear TE of $\alpha -{\mathrm{RuCl}}_3$ perpendicular to the $ab$ plane, $\mathrm{\Delta }{L}_{c^{*}}/L_{c^{*},300\text{K}}$, as function of temperature for both zero and finite applied magnetic fields $\vec{H}\parallel$ Ru-Ru bonds; (a) full temperature interval up to $300\mathrm{K}$, (b) low-temperature region up to $30\mathrm{K}$.

Figure 2

The linear TE coefficient perpendicular to the $ab$ plane, ${\alpha }_{c^{*}}$, of $\alpha -{\mathrm{RuCl}}_3$ for zero and applied magnetic fields $\vec{H}\parallel$ Ru-Ru bonds. (a) General overview of ${\alpha }_{c^{*}}$ with the focus on the ordered antiferromagnetic zigzag transition and its shift as function of the applied field. (b) A zoom of the data for applied fields close to the critical field ${\mu }_0{H}_{\mathrm{c}1}=7.8\left(2\right)$ T.

Figure 3

Specific-heat coefficient $C_p/T$ of $\alpha -{\mathrm{RuCl}}_3$ as function of temperature for applied magnetic fields $\vec{H}\parallel$ Ru-Ru bonds, together with $C_p/T$ data for nonmagnetic ${\mathrm{RhCl}}_3$ (black stars). The inset shows the magnetic contribution $C_{p,\mathrm{mag}}/T$ of $\alpha -{\mathrm{RuCl}}_3$ on a semilogarithmic scale.

Figure 4

The Grüneisen ratio $\mathrm{\Gamma }$ of $\alpha -{\mathrm{RuCl}}_3$ in a magnetic field along the Ru-Ru bonds. (a) $\mathrm{\Gamma }\left(T\right)$ for different fields up to 13 T. (b) $\mathrm{\Gamma }\left(H\right)$ for $4\mathrm{K}$ and $10\mathrm{K}$ and for two different samples (1 and 2). The dash-dotted line shows the theoretical calculation for $\mathrm{\Gamma }$ at $4\mathrm{K}$ whose overall amplitude has been scaled to match the low-field part of the data. Close to $H_{\mathrm{c}1}$, no results are shown as the spin-wave approximation becomes unreliable, see text.

Figure 5

The linear MS coefficient ${\lambda }_{c^{*}}$ as function of field and different temperatures (sample 1). The inset highlights the kink observed in ${\lambda }_{c^{*}}$ at 2.4 K.

Figure 6

From top to bottom: TE coefficient, Grüneisen ratio, and MS plots for the two sets of microscopic parameters given in Table 1. In the plots where the independent variable is the magnetic field, the vertical gridline represents the critical field, ${\mu }_0{H}_{\mathrm{c}1}=7.8\mathrm{T}$. In all panels, the left vertical axis gives the theory result normalized to $n_{{\mathrm{\Gamma }}_1}$, the right vertical axis shows experimental units using $n_{{\mathrm{\Gamma }}_1}=0.9{\mathrm{GPa}}^{-1}$ and $V/N\simeq 92.8{\text{Å}}^3$, corresponding to a molar volume $V_{\text{m}}\simeq 55.9{\text{cm}}^3/\text{mol}$ [9, 34, 40]. Our semiclassical analysis does not show any of the features interpreted as signatures of an additional regime at intermediate fields above $H_{\mathrm{c}1}$ and thus supports its genuine quantum character.

Figure 7

The $T-H$ phase diagram of $\alpha -{\mathrm{RuCl}}_3$ as determined in this work for $\vec{H}\parallel$ Ru-Ru bonds ($C_p$: yellow squares, TE: Red and orange triangles for samples 1 and 2, respectively; MS: pink and blue diamonds). Additionally, the Grüneisen ratio $\mathrm{\Gamma }(T,H)$ is shown in color scale (sample 1); these data are the same as in Fig. 4. The gray shaded region marks the limits for the color map. In addition to the zigzag-ordered (ZZ) state at low fields and the conventional paramagnetic (CPM) state at high fields, our analysis suggests the presence of a third low-temperature regime that is of genuine quantum nature (QPM). The field-driven low-temperature transition at $H_{\mathrm{c}1}$ is continuous, whereas the one at $H_{\mathrm{c}2}$ is either a crossover or a weak first-order transition.

Figure 8

(a) Temperature dependence of the linear TE coefficient perpendicular to the $ab$ plane, ${\alpha }_{c^{*}}$, for sample 2 of $\alpha -{\mathrm{RuCl}}_3$ for zero and applied magnetic fields $\vec{H}\parallel$ Ru-Ru bonds. The inset shows a zoom of the data for fields close to the critical field ${\mu }_0{H}_{\mathrm{c}1}=7.8\left(2\right)\mathrm{T}$. (b) Grüneisen ratio $\mathrm{\Gamma }\left(T\right)$ of $\alpha -{\mathrm{RuCl}}_3$ of sample 2 parallel to the Ru-Ru bonds.

Figure 9

Field dependence of (a), (b) the linear TE coefficient ${\alpha }_{c^{*},\mathrm{mag}}$ and (c), (d) the Grüneisen ratio $\mathrm{\Gamma }$ for (a), (c) sample 1 and (b), (d) sample 2 in magnetic fields $\vec{H}\parallel$ Ru-Ru bonds for different temperatures between $3.5\mathrm{K}$ and $10\mathrm{K}$.

Figure 10

The Grüneisen ratio $\mathrm{\Gamma }\left(H\right)$ of $\alpha -{\mathrm{RuCl}}_3$ at $T=4\mathrm{K}$ as a function of the magnetic field $\vec{H}\parallel$ Ru-Ru bonds using different approximations for the phononic contribution to the TE for (a) sample 1 and (b) sample 2; for details, see text.