Combined experimental and theoretical study of hydrostatic He-gas pressure effects in $\alpha \text{−}{\mathrm{RuCl}}_3$

We report a detailed experimental and theoretical study on the effect of hydrostatic pressure on various structural and magnetic aspects of the layered honeycomb antiferromagent $\alpha \text{−}{\mathrm{RuCl}}_3$. Through measurements of the magnetic susceptibility $\chi$ performed under almost ideal hydrostatic-pressure conditions by using helium as a pressure-transmitting medium, we find that the phase transition to zigzag-type antiferromagnetic order at $T_N$ = 7.3 K can be rapidly suppressed to about 6.1 K at a weak pressure of about 94 MPa. A further suppression of $T_N$ with increasing pressure is impeded, however, due to the occurrence of a pressure-induced structural transition at $p\ge$ 104 MPa, accompanied by a strong dimerization of Ru-Ru bonds, which gives rise to a collapse of the magnetic susceptibility. Whereas the dimerization transition is strongly first order, the magnetic transition under varying pressure and magnetic field also reveals indications for a weakly first-order transition. We assign this observation to a strong magnetoelastic coupling in this system. Measurements of $\chi$ under varying pressure in the paramagnetic regime ($T>T_N$) and before dimerization ($p<$ 100 MPa) reveal a considerable increase of $\chi$ with pressure. These experimental observations are consistent with the results of ab initio density functional theory (DFT) calculations on the pressure-dependent structure of $\alpha \text{−}{\mathrm{RuCl}}_3$ and the corresponding pressure-dependent magnetic model. We find that pressure strengthens the nearest-neighbor Heisenberg $J$ and off-diagonal anisotropic $\mathrm{\Gamma }$ coupling and simultaneously weakens the Kitaev $K$ and anisotropic ${\mathrm{\Gamma }}^{\prime }$ coupling. Comparative susceptibility measurements on a second crystal showing two consecutive magnetic transitions instead of one, indicating the influence of stacking faults, reveal that by the application of different temperature-pressure protocols the effect of these stacking faults can be temporarily overcome.

Figure 1

Molar magnetic susceptibility of $\alpha \text{−}{\mathrm{RuCl}}_3$ (crystal #1) as a function of temperature at an in-plane field $B=0.1\mathrm{T}$. The full (open) orange circles represent data points taken with decreasing (increasing) temperature. The inset exhibits specific heat data, plotted as $C_p/T$ vs $T$ for the same crystal.

Figure 2

Magnetic susceptibility of $\alpha \text{−}{\mathrm{RuCl}}_3$ (crystal #1). Upper part (a) shows data as a function of temperature at an in-plane field $B=0.1\mathrm{T}$ for various pressure values: $p=0$ (dark-yellow circles), 56.5 MPa (orange circles), 77.5 MPa (red circles), and 95.0 MPa (dark-brown circles). The full (open) symbols represent the data taken with decreasing (increasing) temperature. Lower part (b) shows data as a function of temperature at $p=95.0\mathrm{MPa}$ for various magnetic fields applied parallel to the planes: $B=1\mathrm{T}$ (brown circles), 3 T (black circles), and 5 T (grey circles). The inset exhibits the data taken at 5 T on enlarged scales.

Figure 3

Magnetic susceptibility of $\alpha \text{−}{\mathrm{RuCl}}_3$ (crystal #1) for temperatures 2 K $\le T\le$ 270 K at an in-plane field $B=0.1\mathrm{T}$ for various pressure values: $p=0$ (orange circles), 29.6 MPa (dark-red circles), 110 MPa (dark-green circles), 140 MPa (dark-blue circles), and 200 MPa (brown circles). The full (open) symbols represent the data taken with decreasing (increasing) temperature. The inset shows a blow-up of the data around $T_s$ taken at ambient pressure (orange circles), at 100 MPa (light-green circles) and at 140 MPa (dark-blue circles).

Figure 4

(a) $\mathit{p}\text{−}\mathit{T}$ phase diagram of $\alpha \text{−}{\mathrm{RuCl}}_3$ (crystal #1) including a high-temperature structural transition at $T_s$, the transition to zigzag-type antiferromagnetic order below $T_N$ and the structural transition characterized by the dimerization of Ru-Ru bonds below $T_d$. Whereas the transitions at $T_s$ and $T_d$ are strongly first order, the transition at $T_N$ shows indications for a weak first-order character. The cross-hatched green area at intermediate pressure 95 MPa $\le p\le$ 140 MPa marks a range of inhomogeneous phase coexistence. The black-broken line in (a) represents the solidification line $T_{sol}(p$) of the pressure-transmitting medium helium. (b) Blow-up of the variation of $T_N$ with pressure for $p\le$ 125 MPa. Up (down) triangles in (a) and (b) indicate data points collected upon increasing (decreasing) temperature.

Figure 5

(a) Magnetic susceptibility of $\alpha \text{−}{\mathrm{RuCl}}_3$ crystal #2 at $p=0$ (blue circles) measured during the initial cooling procedure from room temperature down to 2 K and in the subsequent heating run from 2 K up to 20 K. For the subsequent runs at finite pressure of 18.8 MPa (red circles), 37.6 MPa (dark-green circles) and 95 MPa (black circles), the pressure was applied at 20 K and data were taken upon cooling (full symbols) and warming (open symbols). (b) Magnetic susceptibility of crystal #2 at $p=0$ after following different protocols: The blue circles represent the data taken after the initial cool down whereas the orange circles correspond to data obtained after warming the sample to 50 K and applying a pressure of 140 MPa for a limited period of time of approximately 24 hours before releasing the pressure to $p=0$, see text for details.

Figure 6

(a) Crystal structure of $\alpha \text{−}{\mathrm{RuCl}}_3$ in the C2/m space group. Bond names X, Y, and Z are depicted. [(b)–(d)] Calculated ($\text{DFT}+\text{SOC}+\mathrm{U}$) lattice parameters as a function of pressure. (b) Lattice constants $a, b, c$ (left $y$ axis), and the monoclinic angle $\beta$ (right $y$ axis). (c) Ru-Ru interatomic distances for X/Y bonds and Z bonds. (d) Ru-Cl-Ru bond angle around the Z bond and X(Y) bond. $p_c$ denotes the critical pressure at which the structure is found to dimerize.

Figure 7

Nearest-neighbor exchange parameters as a function of pressure obtained with the projED method. The couplings [parametrized as in Eq. (1)] on Z- and X/Y bonds are shown as red and blue circles, respectively, while the $C_3$-symmetrized parameters are illustrated in black.

Figure 8

Magnetic susceptibilities of the ab initio derived $j_{\text{eff}}=1/2$ models at different pressures normalized to $p_c$, the critical pressure above which the dimerization transition occurs. The data are computed via OFTLM (see main text). The temperature axis has been normalized to the Néel temperature at ambient pressure. The arrow for each pressure indicates the Néel temperature, determined by a maximum in $d(\chi ·T)/dT$. Shaded areas indicate estimates of statistical errors ($\pm 1\sigma$).

Figure 9

The temperature derivative of $\chi ·T$ of $\alpha \text{−}{\mathrm{RuCl}}_3$ (crystal #1). Upper part (a) shows data as a function of temperature at an in-plane field $B$ = 0.1 T for various pressure values: $p$ = 0 (dark-yellow circles), 56.5 MPa (orange circles), 77.5 MPa (red circles), and 95.0 MPa (dark-brown circles). The full (open) symbols represent the data taken with decreasing (increasing) temperature. Lower part (b) shows data as a function of temperature at $p$ = 95.0 MPa for various magnetic fields applied parallel to the planes: $B$ = 1 T (brown circles), 3 T (black circles), and 5 T (grey circles). The position of the maximum in $d(\chi ·T$)/$dT$ is used to determine the transition temperature to antiferromagnetic order at $T_N$.

Figure 10

Thermal expansion $\mathrm{\Delta }L\left(T\right)/L_0$ = ($L(T$) - $L$(300 K))/$L$(300 K) for crystal #1 (yellow triangles) and #2 (red triangles) of $\alpha \text{−}{\mathrm{RuCl}}_3$. The full (open) symbols represent the data taken with decreasing (increasing) temperature.