Low-temperature ordered phases of the spin-$\frac{1}{2}$ XXZ chain system ${\mathrm{Cs}}_2{\mathrm{CoCl}}_4$

In this study the magnetic order of the spin-1/2 XXZ chain system ${\mathrm{Cs}}_2{\mathrm{CoCl}}_4$ in a temperature range from 50 mK to 0.5 K and in applied magnetic fields up to 3.5 T is investigated by high-resolution measurements of the thermal expansion and the specific heat. Applying magnetic fields along $a$ or $c$ suppresses $T_{\mathrm{N}}$ completely at about 2.1 T. In addition, we find an adjacent intermediate phase before the magnetization saturates close to 2.5 T. For magnetic fields applied along $b$, a surprisingly rich phase diagram arises. Two additional transitions are observed at critical fields ${\mu }_0{H}_{SF1}\simeq 0.25\mathrm{T}$ and ${\mu }_0{H}_{SF2}\simeq 0.7\mathrm{T}$, which we propose to arise from a two-stage spin-flop transition.

Figure 1

Crystal structure of ${\mathrm{Cs}}_2{\mathrm{CoCl}}_4$ (left) based on Ref. [4], but with the unit cell shifted by (0,0.5,0.5) as in Ref. [5]. Checkerboard planes indicate the orientations of the magnetic easy planes alternating between sites (1 and 3) and (2 and 4). The dominant intrachain exchange $J$ is along $b$ via Co-Cl-Cl-Co paths (sketched in red). Along $c$, the chains form buckled layers (gray) with frustrated interchain couplings $J_{bc}$ (green) between 1(2) and 4(3). Along the stacking direction $a$, there are nonfrustrated interlayer couplings $J_{ac}$ (blue) between 1(3) and 2(4) and also frustrated couplings $J_{ab}$ (yellow) between 1(2) and 3(4), but the latter are only present between every second pair of $\mathit{bc}$ layers. On the right is a schematic representation of the magnetic lattice with equivalently colored couplings $J,J_{bc},J_{ac}$, but without $J_{ab}$. The alternating easy-plane orientations are represented by open and filled symbols.

Figure 2

Specific heat of ${\mathrm{Cs}}_2{\mathrm{CoCl}}_4$ for representative magnetic fields applied along $c$ (left) or $a$ (right). Curves are offset with respect to each other by 2, 2, 1, 4, and 10 J${\mathrm{mol}}^{-1}{\mathrm{K}}^{-1}$ in panels (a)–(e), respectively.

Figure 3

Specific heat of ${\mathrm{Cs}}_2{\mathrm{CoCl}}_4$ in the gapped phase in magnetic fields applied along $a$. The solid line represents a fit of a Schottky contribution given by Eq. (5) and the dashed line is a fit of a nuclear contribution via Eq. (6).

Figure 4

Thermal expansion (left) and magnetostriction (right) of ${\mathrm{Cs}}_2{\mathrm{CoCl}}_4$ in magnetic fields applied along $a$. For clarity, the $\Delta L/L$ curves in panels (a) and (b) are offset with respect to each other by $4\times {10}^{-6}$ and the corresponding temperature and field derivatives $\alpha$ and $\lambda$ in panels (c) and (d) are offset by $2\times {10}^{-6}$ K and $5\times {10}^{-5}$ T, respectively. Bold (thin) lines represent measurements with increasing (decreasing) temperature or field (indicated by arrows) with rates of 3 mK/min or 5 mT/min, respectively. In (c), only $\alpha \left(T\right)$ obtained with increasing temperature are shown.

Figure 5

Temperature hysteresis of the Néel transition of ${\mathrm{Cs}}_2{\mathrm{CoCl}}_4$ in zero magnetic field as observed by measuring the thermal expansion $\Delta L\left(T\right)\parallel \mathit{b}$ while continuously increasing or decreasing the temperature (indicated by arrows) with different sweep rates. Inset: dependence of the hysteresis width on the sweep rate.

Figure 6

Specific heat of ${\mathrm{Cs}}_2{\mathrm{CoCl}}_4$ for different magnetic fields $H\parallel \mathit{b}$. For clarity, the curves are offset with respect to each other by 1 J${\mathrm{mol}}^{-1}{\mathrm{K}}^{-1}$ in panel (a) and by 2 J ${\mathrm{mol}}^{-1}{\mathrm{K}}^{-1}$ in panels (b) and (c). The arrows in (b) indicate two distinct phase transitions, which are present in a field range from 0.5 to 1.8 T.

Figure 7

Thermal expansion (left) and magnetostriction (right) of ${\mathrm{Cs}}_2{\mathrm{CoCl}}_4$ in magnetic fields along $b$. The relative length changes $\Delta L/L$ are displayed in (a) and (c), while (b) and (d) show the corresponding temperature and field derivatives. In (a) the $\Delta L\left(T\right)/L$ curves for different magnetic fields are shifted according to the measured magnetostriction $\Delta L\left(H\right)/L$ at $T=50\mathrm{mK}$ shown in (c). For clarity, the other curves in (c) are offset with respect to each other by ${10}^{-5}$ and the $\alpha \left(T\right)$ curves in (b) by $5\times {10}^{-5}$ K. In all panels, only data obtained with increasing temperature or field are shown.

Figure 8

Low-temperature $H-T$ phase diagrams of ${\mathrm{Cs}}_2{\mathrm{CoCl}}_4$ for magnetic fields along all three crystallographic axes. Shaded areas are guides to the eye. The magnetic-field dependent specific heat $c_p$ at a constant temperature of 0.11 K is also displayed (red lines, right scales). For $H\parallel b$, the occurrence of two spin-flop phases, SF1 and SF2, is indicated by $c_p\left(H\right)$ and for all three field directions $c_p\left(H\right)$ is strongly enhanced in the field range of phase II.

Figure 9

Representative temperature dependences of the specific heat of the various low-temperature phases of ${\mathrm{Cs}}_2{\mathrm{CoCl}}_4$. Shaded labels indicate the respective low-temperature phase at the given magnetic field applied along $b$. Dashed lines are fits of $c_p\left(T\right)/R=A·T^{\alpha }$.

Figure 10

Sketch of the proposed sequence of field-induced phases in ${\mathrm{Cs}}_2{\mathrm{CoCl}}_4$ for $H\parallel b$. The different colors of neighboring spin chains represent their different easy-plane orientations. AF: all spins are nearly (anti)parallel to the field $H$ and to each other. SF1: every second chain transforms to a spin-flop phase, which is stabilized by Dzyaloshinskii-Moriya interactions between spins of neighboring chains. SF2: spin-flop phase in all chains. Phase II could be an incommensurate or nematic phase preceding the fully saturated phase.