Single-crystal study of the honeycomb XXZ magnet ${\mathrm{BaCo}}_2$(${\mathrm{PO}}_4$)${}_2$ in magnetic fields

We present a study of high-quality $\mathrm{Ba}{\mathrm{Co}}_2{\left(\mathrm{P}{\mathrm{O}}_4\right)}_2$ single crystals via magnetization, heat-capacity, thermal-expansion, and magnetostriction measurements. Sharp anomalies in the thermodynamic properties at $T_N=3.4\mathrm{K}$ reveal a long-range antiferromagnetic order in these single-crystalline samples, which is absent in polycrystalline $\mathrm{Ba}{\mathrm{Co}}_2{\left(\mathrm{P}{\mathrm{O}}_4\right)}_2$. The temperature dependent magnetic susceptibilities for in-plane and out-of-plane magnetic fields are strongly anisotropic and reveal a pronounced easy-plane anisotropy. A Curie-Weiss analysis implies strong orbital magnetism, as it is known from the sister compound $\mathrm{Ba}{\mathrm{Co}}_2{\left(\mathrm{As}{\mathrm{O}}_4\right)}_2$ that is discussed as a potential Kitaev spin-liquid material. When applying in-plane magnetic fields at low temperature, $\mathrm{Ba}{\mathrm{Co}}_2{\left(\mathrm{P}{\mathrm{O}}_4\right)}_2$ is driven to another ordered phase at a critical field ${\mu }_0{H}_{C1}\approx 0.11\mathrm{T}$ and then undergoes a further field-induced transition to a highly polarized paramagnetic phase at ${\mu }_0{H}_{C2}\approx 0.3\mathrm{T}$, which is again similar to the case of $\mathrm{Ba}{\mathrm{Co}}_2{\left(\mathrm{As}{\mathrm{O}}_4\right)}_2$. In addition, our lowest-temperature data reveal that the field-induced transitions in $\mathrm{Ba}{\mathrm{Co}}_2{\left(\mathrm{P}{\mathrm{O}}_4\right)}_2$ become dominated by thermally assisted domain-wall motion.

Figure 1

(a) Hydrothermally grown crystals of $\mathrm{Ba}{\mathrm{Co}}_2{\left(\mathrm{P}{\mathrm{O}}_4\right)}_2$. (b) Typical morphology of grown crystals with pinacoid {001} and rhombohedron face (113) plus symmetrically equivalent faces. Crystallographic axes are indicated. (c) A fraction of the crystal structure of $\mathrm{Ba}{\mathrm{Co}}_2{\left(\mathrm{P}{\mathrm{O}}_4\right)}_2$ with honeycomb layers of edge-sharing $\mathrm{Co}{\mathrm{O}}_6$ octahedra (orange) alternatingly stacked along $c$ with layers of Ba atoms (white) and $\mathrm{P}{\mathrm{O}}_4$ groups (green). Structure data taken from Ref. [28], structure model generated with vesta [29].

Figure 2

(a), (b) Magnetic susceptibilities ${\chi }_{ab}$ and ${\chi }_c$ for in-plane and out-of-plane magnetic fields of 0.1 and 1 T. (b) also shows $1/{\chi }_i$ (right axes) together with Curie-Weiss fits (lines) for $T>150\mathrm{K}$. (c) Zero-field heat capacity and (d) relative length change $\mathrm{\Delta }L\left(T\right)/L_0$ of the $c$ axis together with the uniaxial thermal expansion coefficient ${\alpha }_c=1/L_0\partial \mathrm{\Delta }L\left(T\right)/\partial T$.

Figure 3

(a) Magnetization $M\left(T\right)$ and (b) heat capacity versus temperature for different in-plane magnetic fields. The inset of (a): the first derivative $dM\left(T\right)/dT$ of magnetization under magnetic field of 0.05 T. (c) Field dependent $M\left({\mu }_{0}H\right)$ for $T=1.8$ to 3.5 K, (d) also contains the derivative $\partial M/\partial \left({\mu }_{0}H\right)$ with two peaks signaling two critical fields at 0.11 and 0.27 T. (e), (f) Magnetostriction $\mathrm{\Delta }L\left({\mu }_{0}H\right)/L_0$ of the $c$ axis together with the expansion coefficient $\lambda =1/L_0\partial \mathrm{\Delta }L/\partial \left({\mu }_{0}H\right)$ at 2.3 and 1.5 K; the data were obtained with increasing (red) and decreasing (blue) field. (g) $\mathrm{\Delta }L\left({\mu }_{0}H\right)/L_0$ at 0.3, 0.5, and 0.8 K, where each data set was obtained after zero-field cooling from 6 K with increasing and decreasing field. Each curve ends with a remnant zero-field expansion. (h) Thermal expansion $\mathrm{\Delta }L\left(T\right)/L_0$ (red line) measured after a magnetostriction cycle at 0.3 K, which induces a remnant zero-field expansion (marked by the arrow) in comparison to $\mathrm{\Delta }L\left(T\right)/L_0$ ($\circ$) measured after zero-field cooling.

Figure 4

Phase diagram of $\mathrm{Ba}{\mathrm{Co}}_2{\left(\mathrm{P}{\mathrm{O}}_4\right)}_2$ for in-plane magnetic fields obtained from heat capacity ($C_p$), uniaxial expansion ($\mathrm{\Delta }L$) and magnetization ($M$) measurements. Transition temperatures and fields are defined by the peak positions of $C_p\left(T\right)$ and of the first derivatives of $\mathrm{\Delta }L(T,{\mu }_{0}H)$ or $M(T,{\mu }_{0}H)$ with respect to $T$ or ${\mu }_{0}H$, respectively.