Comprehensive cluster-theory analysis of the magnetic structures and excitations in ${\mathrm{CoCl}}_2·2{\mathrm{H}}_2\mathrm{O}$

The magnetic properties of ${\mathrm{CoCl}}_2·2{\mathrm{H}}_2\mathrm{O}$ are analyzed in the mean-field/random-phase approximation using a basis of clusters with four spins along the $c$-axis chains of Co ions. The model gives a unifying account of the bulk properties, the spin waves, and the higher-order cluster-spin excitations. The theory describes accurately the neutron scattering measurements of the excited doublet of the $S=3/2{\mathrm{Co}}^{++}$ ions in both the antiferromagnetic and the paramagnetic phases. The theory has been applied by Larsen et al. [Phys. Rev. B 96, 174424 (2017)] for analyzing the quantum phase transition at a transverse field of 160 kOe and is found to agree closely with their observations.

Figure 1

Data from the MARI experiment showing the $Q$-integrated intensity vs energy at 5 K and 100 K. The black circles are the results obtained when integrating the scattering intensity with respect to $Q$ between 0 and 4 ${\AA }^{-1}$, whereas the green triangles are the results obtained, when integrating from 0 to 9 ${\AA }^{-1}$. The blue solid lines are the calculated results obtained using a cluster basis of three spins along the Co chains, when assuming the background scattering shown by the dashed brown lines.

Figure 2

The red circles, the blue squares, and the black diamonds show, respectively, the experimental results for the $zz$, the $xx$, and the $yy$ component of the susceptibility tensor in the paramagnetic phase of ${\mathrm{CoCl}}_2·2{\mathrm{H}}_2\mathrm{O}$ measured by Narath [9]. The heavy solid lines with the same colors as the corresponding experimental points, are the results obtained by a perturbative MF modification of the 1D chain susceptibilities. The thin solid lines are the MF results obtained when using the four-spin chain clusters as basis, whereas the dashed lines are the simple MF results assuming a single spin basis.

Figure 3

Magnetization curves of ${\mathrm{CoCl}}_2·2{\mathrm{H}}_2\mathrm{O}$ in the antiferromagnetic phase at 1.3 K measured by Mollymoto et al. [23]. The experimental results are shown by the dashed lines and the thin vertical lines indicate the critical fields. The colors red, blue, and black denote the $zz,xx$, and $yy$ components, respectively, and the solid lines are the calculated results.

Figure 4

The spin wave energies in the antiferromagnetic phase of ${\mathrm{CoCl}}_2·2{\mathrm{D}}_2\mathrm{O}$ determined by neutron scattering at temperatures below 7 K. The circles denote the results obtained by Montfrooij et al. [21]. The results reported in Refs. [19, 20] are shown by squares and triangles, respectively. Open symbols refer to the $x$ mode and filled ones to the $y$ mode. The results of the fitting procedure to the excitation energies for the $x$ and $y$ modes are given by the blue and black lines, respectively.

Figure 5

The resonance frequencies in ${\mathrm{CoCl}}_2·2{\mathrm{H}}_2\mathrm{O}$ at 1.6 and 6 K observed by Torrance and Tinkham in far-infrared transmission experiments [5]. The results are plotted against a field applied along the $b\left(z\right)$ axis. The excitation energies calculated using the four-spin cluster model are shown by the solid lines. The transverse-mode labels $x$ and $y$ are only strictly valid for the spin wave modes (the two lowest lying modes) at zero field. Else, all modes labeled by $x$ contain a minor elliptical $y$ component, and the $x$ component is also the dominant one in the $y$ mode at fields above 10 kOe. The green horizontal line indicates the optical phonon branch, which is observed because this phonon interact with the magnetic excitations as discussed in Refs. [5, 24].

Figure 6

The transverse correlation function at 1.3 K calculated along $Q=(h0h/2)$ plotted on a logarithmic scale. The figure to the left shows the zero-field result, and the figure to the right is the calculated result at the $x$-axis field $H=0.75H_c^x$, corresponding to 120 kOe in the experiment and 137 kOe in the model calculation.

Figure 7

Inelastic scans at $\left(030\right)$ in ${\mathrm{CoCl}}_2·2{\mathrm{D}}_2\mathrm{O}$ at various fields along the hard $y$ axis at 1.5 K. The circles are the experimental points and the blue lines are the results obtained when assuming the same experimental resolution and the same background, the red lines, in all cases. The experimental results nominally at 14.35 T are the average of those obtained at fields 14, 14.2, 14.4, and 14.8 T reducing the scatter by a factor of 2.

Figure 8

Inelastic neutron scattering results obtained at $\left(211\right)$ in ${\mathrm{CoCl}}_2·2{\mathrm{D}}_2\mathrm{O}$ at fields applied along the $x$ axis at 1.5 K. The signatures are the same as used in Fig. 7. The calculations are performed at fields determined so that $H/H_c^x$ is the same as the experimental values.

Figure 9

The energy variation of the excitations at $\left(211\right)$ in ${\mathrm{CoCl}}_2·2{\mathrm{D}}_2\mathrm{O}$, when a field is applied along $x$. The triangles denote the positions of the extra peaks observed experimentally, which may derive from a combination of the $x$-polarized mode and an optical phonon mode. The position of the optical mode is shown by the thin green line. The cross section of the $x$-polarized mode becomes very small (dashed line) above 150 kOe. The dashed blue line is the calculated result scaled so that the critical field coincides with the experimental one, $H_c^{}=160.5$ kOe [1].