High-field phase diagram of the chiral-lattice antiferromagnet $\mathrm{Sr}\left(\mathrm{TiO}\right){\mathrm{Cu}}_4({\mathrm{PO}}_4{)}_4$

We studied the high-field phase diagram of a chiral-lattice antiferromagnet $\mathrm{Sr}\left(\mathrm{TiO}\right){\mathrm{Cu}}_4{\left({\mathrm{PO}}_4\right)}_4$ by means of ultrasound, dielectric, and magnetocaloric-effect measurements. These experimental techniques reveal two new phase transitions at high fields, which have not been resolved by previous magnetization experiments. Specifically, the $c_{66}$ acoustic mode shows drastic changes with hysteresis for magnetic fields applied along the $c$ axis, indicating a strong magnetoelastic coupling. Combined with cluster mean-field theory, we discuss the origin of these phase transitions. By considering the chiral-twist effect of ${\mathrm{Cu}}_4{\mathrm{O}}_{12}$ cupola units, which is inherent to the chiral crystal structure, the phase diagram is reasonably reproduced. The agreement between experiment and theory suggests that this material is a unique quasi-two-dimensional spin system with competing exchange interactions and chirality, leading to a rich phase diagram.

Figure 1

(a) Crystal structure of SrTCPO with magnetic interactions; nearest-neighbor exchange ($J_1$), next-nearest-neighbor exchange ($J_2$), intercupola exchanges ($J^{\prime }$ and $J^{″}$), and DM vector ($D$). $\theta$ is the angle of the DM vector from the $c$ axis. The site index $\ell$ (0–15) is denoted on each ${\mathrm{Cu}}^{2+}$. (b) In-plane interactions of SrTCPO with the twist angle $\varphi$. (c, d) Strained unit cells with (c) ${\epsilon }_{xy}$ and (d) ${\epsilon }_{xx}$. The shorter bonds are shown by thicker lines.

Figure 2

(a–e) Temperature dependence of the elastic constants. Solid lines show the results at zero field. The gray dashed line shows the fitted background due to the anharmonic phonon contribution based on Ref. [31]. (f–j) Relative change of the elastic constants $\mathrm{\Delta }c/c$ near $T_{\mathrm{N}}$. (f) Results for zero and applied fields ($\mathbf{B}\parallel \mathbf{k}\parallel \left[100\right]$) as labeled for each curve. The inset shows the enlarged result at zero field.

Figure 3

Temperature dependence of the specific heat divided by temperature $C/T$ at zero field. A broad hump is indicated by the arrow.

Figure 4

(a–c) Relative changes of the elastic constants $\mathrm{\Delta }c/c$ at 1.8 K as a function of magnetic field for the field directions (a) $\mathbf{B}\parallel \left[001\right]$, (b) $\mathbf{B}\parallel \left[100\right]$, and (c) $\mathbf{B}\parallel \left[110\right]$. Enlarged results of (d) $\mathrm{\Delta }c/c$ and (e) the relative change of the acoustic attenuation of the $c_{66}$ mode ($f=122$ MHz) with $\mathbf{B}\parallel \left[001\right]$. The dashed lines show results obtained using lower maximum-field pulses. The results are shifted for clarity.

Figure 5

Dielectric constant along the [100] direction ${\varepsilon }_{\left[100\right]}$. The magnetic fields are applied along the (a) [001] and (b) [100] directions. The inset shows the results obtained under static fields along the [010] direction at selected temperatures.

Figure 6

MCE curves obtained for the field directions (a, d) $\mathbf{B}\parallel \left[001\right]$, (b, e) $\mathbf{B}\parallel \left[100\right]$, and (c, f) $\mathbf{B}\parallel \left[110\right]$. (a–c) Dashed (solid) lines represent the data for field up (down) sweeps. (d–f) Down-sweep results are shown with the anomalies detected in the elastic and dielectric constants. Symbols are summarized in Fig. 6. The dotted lines show guides for the phase boundary and crossover line (see text for details).

Figure 7

Theoretical magnetic-field-temperature phase diagram for (a) $\mathbf{B}\parallel \left[001\right]$, (b) $\mathbf{B}\parallel \left[100\right]$, and (c) $\mathbf{B}\parallel \left[110\right]$. Here, the results for the effective Hamiltonian [Eq. (1)] with the parameters $J_1=0.6$, $J_2=1/6$, $J^{\prime }=1/2$, $J^{″}=1/100$, $D=0.7$, $\theta ={90}^{\circ }$, and $\varphi =1^{\circ }$ are shown. The cyan dashed line in panel (b) shows the crossover field extracted from an inflection point of $dm/dB$. For the representative spin structures of the proposed phases, see Appendix.

Figure 8

Contour plots of the theoretically calculated magnetic entropy, $S$, for (a) $\mathbf{B}\parallel \left[001\right]$, (b) $\mathbf{B}\parallel \left[100\right]$, and (c) $\mathbf{B}\parallel \left[110\right]$. The used parameters are the same as in Fig. 7. The white solid lines and the cyan dashed line show the phase boundaries and crossover extracted from the phase diagram (Fig. 7), respectively.

Figure 9

Second derivatives of the magnetic energy with respect to strain as a function of the magnetic field along (a) $\mathbf{B}\parallel \left[001\right]$, (b) $\mathbf{B}\parallel \left[100\right]$, and (c) $\mathbf{B}\parallel \left[110\right]$. The derivatives are taken at $T=0$ for the strains ${\epsilon }_{xy}$ (upper panel) and ${\epsilon }_{xx}$ (lower panel). For details, see the main text.

Figure 10

Second derivatives of the magnetic energy near the phase III-II boundary ($T=0.23$) for $\mathbf{B}\parallel \left[001\right]$. The derivatives are taken for the strains ${\epsilon }_{xy}$ (upper panel) and ${\epsilon }_{xx}$ (lower panel).

Figure 11

Spin configurations with the following parameters: (a) $B=0.0$, $T=0.00$, (b) $\mathbf{B}\left|\right|\left[001\right]B=1.0$, $T=0.00$, (c) $\mathbf{B}\left|\right|\left[001\right]B=1.8$, $T=0.00$, (d) $\mathbf{B}\left|\right|\left[001\right]B=1.3$, $T=0.23$, (e) $\mathbf{B}\left|\right|\left[100\right]B=0.5$, $T=0.00$, (f) $\mathbf{B}\left|\right|\left[100\right]B=1.5$, $T=0.00$, (g) $\mathbf{B}\left|\right|\left[100\right]B=2.2$, $T=0.00$, (h) $\mathbf{B}\left|\right|\left[110\right]B=0.5$, $T=0.00$, and (i) $\mathbf{B}\left|\right|\left[110\right]B=1.5$, $T=0.00$.