Dimensionality and irreversibility of field-induced transitions in ${\mathrm{SrDy}}_2{\mathrm{O}}_4$

Low-temperature ultrasound velocity measurements are presented on the frustrated spin system ${\mathrm{SrDy}}_2{\mathrm{O}}_4$ that allow us to define high-resolution phase diagrams with the magnetic field applied along all three principal axes. For $H\left|\right|b$, a region of field-induced long-range order is delimited by a dome of first-order phase transitions. An unusual magnetization process is observed with significant irreversibility at very low temperatures when passing between the low-field spin liquid phase and the long-range ordered phase which we attribute to large energy barriers. For $H\left|\right|c$, the system appears to remain effectively one dimensional, exhibiting two transitions as a function of magnetic field, but no finite-temperature long-range order.

Figure 1

Structure of Dy ions in ${\mathrm{SrDy}}_2{\mathrm{O}}_4$, with the shape of the ellipsoids illustrating the easy axes of the two sites. Site Dy1 (red) has its easy axis along $c$ and site Dy2 (blue) has its easy axis along $b$.

Figure 2

(a) Relative change in sound velocity as a function of $H\left|\right|b$. Dotted lines are for decreasing field, and solid lines are for increasing field. Temperatures labels are given in units of mK. (b) Relative change in sound velocity as a function of $T$ for various fields $H\left|\right|b$. Dotted lines are for FC and solid lines are for FCW. (c) Temperature scans of $\mathrm{\Delta }v/v$ for $H\left|\right|b$ showing the difference between FCW (solid lines) and ZFCW (dotted lines). Curves have been staggered for ease of viewing.

Figure 3

Surface plots of the field derivative of sound velocity $(\partial v/\partial H)/v$ as a function of $H_b$ and temperature for (a) increasing field and (b) decreasing field. In addition to a shift of the position of the dome of LRO (due to hysteresis at the first-order phase transitions), the “up” curves demonstrate a great deal of increased structure in the vicinity of $H_{b1}$, with this line of transitions appearing to bifurcate into three separate transitions. (c) Phase diagram for $H\left|\right|b$. Blue circles are the two initial jumps seen when raising the magnetic field. Black triangles present the first-order phase transitions that delimit what is likely the $\uparrow \uparrow \downarrow$ LRO phase, with the orientation of the triangles indicating the direction of field or temperature sweeps. Red squares illustrate the crossover to full polarization (of Dy2). The hatched region roughly shows where slow (glassy) dynamics is observed.

Figure 4

Relative change in sound velocity for $H\left|\right|c$ (a) as a function field and (b) as a function of temperature. Solid lines correspond to increasing field or temperature and dotted lines correspond to decreasing field or temperature. Curves have been staggered for ease of viewing. (c) Phase diagram for $H\left|\right|c$. The orientation of the triangles indicates the direction of the field sweeps. The red squares show a crossover to a ferromagnetic (fully polarized) phase.