Neutron scattering studies of the spin ices ${\mathrm{Ho}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ and ${\mathrm{Dy}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ in applied magnetic field

Neutron diffraction has been used to investigate the magnetic correlations in single crystals of the spin ice materials ${\mathrm{Ho}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ and ${\mathrm{Dy}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ in an external magnetic field applied along either the $\left[001\right]$ or $\left[1\overline{1}0\right]$ crystallographic directions. With the field applied along $\left[001\right]$ a long range ordered ground state is selected from the spin ice manifold. With the field applied along $\left[1\overline{1}0\right]$ the spin system is separated into parallel $\left(\alpha \right)$ and perpendicular $\left(\beta \right)$ chains with respect to the field. This leads to partial ordering and the appearance of quasi-one-dimensional magnetic structures. In both field orientations this frustrated spin system is defined by the appearance of metastable states, magnetization plateaus and unusually slow, field regulated dynamics.

Figure 1

Spin ice properties with the field applied along $\left[001\right]$: Expected “$Q=0$” ordered structure parallel to the applied field (top) and projected along the field direction (bottom). Each tetrahedron is in the same ice rule obeying local ground state. The field is indicated by the arrow, canting of spins above and below the plane indicated by $+$ and $-$, and the numbers give the coordinate along the projection axis. Only four tetrahedra of the pyrochlore unit cell are shown so each axis runs 0–0.75.

Figure 2

The phase diagram of the near neighbor spin ice model with the field applied along $\left[001\right]$ (Ref. 11). With the field applied in this direction the model shows unusual liquid-gas-type behavior. A line of first order phase transitions from the gaslike region (G) to the liquidlike region (L) is terminated by a critical point where the transition becomes continuous.

Figure 3

Spin ice properties with the field applied along $\left[1\overline{1}0\right]$: The field separates the spin system into two sets of chains, parallel to the field (dark spins, $\alpha$ chains) and perpendicular to the field (light spins, $\beta$ chains). Two ordered structures are illustrated depending on the relative orientation of the $\beta$ chains. The $Q=0$ structure ($\beta$ chains parallel) is shown with vertical field (top) and projected along the field (second from top). The same views of the $Q=X$ structure ($\beta$ chains antiparallel) are shown beneath. Again the $Q=0$ structure has the same spin configuration on each tetrahedron but the ferromagnetic component is now at 45° to the field. The antiparallel $\beta$ chains of the $Q=X$ structure lower the symmetry and this allows magnetic Bragg scattering at both $Q=0$ and $Q=X$ positions.

Figure 4

The reciprocal space of ${\mathrm{Dy}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ in the $h,k,0$ (top) and $h,h,l$ (bottom) planes as studied on PRISMA. The $\mathbf{Q}=0$ and $\mathbf{Q}=X$ positions are indicated by $\mathtt{o}$ and $\mathtt{x}$, respectively. In both diagrams the area of the scattering plane that lies within the first aluminum powder line and was included in a mapping is indicated by the curving shells. With the field along $\left[001\right]$ ($h,k,0$ scattering plane), single setting scans were measured at intermediate field points. In this case 16 detectors cover the region between the two radial detector trajectories enclosing the $h,0,0$ axis. With the field along $\left[1\overline{1}0\right]$ ($h,h,l$ scattering plane), rocking scans were measured. In this case 80 detector trajectories cover the region between the two radial detector trajectories enclosing the $0,0,l$ axis. This gives vastly improved reciprocal space coverage compared to a single setting scan such as that indicated about the $h,h,0$ axis, where only 16 detector trajectories cover the same area of reciprocal space.

Figure 5

${\mathrm{Dy}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ $(B\Vert [001\left]\right)$: scattering intensity in the $h,k,0$ plane recorded on PRISMA at $0.05\mathrm{K}$ in zero field (top), and in $1\mathrm{T}$ (bottom). In zero field there is strong diffuse scattering characteristic of the short range ordered spin ice state [e.g., at (3,0,0) and (3,3,0)]. In $1\mathrm{T}$ the diffuse scattering disappears entirely, giving way to magnetic Bragg scattering at $Q=0$ positions. Note that the intensity scale is cut off at 2 to reveal the presence and/or absence of diffuse scattering; the maximum intensity of a Bragg peak is much greater.

Figure 6

${\mathrm{Dy}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ $(B\Vert [001]$, $T=0.05\mathrm{K})$: Integrated intensity of the $(2,0,0)$ peak on the first field cycle and the $(4,\overline{2},0)$ on the second cycle (inset) with the field applied along $\left[001\right]$ [the intensity of $(2,0,0)$ has been divided by 2]. Both peaks are purely magnetic scattering. The history dependence and existence of metastable states during the magnetization process is clearly seen by the different hysteresis in the two cycles and the plateaus. Lines are to guide the eye only; filled symbols indicate the falling field leg.

Figure 7

${\mathrm{Ho}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ $(B\Vert [001\left]\right)$: development of $\mathbf{Q}=0$ magnetization at different temperatures, with the field applied along $\left[001\right]$. As the temperature decreases increasing hysteresis and metastable states appear in the magnetization, although the final ordered state is always the same. Lines are to guide the eye only, filled symbols indicate the falling field leg, and successive higher temperature loops are shifted by $-1\mathrm{T}$ and $+0.5{\mu }_{\mathrm{B}}{\mathrm{atom}}^{-1}$.

Figure 8

${\mathrm{Ho}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ $(B\Vert [001\left]\right)$: the intensity of the diffuse scattering, at the $(1,0,0)$ position as a function of rising (top) and falling (middle) field, at $0.05\mathrm{K}$. The $(2,0,0)$ peak intensity is shown as a comparison (it has been divided by 100). As the fully ordered structure develops the sample passes through a partially ordered regime in which both Bragg scattering and diffuse scattering from the short range ordered spin ice state are observed (between 0.3 and $0.7\mathrm{T}$). At the end of the field cycle the sample is frozen in a partially ordered state with a remnant magnetization. As the temperature is raised above $0.5\mathrm{K}$, the remnant magnetization decays and the diffuse scattering characteristic of the zero field spin ice state is re-established (bottom). Lines are to guide the eye only; filled symbols indicate the falling field leg.

Figure 9

${\mathrm{Dy}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ $(B\Vert [1\overline{1}0\left]\right)$: Scattering intensity in the $h,h,l$ plane recorded on PRISMA at $0.05\mathrm{K}$ in zero field (top), and in $1.5\mathrm{T}$ (bottom). In zero field the diffuse scattering is characteristic of the short range ordered spin ice state (Ref. 32). In $1.5\mathrm{T}$ both Bragg [at $Q=0$ positions such as $(0,0,2)$] and diffuse [at $Q=X$ positions such as $(0,0,3)$] magnetic scattering is observed indicating the coexistence of long and short range magnetic order. Note that the intensity scale is cut off at 2 to reveal the presence and/or absence of diffuse scattering; the maximum intensity of a Bragg peak is much greater.

Figure 10

${\mathrm{Dy}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ $(B\Vert [1\overline{1}0\left]\right)$: the amplitude of the $Q=X$ feature at $(0,0,3)$ during hysteresis loops at 0.05 and $0.3\mathrm{K}$ in the second experiment. The degree of correlation in the $\beta$ chains is temperature dependent. At $0.3\mathrm{K}$ they become more strongly correlated. Lines are to guide the eye only; filled symbols indicate the falling field leg.

Figure 11

${\mathrm{Dy}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ $(B\Vert [1\overline{1}0\left]\right)$: Integrated intensity of $(0,0,2)$ with the field applied along $\left[1\overline{1}0\right]$ and the coarse field stepping at $0.05\mathrm{K}$ (top), fine field stepping at $0.05\mathrm{K}$ (middle), and fine field stepping at 0.05 and $0.3\mathrm{K}$ (the loops at $0.05\mathrm{K}$ were recorded in separate experiments and the intensities from the first are scaled to compare with the second). At the same temperature, slower sweep rates smooth the magnetization process (top and middle). At similar sweep rates there is a change in the dynamical regime between 0.3 and $0.05\mathrm{K}$ as shown by the presence and/or absence of steps at $0.05–0.3\mathrm{K}$ (bottom). Lines are to guide the eye only; filled symbols indicate the falling field leg.

Figure 12

${\mathrm{Ho}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ $(B\Vert [1\overline{1}0\left]\right)$: Hysteresis loop at $0.05\mathrm{K}$. The amplitude of the diffuse feature at $(0,0,1)$ relates to the degree of correlation in the $\beta$ chains (top). The $(0,0,2)$ peak (bottom) is directly related to the order in the $\alpha$ chains. As the $\alpha$ chains are ordered by the field, correlations are produced in the $\beta$ chains. The $\beta$ chains evolve toward an increasingly correlated state but are only able to do so when the field produces dynamics in the $\alpha$ chains. Only the central region of the hysteresis loop is shown; there are no intensity changes in the sections between 1.5 and $2.5\mathrm{T}$ or $-1.5$ and $-2.5\mathrm{T}$. Lines are to guide the eye only; filled symbols indicate the falling field leg.

Figure 13

${\mathrm{Ho}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ $(B\Vert [1\overline{1}0\left]\right)$: temperature scan in a field of $2.5\mathrm{T}$. The amplitude of the diffuse feature at $(0,0,1)$ was fitted as a flat line $(0.05–0.3\mathrm{K})$, single Gaussian $(0.3–0.6\mathrm{K})$, or triangular peak $(0.6–1.2\mathrm{K}$ and all points on the falling $T$ scan). For $T<J_{\mathrm{eff}}$ increasing the temperature activates the dynamics of the $\beta$ chains and also causes their level of correlation to increase. As $T$ approaches $J_{\mathrm{eff}}$ the activated dynamics begins to destroy the correlations; this is evident in the down turn of $(0,0,1)$ intensity at $1.1\mathrm{K}$. Lines are to guide the eye only; filled symbols indicate the falling temperature leg.