Magnetic order, magnetic correlations, and spin dynamics in the pyrochlore antiferromagnet Er${}_2$Ti${}_2$O${}_7$

Er${}_2$Ti${}_2$O${}_7$ is believed to be a realization of an $\mathit{XY}$ antiferromagnet on a frustrated lattice of corner-sharing regular tetrahedra. It is presented as an example of the order-by-disorder mechanism in which fluctuations lift the degeneracy of the ground state, leading to an ordered state. Here we report detailed measurements of the low-temperature magnetic properties of Er${}_2$Ti${}_2$O${}_7$, which displays a second-order phase transition at $T_{\mathrm{N}}\simeq 1.2$ K with coexisting short- and long-range orders. Magnetic susceptibility studies show that there is no spin-glass-like irreversible effect. Heat capacity measurements reveal that the paramagnetic critical exponent is typical of a 3-dimensional $\mathit{XY}$ magnet while the low-temperature specific heat sets an upper limit on the possible spin-gap value and provides an estimate for the spin-wave velocity. Muon spin relaxation measurements show the presence of spin dynamics in the nanosecond time scale down to 21 mK. This time range is intermediate between the shorter time characterizing the spin dynamics in Tb${}_2$Sn${}_2$O${}_7$, which also displays long- and short-range magnetic order, and the time scale typical of conventional magnets. Hence the ground state is characterized by exotic spin dynamics. We determine the parameters of a symmetry-dictated Hamiltonian restricted to the spins in a tetrahedron, by fitting the paramagnetic diffuse neutron scattering intensity for two reciprocal lattice planes. These data are recorded in a temperature region where the assumption that the correlations are limited to nearest neighbors is fair.

Figure 1

The network of corner-sharing regular tetrahedra formed by the rare-earth atoms in the pyrochlore structure in which Er${}_2$Ti${}_2$O${}_7$ crystallizes. The axis of trigonal symmetry at the position of a rare earth is one of the cube diagonals. There are two types of tetrahedra in the network, which differ by their orientation: Type B is rotated by 90${}^{\circ }$ about the cubic axes with respect to type A. We distinguish the two sets by two colors in the drawing. Since each rare earth is at a corner shared by two tetrahedra, one of each kind, either the set of the four corners of all the A tetrahedra or the set of the four corners of all the B tetrahedra is sufficient to describe the Er${}^{3+}$ lattice.

Figure 2

Neutron intensity for the (2,2,0) Bragg peak of Er${}_2$Ti${}_2$O${}_7$ plotted as a function of $|q-q_{(2,2,0)}|$. The data are from Ruff et al. (Ref. 20). The instrumental resolution is determined from the reflection measured under a 3 T magnetic field. The line for $|q-q_{(2,2,0)}|\ge 0.015$ Å${}^{-1}$ is described in the main text.

Figure 3

An example of an x-ray diffraction pattern recorded at room temperature for an Er${}_2$Ti${}_2$O${}_7$ powder obtained after crushing part of a crystal. The radiation used is Co K${}_{\alpha }$. The line at the bottom of the graph shows the difference between the data and the refinement model.

Figure 4

Specific heat versus temperature for three samples cut from an Er${}_2$Ti${}_2$O${}_7$ crystal prepared as indicated in the main text. For the first sample, no subsequent treatment has been performed, while the other two result from postgrowth heat treatments as follows. Case 1: Heat treatment for 7 days under oxygen at 1150 ${}^{\circ }$C, followed by a slow cooling down to 400 ${}^{\circ }$C. Case 2: Heat treatment for 7 days under argon at 1150 ${}^{\circ }$C, followed by a slow cooling down to 400 ${}^{\circ }$C.

Figure 5

Inverse of the magnetic susceptibility of an Er${}_2$Ti${}_2$O${}_7$ crystal versus temperature in a large temperature range. The solid line results from a fit with the Curie-Weiss law. A field of 1 mT is applied along a diagonal of the cubic crystal structure. The inset shows the low-temperature range of the data.

Figure 6

The magnetic susceptibility of an Er${}_2$Ti${}_2$O${}_7$ crystal versus temperature in the low-temperature region. The same magnetic response is observed using either the field- or zero-field-cooling procedure, fc and zfc, respectively. A field of 1 mT is applied along a diagonal of the cubic crystal structure. In the zfc mode the sample was warmed to 300 K and then cooled to 2 K before applying the magnetic field. In the fc mode the field was applied at 300 K and the sample was subsequently cooled down.

Figure 7

Low-temperature specific heat of Er${}_2$Ti${}_2$O${}_7$. The open symbols show our experimental data compared to literature results from Blöte et al. (Ref. 29) and Siddharthan et al. (Ref. 40). Note the large overlap between our results obtained with the PPMS and the homemade calorimeter inserted in a dilution refrigerator. The filled symbols present the electronic specific heat deduced from the data of Siddharthan et al. and this work. The dashed lines show the contributions from the nuclear specific heat and the full line results from a fit of the low-temperature electronic specific heat to the $C_{\mathrm{elec}}=\mathcal{B}{T}^3$ law, with $\mathcal{B}=2.50\left(15\right)$ J K${}^{-4}$ mol${}^{-1}$. The inset displays the very low temperature details. The solid lines result from fits as explained in the main text.

Figure 8

Zero-field specific heat of Er${}_2$Ti${}_2$O${}_7$ versus the reduced temperature parameter $\tau$ in the paramagnetic regime for three data sets. The full (dotted) line is the prediction of Eq. (14) for $\alpha =-0.015$ ($-0.134$). For $\alpha =-0.015$ we find $C_{\mathrm{sh}}=1.7\left(1\right)$ J K${}^{-1}$ mol${}^{-1}$. The critical regime is observed up to $\tau$ $\simeq 0.2$.

Figure 9

The electronic specific heat of Er${}_2$Ti${}_2$O${}_7$ in an extended temperature range. In the inset are displayed the total specific heat ($C_{\mathrm{p}}$) of Er${}_2$Ti${}_2$O${}_7$ and the sum of the estimated nuclear and lattice contributions.

Figure 10

The variation of the electronic entropy $\Delta S_{\mathrm{elec}}\left(T\right)$ of Er${}_2$Ti${}_2$O${}_7$ versus temperature $T$. The solid line is computed assuming two singlet states at energies corresponding to 0 and 2.85 K and two doublets at 74 and 85.8 K. A signature of the two doublets has been seen by inelastic neutron scattering (Ref. 19). The ground-state doublet has been split to account for the magnetic order of Er${}_2$Ti${}_2$O${}_7$ below $T_{\mathrm{N}}$, with the splitting taken as a fitting parameter. Admittedly, this is a very rough description of the physics. The dashed line is the result of the computation of $\Delta S_{\mathrm{elec}}\left(T\right)$ when only the first two levels are taken into account.

Figure 11

Temperature dependence of the specific heat of a Er${}_2$Ti${}_2$O${}_7$ single crystal for different magnetic field intensities applied along $\left[110\right]$. The maximum of the specific-heat peak moves to lower temperatures as the field increases up to 1.7 T. No peak is observed when the field strength is above 1.7 T.

Figure 12

The phase diagram derived from specific-heat measurements for the three main crystal directions of cubic Er${}_2$Ti${}_2$O${}_7$. The dashed-dotted lines are guides to the eye.

Figure 13

$\mu$SR spectra of a Er${}_2$Ti${}_2$O${}_7$ crystal taken at S$\mu$S with the initial muon beam polarization parallel to the $\left[111\right]$ crystal direction. The spectra were recorded in zero and various longitudinal fields as indicated in the figure.

Figure 14

Top two panels: Magnetic diffuse neutron scattering intensity recorded for a crystal of Er${}_2$Ti${}_2$O${}_7$ in the reciprocal $(h,k,0)$ and $(h,k,k)$ planes at 2.00 (3) and 1.47 (3) K, respectively. The positions in the reciprocal lattice are in $2\pi /a$ units, where $a$ is the lattice parameter of the cubic unit cell. These maps are obtained as explained in the main text. To preserve the maps appearance, pixels with off-scale intensities, e.g., pixels influenced by Bragg reflections and critical scattering, as well as pixels located near the origin of the reciprocal lattice have been graphically eliminated: They are represented in white color. Bottom two panels: $(h,k,0)$ and $(h,k,k)$ magnetic correlation maps computed with the tetrahedron model explained in the main text. The comparison between the theoretical and experimental maps displayed above enables us to derive information on the Er${}_2$Ti${}_2$O${}_7$ interaction constants. The lines drawn in the $(h,k,0)$ maps indicate the position of the cuts shown in Fig. 16.

Figure 15

Magnetic scattering intensity versus wave vector in the vicinity of the two reciprocal lattice positions ${\mathbf{q}}_{(h,k,l)}$ where $(h,k,l)=(1,1,1)$ and (2,2,0), respectively. The wave vector unit is $2\pi /a$ where $a$ is the cube edge. The data are obtained from the intensity differences shown in Fig. 14 by averaging the data at reciprocal space points located at equal distance from the two lattice positions. Two requirements were considered when choosing these two positions: They do not lie close to the boundary of the maps and the magnetic intensity is relatively important. The lines are results from fits of Eq. (15) to the data.

Figure 16

Cuts of the $(h,k,0)$ maps shown in Fig. 14 along the [1, 1,0] and [0, 1,0] directions. The circles represent the experimental data and the lines the results of the model. The origin of the horizontal axis is taken at the origin of the reciprocal lattice.