Field evolution of the magnetic structures in ${\text{Er}}_2{\text{Ti}}_2{\text{O}}_7$ through the critical point

We have measured neutron-diffraction patterns in a single-crystal sample of the pyrochlore compound ${\text{Er}}_2{\text{Ti}}_2{\text{O}}_7$ in the antiferromagnetic phase $\left(T=0.3\text{K}\right)$, as a function of the magnetic field, up to 6 T, applied along the [110] direction. We determine all the characteristics of the magnetic structure throughout the critical point at $H_c=2\text{T}$. As a main result, all Er moments align along the field at $H_c$ and their values reach a minimum. Using a four-sublattice self-consistent calculation, we show that the evolution of the magnetic structure and the value of the critical field are rather well reproduced using the same anisotropic exchange tensor as that accounting for the local paramagnetic susceptibility. In contrast, an isotropic exchange tensor does not yield the correct moment variations through the critical point. The model also accounts semiquantitatively for other experimental data obtained in previous works, such as the field dependence of the heat capacity, energy of the dispersionless inelastic modes and transition temperature.

Figure 1

Integrated intensities versus the field $H$ applied along [110] of some typical Bragg peaks measured at 0.3 K in ${\text{Er}}_2{\text{Ti}}_2{\text{O}}_7$.

Figure 2

(Color online) Zero-field configuration in the ${\psi }_2$ state in the $\left(\overline{1}10\right)$ (left) and (110) (right) plane. The [110] incipient field direction is indicated by a green arrow. The local trigonal axis (hard magnetic axis) for the ${\alpha }_1$ (respectively, ${\alpha }_2$) moments is [111] (respectively, $\left[\overline{1}\overline{1}1\right]$); for the ${\beta }_3$ (respectively, ${\beta }_4$) moments, it is $\left[\overline{1}1\overline{1}\right]$ (respectively, $\left[1\overline{1}\overline{1}\right]$). This configuration corresponds to that of the single domain occurring above 0.1 T. The upper (respectively, lower) hemisphere referred to in the text is that containing the positive (respectively, negative) part of the [001] axis.

Figure 3

(Color online) Evolution of the spin structure inside a tetrahedron for selected fields. The flat disks visualize the easy (111) planes. The $\alpha$ moments at the bottom of a tetrahedron are shown by red arrows and the $\beta$ moments at the top by blue arrows with lengths proportional to the moment values.

Figure 4

(Color online) Variation with the field, applied along [110], of the moment magnitude (a) and orientation with respect to the local trigonal axis (b) and to the field direction (c and d), for the 4 Er sites in ${\text{Er}}_2{\text{Ti}}_2{\text{O}}_7$ at 0.3 K. Our convention for the angle with the field direction is that it is negative if the moment lies in the lower hemisphere. Solid lines in (a)–(c) are self-consistent calculations with ${\lambda }_{\perp }=-0.51\text{T}/{\mu }_B$ and ${\lambda }_{\parallel }=-0.06\text{T}/{\mu }_B$; solid lines in (d) are calculations with ${\lambda }_X=-0.54\text{T}/{\mu }_B$, ${\lambda }_Y=-0.48\text{T}/{\mu }_B$, and ${\lambda }_{\parallel }=-0.06\text{T}/{\mu }_B$; dashed lines are interpolated ones.

Figure 5

(Color online) Low-temperature region of the thermal variation in the longitudinal and transverse susceptibilities in ${\text{Er}}_2{\text{Ti}}_2{\text{O}}_7$. The data are taken from Ref. 8 and the lines are calculated using the self-consistent model described in the text.

Figure 6

(Color online) Evolution of the Er magnetic moments with the magnetic field at 0.3K, as calculated from the self-consistent calculation with ${\lambda }_X=-0.54\text{T}/{\mu }_B$, ${\lambda }_Y=-0.48\text{T}/{\mu }_B$, and ${\lambda }_{\parallel }=-0.06\text{T}/{\mu }_B$. Arrows are proportional to the moment values. Each field step corresponds to 0.48 T.

Figure 7

(Color online) Calculated lowest excitation single ion energies for the four sites in ${\text{Er}}_2{\text{Ti}}_2{\text{O}}_7$, in the AF ordered phase at 0.05 K as a function of the applied magnetic field. The square dots are the (estimated) experimental values from Ref. 14, for fields where a clearly nondispersive level is observed in the inelastic neutron spectra. Insert: calculated thermal variation of the heat capacity for $H=1.5$, 3, and 7 T, obtained from the single-ion energies.

Figure 8

(Color online) Calculated thermal variation in the ${\alpha }_1$ Er moment in the presence of a magnetic field applied along [110], for different field values below 1.3 T, upper limit of validity of our model in the AF phase. Insert: $\left(H,T_{\text{N}}\right)$ phase diagram obtained from the break-points of the ${\alpha }_1$-site $m\left(T\right)$ curves. The dashed line represents the law $H=H_c\left[1-{\left(\frac{T_{\text{N}}}{T_{\text{N}}^0}\right)}^2\right]$ with $H_c=1.85\text{T}$ and $T_{\text{N}}^0=1.48\text{K}$.