High-field magnetic structure of the triangular antiferromagnet $\mathrm{RbFe}({\mathrm{MoO}}_4{)}_2$

The magnetic $H-T$ phase diagram of a quasi-two-dimensional antiferromagnet $\mathrm{RbFe}{\left({\mathrm{MoO}}_4\right)}_2$ $(S=5/2)$ with an equilateral triangular lattice structure is studied with ${}^{87}\mathrm{Rb}$ NMR and neutron-diffraction techniques. This combination of experimental techniques allows us to determine the ordered components of the magnetic moments on the ${\mathrm{Fe}}^{3+}$ ions within various high-field phases—the Y, UUD, V, and fan structures, stabilized in the compound by the in-plane magnetic field. It is also established that the transition from the V to the fan phase is of first order, whereas the transition from the fan phase to the polarized paramagnetic phase is continuous. An analysis of the NMR spectra shows that the high-field fan phase of $\mathrm{RbFe}{\left({\mathrm{MoO}}_4\right)}_2$ can be successfully described by a periodic commensurate oscillation of the magnetic moments around the field direction in each ${\mathrm{Fe}}^{3+}$ layer combined with an incommensurate modulation of the magnetic structure perpendicular to the layers.

Figure 1

(a) Illustration of the spin arrangements for two degenerate magnetic structures selected (in the framework of a classical model) from the manifold of magnetic phases. Here the total magnetization is arbitrarily set to $0.8M_{\mathrm{sat}}$. The structure represented by the solid arrows corresponds to the V phase. (b) Magnetic structures expected for the XY TLAF model which includes spin fluctuations.

Figure 2

Crystal structure of $\mathrm{RbFe}{\left({\mathrm{MoO}}_4\right)}_2$. The smaller spheres mark the positions of the ${\mathrm{Fe}}^{3+}$ magnetic ions, the larger spheres are at the positions of the ${\mathrm{Rb}}^{+}$ ions, while the (${\mathrm{MoO}}_4{)}^{2-}$ complexes are not shown. The crystal symmetry dictates the following main exchange bonds within one cell: in-plane bonds $J$, interplane bonds $J^{\prime }, J_a, J_b$.

Figure 3

Magnetic structures and their propagation vectors [5] expected in $\mathrm{RbFe}{\left({\mathrm{MoO}}_4\right)}_2$ for the field $H$ applied within the basal plane. The arrows show the spin directions of the three neighboring ${\mathrm{Fe}}^{3+}$ ions within a triangular plane. The spins from three adjacent planes are shown with different colors and line styles. The sequence of structures shown corresponds to an increasing magnetic field.

Figure 4

Neutron diffraction intensity map of the $\left(hhl\right)$ plane measured in $\mathrm{RbFe}{\left({\mathrm{MoO}}_4\right)}_2$ single-crystal sample at $T=1.4$ K in a field of 3 T applied perpendicular to the $c$ axis. Both nuclear reflections $\left(\overline{1}\overline{1}\overline{1}\right), \left(\overline{1}\overline{1}0\right), \left(\overline{1}\overline{1}1\right)$ and magnetic reflections $\left(\overline{\frac{1}{3}}\overline{\frac{1}{3}}{q}_z\right), \left(\overline{\frac{2}{3}}\overline{\frac{2}{3}}{q}_z\right)$ with $q_z\approx \pm \frac{1}{2}$ are captured within the covered reciprocal space.

Figure 5

Magnetic phase diagram of $\mathrm{RbFe}{\left({\mathrm{MoO}}_4\right)}_2$, with the magnetic phases labeled in accordance with the notations used in Fig. 3. The open red circles mark the transitions to the PM phase obtained from the neutron-diffraction experiments on a sample from batch I. The solid circles show the transitions to the PM phase obtained from the NMR experiments on samples from batches I and II; the symbols are colored red and black, respectively. A black star shows the $T_{\mathrm{N}}$ measured in samples of batch II from the temperature dependence of the magnetization at ${\mu }_{0}H=0.1$ T. The solid black lines show the phase boundaries obtained with different experimental techniques from Ref. [6] on samples from batch I. Dash-dotted lines show the temperature and field scans made in the NMR study. The wave vectors of the magnetic phases are given from Ref. [5]. The vertical and horizontal pink bars mark the transition regions where the magnetic structure is changing from one phase to another.

Figure 6

${}^{87}\mathrm{Rb}$ NMR spectra measured at $T=1.45$ K in the range from ${\mu }_0{H}_{\mathrm{sat}}/3$ to 17 T with the field applied perpendicular to the $c$ axis of the crystal on the samples from batch II. Lines measured at different frequencies are offset for clarity. Red lines are the spectra measured in a field-polarized PM phase. Blue lines are the spectra obtained in fields below $H_{\mathrm{sat}}$ where the fan phase is expected. The spectra in low fields shown in green are obtained in the commensurate phases, UUD or V. Magenta lines show the spectra within the field range of the transition between the commensurate and incommensurate phases. Dashed lines are guides to the eye.

Figure 7

Temperature evolution of the ${}^{87}\mathrm{Rb}$ NMR spectra measured at (a) $\nu =139.3$ MHz ($\nu /\gamma \approx 10$ T) and (b) $\nu =195$ MHz ($\nu /\gamma \approx 14$ T). Dashed lines are guides to the eye. Colors are used to indicate the spectra obtained in different magnetic phases: PM (red), fan (blue), V (green), and intermediate (magenta). The dotted lines are the NMR spectra computed for the fan structure model with the magnetic moments of ${\mathrm{Fe}}^{3+}$ ions shown in (b) near the spectra. The sample used is from batch II.

Figure 8

Temperature dependences of (a) the spin-lattice, $1/T_1$, and (b) the spin-spin, $1/T_2$, relaxation frequencies. The measurements were taken on the central line of the spectrum at $\nu =139.3$ MHz, ${\mu }_{0}H=\nu /\gamma \approx 10$ T and $\nu =195$ MHz, ${\mu }_{0}H=\nu /\gamma \approx 14$ T. The points corresponding to the measurements on the sample from batches I and II are shown with the blue circles and red squares, respectively. The insets show the same dependencies on a logarithmic scale.

Figure 9

Temperature dependence of $1/T_1$ and FWHM of the NMR line (circles) at (a) 14 and (b) 16 T. The square symbols show the temperature dependence of the integrated intensity of (1/3,1/3, $\approx 0.5$) neutron-diffraction peak measured at the same fields. The samples used for the measurements were from batch I.

Figure 10

(a) Magnetic neutron-diffraction intensity maps of the $\left(0kl\right)$ plane observed in $\mathrm{RbFe}{\left({\mathrm{MoO}}_4\right)}_2$ single-crystal sample in different fields applied perpendicular to the $c$ axis. The magnetic component was isolated by subtracting a high-field background signal corresponding to the polarized PM phase. (b) Field evolution between 13 and 17 T of the magnetic intensity for the reciprocal space cuts along the $c^{*}$ axis taken at the $h=k=\overline{1}/3$ positions. (c) Field dependence of the total magnetic intensity averaged over the $(\overline{1}/3\overline{1}/31/2)$ and $(\overline{1}/3\overline{1}/3\overline{1}/2)$ peaks. For all three panels the data shown are measured at $T=1.4$ K.

Figure 11

Rb NMR spectra measured at $T=1.5$ K in the UUD, V, and PM phases in the sample from batch II (black lines). Blue dashed lines show the computed central lines of spectra for the UUD, V, and PM structures with the values of the ordered moments on the ${\mathrm{Fe}}^{3+}$ ions, $\mu$, and the FWHM of the individual NMR lines, $\delta$, stated for each spectrum. These parameters provide a good match to the shape of the experimental spectra. The red lines show the computed spectra shifted by approximately 0.01 T so that they best coincide with the positions of the experimental lines.

Figure 12

${}^{87}\mathrm{Rb}$ NMR spectra measured in the fan phase for a sample from batch II in fields of ${\mu }_{0}H=12.5$, 13.5, and 14.5 T. The experimental data are shown with black lines while the red lines represent the fits. The ordered moments of the ${\mathrm{Fe}}^{3+}$ ions, $\mu$, and the width of the individual NMR lines, $\delta$, used for the computation of the spectra are given near the NMR lines. The spectra are consecutively offset for clarity.

Figure 13

Field dependence of the distances between the maxima of the NMR spectra, ${\mu }_0\delta H\left(h\right)$, measured at $T=1.5$ K on samples of batch I (red and magenta symbols) and batch II (black symbols). $h=H/H_{\mathrm{sat}}$. ${\mu }_0{H}_{\mathrm{sat}}=17$, and 16.25 T for samples from batch I and II, respectively. Red symbols show the results of previous work [25]. Solid lines show the ${\mu }_0\delta H$ obtained from the NMR spectra computed for the commensurate magnetic structures Y, UUD, and V with the magnetic moment of the ${\mathrm{Fe}}^{3+}$ ions $\mu =5{\mu }_{\mathrm{B}}$. Dash-dotted lines indicate the phases boundaries. The field regions of mixed magnetic phases are shown in gray.

Figure 14

Field dependence of the ordered component of the magnetic moment $\mu \left(h\right)$ at $T=1.5$ K, $h=H/H_{\mathrm{sat}}$. The values of $\mu$ within the fan phase were obtained from the fits to the NMR spectra as described in the main text. Different colors show the results obtained on the samples from different batches with the color scheme identical to Fig. 13. Open circles show the values of the ordered components of magnetic moments from Refs. [5, 7], with the exact temperature of the neutron-diffraction experiments given near the symbols.

Figure 15

(a) Schematic of the magnetic fan structure given by Eq. (2). (b) Energy dependence vs normalized magnetization, $M/M_{\mathrm{sat}}$, for “I” – the difference between the exchange energy of the PM phase, $J{S}^2$, and the exchange energy of the ground state, $E_0$; “II” – the variation of the exchange energy from one plane to another for the fan structure, $\delta E$; “III” – the difference between the energy of the fan structure, $E$, and $E_0$. The energies were computed in the limit of large spins with the exchange energy given by the first term of Eq. (1). A gray area marks the region of magnetic fields where the transition from the V to fan structure is experimentally observed