Magnetic structure of the triangular antiferromagnet $\mathrm{RbFe}{\left({\mathrm{MoO}}_4\right)}_2$ weakly doped with nonmagnetic ${\mathrm{K}}^{+}$ ions studied by NMR

The magnetic structure of the quasi-two-dimensional antiferromagnet ${\mathrm{Rb}}_{1-x}{\mathrm{K}}_x\mathrm{Fe}{\left({\mathrm{MoO}}_4\right)}_2$ with weakly distorted triangular lattice was studied with ${}^{87}\mathrm{Rb}$ NMR. The samples with $0\le x\le 0.15$ were studied in the field region near $H_{\mathrm{sat}}/3$, where quantum and thermal fluctuations play a decisive role in the formation of the magnetic structure of undoped frustrated triangular $\mathrm{RbFe}{\left({\mathrm{MoO}}_4\right)}_2$. Line-shape analysis reveals that in samples with $x=0,0.025$, and 0.075 the magnetic structure is stabilized by the thermal fluctuations, whereas in the sample with $x=0.15$ the magnetic structure with short-range order within each individual triangular plane and interplanar disorder is realized. The short-range static correlations in this two-dimensional ordered state are in agreement with a theoretical prediction for the $XY$ triangular-lattice antiferromagnet with nonmagnetic impurities by Maryasin and Zhitomirsky [Phys. Rev. Lett. 111, 247201 (2013)].

Figure 1

(a) Magnetic structures expected for the $XY$ TLAF with an in-plane external field as the field increases. (b) Magnetic structures with the total magnetization $M=M_{\mathrm{sat}}/3$. (c) Crystal structure of $\mathrm{RbFe}{\left({\mathrm{MoO}}_4\right)}_2$. The small circles are the positions of ${\mathrm{Fe}}^{3+}$ magnetic ions, the bigger circles (both black and gray, see text) are the positions of ${\mathrm{Rb}}^{+}$ ions, the (${\mathrm{MoO}}_4{)}^{2-}$ complexes are not shown. One magnetic cell of the $uud$ phase is shown. $\mathbf{a},\mathbf{b}$, and $\mathbf{c}$ are the basis vectors describing the crystallographic structure; $\mathbf{a},\mathbf{b}$ are directed along the sides of the triangle while $\mathbf{c}$ is aligned along ${\mathbf{C}}^3$. The translational vectors of the $uud$ structure are $\mathbf{a}+2\mathbf{b}$ and $2\mathbf{a}+\mathbf{b}$. The short arrows depict the expected orientations of magnetic moments in the $uud$ phase with a period of $3c$ for the direction of the magnetic field $\mathbf{H}$ as shown.

Figure 2

${}^{87}\mathrm{Rb}$ NMR spectra of ${\mathrm{Rb}}_{1-x}{\mathrm{K}}_x\mathrm{Fe}{\left({\mathrm{MoO}}_4\right)}_2$, $\nu =75.5$ MHz, (a)–(c) $\mathbf{H}\perp {\mathbf{C}}^3$, (d) $\mathbf{H}\parallel {\mathbf{C}}^3$.

Figure 3

Temperature evolution of ${}^{87}\mathrm{Rb}$ NMR spectra ($m_I=+1/2\leftrightarrow -1/2$ transition) in ${\mathrm{Rb}}_{0.85}{\mathrm{K}}_{0.15}\mathrm{Fe}{\left({\mathrm{MoO}}_4\right)}_2$, $\nu =81$ MHz, $\mathbf{H}\perp {\mathbf{C}}^3$. The spectrum shown with red line corresponds to the maximum of ${}^{87}\mathrm{Rb}$ spin lattice relaxation rate $T_1^{-1}$ shown in Fig. 5.

Figure 4

Field evolution of ${}^{87}\mathrm{Rb}$ NMR spectra in ${\mathrm{Rb}}_{0.85}{\mathrm{K}}_{0.15}\mathrm{Fe}{\left({\mathrm{MoO}}_4\right)}_2$ at temperature $\approx 1.47$ K, $\mathbf{H}\perp {\mathbf{C}}^3$. The spectrum shown with red line corresponds to the maximum of ${}^{87}\mathrm{Rb}$ spin lattice relaxation rate $T_1^{-1}$ shown in Fig. 5. The green colored spectra correspond to paramagnetic state ($T=10$ K).

Figure 5

(a) Temperature dependence of ${}^{87}\mathrm{Rb}$ spin lattice relaxation rate $T_1^{-1}$ in ${\mathrm{Rb}}_{0.85}{\mathrm{K}}_{0.15}\mathrm{Fe}{\left({\mathrm{MoO}}_4\right)}_2$ at field 5.82 T, $\mathbf{H}\perp {\mathbf{C}}^3$. (b) Field dependence of ${}^{87}\mathrm{Rb}$ spin lattice relaxation rate $T_1^{-1}$ in ${\mathrm{Rb}}_{0.85}{\mathrm{K}}_{0.15}\mathrm{Fe}{\left({\mathrm{MoO}}_4\right)}_2$ at temperature 1.47 K, $\mathbf{H}\perp {\mathbf{C}}^3$.

Figure 6

Simulated NMR spectra corresponding to the $uud$ structure ($\alpha =0^{\circ }$, left panels) and the $fan$ structure ($\alpha ={90}^{\circ }$, right panels), $\mathbf{H}\perp {\mathbf{C}}^3$. For comparison experimental ${}^{87}\mathrm{Rb}$ NMR spectra (black lines) at frequency 81 MHz, $\mathbf{H}\perp {\mathbf{C}}^3$, and two temperatures are shown for (e),(g) $x=0$ and (h),(f) $x=0.15$. Simulated spectra data: (a) 3D $uud$ with periodicity $2c$ or $3c$ in the $C^3$ direction, $\mu =5{\mu }_B$, individual linewidth $\delta =20$ mT; (b) 3D $fan$ with periodicity $2c$ or $3c$ in the $C^3$ direction, $\mu =5{\mu }_B$, $\delta =20$ mT; (c) 2D $uud$, $\mu =5{\mu }_B$, $\delta =20$ mT; (d) 2D $fan$, $\mu =5{\mu }_B$, $\alpha ={90}^{\circ }$ (blue line), $\alpha ={90}^{\circ }\pm {24}^{\circ }$ (magenta line), $\delta =20$ mT; (e) 3D $uud$, $\mu =5{\mu }_B$, $\delta =6$ mT; (f) 2D $fan$, $\mu =3.7{\mu }_B$, $\alpha ={90}^{\circ }\pm {24}^{\circ }$, $\delta =20$ mT; (g) 3D $uud$, $\mu =4.2{\mu }_B$, $\delta =7$ mT; (h) 2D $fan$, $\mu =3.3{\mu }_B$, $\alpha ={90}^{\circ }\pm {22}^{\circ }$, $\delta =20$ mT.