Neutron diffraction study on the two-dimensional Ising system $\text{KEr}{\left({\text{MoO}}_4\right)}_2$

The magnetic properties of the two-dimensional Ising antiferromagnet $\text{KEr}{\left({\text{MoO}}_4\right)}_2$ have been investigated below and above transition temperature $T_N\sim 0.95\text{K}$ in zero field and in fields up to 6.5 T by means of elastic neutron-diffraction, heat-capacity, and magnetization measurements. The low-temperature signal recorded at 0.34 K by neutron diffraction is explained within a noncollinear magnetic structure model. However, additional contribution is also present when applying the external magnetic field along the $c$ axis even at temperatures well above the magnetic transition temperature $T_N$. Various explanations are discussed.

Figure 1

(Color online) The crystal structure of $\text{KEr}{\left({\text{MoO}}_4\right)}_2$: (a) a picture (left) shows the $1\times 1\times 2\text{unit}$ cell seen from [100] direction; the largest spheres (light gray) stand for potassium K atoms, the 2nd largest (blue) for Er atoms located in the octahedral oxygen environment, Mo atoms (green) are surrounded in tetrahedral oxygen arrangement, and the smallest (red) stand for O oxygen atoms. (b) A picture (top right) displays an octahedra: eight nearest neighbors oxygen atoms surrounded around the central Er atom (blue) with typical distances $d\left(\text{Er-O}1\right)=2.319\text{Å}$, $d\left(\text{Er-O}2\right)=2.301\text{Å}$, $d\left(\text{Er-O}3\right)=2.439\text{Å}$, and angles $\text{O}1\text{-Er-O}1=84.73°$, $\text{O}2\text{-Er-O}2=73.3°$, $\text{O}3\text{-Er-O}3=116.3°$, and $\text{O}1\text{-Er-O}2=109.0°$. (c) A picture (bottom right) shows nearest neighbor of molybdenum atoms, typical distances, and angles between oxygen and molybdenum atoms are listed: $d\left(\text{Mo-O}4\right)=2.0174\text{Å}$, $d\left(\text{Mo-O}3\right)=2.2112\text{Å}$, $d\left(\text{Mo-O}2\right)=1.8243\text{Å}$, $\text{O}3\text{-Mo-O}4=86.93°$, $\text{O}2\text{-Mo-O}4=118.57°$, $\text{O}2\text{-Mo-O}2=51.49°$, and $\text{Mo-O}2\text{-Mo}=127.6°$. Atoms not drawn to scale.

Figure 2

(Color online) The field dependence of the magnetic moment for the $\text{KEr}{\left({\text{MoO}}_4\right)}_2$ single crystal with magnetic field $B\parallel a$ axis at various temperatures from 2 to 30 K.

Figure 3

(Color online) The field dependence of the magnetic moment for $\text{KEr}{\left({\text{MoO}}_4\right)}_2$ single crystal with magnetic field $B\parallel b$ axis at different temperatures of 2, 3, 4.2, 6, 8, and 25 K.

Figure 4

(Color online) The field dependence of the longitudinal magnetic moment for $\text{KEr}{\left({\text{MoO}}_4\right)}_2$ single crystal with magnetic field $B\parallel c$ direction at different temperatures of 2, 4.2, 6, 8, and 10 K.

Figure 5

(Color online) The molar susceptibility ${\chi }_{mol}$ and the inverse molar susceptibility ${\chi }_{mol}^{-1}$ of $\text{KEr}{\left({\text{MoO}}_4\right)}_2$ single-crystal measured in external magnetic field of 1 T for $a$, $b$, and $c$ axes, respectively. Solid lines represent best fits to a Curie-Weiss law.

Figure 6

Specific-heat capacity ${\text{C}}_p/T$ vs $T$ of $\text{KEr}{\left({\text{MoO}}_4\right)}_2$ measured by adiabatic method in zero field.

Figure 7

(Color online) The noncollinear four lattice magnetic structure 6AF-12 suggested for $\text{KEr}{\left({\text{MoO}}_4\right)}_2$ by Anders et al. to be the ground-state magnetic structure. The figure displays $1.5\times 1.5\times 1\text{unit}$ cell seen from [010] direction and showing a doubling along the $a$ and the $b$ axes. The magnetic model expects magnetic moments (in picture drawn as arrows) on the erbium atoms only. For clarity, we mark the structurally equivalent atom positions with spheres of the same color (e.g., for Er1-cyan, Er2-black, Er3-violet, Er4-red, and the shortest distance between erbium atoms is displayed by the two-color line).

Figure 8

(Color online) Neutron-diffraction experiment data taken at the E10 instrument within the $a\text{−}c$ crystal orientation performed in the temperature range between 0.35 and 1.2 K in zero magnetic field. The temperature dependence of the integrated intensity at the forbidden nuclear reflection (001) (○-open points) with the best fit to formula $I=I_0{\left(1-T/T_N\right)}^{2\beta }$ shown by the full line. The inset displays the omega scan at (001) reciprocal node at the temperature of 0.35 (◻) and 1.2 K (◼), magenta dotted line is the best fit of data to a Gaussian profile.

Figure 9

(Color online) Magnetic model derived from our data. Erbium magnetic moments having $\left(8.64,0.0,0.53\right){\mu }_B$ Cartesian components constitute the magnetic unit cell of equal size with the chemical one. The existence of a large $x$ and smaller $z$ components assure the existence of forbidden reflections of the $\left(0K0\right)$, $K=\text{odd}$ and the $\left(00L\right)$, $L=\text{odd}$ types below the transition temperature. The structure is seen from [010] direction.

Figure 10

(Color online) The field dependence of integrated intensity at the (030) forbidden nuclear reflection in field $B\parallel c$ axis at various temperatures between 0.46 and 6.0 K. Data were taken with field increasing up to 6.5 T and decreasing down to 0 T. The lines are guides to the eye.

Figure 11

The field dependence of integrated intensity at (020) (◁—right scale), (110), and (200) nuclear reflections at temperature of 1.8 K in magnetic field $B\parallel c$ axis. The lines are guides to the eye.

Figure 12

(Color online) The field dependencies of (a) integrated intensity, (b) position, and (c) FWHM for (020) nuclear reflection at low (○) and high temperatures (▲, △) for a magnetic field $B\parallel a$-axis ramped up (▲) and down (△). The lines are guides to the eye.

Figure 13

(Color online) The field dependencies of (a) integrated intensity, (b) position, and (c) FWHM for (030) forbidden nuclear reflection at 0.34 (○) and 2.6 K [for field ramped up (▲) and down (△)] for a magnetic field $B\parallel a$ axis. The lines are guides to the eye.

Figure 14

(Color online) The field dependence $\left(B\parallel c\right)$ of the integrated intensity at the (030) forbidden nuclear reflection measured by the synchrotron scattering technique at the MAGS beam line at temperatures of 2, 3, 4, 6, 8, 10, and 12 K.