Strong decoupling between magnetic subsystems in the low-dimensional spin-$\frac{1}{2}$ antiferromagnet ${\mathrm{SeCuO}}_3$

The search for and understanding of low-dimensional magnetic materials is essential for both fundamental and technological purposes. Here we propose a combined experimental and theoretical investigation of such a system, namely the monoclinic phase of ${\mathrm{SeCuO}}_3$. This low-dimensional spin-$\frac{1}{2}$ antiferromagnet appears to be based on two decoupled magnetic subsystems which respond differently to applied magnetic field in the antiferromagnetic phase. From our results we are able to propose a zero-field magnetic structure as well as a more exotic finite magnetic field structure, to be tested by future experiments. This finding is based on torque magnetometry measurements on the one side, and the use of a refined phenomenological model and state-of-the-art density functional theory calculations on the other. The existence of such systems opens a way to very exciting physics with the possibility to control separately two magnetic subsystems in one material.

Figure 1

Two magnetically inequivalent tetramers in ${\mathrm{SeCuO}}_3$ as proposed in Ref. [7]. Cu1 atoms are shown in orange color, Cu2 in blue, and O atoms in red. Se atoms are not shown for simplicity.

Figure 2

Angular dependence of torque $\tau$ measured in three crystal planes in different magnetic fields. (a) The torque measured in the $ac$ plane. (b) Full symbols: The dependence of torque amplitude ${\tau }_0$ on $H^2$ in the $ac$ plane [see (a)]. Solid line is fit to the experimental data for $H>H_{\mathrm{SF}}$. Dashed line represents expected $H^2$ dependence of the torque amplitude for an antiferromagnet with no reorientation (extrapolation of $H\ll H_{\mathrm{SF}}$ data using Eq. (3) and low-field anisotropy data [10]). Empty symbols represent simulation (see Sec. 3b). (c) Angular dependence of torque measured in the plane spanned by $b$ and ${\left[\overline{1}01\right]}^{*}$ axes, and (d) the plane spanned by $b$ and ${\left[101\right]}^{*}$ axes. In (d) the torque is multiplied by $2M_{\mathrm{mol}}/\left(m{H}^2\right)$ resulting in practically the same curve for all applied fields. This is consistent with no reorientation of spins in this plane [see Eq. (2)]. The angles corresponding to the specific crystal directions are pointed by arrows.

Figure 3

The MAE shape in ${\mathrm{SeCuO}}_3$ obtained from the torque measurements in this work. Red lines represent the magnetic axes which are also extrema of the anisotropy energy. Easy axis direction (a global minimum) is along the ${〈\overline{1}01〉}^{*}$ axis. Polar and azimuthal angles $\theta$ and $\varphi$ used in the spherical coordinate system throughout the paper are also shown, as well as the $\psi$ and $\xi$ defining the direction of magnetic field $\mathit{H}$.

Figure 4

A simulation of the magnetization dependence on magnetic field when $H$ is parallel to the easy axis compared to the experimental result at $T=2$ K published in Ref. [7] (empty blue squares). Inset: The dependence of the spin axis direction on applied magnetic field obtained from simulation. Simulation 1 allows all spins to rotate simultaneously. Simulation 2 allows only a fraction of the spins to rotate, as described in the main text. Solid red line (2a) represents the result for $H\parallel$ easy axis, while dashed line (2b) simulates a misorientation of the sample by ${10}^{\circ }$, to mimic probable misorientation in our experiment.

Figure 5

The torque measured in the $ac$ plane in different magnetic fields compared to two different simulations, as described in the main text. (a) A simulation with reorientation of all spins. (b) A simulation allowing the site-specific reorientation. Bottom right panel: The angle-dependent reorientation of the spin axis obtained from the simulation shown in the laboratory coordinate system. Double-headed arrow represents the direction of the spin axis, i.e., the Néel vector. In the case of partial spin reorientation (see text) the plotted arrows correspond to subsystem 2 while for subsystem 1 the spin axis remains in the easy axis direction. The plane of rotation of the magnetic field is shown as a dark square in the accompanying coordinate system. The angle is measured with respect to the ${\left[\overline{1}01\right]}^{*}$ axis, while in other panels a goniometer angle is shown.

Figure 6

The torque measured in the plane spanned by $b$ and ${\left[\overline{1}01\right]}^{*}$ axes compared to the results of simulation, as described in the main text. Dotted lines represent a perfect orientation (2a), and solid lines simulate a slightly misoriented sample (2b). Only site-specific simulation is shown. Bottom right panel: Angle-dependent spin axis reorientation for perfect orientation of the sample shown in the laboratory coordinate system. Double-headed arrow represents the direction of the spin axis, i.e., the Néel vector. In the case of partial spin reorientation (see text) plotted arrows correspond to subsystem 2 while for subsystem 1 the spin axis remains in the easy axis direction. The plane of rotation of the magnetic field is shown as a dark square in the accompanying coordinate system. The angle is measured with respect to the $b$ axis, while in other panels the goniometer angle is shown.

Figure 7

The torque measured in the plane spanned by $b$ and ${\left[101\right]}^{*}$ axes compared to the results of the site-specific simulation (sim.), as described in the main text.

Figure 8

(a) Antiferromagnetic order considered in our DFT calculations. The ${\mathrm{Cu}}^{2+}$ sites are depicted as filled and empty circles, representing up-spin and down-spin, respectively. (b) Orbital moment of one tetramer. Red and blue colors show the maximum and minimum of the orbital moment, respectively. The arrow on the right side shows the vector sum of orbital moments represented by brown and blue vectors on Cu1 and Cu2 sites, respectively.

Figure 9

(a) Experimental MAE. (b) Contributions MAE1 (MAE2) to the total MAE (top) determined substituting Cu2 (Cu1) by Zn atoms and considering a $U_{\mathrm{eff}}=5\mathrm{eV}$ correction for Cu sites.

Figure 10

(a) Experimental MAE. (b) MAE considering $U_{\mathrm{eff}}=0\mathrm{eV}$ for Cu1 and $U_{\mathrm{eff}}=5\mathrm{eV}$ for Cu2. (c) Similar to previous, except a $U=5\mathrm{eV}$ and $J=0.5\mathrm{eV}$ correction on Cu2. (d) A detailed comparison between the experimental MAE (solid lines) and theoretical MAE shown in (c) (dotted lines) in the $ac$ plane.

Figure 11

The magnetic structure in AFM state in zero field (a) and (c) and in field $H\ge H_{\mathrm{SF}}$ applied along the easy axis (b) and (d) obtained in this work. (a) and (b) show the $ac$ plane to which the spins are confined. In (a) and (b) the directions of the easy ${\left[\overline{1}01\right]}^{*}$ and intermediate [101] axes obtained from experiment are also shown.

Figure 12

Projected density of states (PDOS) on (a) Cu2 and (b) Cu1 sites considering a $U_{\mathrm{eff}}=5\mathrm{eV}$, and on (c) Cu1 only at the GGA level. Projection axes for each Cu site are represented on the scheme in (a). Insets zoom in on the spin minority states over the range of 2 eV below the Fermi energy. Pink stars correspond to first excitations allowed with the excited states.