Structural domain and finite-size effects of the antiferromagnetic $\text{S}=1/2$ honeycomb lattice in ${\text{InCu}}_{2/3}{\text{V}}_{1/3}{\text{O}}_3$

The compound ${\text{InCu}}_{2/3}{\text{V}}_{1/3}{\text{O}}_3$ has been studied by spectroscopy in the IR, near-infrared, and visible regions. X-ray and neutron diffraction indicate two-dimensional (2D) structural order of magnetic ${\text{Cu}}^{2+}$ $\left(S=1/2\right)$ ions on a honeycomb lattice with nonmagnetic ${\text{V}}^{5+}$ ions complementing the hexagonal layer. Superstructure reflections observed by neutron diffraction reveal the presence of this 2D structural Cu/V order in domains of approximately $300\text{Å}$ with a disordered stacking along the $c$ axis. Specific-heat, thermal-expansion, and neutron-diffraction experiments gave no signs of a three-dimensional long-range magnetic order. Strong 2D magnetic correlations seem to persist down to low temperatures, providing evidence of an incipient Néel-type ground state on this hitherto rarely studied lattice geometry in ${\text{InCu}}_{2/3}{\text{V}}_{1/3}{\text{O}}_3$. Magnetic correlations in these structural domains are compared with continuous time quantum-Monte Carlo calculations of the specific heat and in particular of the spin-spin-correlation length.

Figure 1

(Color online) Perspective view of the crystal structure of ${\text{InCu}}_{2/3}{\text{V}}_{1/3}{\text{O}}_3$ with $\left[{\text{InO}}_6\right]$ and $\left[{\text{MO}}_5\right]$ polyhedra (left). On the right (top), a $\left[{\text{MO}}_5\right]$ entity is shown with the thermal ellipsoids given with 75% probability [single-crystal data at 293 K (Ref. 18)]. Below the in-plane Cu/V order representing the honeycomb structure (dashed lines) without the oxygen atoms is given with the respective unit cell for $P{6}_3/mmc$ (reference axis, line) and $p31m$ (bold line). Note that there are two distinct crystallographic sites for copper and vanadium. The lattice parameters are $a_{p31m}=\sqrt{3}{a}_{P{6}_3/mmc}\approx 5.8\text{Å}$.

Figure 2

(Color online) Typical Rietveld refinement of powder x-ray diffraction data at 140 K (above) and neutron-diffraction data measured on the 3T.2 beamline (LLB) at 10 K. Refinements were carried out in the average structure model in space-group $P{6}_3/mmc$ (see Table ).

Figure 3

Comparison of calculated normal modes for $\left[{\text{MnO}}_5\right]$ (triangles, taken from Ref. 24) and observed normal modes for ${\text{InCu}}_{2/3}{\text{V}}_{1/3}{\text{O}}_3$ (circles and squares). The assignment given in brackets is tentative. The measured infrared spectrum is shown in the inset.

Figure 4

(Color online) Electronic spectrum of ${\text{InCu}}_{2/3}{\text{V}}_{1/3}{\text{O}}_3$ showing the two principal transitions. A Gaussian fit is given for the two members of the ${{}^{2}E}^{\prime }$ (green triangles) and the ${{}^{2}E}^{″}$ transition (blue line) as well as the sum displayed in red.

Figure 5

(Color online) Neutron powder-diffraction pattern of ${\text{InCu}}_{2/3}{\text{V}}_{1/3}{\text{O}}_3$ at 1.5 K measured on instrument G4.1 (LLB). Superstructure reflections can be seen at $2\theta \approx 28°$, $50°$ (overlapping with a 3D Bragg peak), $58°$, and $79.5°$. These are marked with 2D Miller indices (10), (11), (20), and (21) according to the model shown in Fig. 1 and Table . The peak marked by an asterisk originates from the sample holder.

Figure 6

(Color online) Neutron powder-diffraction patterns of ${\text{InCu}}_{2/3}{\text{V}}_{1/3}{\text{O}}_3$ measured on instrument G4.1 (LLB). For the (002) and the (10) reflection data are shown for 1.5, 50, and 160 K, indicating the temperature independent nature of the 2D superstructure reflections.

Figure 7

(Color online) Warren line shape fits to the data using Eq. (3) for the 2D superstructure reflections (10) and (20) at 1.5 K with $L_0=250\text{Å}$ (blue dotted line), $L_0=300\text{Å}$ (red line), and $L_0=500\text{Å}$ (green dashed line). The instrumental resolution of the G4.1 instrument is indicated by the gray horizontal lines.

Figure 8

Normalized temperature dependence of the lattice parameters and the cell volume of ${\text{InCu}}_{2/3}{\text{V}}_{1/3}{\text{O}}_3$ obtained by temperature-dependent x-ray powder diffraction.

Figure 9

(Color online) The temperature dependence of the thermal-expansion coefficients for ${\text{InCu}}_{2/3}{\text{V}}_{1/3}{\text{O}}_3$ determined from the x-ray diffraction data: ${\alpha }_a$ (blue), ${\alpha }_c$ (green, dashed), and ${\alpha }_{\text{V}}$ (violet, dotted) in comparison with the thermal-expansion measurement $1/L\cdot \partial L/\partial T$ (red) are shown (left axis). The experimental specific-heat data, $c_p\left(T\right)$ (black circles, home-built calorimeter data), are included as well (right axis).

Figure 10

(Color online) The temperature dependence of the measured specific heat divided by $T^2$ (black circles, PPMS data) of ${\text{InCu}}_{2/3}{\text{V}}_{1/3}{\text{O}}_3$ is shown in (a). The red line indicates the calculated phononic contribution applied to extract the magnetic specific heat, $c_m/T^2$ (green circles). Inset (b): the low-temperature regime of the specific heat divided by $T^2$ (circles) and the applied linear fit (red line) is depicted. In (c) the calculated magnetic part of the specific heat divided by $T$ $\left(c_m/T\right)$ obtained from QMC methods (blue triangles) is compared with the phonon corrected experimental data (circles). (Note that $c_m\propto A{T}^2$ cannot be discerned from this plot, see also Ref. 31.) The magnetic part of the entropy is plotted in inset (d) for the phonon corrected experimental data (line) and the results of QMC calculations (blue triangles).

Figure 11

(Color online) Main panel: Susceptibility as measured (${\chi }_{\text{meas}}\left(T\right)$, circles), correction for paramagnetic impurities at defect sites (${\chi }_{\text{meas}}\left(T\right)$-${\chi }_{\text{CW}}\left(T\right)={\chi }_{\text{corr}}\left(T\right)$, red line), and the calculated susceptibility from QMC methods for $J_{\text{AF}}=120\text{K}$ (triangles, blue). A logarithmic plot of the spin-spin-correlation length in units of shortest Cu-Cu contacts from QMC calculations for two different systems is shown in the inset. Note that $\xi \left(T\right)$ diverges above $J/T=1$, e.g., below 120 K and approaches a finite value below $\approx 50\text{K}$, according to QMC calculations for a system of $32\times 64$ spins.