Ground state and low-temperature magnetism of the quasi-two-dimensional honeycomb compound ${\mathrm{InCu}}_{2/3}{\mathrm{V}}_{1/3}{\mathrm{O}}_3$

We report a combined ${}^{115}\mathrm{In}$ nuclear quadrupole resonance, ${}^{51}\mathrm{V}$ nuclear magnetic resonance, and muon spin-relaxation spectroscopic study of the low-temperature magnetic properties of ${\mathrm{InCu}}_{2/3}{\mathrm{V}}_{1/3}{\mathrm{O}}_3$, a quasi-two-dimensional (2D) compound comprising in the spin sector a honeycomb lattice of antiferromagnetically coupled spins $S=1/2$ associated with ${\mathrm{Cu}}^{2+}$ ions. Despite substantial experimental and theoretical efforts, the ground state of this material has not been ultimately identified. In particular, two characteristic temperatures of about 40 and 20 K manifesting themselves as anomalies in different magnetic measurements are discussed controversially. A combined analysis of the experimental data complemented with theoretical calculations of exchange constants enabled us to identify, below 39 K, an “intermediate” quasi-2D static spin state. This spin state is characterized by a staggered magnetization with a temperature evolution that agrees with the predictions for the 2D $\mathit{XY}$ model. We observe that this state gradually transforms at 15 K into a fully developed 3D antiferromagnetic Néel state. We ascribe such an extended quasi-2D static regime to an effective magnetic decoupling of the honeycomb planes due to a strong frustration of the interlayer exchange interactions, which inhibits long-range spin-spin correlations across the planes. Interestingly, we find indications of the topological Berezinsky-Kosterlitz-Thouless transition in the quasi-2D static state of the honeycomb spin-1/2 planes of ${\mathrm{InCu}}_{2/3}{\mathrm{V}}_{1/3}{\mathrm{O}}_3$.

Figure 1

High-temperature ${}^{115}\mathrm{In}$ NQR spectrum of ${\mathrm{InCu}}_{2/3}{\mathrm{V}}_{1/3}{\mathrm{O}}_3$ (data points). The solid line is a fit to the data according to Hamiltonian (1). The positions of the lines corresponding to two nonequivalent In sites in the crystal structure (In1 and In2) are indicated by arrows (see the text for details). For technical reasons, all pairs of transitions corresponding to In1 and In2 were measured separately and the intensities of the respective parts of the spectrum were each normalized to unity.

Figure 2

Two crystallographic In1 and In2 positions in ${\mathrm{InCu}}_{2/3}{\mathrm{V}}_{1/3}{\mathrm{O}}_3$ with (a) high and (b) low local environment symmetry, respectively.

Figure 3

${}^{115}\mathrm{In}$ NQR spectra corresponding to the ($\pm \frac{3}{2}\leftrightarrow \pm \frac{5}{2}$) transition (cf. Fig. 1) at temperatures of 100 K (triangles) and 37 K (circles).

Figure 4

Temperature dependence of the ${}^{115}\mathrm{In}$ NQR spectra corresponding to the ($\pm \frac{3}{2}\leftrightarrow \pm \frac{5}{2}$) transition. Arrows labeled $\mathrm{\Delta }{H}_1$ and $\mathrm{\Delta }{H}_2$ indicate pairs of lines split by a strong internal magnetic field (see the text for details).

Figure 5

Decay of the echo intensity as a function of the time delay $\tau$ between the pulses at temperatures of 70 K (triangles), 40 K (circles), 20 K (squares), and 15 K (stars). The inset depicts the $T$ dependence of the stretching parameter $b$ (see the text for details).

Figure 6

Temperature dependence of the ${}^{115}\mathrm{In}$ spin-lattice relaxation rate $T_1^{-1}$ measured at the $\pm \frac{3}{2}\leftrightarrow \pm \frac{5}{2}$ transition.

Figure 7

Temperature dependence of the ${}^{51}\mathrm{V}$ spin-lattice relaxation rate measured in a magnetic field of 3 T (symbols). Characteristic temperatures $T^{*}=15$ K and $T^{**}=39$ K are indicated by the arrows (cf. Fig. 6). Inset: $T$ dependence of the stretching parameter $b$ of the decay of the nuclear magnetization (symbols). Solid lines are guides for the eye (see the text for details).

Figure 8

Transverse field $\mu \mathrm{SR}$ results of ${\mathrm{InCu}}_{2/3}{\mathrm{V}}_{1/3}{\mathrm{O}}_3$. Temperature dependence of the magnetic volume fraction determined from the amplitude fraction of muon spins precessing with the Larmor frequency ${\omega }_{\mu }={\gamma }_{\mu }{B}_{\mathrm{ext}}$ corresponding to the external field. The solid line is a guide for the eye. Inset: Typical time-dependent muon spin polarization at 60 K (red dots) and at 35 K (black triangles). The solid lines are results of a least-squares nonlinear fit analysis of the data as described in the text.

Figure 9

Typical zero-field $\mu \mathrm{SR}$ time spectra of ${\mathrm{InCu}}_{2/3}{\mathrm{V}}_{1/3}{\mathrm{O}}_3$ at 42.3 K (blue triangles), 40.2 K (red dots), and 1.6 K (black squares). The solid lines are results of a least-squares nonlinear fit analysis of the data as described in the text.

Figure 10

Temperature dependence of the spontaneous muon spin precession frequencies ${\nu }_1$ (black dots) and ${\nu }_2$ (red triangles) in the magnetically ordered phase of ${\mathrm{InCu}}_{2/3}{\mathrm{V}}_{1/3}{\mathrm{O}}_3$ below 40 K. The global fit using an order parameter function given by Eq. (4) with $T_{\mathrm{N}}=39.4\left(1\right)$ K and $\beta =0.254\left(5\right)$ is described in the text.

Figure 11

Temperature dependence of the longitudinal relaxation rate ${\lambda }_{\mathrm{l}}$ in ${\mathrm{InCu}}_{2/3}{\mathrm{V}}_{1/3}{\mathrm{O}}_3$. In the magnetically ordered phase below 40 K, in general, a weak monotonous decrease of ${\lambda }_{\mathrm{l}}$ is observed superimposed by a small peak at $\approx \left(10\right)$ K. The solid and dashed lines are guides to the eye.

Figure 12

Temperature dependence of the static linewidths ${\lambda }_{\mathrm{t},1}$ (black dots) and ${\lambda }_{\mathrm{t},2}$ (red triangles) in the magnetically ordered phase of ${\mathrm{InCu}}_{2/3}{\mathrm{V}}_{1/3}{\mathrm{O}}_3$ below 40 K.

Figure 13

Left panel: GGA band structure of ${\mathrm{InCu}}_{2/3}{\mathrm{V}}_{1/3}{\mathrm{O}}_3$. The radii of the green circles denote the weight of the half-filled Cu $d_{3z^2-r^2}$ orbital. Red solid lines are eigenvalues of the tight-binding Hamiltonian constructed from the Wannier projections. The notation of the $k$ points is $\mathrm{\Gamma }=[0,0,0], X$ = $[0,\frac{1}{3},0], S$ = $[-\frac{1}{4},\frac{1}{4},0], R$ = $[-\frac{1}{4},\frac{1}{4},\frac{1}{2}], A$ = $[0,\frac{1}{3},\frac{1}{2}], Z$ = $[0,0,\frac{1}{2}]$, and $Y$ = $[-\frac{1}{2},0,0]$ in terms of the reciprocal lattice vectors of the conventional ($C$-centered) unit cell. Right panel: A sketch of the spin model of ${\mathrm{InCu}}_{2/3}{\mathrm{V}}_{1/3}{\mathrm{O}}_3$ with the nearest-neighbor exchange $J_1$ (black cylinders) and the interlayer exchange $J_{\text{il1}}$ (red lines). Both exchanges are antiferromagnetic. Shaded triangles illustrate the magnetic frustration and are a guide to the eye. The spin model picture has been created using vesta [48].

Figure 14

Cu-centered Wannier functions for the $|3z^2-r^2\rangle$ states in ${\mathrm{InCu}}_{2/3}{\mathrm{V}}_{1/3}{\mathrm{O}}_3$: (a) lateral and (b) top views. The Cu–O–V–O–Cu paths facilitating a sizable antiferromagnetic $J_1$ exchange are highlighted in the right plot.

Figure 15

Upper panel: ${}^{115}\mathrm{In}$ NQR line splitting (left scale) due to local internal magnetic field as a function of temperature [open circles correspond to the splitting $\mathrm{\Delta }{H}_1$ and squares correspond to the splitting $\mathrm{\Delta }{H}_1$ (cf. Fig. 4)] and ${\mu }_{SR}$ frequency (right scale) as a function of temperature (triangles). Lower panel: ${\mu }_{SR}$ frequency as a function of reduced temperature $(1-T/T_{\mathrm{N}})$. The solid lines are the fit to the power-law function given by Eq. (4).

Figure 16

$T$ dependence of the width of the left and right peaks in the central part of the ${}^{115}\mathrm{In}$ NQR spectrum (Fig. 3) corresponding to the In sites exposed to a small internal field. (Solid lines are guides for the eye.)

Figure 17

${}^{115}\mathrm{In}$ spin-lattice relaxation rate as a function of the reduced temperature $(T-T_{\mathrm{N}})/T_{\mathrm{N}}$ (circles) and the fit according to Eq. (8) (solid line).

Figure 18

NQR transition frequencies as a function of the EFG asymmetry parameter $\eta$ numerically calculated using Hamiltonian (1).

Figure 19

(a), (b) NQR transition frequencies as a function of local magnetic field for the nuclear spin $I$ = $\frac{9}{2}$ in an asymmetric electric field with (a) $\eta =0.1$ and (b) $\eta =0.05$. The magnetic field $h$ is applied along the $z$ symmetry axis of EFG. (c) Comparison of the calculated frequencies for the $\pm \frac{3}{2}\leftrightarrow \pm \frac{5}{2}$ transition for $h\parallel z$ with the position of the experimentally observed NQR signals at $T=10\mathrm{K}$ indicated by horizontal dashed lines (cf. Fig. 4). (d) Calculated frequencies for $h\perp z$. Red solid triangles in (c) and (d) correspond to $\eta =0.05$ and black circles correspond to $\eta =0.1$.