Two types of alternating spin-$\frac{1}{2}$ chains and their field-induced transitions in $\varepsilon \text{−}{\mathrm{LiVOPO}}_4$

Thermodynamic properties, ${}^{31}\mathrm{P}$ nuclear magnetic resonance (NMR) measurements, and density-functional band-structure calculations for $\varepsilon \text{−}{\mathrm{LiVOPO}}_4$ are reported. This quantum magnet features a singlet ground state and comprises two types of alternating spin-$\frac{1}{2}$ chains that manifest themselves by the double maxima in the susceptibility and magnetic specific heat, and by the two-step magnetization process with an intermediate $\frac{1}{2}$-plateau. From thermodynamic data and band-structure calculations, we estimate the leading couplings of $J_1\simeq 20$ K and $J_2\simeq 60$ K and the alternation ratios of ${\alpha }_1=J_1^{\prime }/J_1\simeq 0.6$ and ${\alpha }_2=J_2^{\prime }/J_2\simeq 0.3$ within the two chains, respectively. The zero-field spin gap ${\mathrm{\Delta }}_0/k_{\mathrm{B}}\simeq 7.3$ K probed by thermodynamic and NMR measurements is caused by the $J_1\text{−}{J}_1^{\prime }$ spin chains and can be closed in the applied field of ${\mu }_0{H}_{\mathrm{c}1}\simeq 5.6$ T, giving rise to a field-induced long-range order. The NMR data reveal predominant three-dimensional spin-spin correlations at low temperatures. Field-induced magnetic ordering transition observed above $H_{\text{c}1}$ is attributed to the Bose-Einstein condensation of triplons in the sublattice formed by the $J_1\text{−}{J}_1^{\prime }$ chains with weaker exchange couplings.

Figure 1

(a) Crystal structure of $\varepsilon \text{−}{\mathrm{LiVOPO}}_4$ projected onto the $ac$ plane. The deep blue and light blue solid circles represent the V(1) and V(2) sites, respectively. Chain 1 with the couplings $J_1$ and $J_1^{\prime }$ is formed by V(1) and chain 2 is formed by V(2) atoms with the couplings $J_2$ and $J_2^{\prime }$ via the extended V-$\mathrm{O}...\mathrm{O}$-V paths. These chains are nearly orthogonal to each other, whereas the structural chains are parallel comprising both V1 and V2 atoms at the same time. These chains run perpendicular to the $ac$ plane. (b) A segment of the chain 1 formed by the V(1)${\mathrm{O}}_6$ octahedra with the intrachain couplings $J_1$ and $J_1^{\prime }$. The distances $d_1$ and $d_1^{\prime }$ are the lateral displacements of the ${\mathrm{VO}}_6$ octahedra in chain 1. (c) A section of the structural chain with the V(1)-O-V(2) paths along the $b$ axis. (d) An empirical/qualitative sketch of the spin model with all possible exchange interactions involving chain 1 ($J_1$, $J_1^{\prime }$) and chain 2 ($J_2,J_2^{\prime }$).

Figure 2

X-ray powder diffraction patterns of $\varepsilon \text{−}{\mathrm{LiVOPO}}_4$ at two different temperatures ($T=300$ and 15 K). The solid lines denote the Rietveld refinement fit of the data. The Bragg-peak positions are indicated by green vertical bars and bottom blue line indicates the difference between the experimental and calculated intensities.

Figure 3

Lattice parameters (a) $a$, (b) $b$, (c) $c$, (d) $\alpha$, (e) $\beta$, and (f) $\gamma$ vs temperature. The unit-cell volume ($V_{\mathrm{cell}}$) is plotted as a function of temperature in right $y$ axis of (c) and the solid line represents the fit using Eq. (1).

Figure 4

Upper panel: $\chi \left(T\right)$ measured in ${\mu }_{0}H=0.5$ T. The two shoulders of the broad maximum are indicated by the vertical arrows. Inset: low-$T\chi \left(T\right)$ measured in different applied fields. Lower panel: $1/\chi$ vs $T$ and the solid line is the CW fit using Eq. (3).

Figure 5

Magnetization vs field measured at $T=1.5$ K using pulsed magnetic field. Inset: $dM$/$dH$ vs $H$ to highlight the critical fields $H_{\mathrm{c}1}$, $H_{\mathrm{c}2}$, and $H_{\mathrm{c}3}$.

Figure 6

Upper panel: specific heat $C_{\mathrm{p}}$ vs $T$ of $\varepsilon \text{−}{\mathrm{LiVOPO}}_4$ in zero applied field. The dashed line stands for the phonon contribution to the specific heat ($C_{\mathrm{ph}}$) using Debye fit [Eq. (4)]. The solid line is the magnetic contribution to the specific heat ($C_{\mathrm{mag}}$). Inset: $C_{\mathrm{mag}}$ vs $T$ in the low-temperature region. The solid line is an exponential fit as described in the text. Lower panel: $C_{\mathrm{mag}}/T$ and $S_{\mathrm{mag}}$ vs $T$ in the left and right $y$ axes, respectively. Inset: $C_{\mathrm{p}}/T$ vs $T$ in different magnetic fields in the low-temperature regime.

Figure 7

$C_{\mathrm{p}}/T$ vs $T$ measured in different applied magnetic fields up to 14 T.

Figure 8

Typical ${}^{31}\mathrm{P}$ field-sweep NMR spectra of $\varepsilon \text{−}{\mathrm{LiVOPO}}_4$ measured at 24.96 MHz at different temperatures. The dashed lines track the shifting of NMR line positions. The vertical line corresponds to the ${}^{31}\mathrm{P}$ reference line ($H_{\mathrm{ref}}$) measured for ${\mathrm{H}}_3{\mathrm{PO}}_4$. The solid line is the fit of the spectra at $T=50$ K using a double Gaussian function.

Figure 9

Upper panel: temperature-dependent ${}^{31}\mathrm{P}$ NMR shift $K\left(T\right)$ measured at 24.96 MHz as a function of temperature for both P(1) and P(2) sites. Lower panel: $K$ vs $\chi$ (measured at 1.5 T) with temperature as an implicit parameter. The solid lines are the linear fits.

Figure 10

Upper panel: $K$ vs $T$ for both P(1) and P(2) sites. The solid lines are the fits below 5 K by $K=K_0+m{T}^{(d/2-1)}{e}^{-\mathrm{\Delta }/k_{\mathrm{B}}T}$, fixing $\mathrm{\Delta }/k_{\mathrm{B}}\simeq 5.5$ K. Lower panel: $(K-K_0)e^{5.5/T}$ vs $T$ is shown in the low-temperature regime. The solid lines are the fits with $d\simeq 2.83$ and 2.73 for the P(1) and P(2) sites, respectively.

Figure 11

Upper panel: recovery of the longitudinal magnetization as a function of waiting time $t$ at three different temperatures for the P(1) site. Solid lines are the fits using Eq. (7). Lower panel: temperature variation of $1/T_1$ measured in different magnetic fields. For ${\mu }_{0}H=1.4$ T, measurements are done on both P(1) and P(2) sites while for other fields, only P(1) site is probed. The solid line represents the fit using $1/T_1\propto T^{\alpha }$ to the 10 T data at low temperatures with $\alpha \simeq -0.6$.

Figure 12

$1/T_1{e}^{5.5/T}$ vs $T$ in the log-log plot. The solid lines are the fits, as described in the text with $d\simeq 2.74$ and 3.2, for the P(1) and P(2) sites, respectively.

Figure 13

Temperature-dependent susceptibility (top) and field-dependent magnetization (bottom) of $\varepsilon \text{−}{\mathrm{LiVOPO}}_4$ fitted using the model of two nonequivalent alternating spin-$\frac{1}{2}$ chains, as explained in the text. See Table 3 for the fitting parameters.

Figure 14

$H\text{−}T$ phase diagram of $\varepsilon \text{−}{\mathrm{LiVOPO}}_4$ obtained using the data points from $C_{\mathrm{p}}\left(T\right)$ measurements. The solid line corresponds to Eq. (10). Inset: $T_{\mathrm{N}}$ vs ($H-H_{\mathrm{c}1}{)}^{1/1.5}$ to highlight the low-temperature linear regime and the agreement of the simulated curve with the experimental data points.