Structural and magnetic properties of the quantum magnet ${\mathrm{BaCuTe}}_2{\mathrm{O}}_6$

We investigate the structural and magnetic properties of the quantum magnet $\mathrm{Ba}\mathrm{Cu}{\mathrm{Te}}_2{\mathrm{O}}_6$. This compound is synthesized in powder and single crystal form for the first time. Synchrotron x-ray and neutron diffraction reveal a cubic crystal structure $\left(P{4}_{1}32\right)$ where the magnetic ${\mathrm{Cu}}^{2+}$ ions form a complex network. Heat capacity and static magnetic susceptibility measurements suggest the presence of antiferromagnetic interactions with a Curie-Weiss temperature of $\approx -33$ K, while long-range magnetic order occurs at the much lower temperature of $\approx 6.3$ K. The magnetic structure, solved using neutron diffraction, reveals antiferromagnetic order along chains parallel to the $a,$ $b,$ and $c$ crystal axes. This is consistent with the magnetic excitations which resemble the multispinon continuum typical of the spin-1/2 Heisenberg antiferromagnetic chain. A consistent intrachain interaction value of $\approx 34$ K is achieved from the various techniques. Finally the magnetic structure provides evidence that the chains are coupled together in a noncollinear arrangement by a much weaker antiferromagnetic, frustrated hyperkagome interaction.

Figure 1

(a) Laue diffraction pattern of a single crystal of $\mathrm{Ba}\mathrm{Cu}{\mathrm{Te}}_2{\mathrm{O}}_6$ with the x-ray beam parallel to the [1 1 0] direction. From the figure it can be deduced that the illuminated surface of the crystal is single phased. (b) Crystal III which was used for the inelastic neutron scattering measurements due to its size and quality.

Figure 2

Crystal structure of $\mathrm{Ba}\mathrm{Cu}{\mathrm{Te}}_2{\mathrm{O}}_6$ using the lattice parameters and atomic coordinates extracted from the analysis of the high-resolution synchrotron powder x-ray diffraction data at room temperature. The ions ${\mathrm{Ba}}^{2+}$, ${\mathrm{Cu}}^{2+}$, ${\mathrm{Te}}^{4+}$, and ${\mathrm{O}}^{2-}$ are shown as green, blue, gold, and red spheres, respectively, and the coordination of ${\mathrm{Te}}^{4+}$ and ${\mathrm{Cu}}^{2+}$ by ${\mathrm{O}}^{2-}$ are shown by the polyhedra. The figure was prepared using the VESTA software [30].

Figure 3

The magnetic exchange interactions $J1$, $J2$, and $J3$ between the ${\mathrm{Cu}}^{2+}$ ions in $\mathrm{Ba}\mathrm{Cu}{\mathrm{Te}}_2{\mathrm{O}}_6$ are presented from left to right. The first neighbor interactions, $J1$ (black dashed lines), form isolated ${\mathrm{Cu}}^{2+}$ triangles. The second neighbor interactions, $J2$ (red dashed lines), lead to a 3D network of corner sharing triangles, known as the hyperkagome lattice. The third neighbor interactions $J3$ (green lines) form chains along the $a$, $b$, and $c$ axes. Only the ${\mathrm{Cu}}^{2+}$ ions are shown (blue spheres) and the light gray lines indicate the boundaries of one unit cell. The figure was prepared by the Crystal Maker software [31].

Figure 4

(a) Temperature dependence of the static magnetic susceptibility ${\chi }_{\text{dc}}$ measured on a powder sample of $\mathrm{Ba}\mathrm{Cu}{\mathrm{Te}}_2{\mathrm{O}}_6$ in an applied magnetic field $H=0.1$ T. The data in the range 7 K to 400 K were fitted by Eq. (1) which includes the $S\text{−}1/2$ HAFC model. (b) Inverse susceptibility as a function of temperature. The data in the range 100 K to 400 K were fitted to the Curie-Weiss Law.

Figure 5

(a) Heat capacity data collected as a function of temperature from a powder sample of $\mathrm{Ba}\mathrm{Cu}{\mathrm{Te}}_2{\mathrm{O}}_6$ in zero applied magnetic field. The data were fitted with a linear combination of three Debye terms above $T=40$ K to model the lattice heat capacity (solid green line). The phononic parameters obtained from fitting Eq. (4) to the data are: $c_1=2.2\left(2\right)$ with $T_{\text{D}i}=128\left(5\right)$, $c_2=2.96\left(6\right)$ with $T_{\text{D}i}=333\left(15\right)$, and $c_3=37\left(3\right)$ with $T_{\text{D}i}=1905\left(2\right)$. The magnetic heat capacity $C_{\text{mag}}$ was obtained by subtracting the phononic contribution from the data (solid black line). (b) The low temperature $C_{\text{mag}}/T$ is given by the yellow symbols (left-hand $y$ axis) as a function of temperature and reveals a $\lambda$ anomaly at $T_N=6.32$ K. The green dashed line is a fit of $C_{\text{mag}}$ below $T_N$, by the power law $C_{\text{mag}}=A\left(T^3\right)$. The temperature dependence of the magnetic entropy is given by the black dots (right-hand $y$ axis).

Figure 6

Rietveld refinement of the magnetic diffraction pattern collected on (a) detector banks 1 and 10 showing the high $d$-spacing peaks, and (b) detector banks 2 and 9 showing the low $d$-spacing peaks, of the WISH diffractometer for a powder sample of $\mathrm{Ba}\mathrm{Cu}{\mathrm{Te}}_2{\mathrm{O}}_6$. The black circles give the result of subtracting the 8 K dataset from the 2 K dataset, the red line gives the refined fit of the data assuming the IR $2\times {\mathrm{\Gamma }}_2^1$ $({\chi }^2=3.1)$ while the blue line gives the difference between theory and experiment.

Figure 7

Temperature dependence of the magnetic Bragg peaks obtained by subtracting the neutron diffraction pattern measured at 8 K from the $T=2$, 3, 4, 5, and 6 K neutron diffraction patterns. The datasets were collected from detector banks 1 and 10 of WISH. In the inset the fitted ordered moment ${\mathrm{m}}_{\text{ord}}$ is presented as a function of temperature. The values were found after refining the data with the $2\times {\mathrm{\Gamma }}_2^1$ IR.

Figure 8

Visualization of the magnetic structure of $\mathrm{Ba}\mathrm{Cu}{\mathrm{Te}}_2{\mathrm{O}}_6$ found by fitting the diffraction patterns measured at 2 K by the $2\times {\mathrm{\Gamma }}_2^1$ IR. Only the ${\mathrm{Cu}}^{2+}$ ions are shown represented by the blue spheres, while the red arrows indicate the directions of the ordered spin moments. (a) includes the first $J1$ (dotted black lines), and third $J3$ (green lines), nearest neighbor interactions while (b) includes the second $J2$ (dotted red line), and third $J3$ (green lines) nearest neighbor interactions. To the right of panel (a) [(b)], the triangles formed by $J1$ $\left[J2\right]$ are presented. This figure was produced using the VESTA software [30].

Figure 9

Inelastic neutron scattering spectra as a function of energy and wave-vector transfer. The data were measured using the LET ToF spectrometer on a powder sample at $T=8$ K (a) and $T=1.8$ K (b) with an incident energy, $E_{\text{i}}=14.10$ meV. No background has been subtracted from the data. (c) Energy cut through the data integrated over the wave-vector range 1.0 ${\text{Å}}^{-1}<\left|Q\right|<2.0{\text{Å}}^{-1}$ and plotted as a function of energy. A broad maximum at $\approx 4$ meV at $T=8$ K (yellow symbols) becomes sharper at 4.5 meV in the ordered state at $T=1.8$ K (blue symbols). With increasing energy, the signal drops gradually and seems unaffected by the temperature.

Figure 10

Inelastic neutron scattering plotted as a function of wave-vector transfer in the [H,H,L] plane at a constant energy transfer of 2.0 meV. The data was measured using the LET ToF spectrometer on a single crystal sample at $T=1.8$ K (a) and $T=8$ K (b) using an incident energy of $E_{\text{i}}=4.81$ meV. The data was integrated over the energy transfer range 1.8 $\mathrm{meV}<E<2.2$ meV and over wave vector in the vertical out-of-plane direction by $[\pm 0.2,\mp 0.2,0]$. No background subtraction has been performed and only a subtle smoothing is implemented.

Figure 11

(a)–(d) Inelastic neutron scattering measured on a single crystal sample plotted as a function of energy and wave-vector transfer without background subtraction. The data were collected on the LET ToF spectrometer using an incident neutron energy of $E_{\text{i}}=13.75$ meV (energy resolution, 0.78 meV). (a) and (b) show the spectra along $[H,H,0]$ collected at $T=1.8$ K and $T=8$ K, respectively. The data have been integrated by $\pm 0.2$ r.l.u. over the $[0,0,L]$ direction and by $\pm 0.5$ r.l.u. over the $[K,-K,0]$ vertical out-of-plane direction. (c) and (d) show the spectra along $[0,0,L]$ collected at $T=1.8$ K and $T=8$ K, respectively. The data have been integrated by $\pm 0.2$ r.l.u. over the $[H,H,0]$ and $[K,-K,0]$ directions. The ranges of integration were selected to achieve an adequate resolution and intensity. (e) Resolution corrected dynamical structure factor calculated via the algebraic Bethe-Ansatz for $J3=2.90\left(6\right)$ meV. (f) Cuts along energy through the data collected with $E_{\text{i}}=13.75$ meV and $T=1.8$ K at $L=-2.5$ and $-3.5$ with integration over $L$ of $\pm 0.05$, and over $[H,H,0]$ and $[K,-K,0]$ of $\pm 0.2$ r.l.u. The red line gives the fit of a Gaussian to the $L=-3.5$ cut in order to extract the maximum of the lower boundary of the spinon continuum (red line). The magenta line gives the related cut through the theoretical dynamical structure factor calculated via the algebraic Bethe-Ansatz for $J3=2.90\left(6\right)$ meV at $L=-3.5$. The fits include a sloping background and the elastic incoherent scattering peak.

Figure 12

(a) Structural Rietveld refinement (black line) of the synchrotron x-ray diffraction pattern (red circles) collected for a crushed single crystal of $\mathrm{Ba}\mathrm{Cu}{\mathrm{Te}}_2{\mathrm{O}}_6$ at room temperature along with the difference between the fit and data (blue line). The green crosses indicate the Bragg peak positions for the two included phases, $\mathrm{Ba}\mathrm{Cu}{\mathrm{Te}}_2{\mathrm{O}}_6$ (top row) and diamond powder (bottom row). The refinement confirms the cubic space group ($P{4}_{1}32$) with lattice parameter $a=12.8330\left(2\right)$ Å, achieved with ${\chi }^2=15.0$. (b) A small region of the diffraction pattern at low $d$ spacing showing the goodness of the refinement.

Figure 13

(a) Rietveld refinement of the neutron powder diffraction pattern of the $\mathrm{Ba}\mathrm{Cu}{\mathrm{Te}}_2{\mathrm{O}}_6$ powder sample prepared by solid state reaction, collected at $T=15$ K on the SPODI diffractometer. The red open circles refer to the experimental data, and the black and blue lines indicate the refined fit of the data and the difference between fit and experiment, respectively. The green crosses denote the Bragg peak positions for the three included phases, $\mathrm{Ba}\mathrm{Cu}{\mathrm{Te}}_2{\mathrm{O}}_6$, Cu container, and the Al cryostat. The refinement confirms the cubic space group $P{4}_{1}32$ with lattice parameter $a=12.8328\left(1\right)$ Å. (b) A small region of the diffraction pattern at low $d$ spacing showing the goodness of the refinement.

Figure 14

(a) Neutron powder diffraction data collected above and below the magnetic phase transition at $T=8$ K (red points) and 2 K (black points), respectively, on the WISH diffractometer plotted as a function of $d$ spacing. The difference pattern (dataset at 2 K with the 8 K dataset subtracted) is also plotted as shown by the blue symbols (the $y$ scale is shifted for clarity and shown on the right-hand side). The peaks are identified by their Miller indices. (a) shows the pattern at high $d$ spacing collected from detector banks 2 and 9 while the lower $d$-spacing range is shown by (b) detector banks 3 and 8 and (c) detector banks 4 and 7.

Figure 15

Rietveld refinement of the magnetic diffraction pattern collected on (a) detector banks 1 and 10 showing the high $d$-spacing peaks and (b) detector banks 2 and 9 showing the low $d$-spacing peaks of the WISH diffractometer for a powder sample of $\mathrm{Ba}\mathrm{Cu}{\mathrm{Te}}_2{\mathrm{O}}_6$. The black circles give the result of subtracting the 8 K dataset from the 2 K dataset, the red line gives the refined fit of the data assuming the IR $1\times {\mathrm{\Gamma }}_1^1$ $({\chi }^2=8.5)$, while the blue line gives the difference between theory and experiment.