Magnetic properties and heat capacity of the three-dimensional frustrated $S=\frac{1}{2}$ antiferromagnet ${\mathrm{PbCuTe}}_2{\mathrm{O}}_6$

We report magnetic susceptibility $\left(\chi \right)$ and heat capacity $\left(C_p\right)$ measurements along with ab initio electronic structure calculations on ${\mathrm{PbCuTe}}_2{\mathrm{O}}_6$, a compound made up of a three-dimensional (3D) network of corner-shared triangular units. The presence of antiferromagnetic interactions is inferred from a Curie-Weiss temperature $({\theta }_{\mathrm{CW}})$ of about $-22$ K from the $\chi \left(T\right)$ data. The magnetic heat capacity $C_m$ data show a broad maximum at $T^{\max }\simeq 1.15$ K (i.e., $T^{\max }/{\theta }_{\mathrm{CW}}\simeq 0.05)$, which is analogous to the the observed broad maximum in the $C_m/T$ data of a hyper-kagome system, ${\mathrm{Na}}_4{\mathrm{Ir}}_3{\mathrm{O}}_8$. In addition, $C_m$ data exhibit a weak kink at $T^{*}\simeq 0.87$ K. While the $T^{\max }$ is nearly unchanged, the $T^{*}$ is systematically suppressed in an increasing magnetic field $\left(H\right)$ up to 80 kOe. For $H\ge 80$ kOe, the $C_m$ data at low temperatures exhibit a characteristic power-law $\left(T{}^{\alpha }\right)$ behavior with an exponent $\alpha$ slightly less than 2. Hopping integrals obtained from the electronic structure calculations show the presence of strongly frustrated 3D spin interactions along with non-negligible unfrustrated couplings. Our results suggest that ${\mathrm{PbCuTe}}_2{\mathrm{O}}_6$ is a candidate material for realizing a 3D quantum spin liquid state at high magnetic fields.

Figure 1

The Rietveld refinement for the powder x-ray diffraction of ${\mathrm{PbCuTe}}_2{\mathrm{O}}_6$. The open circles indicate the observed data, while the Rietveld refinement fit is shown as a red solid line. The Bragg positions and the difference curve are denoted by vertical blue dashes and green line, respectively. The residual parameters for the Rietveld refinement are $R_p\simeq 3.099\%$, $R_{\mathrm{wp}}\simeq 4.279\%$, $R_{\exp }\simeq 2.024\%$, $R_{\mathrm{Bragg}}\simeq 1.916\%$, and goodness of fit (GOF) $\simeq 2.113,$ respectively.

Figure 2

(a) The crystal structure of ${\mathrm{PbCuTe}}_2$O${}_6.$ (b) The Cu network is built up of corner-shared triangles of two kinds (blue and red) with different Cu-Cu bond lengths 4.37 and 5.60 $\AA$, respectively. (c) Formation of triangles (blue) by the first $nn$ between Cu atoms with a bond length of 4.37 $\AA$. (d) A hyper-kagome network (with red triangles) is formed when only the second $nn$ between the Cu atoms of the bond length 5.60 $\AA$ are considered. (e) The third $nn$ bond makes the uniform chains, which pass along all three crystallographic directions.

Figure 3

(a) The temperature dependence of the magnetic susceptibility $\chi \left(T\right)$ (left $y$ axis) is shown. The $1/(\chi -{\chi }_0)$ data (right $y$ axis) is fitted to the inverse-Curie-Weiss law. Inset shows the variation of magnetic susceptibility with temperature (note the logarithmic scale) in zero-field-cooled (ZFC) and field-cooled (FC) conditions in a magnetic field of 100 Oe. $\left(b\right)$ Heat capacity $C_p$ vs $T$ in zero field with the fit to the lattice (solid line). Inset shows the low-$T$ data of $C_p$ vs $T$.

Figure 4

$C_m/T$ vs $T$ in various fields. The red line is a fit to the power law $\left(C_m\propto T^{\alpha }\right)$ behavior.

Figure 5

The variation of $T^{*}$ and $T^{\max }$ with the applied magnetic fields. Inset shows the plot of $C_m/T^2$ vs $T$. The black arrow marks indicate the positions of $T^{*}$. Note that the data in the inset is shifted along the $y$ axis for clarity.

Figure 6

(a) Non-spin-polarised band dispersion along various high-symmetry directions. Inset shows the crystal-field splitting corresponding to the square planar environment. (b) Wannier interpolated bands superimposed on the DFT bands.

Figure 7

Effective Cu $d_{x^2-y^2}$ Wannier function plot.