$S=\frac{1}{2}$ quantum critical spin ladders produced by orbital ordering in ${\mathrm{Ba}}_2{\mathrm{CuTeO}}_6$

The ordered hexagonal perovskite ${\mathrm{Ba}}_2{\mathrm{CuTeO}}_6$ hosts weakly coupled $S=\frac{1}{2}$ spin ladders produced by an orbital ordering of ${\mathrm{Cu}}^{2+}$. The magnetic susceptibility $\chi \left(T\right)$ of ${\mathrm{Ba}}_2{\mathrm{CuTeO}}_6$ is well described by that expected for isolated spin ladders with exchange coupling of $J\approx 86\mathrm{K}$ but shows a deviation from the expected thermally activated behavior at low temperatures below $T^{*}\approx 25\mathrm{K}$. An anomaly in $\chi \left(T\right)$, indicative of magnetic ordering, is observed at $T_{\mathrm{mag}}=16\mathrm{K}$. No clear signature of long-range ordering, however, is captured so far in NMR $1/T_1$, specific heat or neutron diffraction measurements at and below $T_{\mathrm{mag}}$. The marginal magnetic transition, indicative of strong quantum fluctuations, is evidence that ${\mathrm{Ba}}_2{\mathrm{CuTeO}}_6$ is in very close proximity to a quantum critical point between magnetically ordered and spin-gapped phases controlled by interladder couplings.

Figure 1

(a) A rough schematic phase diagram at $T=0$ for the two-leg $S=\frac{1}{2}$ antiferromagnetic ladder system with interladder coupling $J_{\mathrm{inter}}$. The spin gap $\mathrm{\Delta }$ is suppressed with increasing $J_{\mathrm{inter}}$ and goes to zero at the critical coupling $J_c$. For $J_{\mathrm{inter}}>J_c$ long-range order is expected to develop below $T_{\mathrm{N}}$. The arrow indicates a possible position for ${\mathrm{Ba}}_2{\mathrm{CuTeO}}_6$ in this phase diagram. (b) The chain and ladder arrangement in ${\mathrm{Ba}}_2{\mathrm{CuTeO}}_6$. The ${\mathrm{Cu}}^{2+}$ chains run along the monoclinic $b$ axis (into the page) and form ladders between layers linked by corner-sharing ${\mathrm{TeO}}_6$ octahedra (a single ladder is shaded and a second outlined). The ladders, indicated by thick gray lines, form a stacked layer within the $ab$ plane. The interladder couplings are indicated by arrows. The inset with label “J-T” indicates the Jahn-Teller distortion direction. (c) The ${\mathrm{Cu}}^{2+}$ chains running along the $b$ axis in ${\mathrm{Ba}}_2{\mathrm{CuTeO}}_6$. These chains can couple to form ladders via superexchange through oxygen ions. The intraladder leg and rung exchange couplings are indicated by $J$ and $J^{\prime }$, respectively. The diagonal intraladder coupling is labeled $j_1$ and the intralayer interladder couplings are labeled $j_2$ and $j_3$. The interlayer coupling between ladders through face-sharing ${\mathrm{TeO}}_6$ octahedra is labeled $j_4$. (d) A single layer of ladders in ${\mathrm{Ba}}_2{\mathrm{CuTeO}}_6$. $J, J^{\prime }, j_2$, and $j_3$ are indicated by arrows. The ladders are shown as thick gray lines.

Figure 2

$\chi \left(T\right)$ for ${\mathrm{Ba}}_2{\mathrm{CuTeO}}_6$ single crystal arrays with the magnetic field of ${\mu }_{0}H=1\mathrm{T}$ applied parallel (upper panel) and perpendicular (lower panel) to the $ab$ plane. The dashed and solid lines indicate chain and ladder model fits, respectively. The inset to the upper panel shows the low temperature region for both orientations along with the high field behavior for $H_{\perp ab}$. The inset to the lower panel shows $M\left(H\right)$ of ${\mathrm{Ba}}_2{\mathrm{CuTeO}}_6$ for $H_{\perp ab}$.

Figure 3

(a) Fitting of experimental ${}^{125}\mathrm{Te}$ NMR spectrum by simulation of the two Te crystallographic sites at 26 K [20]. (b) $K_{\text{iso}}$ (%) vs $\chi$($T$) plots with the temperature as an implicit parameter for both Te sites. (c) Temperature dependence of ${}^{125}\mathrm{Te}$ NMR spectra.

Figure 4

(a) Contour plot of the spin echo intensity of the ${}^{125}\mathrm{Te}$-field-sweep NMR spectra as a function of temperature (NMR frequency: 55.44 MHz) for a polycrystalline sample. The dotted line is a guide to the eye. The individual spectra are plotted in Fig. 3. (b) Semilog plot of $1/T_1$ versus 1/$T$. Solid lines correspond to the behavior $1/T_1=A\exp (-{\mathrm{\Delta }}_R/k_{\mathrm{B}}T$) at high temperatures where ${\mathrm{\Delta }}_R$ is the dynamical spin gap.

Figure 5

(a) The specific heat $C$ of single crystal ${\mathrm{Ba}}_2{\mathrm{CuTeO}}_6$ plotted as $C/T$ vs $T^2$. The dashed line indicates an approximate lattice $T^3$ term as described in the text. The inset shows an enlarged view of the low temperature region. (b) The estimated magnetic specific heat $C_{\text{mag}}\left(T\right)$ for ${\mathrm{Ba}}_2{\mathrm{CuTeO}}_6$ plotted as $C_{\text{mag}}/T$ vs $T$. The inset shows the estimated magnetic entropy $S_{\text{mag}}\left(T\right)$ as a percentage of $R\ln 2$ up to $T=20\mathrm{K}$.