Extreme Suppression of Antiferromagnetic Order and Critical Scaling in a Two-Dimensional Random Quantum Magnet

${\mathrm{Sr}}_2{\mathrm{CuTeO}}_6$ is a square-lattice Néel antiferromagnet with superexchange between first-neighbor $S=1/2$ Cu spins mediated by plaquette centered Te ions. Substituting Te by W, the affected impurity plaquettes have predominantly second-neighbor interactions, thus causing local magnetic frustration. Here we report a study of ${\mathrm{Sr}}_2{\mathrm{CuTe}}_{1-x}{\mathrm{W}}_x{\mathrm{O}}_6$ using neutron diffraction and $\mu \mathrm{SR}$ techniques, showing that the Néel order vanishes already at $x=0.025\pm 0.005$. We explain this extreme order suppression using a two-dimensional Heisenberg spin model, demonstrating that a W-type impurity induces a deformation of the order parameter that decays with distance as $1/r^2$ at temperature $T=0$. The associated logarithmic singularity leads to loss of order for any $x>0$. Order for small $x>0$ and $T>0$ is induced by weak interplane couplings. In the nonmagnetic phase of ${\mathrm{Sr}}_2{\mathrm{CuTe}}_{1-x}{\mathrm{W}}_x{\mathrm{O}}_6$, the $\mu \mathrm{SR}$ relaxation rate exhibits quantum critical scaling with a large dynamic exponent, $z\approx 3$, consistent with a random-singlet state.

Figure 1

2D Heisenberg couplings $J_{ij}{\mathbf{S}}_i·{\mathbf{S}}_j$ in ${\mathrm{Sr}}_2{\mathrm{CuTe}}_{1-x}{\mathrm{W}}_x{\mathrm{O}}_6$. The small black circles represent the $S=1/2$ carrying Cu ions, while red and blue circles correspond to Te and W ions, respectively. The dominant couplings mediated by Te in (a) and W in (b) are first-neighbor $J_1$ (solid red lines) and second-neighbor $J_2$ (solid blue lines), with $J_1\approx J_2\approx 8\mathrm{meV}$ [30, 33]. The couplings $J_1^{\prime }$ and $J_2^{\prime }$ indicated by the thin dashed lines are roughly 10% of the dominant couplings. The first-neighbor coupling $J_1^{\prime \prime }$ on links between Te and W ions, the gray dashed line in (c), is about 4% of $J_1$ [33].

Figure 2

Neutron diffraction results for (a) $x=0$ and 0.02, (b) 0.03 and 0.05 (c) 0.1, and (d) 0.2. The peaks correspond to wave vectors $q=(1/2,1/2,0)$ and $(1/2,1/2,1)$ in the tetragonal magnetic Brillouin zone, indicating dominant Néel AFM order ($x=0$ and 0.02) and short-range correlations ($x\ge 0.03$). Data at $T=40\mathrm{K}$ have been subtracted as background. The $x=0$ and 0.03 values have been shifted vertically for clarity. The curves are Gaussian fits and the green bars indicate the instrumental resolutions.

Figure 3

Time-dependent zero-field $\mu \mathrm{SR}$ spectra for (a) $x=0.05$ and (b) $x=0.1$ samples at different temperatures (the highest and lowest indicated) along with fits to Eq. (1). (c) Temperature dependent $\mu \mathrm{SR}$ asymmetry for $x=0$, 0.05, and 0.1, normalized by the values at $T$ = 30 K. (d) Temperature dependent relaxation rate $\lambda$ for $x=0.05$ and 0.1. The fitted lines correspond to critical scaling, $\lambda \approx T^{-\gamma }$, with $\gamma =0.35\pm 0.03$ ($x=0.05$) and $0.42\pm 0.03$ ($x=0.1$).

Figure 4

(a) Magnetic phase diagram of ${\mathrm{Sr}}_2{\mathrm{CuTe}}_{1-x}{\mathrm{W}}_x{\mathrm{O}}_6$. NAF and CAF denote Néel and columnar AFM correlations, respectively, either short-range (SR) or long-range (LR). The ordering temperature $T_{\mathrm{c}}$ and characteristic short-range correlation temperature $T^{*}$ were determined by $\mu \mathrm{SR}$ measurements, except for $T^{*}$ of the $x=0.2$ sample, which was obtained (Supplemental Material [34]) by neutron diffraction. (b) Transition temperatures of the classical Heisenberg model of coupled layers, determined using Monte Carlo simulations. In the notation of Fig. 1 the 2D couplings are $J_1=J_2=1$, $J_1^{\prime }=J_2^{\prime }=0.1$, and $J_1^{\prime \prime }=0$. Two different interlayer couplings are used; $J_{\perp }={10}^{-2}$ and ${10}^{-3}$. Curves are drawn through the data points as guides to the eye.

Figure 5

Deformation of the order parameter of the classical Heisenberg model with a W-type plaquette impurity as defined in Fig. 1, with the same couplings as in Fig. 4. The deviation $\mathrm{\Delta }m=1-|S_i^z|$, where the $z$ direction is that of the bulk Néel order, is shown vs the distance $r$ from the impurity along the (1,0) lattice direction for several system sizes. The line shows the form $1/r^2$. The inset shows the projection of the spins to the $xy$ spin plane, with the color coding corresponding to $m\in [0.49,1]$. The magnitude of the $xy$ component decays as $1/r$ from its maximal value $\approx 0.87$ closest to the impurity. The behaviors correspond to an angular distortion $\propto 1/r$.