Randomness and frustration in a $S=\frac{1}{2}$ square-lattice Heisenberg antiferromagnet

We explore the interplay between randomness and magnetic frustration in the series of $S=\frac{1}{2}$ Heisenberg square-lattice compounds ${\mathrm{Sr}}_2{\mathrm{CuTe}}_{1-x}{\mathrm{W}}_x{\mathrm{O}}_6$. Substituting W for Te alters the magnetic interactions dramatically, from strongly nearest-neighbor to next-nearest-neighbor antiferromagnetic coupling. We perform neutron scattering measurements to probe the magnetic ground state and excitations over a range of $x$. We propose a bond-disorder model that reproduces ground states with only short-ranged spin correlations in the mixed compounds. The calculated neutron diffraction patterns and powder spectra agree well with the measured data and allow detailed predictions for future measurements. We conclude that quenched randomness plays the major role in defining the physics of ${\mathrm{Sr}}_2{\mathrm{CuTe}}_{1-x}{\mathrm{W}}_x{\mathrm{O}}_6$ with frustration being less significant.

Figure 1

The Heisenberg square lattice and ${\mathrm{Sr}}_2{\mathrm{CuTe}}_{1-x}{\mathrm{W}}_x{\mathrm{O}}_6$. (a) Phase diagram of the $J_1\text{−}{J}_2$ square-lattice Heisenberg model, showing the ferromagnetic, columnar, and Néel AFM states, as well as the frustrated parameter regimes of QSL and valence-bond crystal (VBC) behavior. (b) Magnetic interactions in ${\mathrm{Sr}}_2{\mathrm{CuTe}}_{1-x}{\mathrm{W}}_x{\mathrm{O}}_6$, represented for the three cases where the counterions in each square are both Te, both W, or one of each. The $J_1$ and $J_2$ parameters are those obtained by quantum chemistry calculations [24]; we note that the nearest-neighbor interaction is always small in the presence of W.

Figure 2

Polarized neutron diffraction. Intensities measured for powder samples of ${\mathrm{Sr}}_2{\mathrm{CuTe}}_{1-x}{\mathrm{W}}_x{\mathrm{O}}_6$ with $x=0.2$ (gray diamonds) and $x=0.5$ (black squares). Data for $x=0.5$ are translated upwards by 0.06 arbitrary units (arb. units) for clarity. The inset shows intensities calculated using the random-bond model of Fig. 1; four of these curves are scaled and superimposed on the data in the main panel.

Figure 3

Ground-state spin configurations. (a) Examples of random-bond networks for ${\mathrm{Sr}}_2{\mathrm{CuTe}}_{1-x}{\mathrm{W}}_x{\mathrm{O}}_6$ with $x\approx 0, x=0.4$, and $x\approx 1$; weak interactions are omitted for clarity. Thick gray lines at $x=0.4$ indicate two representative geometrically frustrated paths, with five bonds being the shortest possible. (b) Spin configuration for $x=0.4$, matching the bond network in (a). The color code quantifies the correlations around each square, ${\sum }_{\square }{\mathbf{S}}_i·{\mathbf{S}}_{i+1}$ ($i=\{1,2,3,4\}$ labeling the four sites), which vary from Néel (red) to columnar (blue) character. Because the spins are of fixed size but rotate in three dimensions, shorter arrows indicate an out-of-plane component. (c) Calculated structure factor $S\left(\mathbf{Q}\right)$ showing the evolution from Néel to columnar order. Only for $x=0$ and 1 are the peaks very sharp, as expected for long-range order.

Figure 4

Magnetic excitations in ${\mathrm{Sr}}_2{\mathrm{CuTe}}_{1-x}{\mathrm{W}}_x{\mathrm{O}}_6$. (a)–(e) Powder INS spectra for samples with $x=0$, 0.2, 0.4, 0.5, and 1, normalized to the nuclear Bragg peak for comparison between different $x$ values. (f)–(j) Spectra calculated using the random-bond model. The magnetic form factor for ${\mathrm{Cu}}^{2+}$ was estimated following Ref. [36] and an energy broadening ($dE=1.2$ meV for $x=0$ and 1.8 meV for $x>0$) was convolved with each calculated spectrum to approximate the instrumental resolution. Regions of $\left|\mathbf{Q}\right|$ not covered in experiment are dimmed in the modeled spectra. The dashed lines mark the constant $\left|\mathbf{Q}\right|$ and energy values analyzed in Fig. 5.

Figure 5

Dynamical structure factor. Measured (left) and calculated (right panels) $S\left(\right|\mathbf{Q}|,E)$ at constant $\left|\mathbf{Q}\right|=1.5{\text{Å}}^{-1}$ (top) and constant $E=4\mathrm{meV}$ (bottom panels). The respective integration widths are $\mathrm{\Delta }\left|\mathbf{Q}\right|=0.2{\text{Å}}^{-1}$ and $\mathrm{\Delta }E=2$ meV. The curves are offset along the $y$ axis for clarity.