Magnetic dimerization in the frustrated spin ladder ${\mathrm{Li}}_2{\mathrm{Cu}}_2\mathrm{O}{\left({\mathrm{SO}}_4\right)}_2$

The magnetic properties of ${\mathrm{Li}}_2{\mathrm{Cu}}_2\mathrm{O}{\left({\mathrm{SO}}_4\right)}_2$ are investigated in the framework of density functional theory. In its high-temperature tetragonal structure, this compound appears as a rare material realization of a frustrated spin-1/2 two-leg ladder, where magnetic frustration arises from competing nearest and next-nearest interactions along the legs. Through a large magnetoelastic coupling, the triclinic distortion occurring around 125 K is shown to induce the formation of a staggered dimer structure, lifting most of the magnetic frustration.

Figure 1

(a) Tetragonal crystal structure of ${\mathrm{Li}}_2{\mathrm{Cu}}_2\mathrm{O}{\left({\mathrm{SO}}_4\right)}_2$ at room temperature. Cu are in blue, O in red, S in yellow, and Li in green. (b) Detail of the atomic structure of the chains running along the $c$ axis. (c) Magnetic model deduced from the atomic structure, with the three dominant interactions along the chain: $J_{\perp }$ in green, $J=J_{\times }$ in blue, and $J_2$ in red. (d) Topologically equivalent frustrated two-leg spin ladder.

Figure 2

Detail of the paramagnetic Cu-$3d_{x^2-y^2}$ band manifold around the Fermi level in ${\mathrm{Li}}_2{\mathrm{Cu}}_2\mathrm{O}{\left({\mathrm{SO}}_4\right)}_2$ calculated using GGA-PBE and interpolated with MLWFs (left panel) and corresponding total and partial density of states (right panel). The inset shows one of the MLWFs centered on the Cu site; the large antibonding O-$2p$ tails are clearly visible on neighboring atoms.

Figure 3

(a) Temperature dependence of the magnetic couplings in ${\mathrm{Li}}_2{\mathrm{Cu}}_2\mathrm{O}{\left({\mathrm{SO}}_4\right)}_2$ calculated in GGA+$U$. Data points are represented with dots; the Boltzmann sigmoid fits are represented with lines. (b) Experimental temperature dependence of the interplatelet Cu-O-Cu superexchange angles. (c) Schematic representation of the staggered dimer structure deduced for the triclinic phase (below the transition temperature of $\sim 125$ K). The inequivalent triclinic interactions along the legs $J^a$ and $J^d$, diagonal interactions between the legs $J_{\times }^b$ and $J_{\times }^c$, and between the legs along the rungs $J_{\perp }^a$ and $J_{\perp }^b$, are represented in dark blue, blue, and green, respectively. The same color convention has been used for the superexchange angles. The NNN interaction along the legs $J_2$ is represented in red.

Figure 4

(a) Temperature dependence of the magnetic susceptibility of ${\mathrm{Li}}_2{\mathrm{Cu}}_2\mathrm{O}{\left({\mathrm{SO}}_4\right)}_2$. The experimental points are shown in green. The calculations performed with the fixed couplings corresponding to the high-temperature (300 K) structure are shown in orange, with the fixed couplings corresponding to the low-temperature (2 K) structure in dark blue, and with temperature-dependent couplings shown by a thick purple line. (b) Temperature dependence of the spin correlations $\langle {\mathbf{S}}_i·{\mathbf{S}}_j\rangle$ calculated using Lanczos diagonalization. The labels are the associated exchange interactions and the colors follow the convention used in Fig. 3.