Spin waves and magnetic interactions in $\mathrm{Li}{\mathrm{Cu}}_2{\mathrm{O}}_2$

The quasi-one-dimensional helimagnet $\mathrm{Li}{\mathrm{Cu}}_2{\mathrm{O}}_2$ was studied by single crystal inelastic neutron scattering. The dispersion relation of spin wave excitations was measured in the vicinity of the principal magnetic Bragg reflection. A spin wave theoretical analysis of the data yields an estimate of the relevant exchange constants and explains the mechanism of geometric frustration that leads to helimagnetism. It is found that the simple antiferromagnetic $J_1\text{−}{J}_2$ model that was previously proposed is inadequate for $\mathrm{Li}{\mathrm{Cu}}_2{\mathrm{O}}_2$. The experimental findings are generally in a qualitative agreement with first principles calculations of [A. A. Gippius, E. N. Morozova, A. S. Moskvin, A. V. Zalessky, A. A. Bush, M. Baenitz, H. Rosner, and S.-L. Drechsler, Phys. Rev. B 70, 020406 (2004)], though certain important discrepancies remain to be explained.

Figure 1

(a) A schematic view of exchange interactions between magnetic ${\mathrm{Cu}}^{2+}$ ions in $\mathrm{Li}{\mathrm{Cu}}_2{\mathrm{O}}_2$. (b) An equivalent Bravais lattice of spins obtained by displacing every other ${\mathrm{Cu}}^{2+}$ ion in the original non-Bravais lattice by the vector $\mathbf{d}$. It is assumed that the spin Hamiltonian remains intact upon this transformation.

Figure 2

A constant-$E$ scan along the $(-0.5,k,0)$ reciprocal-space rod measured in $\mathrm{Li}{\mathrm{Cu}}_2{\mathrm{O}}_2$ at $T=1.5\mathrm{K}$ at an energy transfer of $5\mathrm{meV}$ (symbols) using Setup 1. The lines are a simulation based on the measured spin wave dispersion relation and the known resolution function of the instrument.

Figure 3

Typical constant-$q$ scans measured in $\mathrm{Li}{\mathrm{Cu}}_2{\mathrm{O}}_2$ (symbols) at $T=1.5\mathrm{K}$. The lines are as in Fig. 2. The shaded area in the lower scan is an “accidental Bragg” spurious peak originating from $2k_i$ scattering in the monochromator, (0, 2, 0) Bragg scattering in the sample, and inelastic thermal-diffuse scattering in the analyzer. The spurious peak appears much narrower than the experimental resolution (about $2\mathrm{meV}$ FWHM).

Figure 4

Spin wave dispersion measured in $\mathrm{Li}{\mathrm{Cu}}_2{\mathrm{O}}_2$ at $T=1.5\mathrm{K}$ (symbols). Shaded and open circles are data points obtained from constant-$E$ and constant-$q$ scans, respectively. The lines are a fit to the data, as described in the text. Heavy solid, heavy dashed, and thin dotted lines correspond to Models 1, 2, and 3, respectively. In both panels the ellipses represent the FWHM of the instrument resolution function in the appropriate projection (open ellipse) and section (shaded ellipse).

Figure 5

Typical scans measured in $\mathrm{Li}{\mathrm{Cu}}_2{\mathrm{O}}_2$ using Setup 2 (symbols). The lines are as in Fig. 2.

Figure 6

(Color online) (a) Dispersion of all 12 spin wave branches along the $(0.5,k,0)$ reciprocal-space rod in twinned $\mathrm{Li}{\mathrm{Cu}}_2{\mathrm{O}}_2$ crystals calculated using Model 1. (b) The corresponding structure factors scaled by energy transfer. In both panels the out-of-plane modes are plotted in red. Modes shown in blue and green lines are polarized in the plane of spin rotation. Thick and thin lines of all types correspond to the first and second terms in Eq. (2a), respectively. Solid and dashed lines refer to type-A and type-B crystallographic domains. Symbols are as in the top panel of Fig. 4.