Magnetic structure and spin waves in the frustrated ferro-antiferromagnet ${\mathrm{Pb}}_2\mathrm{VO}{\left({\mathrm{PO}}_4\right)}_2$

Single crystal neutron diffraction, inelastic neutron scattering, and electron spin resonance experiments are used to study the magnetic structure and spin waves in ${\mathrm{Pb}}_2\mathrm{VO}{\left({\mathrm{PO}}_4\right)}_2$, a prototypical layered $S=1/2$ ferromagnet with frustrating next-nearest neighbor antiferromagnetic interactions. The observed excitation spectrum is found to be inconsistent with a simple square lattice model previously proposed for this material. At least four distinct exchange coupling constants are required to reproduce the measured spin wave dispersion. The degree of magnetic frustration is correspondingly revised and found to be substantially smaller than in all previous estimates.

Figure 1

(a) A single vanadophosphate layer in the crystal structure of ${\mathrm{Pb}}_2\mathrm{VO}{\left({\mathrm{PO}}_4\right)}_2$ showing the crystallographic unit cell and the four distinct $V^{4+}$ ions that it contains. (b) The five distinct nearest-neighbor and next-nearest-neighbor V-V bonds in each such layer (lines). Arrows represent the spin orientation in the ordered state as deduced from our single crystal diffraction data.

Figure 2

Measured (solid symbols) and calculated (open symbols) squared structure factors of magnetic Bragg reflections in ${\mathrm{Pb}}_2\mathrm{VO}{\left({\mathrm{PO}}_4\right)}_2$ plotted against momentum transfer.

Figure 3

Measured temperature dependence of the (1,0,0) magnetic Bragg peak intensity (symbols) and an empirical power law fit to the data as described in the text (solid line).

Figure 4

(a)–(e): False color plots of inelastic neutron scattering intensities measured in ${\mathrm{Pb}}_2\mathrm{VO}{\left({\mathrm{PO}}_4\right)}_2$ at $T<0.1$ K on the MACS spectrometer. Each panel corresponds to a different energy transfer. (f)–(j): Spin wave theory simulations using fitted parameter values in Table 1 and a convolution with the calculated instrument resolution as described in the text. The red rectangle shows the Brillouin zone used in the fit.

Figure 5

(a)–(c) False color plots of inelastic neutron scattering intensities measured at $T<0.1$ K on IN12 [24]. Red symbols are peak positions in individual scans as obtained through the procedure described in the text. Solid lines show the spin wave dispersion calculated using parameter values in Table 1. (d) Constant-Q scans at the Brillouin zone center and boundary. Solid lines are fits as described in the text.

Figure 6

The frequency-field diagram of ESR excitations measured at $T=1.5$ K for different orientations of magnetic field. Solid lines are the fit results (see text for the details) and the dashed line is a guide to the eye. The zero-field gap as observed through inelastic neutron scattering is shown by a gray square.