Spin excitation spectrum in striped bilayer compounds

The spin dynamics of bilayer cuprate compounds are studied in a basic model. The magnetic spectral properties are calculated in linear spin-wave theory for several stripe configurations which differ by the relative location of the stripes in the layers. We focus on the bilayer splitting of the magnon bands near the incommensurate low energy peaks as well as near the $\pi$ resonance, distinguishing between the odd and even channel. We find that an x-shaped dispersion near the $\pi$ resonance is generic for stripes. By comparison of our results to neutron scattering data for ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+x}$ we conclude that the stripe model is consistent with characteristic features of bilayer high-$T_c$ compounds.

Figure 1

Classical ground states for bilayer systems with parallel (upper row), shifted parallel (middle row), and perpendicular hole stripes (lower row) for a stripe spacing $p=4$. The exchange couplings of the simple model are illustrated in the lower row: AF couplings $J>0$ for nearest neighbors within the domains (bold dashed), $\lambda J$ between nearest neighbors across a hole stripe (zig-zag) and couplings $\mu J$ between spins one above the other (dashed). Frustration of exchange coupling may lead to a canting of spins (calculated for $\mu =0.09$ and $\lambda =0.07$ in the middle and bottom row, respectively). Possible magnetic unit cells are outlined by gray lines, identical gray levels of spins correspond to identical canting angles.

Figure 2

Classical ground-state phase diagrams for shifted parallel and perpendicular stripes. For $\mu <{\mu }_c\left(\lambda \right)$ the ground states show a canted planar spin pattern illustrated in Fig. 1. For $\mu >{\mu }_c\left(\lambda \right)$ the topology of the ground states changes into a collinear pattern where spins lying on top of each other are strictly antiparallel and nearest neighbors across a stripe are parallel. For $\lambda \to {\lambda }_c$ (${\lambda }_c\approx 0.59$ for shifted parallel and ${\lambda }_c\approx 0.35$ for perpendicular stripes) ${\mu }_c$ goes to infinity. Above ${\lambda }_c$ the ground states are always planar.

Figure 3

Upper row: Energy per lattice site in units of $J{S}^2$ as a function of $\mu$ for $\lambda =0.1$. For both stripe configurations the energies of the canted planar and the collinear spin pattern are plotted. The curves intersect at $\mu ={\mu }_c$ where the topology of the ground states changes. Lower row: Relative values of the tilting angles of the spins in the planar configuration as a function of the interlayer coupling $\mu$ for $\lambda =0.1$.

Figure 4

Band structure and spectral weight along the $(H,0.5,L^{\pm })$ and $(0.5,H,L^{\pm })$ directions for parallel stripes lying on top of each other and couplings $\lambda =0.15$ and $\mu =0$,$0.08$. The last row shows the band structure of a twinned sample (see text). $L^{+}$ corresponds to even, $L^{-}$ to odd excitations. Darker and larger points correspond to a larger weight of the inelastic structure factor.

Figure 5

Band structure and spectral weight along the $(H,0.5,L^{\pm })$ and $(0.5,H,L^{\pm })$ directions for shifted parallel stripes and couplings $\lambda =0.15$ and $\mu =0,0.08$. The last row shows the resulting band structure of a twinned sample.

Figure 6

Band structure and spectral weight for even (left panel) and odd (right panel) excitations along $(H,0.5,L^{\pm })$ direction for perpendicular stripes with couplings $\lambda =0.15$ across the stripes and interlayer couplings $\mu =0.04$ (upper row) and $\mu =0.08$ (lower row).

Figure 7

Schematic illustration of the band splitting in the vicinity of the antiferromagnetic wave vector $(1∕2,1∕2)$ along the $(H,1∕2)$ direction. In the cases of parallel stripes the band structures for twinned samples are shown. Even and odd bands are gathered together.

Figure 8

Splitting of the resonance energy as a function of the interlayer coupling $\mu$ for $\lambda =0.15$. In the odd channel $\left(L=L^{-}\right)$ the spectral weight is concentrated at ${\omega }_{\pi }^{-}$ and no intensity is found at ${\omega }_{\pi }^{+}$, in the even channel $\left(L=L^{+}\right)$ no excitations at ${\omega }_{\pi }^{-}$ are observable and the spectral weight is concentrated at ${\omega }_{\pi }^{+}$. For shifted and perpendicular stripes in both channels a small intensity is found at the intermediate energy ${\omega }_{\pi }^0$.

Figure 9

Band structure in the collinear regime $\mu >{\mu }_c\left(\lambda \right)$ along the direction $(H,1∕2,L^{\pm })$ for shifted parallel and perpendicular stripes with $\lambda =0.07$ and $\mu =0.50$.