Raman response of magnetic excitations in cuprate ladders and planes

A unified picture for the Raman response of magnetic excitations in cuprate spin-ladder compounds is obtained by comparing calculated two-triplon Raman line shapes to those of the prototypical compounds $\mathrm{Sr}{\mathrm{Cu}}_{2}{\mathrm{O}}_{3}$ (Sr123), ${\mathrm{Sr}}_{14}{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}$ (Sr14), and ${\mathrm{La}}_{6}{\mathrm{Ca}}_{8}{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}$ (La6Ca8). The theoretical model for the two-leg ladder contains Heisenberg exchange couplings ${J}_{\ensuremath{\Vert}}$ and ${J}_{\ensuremath{\perp}}$ plus an additional four-spin interaction ${J}_{\mathrm{cyc}}$. Within this model Sr123 and Sr14 can be described by $x\ensuremath{\mathrel{:=}}{J}_{\ensuremath{\Vert}}∕{J}_{\ensuremath{\perp}}=1.5$, ${x}_{\mathrm{cyc}}\ensuremath{\mathrel{:=}}{J}_{\mathrm{cyc}}∕{J}_{\ensuremath{\perp}}=0.2$, ${J}_{\ensuremath{\perp}}^{\mathrm{Sr}123}=1130\phantom{\rule{0.3em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}1}$ and ${J}_{\ensuremath{\perp}}^{\mathrm{Sr}14}=1080\phantom{\rule{0.3em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}1}$. The couplings found for La6Ca8 are $x=1.2$, ${x}_{\mathrm{cyc}}=0.2$, and ${J}_{\ensuremath{\perp}}^{\mathrm{La}6\mathrm{Ca}8}=1130\phantom{\rule{0.3em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}1}$. The unexpected sharp two-triplon peak in the ladder materials compared to the undoped two-dimensional cuprates can be traced back to the anisotropy of the magnetic exchange in rung and leg direction. With the results obtained for the isotropic ladder, we calculate the Raman line shape of a two-dimensional square lattice using a toy model consisting of a vertical and a horizontal ladder. A direct comparison of these results with Raman experiments for the two-dimensional cuprates ${\mathrm{R}}_{5}\mathrm{Cu}{\mathrm{O}}_{4}(\mathrm{R}=\mathrm{La},\mathrm{Nd})$, ${\mathrm{Sr}}_{2}\mathrm{Cu}{\mathrm{O}}_{2}{\mathrm{Cl}}_{2}$, and $\mathrm{Y}{\mathrm{Ba}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{6+\ensuremath{\delta}}$ yields a good agreement for the dominating two-triplon peak. We conclude that short-range quantum fluctuations are dominating the magnetic Raman response in both ladders and planes. We discuss possible scenarios responsible for the high-energy spectral weight of the Raman line shape, i.e., phonons, the triple-resonance, and multiparticle contributions.