One- and Two-Triplon Spectra of a Cuprate Ladder

We have performed inelastic neutron scattering on the near ideal spin-ladder compound ${\mathrm{La}}_4{\mathrm{Sr}}_{10}{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}$ as a starting point for investigating doped ladders and their tendency toward superconductivity. A key feature was the separation of one-triplon and two-triplon scattering. Two-triplon scattering is observed quantitatively for the first time and so access is realized to the important strong magnetic quantum fluctuations. The spin gap is found to be $26.4\pm 0.3\mathrm{meV}$. The data are successfully modeled using the continuous unitary transformation method, and the exchange constants are determined by fitting to be $J_{\mathrm{leg}}=186\mathrm{meV}$ and $J_{\mathrm{rung}}=124\mathrm{meV}$ along the leg and rung, respectively; a substantial cyclic exchange of $J_{\mathrm{cyc}}=31\mathrm{meV}$ is confirmed.

Figure 1

Upper part: Arrangement of ladder planes in ${\mathrm{La}}_4{\mathrm{Sr}}_{10}{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}$. Big spheres represent the copper ions, small spheres oxygen atoms. The dashed rectangle displays unit cell in the (010) crystallographic plane for the $a$ axis and half the unit cell in the $c$ axis. Lower part: $S=1$ excitation spectrum. Gap from the $S=0$ ground state to the $S=1$ triplon band (red solid curve). Two-triplon continuum states bounded by the two blue dashed curves can be constructed via the scattering of the two triplons illustrated by the arrows. Typical wave-vector (horizontal) and energy (vertical) cuts are indicated.

Figure 2

(a) Cut along the rung wave vector shows the intensity modulation. Upper part: one-triplon scattering. Cut for energy of $30\pm 5\mathrm{meV}$ and leg wave vector of $0.45\pm 0.05$ in the rung direction. Lower part: two-triplon cut for energy of $300\pm 10\mathrm{meV}$ and leg wave vector of $0.25\pm 0.05$. Solid line described in text. (b) Typical wave-vector cut through one-triplon along $Q_c$ for energy of $130\pm 5\mathrm{meV}$. The solid line is a Gaussian fit to extract peak position. (c) Energy cut for $0.48<Q_c<0.52$ to determine the energy gap. Light gray points are the corresponding cut for a hole-doped ladder. (d) Experimental points from neutron data. Theoretical one-triplon dispersion curves shown by solid red and black curves. Black curve includes interladder coupling. Theoretical lower boundary of two-triplon scattering shown by solid blue curve. Dashed lines show respective scattering without cyclic exchange.

Figure 3

(a) Measured one-triplon data with nonmagnetic background subtracted. The white curve gives the theoretically calculated one-triplon dispersion curve. (b) Calculated one-triplon scattering with corrections for instrumental resolution and magnetic form factor. (c) Measured two-triplon data with nonmagnetic background subtracted. The white curve gives the lower boundary of the two-triplon dispersion curve. (d) Calculated two-triplon scattering.

Figure 4

(a) Energy cut for two triplon at $Q_c=0.5\pm 0.1$. The solid line is the two-triplon model. Intensities had to be increased by 15% to match data. Light gray points are the corresponding cut for a hole-doped ladder. The thin dashed line is the effect of absent cyclic exchange for fixed $J_{\mathrm{rung}}$. (b) Relationship of ${\omega }_{\mathrm{onset}}$ and $J_{\mathrm{rung}}$ with respect to $x_{\mathrm{cyc}}$. For an onset energy of 225 meV, $J_{\mathrm{rung}}=124\mathrm{meV}$ and $x_{\mathrm{cyc}}=0.25$ are needed (see text). $J_{\mathrm{cyc}}=x_{\mathrm{cyc}}{J}_{\mathrm{rung}}$. (c) Frustration of bound mode by cyclic exchange. Binding energy ${\omega }_{\mathrm{binding}}$ is the difference between ${\omega }_{\mathrm{onset}}$ and energy of the bound mode.