Magneto-optic far-infrared study of ${\mathrm{Sr}}_{14}{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}$: Triplet excitations in chains

Using far-infrared spectroscopy, we have studied the magnetic field and temperature dependence of the spin gap modes in the chains of ${\mathrm{Sr}}_{14}{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}$. Two triplet modes $T_1$ and $T_2$ were found in the center of the Brillouin zone at ${\Delta }_1=9.65\mathrm{meV}$ and ${\Delta }_2=10.86\mathrm{meV}$ in zero magnetic field. The $T_1$ mode was excited when the electric field vector $\mathbf{E}$ of the light was polarized along the $b$ axis (perpendicular to the planes of chains and ladders) and $T_2$ was excited for $\mathbf{E}\Vert \mathbf{a}$ (perpendicular to the chains and along the rungs). Up to the maximum magnetic field of $18\mathrm{T}$, applied along the chains, the electron $g$ factors of these two modes were similar, $g_{1c}=2.049$ and $g_{2c}=2.044$. Full linewidth at half maximum for both modes was $1{\mathrm{cm}}^{-1}$ $\left(0.12\mathrm{meV}\right)$ at $4\mathrm{K}$ and increased with $T$. The temperature dependence of mode energies and line intensities was in agreement with the inelastic neutron scattering results from two groups [M. Matsuda et al., Phys. Rev. B 59, 1060 (1999); L. P. Regnault et al., ibid 59, 1055 (1999)]. The $T_1$ mode has not been observed by inelastic neutron scattering in the points of the $k$ space equivalent to the center of the Brillouin zone. Our study indicates that the zone structure model of magnetic excitations of ${\mathrm{Sr}}_{14}{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}$ must be modified to include a triplet mode at $9.65\mathrm{meV}$ in the center of the magnetic Brillouin zone.

Figure 1

Differential absorption spectra of triplet $T_1$ in magnetic field applied in the $c$ axis direction with light $k$ vector $\mathbf{k}\Vert \mathbf{c}$ and polarization ${\mathbf{E}}_1\Vert \mathbf{b}$ at $4.4\mathrm{K}$. The spectra are shifted in the vertical direction by $0.2B_0^i$. Numbers beside absorption lines give values of the applied field $B_0^i$ and reference field $B_0^j$. Dashed lines are eye guides. The shaded area covers the spectral region of strong phonon absorption.

Figure 2

Differential absorption spectra of triplet $T_2$ in magnetic field applied in the $c$ axis direction with light $k$ vector $\mathbf{k}\Vert \mathbf{c}$ and polarization ${\mathbf{E}}_1\Vert \mathbf{a}$ at $4.4\mathrm{K}$ measured using the $ab$-plane crystal. The spectra are offset in vertical direction by magnetic field value $B_0$. The inset shows differential absorption spectra for the $ac$-plane crystal in Faraday (dashed line) and Voigt (solid line) configurations. Triplet $T_2(+)$ is not infrared active when ${\mathbf{B}}_0\Vert \mathbf{b}$, as shown by the dashed line for an arbitrary chosen $B_0=5\mathrm{T}$.

Figure 3

Magnetic field ${\mathbf{B}}_0\Vert \mathbf{c}$ dependence of line areas and positions. (a) Line areas of triplet modes (left axis) and FIR absorption spectra (right axis) at temperature of $4.4\mathrm{K}$ for ${\mathbf{E}}_1\Vert \mathbf{a}$ and ${\mathbf{E}}_1\Vert \mathbf{b}$ polarizations. The solid line is the dynamic Dzyaloshinskii-Moriya interaction model fit of the $T_1$ absorption line area. (b) $T_1$ and $T_2$ line positions as a function of magnetic field $B_0$. The inset schematically shows the observed transitions.

Figure 4

Magnetic field dependence of the paramagnetic transition energy in ${\mathbf{B}}_0\Vert \mathbf{c}$ orientation. The line is a fit to the data yielding $g$ factor $g_c=2.038\pm 0.016$. The inset demonstrates the paramagnetic transition signal in differential absorption spectrum. The linewidth is resolution limited by instrument to $0.3{\mathrm{cm}}^{-1}$.

Figure 5

Temperature dependence of triplet modes: (a) $T_2(+)$ in $10\mathrm{T}$ field, (b) $T_1$ in zero magnetic field, (c) line positions, and (d) observed normalized transition intensities; calculated singlet-triplet transition line area assuming two triplets (solid line) or four triplets in the folded zone structure (dashed line). The INS data, Ref. 9, are at $\mathbf{k}=(2,0,-0.1)$.

Figure 6

Dispersion curves of the magnetic excitations in the chains (Refs. 9, 10) and the observed triplet modes from FIR. The dash-dotted lines are extra dispersion curves, derived from data by Regnault et al. (Ref. 10), after doubling the unit cell in the $c$ axis direction. INS energy resolution is shown by error bars; FIR transitions have linewidth less than the size of experimental points on the graph.