Charge-Order-Induced Sharp Raman Peak in ${\mathrm{S}\mathrm{r}}_{14}{\mathrm{C}\mathrm{u}}_{24}{\mathrm{O}}_{41}$

In the two-leg $S=1/2$ ladders of ${\mathrm{S}\mathrm{r}}_{14}{\mathrm{C}\mathrm{u}}_{24}{\mathrm{O}}_{41}$, a modulation of the exchange coupling arises from the charge order within the other structural element, the ${\mathrm{C}\mathrm{u}\mathrm{O}}_2$ chains. In general, breaking translational invariance by modulation causes gaps within the dispersion of elementary excitations. We show that the gap induced by the charge order can drastically change the magnetic Raman spectrum, leading to the sharp peak observed in ${\mathrm{S}\mathrm{r}}_{14}{\mathrm{C}\mathrm{u}}_{24}{\mathrm{O}}_{41}$. This sharp Raman line gives insight into the charge-order periodicity and hence into the distribution of carriers. The much broader spectrum of ${\mathrm{L}\mathrm{a}}_6{\mathrm{C}\mathrm{a}}_8{\mathrm{C}\mathrm{u}}_{24}{\mathrm{O}}_{41}$ reflects the response of an undoped ladder in the absence of charge order.

Figure 1

Scheme [24] of the superstructure along the chains and the ladders ($c$ axis). Ten chain units (top row) match seven ladder units (bottom row) inducing a modulation in the ladders with wave vector $Q_{\mathrm{S}}=c_{\mathrm{l}\mathrm{a}\mathrm{d}\mathrm{d}\mathrm{e}\mathrm{r}}/c_{\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{i}\mathrm{n}}=10/7=3/7+1$ (in r.l.u. of the ladder) [13]. In ${\mathrm{S}\mathrm{r}}_{14}{\mathrm{C}\mathrm{u}}_{24}{\mathrm{O}}_{41}$ the charge order (CO) implies an additional superstructure with $Q_{\mathrm{C}\mathrm{O}}=2/10$ (in r.l.u. of the chain) [4, 5, 6, 7], corresponding to a periodicity of $5c_{\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{i}\mathrm{n}}$. It is visualized (middle row) as two units of “spin–hole–spin–hole–hole” per seven rungs (grey squares denote the six holes per f.u.). This superstructure induces an additional modulation in the ladder with $Q_{\mathrm{C}\mathrm{O}}=c_{\mathrm{l}\mathrm{a}\mathrm{d}\mathrm{d}\mathrm{e}\mathrm{r}}/(5c_{\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{i}\mathrm{n}})=2/7$ (in r.l.u. of the ladder).

Figure 2

Dispersion of the elementary triplets with (thick line) and without (thin line) a modulation of ${\alpha }_{2/7}=0.15$ [cf. Eq. (5)] with $Q_{\mathrm{C}\mathrm{O}}=2/7$ (arrows). Some higher-order contributions are denoted by dashed arrows.

Figure 3

Thick lines: Raman line shapes in leg-leg polarization for $J_{\parallel }/J_{\perp }=1.25$, $J_{\mathrm{c}\mathrm{y}\mathrm{c}}/J_{\perp }=0.2$, and $J_{\perp }=1100{\mathrm{c}\mathrm{m}}^{-1}$ [20, 22] without modulation (bottom), with the structural modulation $Q_S=3/7$ and $6/7$ appropriate for ${\mathrm{L}\mathrm{a}}_6{\mathrm{C}\mathrm{a}}_8{\mathrm{C}\mathrm{u}}_{24}{\mathrm{O}}_{41}$ (middle), and with the additional charge-order modulation of ${\mathrm{S}\mathrm{r}}_{14}{\mathrm{C}\mathrm{u}}_{24}{\mathrm{O}}_{41}$ (top). Thin lines: Raman data from Ref. 14, $T=20\mathrm{K}$. A modulation-induced gap in the dispersion at ${\omega }_g$ (see Fig. 2) causes a Raman feature at $2{\omega }_g$. Additional features arise due to backfolding, e.g., the small peak at $2200{\mathrm{c}\mathrm{m}}^{-1}$ corresponds to the $S=0$ two-triplet bound state at $k=2/7$ [25].