Strain amplification of the 4k$_F$ chain charge instability in Sr$_{14}$Cu$_{24}$O$_{41}$

We have used resonant soft x-ray scattering (RSXS) to study the misfit strain in Sr$_{14}$Cu$_{24}$O$_{41}$ (SCO), a cuprate that contains both doped spin ladders and spin chains, as well as a "control" sample without holes, La$_{6}$Ca$_{8}$Cu$_{24}$O$_{41}$ (LCCO). The misfit strain wave in SCO is strongly temperature (T)-dependent and is accompanied by a substantial hole modulation. In LCCO the strain wave is weaker, shows no hole modulation, and is T-independent. The observed strain wave vector, $L_c=0.318$, is close to the 4k$_F$ instability of the chain. Our results indicate that the chain charge order observed in SCO by several groups is a 4k$_F$ charge density wave (CDW) amplified by the misfit strain in this material. This demonstrates a new mechanism for CDW formation in condensed matter and resolves several contraversies over the transport properties of SCO.