Quantitative characterization of the nanoscale local lattice strain induced by Sr dopants in La$_{1.92}$Sr$_{0.08}$CuO$_4$

The nanometer scale lattice deformation brought about by the dopants in high temperature superconducting cuprate La$_{2-x}$Sr$_x$CuO$_4$(x=0.08) was investigated by measuring the associated X-ray diffuse scattering around multiple Bragg peaks. A characteristic diffuse scattering pattern was observed, which can be well described by continuum elastic theory. With the fitted dipole force parameters, the acoustic type lattice deformation pattern was re-constructed and found to be of similar size to lattice thermal vibration at 7 K. Our results address the long-term concern of dopant introduced local lattice inhomogeneity, and show that the associated nanometer scale lattice deformation is marginal and cannot, alone, be responsible for the patched variation in the spectral gaps observed with scanning tunneling microscopy in the cuprates.