We succeeded in growing sizable single crystals of La2-xSrxCuO4 (LSCO) having a gradient in the Sr concentration along the axis of crystal growth. These crystals were completely characterized and the gradient of x (x ∼0.03/cm), extending from x = 0.00 to 0.30, was determined from the positional dependences of local lattice constants and by inductively coupled plasma analysis. In accordance with the gradient of x, a gradient of hole concentration was also confirmed from the positional dependences of resistivity and superconducting transition temperatures. A real space boundary between the superconducting and the insulating phases was first observed as the boundary of the vortex state by scanning magnetic fields along the gradient of x using a scanning superconducting quantum interference device (SQUID) microscope. Precise measurement of the diamagnetic susceptibility of LSCO as a function of doping yielded a phase diagram that is consistent with one obtained from samples with constant x. We used these crystals for the phonon measurement with a synchrotron X-ray beam and succeeded in determining the doping dependence of the dispersion relation by changing the position of the incident beam spot on the crystals with the same experimental setup.