LAUSR

Electron–boson spectral density functions (EBSDFs) can be obtained from measured spectra using various spectroscopic techniques, including optical spectroscopy. EBSDFs, known as glue functions, are suggested to have a magnetic origin. Here, we investigated EBSDFs obtained from the measured optical spectra of hole-doped cuprates with wide doping levels, from underdoped to overdoped cuprates. The average frequency of an EBSDF provides the timescale for the spin fluctuations to form Cooper pairs. This timescale is directly associated with retarded interactions between electrons. Using this timescale and Fermi velocity, a reasonable superconducting coherence length, which reflects the size of the Cooper pair, can be extracted. The obtained coherence lengths were consistent with those measured via other experimental techniques. Therefore, the formation of Cooper pairs in cuprates can be explained by spin fluctuations, the timescales of which appear in EBSDFs. Consequently, EBSDFs provide crucial information on the timescale of the microscopic mechanism of Cooper pair formation.