Soon after the discovery of high-T_c cuprate superconductors (HTSC), we started Muon Spin Relaxation (MuSR) measurements of the magnetic field penetration depth λ and found a nearly linear relationship between the superfluid density (SFD) n_s/m* ∝ 1/λ² and T_c in HTSC [1] and other unconventional superconductors (UCSC) [2,3]. We then derived effective Fermi energy using the SFD [2] and compared the results with BEC-BCS crossover [2,3]. In this talk, we examine correlations among key energy-scale parameters, including T_c, SFD n_s/m*, effective Fermi energy ε_F, excitation energy of the magnetic resonance mode (MRM) [3], onset temperatures of the Nernst effect and the light-induced transient superconductivity [4], and the spin fluctuation energy scale ℏω_sf [3]. We discuss effects of competing order to suppress superconducting T_c by resorting to analogy between rotons in the superfluid ⁴He and the MRM in UCSC [3,4], and draw attention to the developments of “local phase coherence” among preformed pairs in the underdoped cuprates [4,5] well above Tc but well below the pair-formation temperature T*. In the overdoped / pressurized side of HTSC and other UCSC, SFD and heat capacity show signatures of coexisting unpaired fermions and condensed bosons in the ground state, which is very different from simple BCS expectations. We will discuss a microscopic phase separation (Swiss Cheese Model) and a resonance of spin (ℏω_sf) and charge (ε_F) energy scales to account for this behavior [5].

[1] Y.J. Uemura et al., Phys. Rev. Lett. 62 (1989) 2317

[2] Y.J. Uemura et al., Phys. Rev. Lett. 66 (1991) 2665

[3] Y.J. Uemura, J. Phys.: Condens. Matter 16 (2004) S4515

[4] Y.J. Uemura, Phys. Rev. Materials 3, 104801 (2019)