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We used high-resolution angle-resolved photoemission, which can directly probe how electrons move in materials8, to investigate hole-doped materials at various dopings in five different families of cuprates. These included (La2−xSrx)CuO4 (LSCO) and (La2−xyNdySrx)CuO4 (Nd-LSCO), Bi2Sr2CaCu2O8 (Bi2212), Bi2Sr2CuO6 (Bi2201), (Ca2−xNax)CuO2Cl2 (Na-CCOC) and Tl2Ba2CuO6 (Tl2201). The LSCO system in particular covers the entire doping range (0 < x ≤ 0.3) over which the physical properties vary from insulator (0 ≤ x < 0.03) to superconductors (0.05 < x < 0.25) to overdoped non-superconducting metal (x > 0.25). Apart from the Na-CCOC data taken at the Stanford Synchrotron Radiation Laboratory, all samples were measured at the Advanced Light Source at Lawrence Berkeley National Laboratory (experimental details are presented elsewhere9,10).

Figure 1a shows the energy–momentum (dispersion) curves of the LSCO system with various dopings (0 < x ≤ 0.3), measured along the (0,0)–(π,π) direction in the Brillouin zone in reciprocal space. This diagonal direction is special in these superconductors because the anisotropic superconducting gap11, as well as the normal-state pseudogap12,13, is zero along this so-called nodal direction. Figure 1a shows that, for all dopings, there is a kink at an energy of about 70 meV that separates the dispersion into a high-energy part (that is, further from the Fermi energy) and a low-energy part (that is, closer to the Fermi energy) with different slopes. Whereas the high-energy dispersion varies with doping, the dispersion converges within about 50 meV of the Fermi energy, revealing a behaviour that is independent of doping. Correspondingly, a decrease is seen in the electron-scattering rate at an energy of about 70 meV, as indicated in Fig. 1b for the LSCO (x = 0.063) sample.

Figure 1: Electron dynamics in the (La2−xSrx)CuO4 (LSCO) system.
figure 1

a, Dispersion energy, E, as a function of momentum, k, of LSCO with various dopings (where x is between 0.03 (black circles, right curve) and about 0.30 (red circles, left) measured at a temperature of 20 K along the (0,0)–(π,π) nodal direction. The dispersion is obtained by fitting momentum-distribution curves (MDCs), which represent the photoelectron intensity as a function of momentum, for a given energy. The arrow indicates the position of the kink that separates the dispersion into high-energy and low-energy parts with different slopes. EF and kF, Fermi energy and Fermi momentum, respectively. b, Scattering rate as measured by MDC width (full width at half-maximum) of the LSCO (x = 0.063) sample measured at 20 K. The MDC width is proportionally related to the scattering rate of electrons. The arrow indicates a decrease at an energy of about 70 meV.

The electron velocity can be extracted quantitatively from the slope in dispersion, as v = ∂ε/∂k. We have obtained the low-energy velocity (the Fermi velocity) and the high-energy velocity as a function of doping for all five families of materials (see supplementary information). The Fermi velocity is nearly constant for all materials and dopings within an experimental error of about 20%. In contrast, the high-energy velocity varies strongly with doping.

This invariance of nodal Fermi velocity in cuprates is surprising, given the range of variation in many other physical properties2,3,4,5,6,7. This universal behaviour, together with the ubiquitous existence of a kink in the dispersion and a decrease in the scattering rate, are puzzles that require answers before the mystery of high-temperature superconductivity can be solved.