Bare electron dispersion from experiment: Self-consistent self-energy analysis of photoemission data

Performing an in-depth analysis of the photoemission spectra along the nodal direction of the high-temperature superconductor Bi-2212 we developed a procedure to determine the underlying electronic structure and established a precise relation of the measured quantities to the real and imaginary parts of the self-energy of electronic excitations. The self-consistency of the procedure with respect to the Kramers-Kronig transformation allows us to draw conclusions on the applicability of the spectral function analysis and on the existence of well-defined quasiparticles along the nodal direction even for the underdoped Bi-2212 in the pseudogap state. The analysis of the real part of the self-energy ${\Sigma }^{\prime }\left(\omega \right)$ for an overdoped and underdoped Bi-2212 helps to distinguish the $70\mathrm{meV}$ “kink” from ${\Sigma }^{\prime }\left(\omega \right)$ maximum and conclude about doping dependence of the kink strength.

Figure 1

(Color online.) Bare band dispersion (solid line) and renormalized dispersion (points) on top of the spectral weight of interacting electrons. Though intended to be general, this sketch represents the nodal direction of an underdoped Bi-2212.

Figure 2

(Color online.) The bare band dispersion along the nodal direction of an underdoped Bi(Pb)-2212 (solid parabola) on top of its spectral weight at $130\mathrm{K}$ measured by ARPES. MDC (or renormalized) dispersion shown by solid white (red) line.

Figure 3

(Color online.) Real and imaginary parts of the self-energy extracted from the experiment with the described procedure. A complete coincidence between the corresponding parts of the self-energy calculated from the two different experimental functions, the MDC dispersion and MDC width, demonstrates the full self-consistency of the ARPES data treated within the self-energy approach.

Figure 4

(Color online.) Real and imaginary parts of the self-energy extracted from the experiment with the described procedure.

Figure 5

(Color online.) The results of the fitting procedure for Bi(Pb)-2212 OD75: (a) real (odd curve) and imaginary (even curves) parts of the self-energy; (b) the experimental (solid line) and bare (dashed line) dispersions on top of the experimentally measured quasiparticle spectral weight.

Figure 6

(Color online.) The real parts of the self-energy for UD77 and OD75 samples at $130\mathrm{K}$: solid lines show the result of fitting these real parts to Eq. (6) in a frequency range $0.17\mathrm{eV}<\omega <0$ for UD77 and $0.12\mathrm{eV}<\omega <0$ for OD75; looking down arrows mark ${\Sigma }^{\prime }\left(\omega \right)$ maxima; the dashed line denotes the $70\mathrm{meV}$ “kink” energy.

Figure 7

(Color online.) Real (thin blue) and imaginary (thick red) parts of the self-energy related by Kramers-Kronig (KK) transform: ${\Sigma }^{\prime }=\mathrm{KK}{\Sigma }^{″}$, for three models of ${\Sigma }^{″}$ tails.

Figure 8

(Color online.) Illustration of the fitting procedure: real parts of the self-energy ${\Sigma }_{\mathrm{disp}}^{\prime }\left(\omega \right)$ (filled squares) obtained by (4) and ${\Sigma }_{\mathrm{KK}}^{\prime }\left(\omega \right)$ (open circles) by Eqs. (5, A1); the difference $\Delta {\Sigma }^{\prime }\left(\omega \right)={\Sigma }_{\mathrm{KK}}^{\prime }\left(\omega \right)-{\Sigma }_{\mathrm{disp}}^{\prime }\left(\omega \right)$ (small crosses) is fitted to $R^{\prime }\left(\omega \right)$ (corresponding solid line), the contribution of overall resolution determined by Eqs. (A12, A13). In the first three panels ${\omega }_0=-0.9$ but different $n=3$, 4, and 6 in Eq. (A10) are compensated by different ${\omega }_c=0.34$, 0.45, and $0.52\mathrm{eV}$, respectively. The last two panels, the “best” fitting results for slightly different ${\omega }_0$’s.

Figure 9

(Color online.) Possible particle-hole asymmetry effect on ${\Sigma }^{″}\left(\omega \right)$ (red/thick lines) and ${\Sigma }^{\prime }\left(\omega \right)$ (blue/thin lines): low-energy (dashed lines, ${\omega }_c=0.1\mathrm{eV}$) and high-energy [solid lines, by Eq. (A14)] contributions shown on the top of the symmetric self-energy (shaded areas).