Extracting the spectral function of the cuprates by a full two-dimensional analysis: Angle-resolved photoemission spectra of ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Cu}{\mathrm{O}}_6$

Recently, angle-resolved photoemission spectroscopy (ARPES) has revealed a dispersion anomaly at high binding energy near $0.3–0.5\mathrm{eV}$ in various families of the high-temperature superconductors. For further studies of this anomaly, we present a new two-dimensional fitting scheme and apply it to high-statistics ARPES data of the strongly overdoped ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Cu}{\mathrm{O}}_6$ cuprate superconductor. The procedure allows us to extract the self-energy in an extended energy and momentum range. It is found that the spectral function of ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Cu}{\mathrm{O}}_6$ can be parametrized using a small set of tight-binding parameters and a weakly momentum-dependent self-energy up to $0.7\mathrm{eV}$ in binding energy and over the entire first Brillouin zone. Moreover, the analysis gives an estimate of the momentum dependence of the matrix element, a quantity, which is often neglected in ARPES analyses.

Figure 1

(Color online) (a) Image plot of the fitted intensity from the 2D analysis of ARPES data of OD Bi2201 along the nodal direction (0,0) to $(\pi ,\pi )$ where the solid line shows a bare dispersion derived from LDA. (b) The fitting parameters for real and imaginary parts of self-energy. (c) The fitting parameters for the matrix element.

Figure 2

(Color online) On the first row, (a)–(c) are the experimental ARPES data along momentum direction as indicated by the band shown in bottom right. On the second row, (d)–(f) are the corresponding image plots of the fitted intensity from the 2D analysis. The thicker color dashed lines are the dispersions generated from the tight-binding parameters from LDA calculation (LDA) given above. The solid color lines are the matrix elements (ME) obtained from the 2D analysis where the smaller white dashed lines are the empirically guessed form of the matrix element in the form of a cosine function (COS). On the third row, (g)–(i) are the MDCs of raw data (black dots) and corresponding image plot of the fitted intensity (blue line). Taken out the matrix element effect and background, (j)–(l) are the corresponding extracted spectral function $\left[\mathcal{A}(\mathbf{k},\omega )\right]$ from the 2D analysis. (m) and (n) are the extracted real and imaginary parts of the self-energy in Eq. (2) for cuts a, b, and c; the extracted values are plotted in colors up to the energy not far from the bottom of the bare band (up to $0.6\mathrm{eV}$ for cut b and $0.25\mathrm{eV}$ for cut c) and in gray at higher energy. (○) shows the band structure generated from the tight-binding parameters and the self-energy along the nodal direction.

Figure 3

(Color online) (a) shows the experimental Fermi surface map of OD Bi2201 where the light polarization $E$ is along the nodal direction, $\Gamma \to Y$. (b) shows the Fermi surface map which is constructed from the extracted spectral function and the empirical guessed matrix element.

Figure 4

(Color online) Measured at photon energy $E_{\nu }=55\mathrm{eV}$. (a) shows the experimental ARPES data along (0,0) to $(\pi ,\pi )$ direction as indicated in (f) where the dash line shows the tight-binding band from LDA calculation used as bare dispersion. (b) shows the corresponding extracted spectral function $\left[\mathcal{A}(\mathbf{k},\omega )\right]$ from the 2D analysis. (c) show the extracted matrix elements of the data taken at $E_{\nu }=55\mathrm{eV}$ (solid line) and $42\mathrm{eV}$ (dash line) where the areas under the graphs between $k<\mid k_F\mid$ are normalized to be the same. (d) and (e) are the extracted $\mathrm{Re}\Sigma$ and $\mathrm{Im}\Sigma$ of the data taken at $E_{\nu }=55\mathrm{eV}$ (solid line) and $42\mathrm{eV}$ (dash line). (f) shows the Fermi surface map.

Figure 5

(Color online) (a) shows the real (solid line) and imaginary (dash line) parts of self-energy used to construct the spectral image shown in (b). In (b), we compare the MDC-peak (fine dash line) and EDC-peak (medium-size dash line) dispersions with the input band dispersion (solid line). We note that $\mathrm{Re}\Sigma$ and $\mathrm{Im}\Sigma$ shown in (a) satisfy the Kramers–Kronig relation where the diamond and circle symbols represent extracted real and imaginary parts of self-energy, respectively, from Figs. 2m, 2n of cut a.