Electrical resistivity and scattering processes in ${(\mathrm{Bi},\mathrm{Pb})}_2{(\mathrm{Sr},\mathrm{La})}_2\mathrm{Cu}{\mathrm{O}}_{6+\delta }$ studied by angle-resolved photoemission spectroscopy

The lifetime of the Bloch states at the Fermi level, the Fermi velocity, and the mean free path of the conduction electrons in ${(\mathrm{Bi},\mathrm{Pb})}_2{(\mathrm{Sr},\mathrm{La})}_2\mathrm{Cu}{\mathrm{O}}_{6+\delta }$ (Bi2201) superconductors are determined by angle-resolved photoemission spectroscopy over a wide range of the hole concentration from heavily overdoped to lightly underdoped conditions. The electrical resistivity in the $\mathrm{Cu}{\mathrm{O}}_2$ plane is quantitatively reproduced by the calculation with the Boltzmann transport equation using the experimentally determined electronic structure. A detailed investigation on the hole concentration and temperature dependence of the lifetime leads us to conclude that the scattering process of conduction electrons in the vicinity of $(\pi ,0)$ is dominated by electron-electron scattering, whereas around the node direction it is dominated by electron-phonon scattering.

Figure 1

Temperature dependence of the electrical resistivity along the $ab$ plane, ${\rho }_{ab}\left(T\right)$, in $\mathrm{OD}7\mathrm{K}$, $\mathrm{OD}21\mathrm{K}$, $\mathrm{OP}35\mathrm{K}$, $\mathrm{UD}27\mathrm{K}$, and $\mathrm{UD}23\mathrm{K}$. The dashed lines represent the extrapolated lines of the ${\rho }_{ab}\left(T\right)$ at high temperature. The characteristic temperature $\left(T^{*}\right)$ at which the slope of ${\rho }_{ab}\left(T\right)$ changes is shown by the arrows.

Figure 2

Angle dependence of (a) Fermi velocity $\left[v_F\left(\theta \right)\right]$, (b) lifetime of Bloch states at the Fermi level $\left[{\tau }_F\left(\theta \right)\right]$, and (c) mean free path of conduction electrons $\left[l_F\left(\theta \right)\right]$ determined by the ARPES measurement at $200\mathrm{K}$ in $\mathrm{OD}7\mathrm{K}$, $\mathrm{OD}21\mathrm{K}$, $\mathrm{OP}35\mathrm{K}$, $\mathrm{UD}27\mathrm{K}$, and $\mathrm{UD}23\mathrm{K}$. The Fermi surface angle $\theta$ is defined as shown in (d). The dashed line shown in (b) represents the hypothetical lifetime $\left[{\tau }_F^{\mathit{hy}}\left(\theta \right)\right]$ calculated with an assumption of the constant mean free path $\left(l_F^{\mathit{const}}\right)$ displayed at (c). (d) Fermi surface in the first Brillouin zone. Antiferromagnetic zone boundary (AFZB) is shown by the dashed lines.

Figure 3

(a) Symmetrization method employed on $I({\mathbf{k}}_F,\omega )$ of $\mathrm{OD}7\mathrm{K}$ at $\theta =0°$. The symmetrized spectrum $I_{\mathit{sym}}({\mathbf{k}}_F,\omega )=I_0\left({\mathbf{k}}_F\right)A({\mathbf{k}}_F,\omega )$ is shown with the measured spectrum $I({\mathbf{k}}_F,\omega )=I_0\left({\mathbf{k}}_F\right)A({\mathbf{k}}_F,\omega )f\left(\omega \right)$ and the flipped spectrum $I({\mathbf{k}}_F,-\omega )=I_0\left({\mathbf{k}}_F\right)A({\mathbf{k}}_F,-\omega )f(-\omega )$. Obtained symmetrized spectra $I_{\mathit{sym}}({\mathbf{k}}_F,\omega )$’s at three ${\mathbf{k}}_F$’s corresponding to $\theta =0°$, 20°, and 45° are shown for (b) $\mathrm{OD}7\mathrm{K}$, (c) $\mathrm{OP}35\mathrm{K}$, and (d) $\mathrm{UD}27\mathrm{K}$. All spectra were measured at $200\mathrm{K}$.

Figure 4

(a) Three EDC’s of $\mathrm{OD}21\mathrm{K}$ (dashed lines) measured at $200\mathrm{K}$ along the node direction, and those EDC’s divided by the FD function with $T=200\mathrm{K}$ (solid lines). The eigenvalues of $\epsilon \left(\mathbf{k}\right)=-40\mathrm{meV}$, $-20\mathrm{meV}$, and $0\mathrm{meV}$ are determined from the peak energies and indicated by the arrows. (b) The comparison of line shape between three spectra shown in (a). The energy of each spectrum is shifted so as its peak is to be located at $0\mathrm{meV}$.

Figure 5

The electrical resistivity calculated by using the FS and $l_F$ determined by the ARPES measurement at $200\mathrm{K}$, ${\rho }_{ab}^{\mathrm{cal}}\left(200\mathrm{K}\right)$. The measured one, ${\rho }_{ab}^{\mathrm{meas}}\left(200\mathrm{K}\right)$, is superimposed.

Figure 6

Temperature dependence of the symmetrized spectrum $I_{\mathit{sym}}({\mathbf{k}}_F,\omega )$ and estimated ${\tau }_F$ for (a) $\mathrm{OD}7\mathrm{K}$, (b) $\mathrm{OP}35\mathrm{K}$, and (c) $\mathrm{UD}23\mathrm{K}$. The upper and middle panels show $I_{\mathit{sym}}({\mathbf{k}}_F,\omega )$ measured at $\theta =40°$ and $\theta =0°$, respectively. The lower panels show the temperature dependence of ${\tau }_F$ at $\theta =0°$, 20°, and 40° determined from the peak width of $I_{\mathit{sym}}({\mathbf{k}}_F,\omega )$.

Figure 7

Temperature dependence of the calculated and measured electrical resistivity ${\rho }_{ab}^{\mathrm{cal}}\left(T\right)$ and ${\rho }_{ab}^{\mathrm{meas}}\left(T\right)$ of $\mathrm{OD}7\mathrm{K}$.