Consistent two-lifetime model for spectral functions of superconductors

Recently it has been found that models with at least two lifetimes have to be considered when analyzing the angle-resolved photoemission data in the nodal region of the cuprates [Kondo et al., Nat. Commun. 6, 7699 (2015)]. In this paper we compare two such models. First we show that the phenomenological model used by Kondo et al. violates the sum rule for the occupation number. Next we consider the recently proposed model of the so-called Dynes superconductors, wherein the two lifetimes measure the strengths of pair-conserving and pair-breaking processes. We demonstrate that the model of the Dynes superconductors is fully consistent with known exact results, and we study in detail the resulting spectral functions. Finally, we show that the spectral functions in the nodal region of the cuprates can be fitted well by the model of the Dynes superconductors.

Figure 1

Spectral function $A_{11}(\mathbf{k},\omega )$ of an electron inside the Fermi sea according to the model (2). The widths of the electronlike branch at $\omega \approx -E_{\mathbf{k}}$ and of the holelike branch at $\omega \approx E_{\mathbf{k}}$, where $E_{\mathbf{k}}$ is the quasiparticle energy Eq. (10), essentially are determined by ${\mathrm{\Gamma }}_1$ and ${\mathrm{\Gamma }}_0$, respectively.

Figure 2

Spectral functions $A_{11}(\mathbf{k},\omega )$ of the Dynes superconductor for an electron inside the Fermi sea with ${\varepsilon }_{\mathbf{k}}=-3\overline{\mathrm{\Delta }}$. The total scattering rate ${\mathrm{\Gamma }}_n=\mathrm{\Gamma }+{\mathrm{\Gamma }}_s$ increases from the left to the right panel. The curves in each panel differ by the strength of the pair-breaking scattering rate $\mathrm{\Gamma }$, whereas ${\mathrm{\Gamma }}_n$ is kept fixed. The color coding is the same in all panels.

Figure 3

Spectral functions $A_{11}(\mathbf{k},\omega )$ of the Dynes superconductor for an electron directly at the Fermi surface ${\varepsilon }_{\mathbf{k}}=0$. The total scattering rate ${\mathrm{\Gamma }}_n=\mathrm{\Gamma }+{\mathrm{\Gamma }}_s$ increases from the left to the right panel. The curves in each panel differ by the strength of the pair-breaking scattering rate $\mathrm{\Gamma }$, whereas ${\mathrm{\Gamma }}_n$ is kept fixed. The color coding is the same in all panels.

Figure 4

Experimentally observed low-temperature symmetrized spectral functions at the Fermi level reported in Ref. [8] for optimally doped Bi2212 at angle $\varphi ={24}^{\circ }$. Also shown are least-squares fits in the region from $-100$ to +100 meV around the Fermi level which make use of Eq. (11) (red solid line) and of Eq. (1) (blue dashed line). The values of the fitting parameters are shown in the main text.

Figure 5

Real and imaginary parts of the function $H\left(\omega \right)$.

Figure 6

Functions $\tilde{\omega }\left(\omega \right),\tilde{\gamma }\left(\omega \right),\tilde{\mathrm{\Omega }}\left(\omega \right)$, and $\tilde{\mathrm{\Gamma }}\left(\omega \right)$ for a Dynes superconductor.