Elusive electron-phonon coupling in quantitative analyses of the spectral function

We examine multiple techniques for extracting information from ARPES data, and test them against simulated spectral functions for electron-phonon coupling. We find that, in the low-coupling regime, it is possible to extract self-energy and bare-band parameters through a self-consistent Kramers-Kronig bare-band fitting routine and verify the momentum independence of the self-energy along the quasiparticle dispersion. We also show that the effective coupling parameters deduced from the renormalization of quasiparticle mass, velocity, and spectral weight are momentum dependent and, in general, distinct from the true microscopic coupling; the latter is thus not readily accessible in the quasiparticle dispersion revealed by ARPES.

Figure 1

(Color online) (a) $A\left(k,\omega \right)$ calculated within ${\text{MA}}^{\left(1\right)}$ for $\Omega =50\text{meV}$ and $\lambda =0.5$; the quasiparticle dispersion ${\epsilon }_k^q$ and the bare band ${\epsilon }_k^b$ are also shown. (b) Quasiparticle and bare-band velocities, $v_k^q$ and $v_k^b$, and (c) corresponding inverse masses, $1/m_k^q$ and $1/m_k^b$, according to the definitions $v_k=\partial {\epsilon }_k/\partial k$ and $1/m_k={\partial }^2{\epsilon }_k/\partial k^2$. (d) Momentum-dependent quasiparticle renormalization as obtained from $v_k^b/v_k^q$, $m_k^q/m_k^b$, and the inverse quasiparticle coherence $1/Z_k^q$, where $Z_k^q={\int }^{q}A\left(k,\omega \right)d\omega$ is the quasiparticle-only integrated spectral weight; in the inset, these quantities are compared near $k=0$ to the renormalization factors $\Omega /W^q$ and $\left(1+\lambda \right)$, obtained from quasiparticle bandwidth $W^q$ and dimensionless coupling $\lambda =g^2/2t\Omega$ in our model.

Figure 2

(Color online) Renormalization parameters defined as in Fig. 1, plotted vs the dimensionless coupling $\lambda =g^2/2t\Omega$. Note that the noise in $v$ and $1/Z$ at $k=k_i$ originates from the determination of the inflection point $k_i$.

Figure 3

(Color online) (a) $A\left(k,\omega \right)$ calculated within ${\text{MA}}^{\left(1\right)}$ for $\Omega =50\text{meV}$ and $\lambda =0.1$; also shown are the $k_m$ path of MDC maxima, as well as the known bare band and the one found through the KKBF analysis. (b) and (c) Real and imaginary part of the self-energy from the model $\left({\Sigma }_{known}\right)$, the bare-band and MDC fitting routine $\left({\Sigma }_{\text{MDC}}\right)$, and the KK transform of ${\Sigma }_{\text{MDC}}^{″}\left({\Sigma }_{\text{KK}}^{\prime }\right)$ and ${\Sigma }_{\text{MDC}}^{\prime }\left({\Sigma }_{\text{KK}}^{″}\right)$. In (c), the MDC ratio results, ${\Sigma }_{ratio}^{″}=-\Delta k_m/A_0$, are also shown.

Figure 4

(Color online) (a) Various estimates for the imaginary part of the self-energy, defined as in the caption of Fig. 3, for $A\left(k,\omega \right)$ calculated within ${\text{MA}}^{\left(1\right)}$ for $\Omega =50\text{meV}$ and $\lambda =0.5$. (b) Deviation (i.e., average of the squared difference at each $k_m$) between estimated and known self-energies vs $\lambda$.