Selective electron-phonon coupling strength from nonequilibrium optical spectroscopy: The case of MgB$_2$

TL;DR

This paper investigates whether nonequilibrium optical spectroscopy can isolate and quantify selective electron–phonon coupling (EPC) to specific phonon modes in superconductors. By combining equilibrium optical spectroscopy, ARPES, and broadband time-resolved optical spectroscopy with an effective three-temperature model, the authors separate the contribution of strongly coupled E$_{2g}$ phonons from the rest of the lattice in MgB$_2$, and demonstrate a two-channel relaxation dynamics driven by selective EPC. They report a partial EPC strength for the E$_{2g}$ modes of about lambda_SCP ≈ 0.56, with the total EPC λ ≈ 1.1, implying that SCPs account for roughly half of the coupling and are essential to the high Tc of MgB$_2$; AlB$_2$ serves as a non-superconducting benchmark lacking such selectivity. The results establish time-resolved spectroscopy as a quantitative tool to resolve and quantify mode-selective EPC, with implications for controlling hot-phonon populations and for understanding superconductivity in anisotropic systems.

Abstract

The coupling between quasiparticles and bosonic excitations rules the energy transfer pathways in condensed matter systems. The possibility of inferring the strength of specific coupling channels from their characteristic time scales measured in nonequilibrium experiments is still an open question. Here, we investigate MgB$_2$, in which conventional superconductivity at temperatures as high as 39 K is mediated by the strong coupling between the conduction electrons and the E$_{2g}$ phonon mode. By means of broadband time-resolved optical spectroscopy, we show that this selective electron-phonon coupling dictates the nonequilibrium optical response of MgB$_2$, at early times ($<$100 fs) after photoexcitation. Furthermore, based on an effective temperature model analysis, we estimate its contribution to the total electron-boson coupling function extracted from complementary equilibrium spectroscopy approaches, namely optical reflectivity and ARPES. The coupling strength with the E$_{2g}$ phonon modes is thus estimated to be $λ\simeq$ 0.56, which is approximately half of the total coupling constant, in agreement with ab-initio calculations from the literature. As a benchmark, broadband time-resolved optical spectroscopy is performed also on the isostructural and non-superconducting compound AlB$_2$, showing that the nonequilibrium optical response relaxes on a slower time scale due to the lack of strongly-coupled phonon modes. Our findings demonstrate the possibility to resolve and quantify selective electron-phonon coupling from nonequilibrium optical spectroscopy.

Figures (3)

• Figure 1: (a) Equilibrium reflectivity of MgB$_{2}$ measured through spectroscopic ellipsometry (light blue line), and ED model fit to the data (black line). The black line is the fit to the data with an ED model and a sum of Lorentz oscillators which are located in the visible region above the plasma edge. The black horizontal bar indicates the photon energy region of the probe pulse in the tr-OS measurements, and the orange arrow the pump photon energy. The energy of the plasma edge is marked by the vertical dashed line. (b) Momentum-integrated Eliashberg function $\alpha^2\bar{F}(\omega)$ (red histograms) of MgB$_2$ used to fit the data reported in (a), and calculated phonon density of states (PDOS, gray shade) taken from Ref. Kong2001 in arbitrary units. (c) Momentum-integrated Eliashberg function (red histograms) of AlB$_2$ used to approximate the calculated one from Ref. Bohnen2001, and calculated PDOS (gray shade) also taken from Ref. Bohnen2001.
• Figure 2: (a) ARPES mapping of the MgB$_{2}$ Fermi surface. (b) ARPES dispersion of the $\sigma$ electronic bands along the $\Gamma-M$ direction; green and orange markers highlight the maxima of the momentum distribution curves (MDCs), obtained by fitting the MDCs for all binding energies (see main text for details). The dashed lines indicate the non-interacting band dispersion. (c) - (d) Real part of the self energy (markers) obtained from the ARPES data (see main text), together with the best fit according to Eq. \ref{['eq:SE']} (black lines) and relevant glue function (histograms), for the two $\sigma$ bands.
• Figure 3: $\delta R(t)/R_{eq}$ intensity maps at incident fluence of 0.36 mJ cm$^{-2}$ of (a) MgB$_{2}$ and (b) AlB$_2$ as a function of pump-probe time delay and probe photon energy. (c, d) Time-resolved line profiles at 1.3 eV (red markers) of the maps reported in (a) and (b), respectively, and relevant fits (black line) via effective-temperature modeling. The insets show the same line profiles up to 0.3 ps overlaid with the cross-correlation signal between pump and probe pulses (gray line), as measured through frequency optical gating technique DalConte2015. (e) Sketch of the temporal evolution of $\textcolor{black}{\delta} T_{el}$, $\textcolor{black}{\delta} T_{SCP}$ and $\textcolor{black}{\delta} T_{ph}$ in MgB$_{2}$ after photoexcitation. The two dynamics are controlled by $\alpha^2\bar{F}_{SCP}$ and $\alpha^2\bar{F}_{ph}$ (red and blue histogram functions). (f) Sketch of the temporal evolution of $\textcolor{black}{\delta} T_{el}$ and $\textcolor{black}{\delta} T_{ph}$, and relevant $\alpha^2\bar{F}_{ph}$ (black histogram function) in AlB$_2$.