Fig. 1
From: Long-living carriers in a strong electron–phonon interacting two-dimensional doped semiconductor
Schematic picture of a coupled electron–phonon system. a A bare electron with a parabolic dispersion, represented by a solid black line (\(\epsilon^0_{\mathrm{k}}\)), interacts with a dispersionless phonon, represented by a dotted line (\(\omega _0\)). The electron decay processes by phonon emission are energetically allowed only for electrons above \(\omega _0\) (red background). Therefore, the situation is completely different for electrons below this energy range, though even there, quantum field laws allow for virtual excitations of phonons. The result is that the coupling produces two excitation branches \(E_1^{{\mathrm{qp}}}\) and \(E_2^{{\mathrm{qp}}}\), whose dispersions are represented by blue and red solid lines, respectively. Fixing the electron momentum at k0 (dashed line in (a)), one obtains a spectral function with two peaks at real frequencies (b) corresponding to the above pair of bands. Looking at the complex frequency plane, these two excitations are traced back to poles of the electron Green’s function and their position in the complex plane determines the renormalized energy (Eqp) and lifetime broadening (\({\mathrm{\Gamma }}^{{\mathrm{qp}}}\)). When most of the spectral function (coherent part) is recovered from the superposition of the separate contributions of the poles, one may say that a multiple quasi-particle picture is valid. c Electrons below \(\omega _0\) are strongly renormalized and tend to localize as they are followed by a dense virtually emitted phonon cloud. d On the contrary, the higher energy band is energetically allowed to emit phonons and acquires a more extended character and a lighter effective mass
