Relaxation of electrons in quantum-confined states in Pb/Si(111) thin films from master equation with first-principles-derived rates

Abstract

Atomically thin films of Pb on Si(111) provide an experimentally tunable system comprising a highly structured electronic density of states. The lifetime of excited electrons in these states is limited by both electron-electron (e-e) and electron-phonon (e-ph) scattering. We employ the description by a master equation for the electronic occupation numbers to analyze the relative importance of both scattering mechanisms. The electronic and phononic band structures, as well as the matrix elements for electron-phonon coupling within deformation potential theory were obtained from density functional calculations, thus taking into account quantum confinement effects. For the relaxation dynamics, the contribution of impact ionization processes to the lifetime is estimated from the imaginary part of the electronic self-energy calculated in the GW approximation. By numerically solving rate equations for the occupations of the Pb-derived electronic states coupled to a phononic heat bath, we are able to follow the distribution of the electronic excitation energy to the various modes of Pb lattice vibrations. While e-e scattering is the dominant relaxation mechanism, we demonstrate that the e-ph scattering is highly phonon-mode-specific, with a large contribution from surface phonons. At electron energies of about 0.3 eV above the Fermi surface, a 'phonon bottleneck' characteristic of relaxation in nanostructures with well-separated electronic states is observed. The time scales extracted from the simulations are compared to data from pump-probe experiments using time-resolved two-photon photoemission.

Electron dynamics in extended systems within real-time time-dependent density functional theory

“…Next, we pursue an alternative route to compute the electron-electron scattering lifetime from first principles, based on equating the scattering term to the imaginary part of the electronic self-energy, Γ nk = −2 Im {Σ(ε nk )} / . 95 Computing the imaginary part of the self energy within the GW framework provides lifetimes, using a procedure described by Ladstädter et al 86 Here we use a computationally more efficient approach by fitting −2 Im {Σ(ε nk )} to a scattering rate of the form α(ε nk − E F ) 2 , predicted by Landau's theory of the Fermi liquid. 95 We compute the imaginary part of the selfenergy by performing a G 0 W 0 calculation where the complex shift η of the Kramers-Kronig transformation 96,97 is set to a value much smaller than what is used in typical GW band structure calculations.…”

“…95 Computing the imaginary part of the self energy within the GW framework provides lifetimes, using a procedure described by Ladstädter et al 86 Here we use a computationally more efficient approach by fitting −2 Im {Σ(ε nk )} to a scattering rate of the form α(ε nk − E F ) 2 , predicted by Landau's theory of the Fermi liquid. 95 We compute the imaginary part of the selfenergy by performing a G 0 W 0 calculation where the complex shift η of the Kramers-Kronig transformation 96,97 is set to a value much smaller than what is used in typical GW band structure calculations. This allows us to accurately resolve the imaginary part of the self-energy near the Fermi energy, see inset of Fig.…”

Lead overlayer dynamics on Ni(111)

“…23 In this regard, Pb films represent an example of a strong electron-phonon (e-ph) interaction in reduced dimensions. 24,25 Pb films on topological insulator substrates have also drawn much attention, [26][27][28][29] being considered the most promising candidate for realizing topological superconductivity. 30,31 Thin lead films on transition metal surfaces are also of considerable interest.…”

Relaxation of photo-excited hot carriers beyond multi-temperature models: A general theory description verified by experiments on Pb/Si(111)

“…For instance, this can be demonstrated in the context of a Hubbard model coupled to lattice vibrations by applying the formalism of non-equilibrium Green functions 9,10 . Moreover, calculations of the self-energy related to electron-phonon interaction 11,12 or microscopically derived rate-equation models 13,14 can be used to incorporate data from firstprinciples calculations. The results of such simulations indicate that overlapping time scales are relevant for realistic systems.…”