Phonon-induced electronic relaxation in a strongly correlated system: The Sn/Si(111) $(\sqrt{3}\times \sqrt{3})$ adlayer revisited

The ordered adsorbate layer Sn/Si(111) ($\sqrt{3}\times \sqrt{3}$) with coverage of one third of a monolayer is considered as a realization of strong electronic correlation in surface physics. Our theoretical analysis shows that electron-hole pair excitations in this system can be long lived, up to several hundred nanoseconds, since the decay into surface phonons is found to be a highly nonlinear process. We combine first-principles calculations with help of a hybrid functional (HSE06) with modeling by a Mott-Hubbard Hamiltonian coupled to phononic degrees of freedom. The calculations show that the Sn/Si(111) ($\sqrt{3}\times \sqrt{3}$) surface is insulating and the two Sn-derived bands inside the substrate band gap can be described as the lower and upper Hubbard band in a Mott-Hubbard model with $U=0.75$ eV. Furthermore, phonon spectra are calculated with particular emphasis on the Sn-related surface phonon modes. The calculations demonstrate that the adequate treatment of electronic correlations leads to a stiffening of the wagging mode of neighboring Sn atoms; thus, we predict that the onset of electronic correlations at low temperature should be observable in the phonon spectrum, too. The deformation potential for electron-phonon coupling is calculated for selected vibrational modes and the decay rate of an electron-hole excitation into multiple phonons is estimated, substantiating the very long lifetime of these excitations.

Figure 1

(a) Top view of Sn/Si(111). Si atoms are shown in yellow, Sn atoms as pink balls. The $\sqrt{3}\times \sqrt{3}$ and $2\sqrt{3}\times \sqrt{3}$ unit cells for antiparallel spin arrangement are indicated by the full pink and the dashed black polygon. (b) Brillouin zone of $\sqrt{3}\times \sqrt{3}$ (full lines, rhombic shape or hexagon) and $2\sqrt{3}\times \sqrt{3}$ (dashed lines). In reciprocal lattice coordinates, we have $\overline{M}=(0,1/2),\overline{K}=(-1/3,1/3),\overline{K^{\prime }}=(1/3,2/3)$. For band plotting later on, the $k$-point path marked by the filled triangle $\overline{\mathrm{\Gamma }}\text{−}\overline{K}\text{−}\overline{M}\text{−}\overline{\mathrm{\Gamma }}$ is used.

Figure 2

Phonon dispersion for $\sqrt{3}\times \sqrt{3}$ Sn/Si(111) obtained with the PBE functional in a non-spin-polarized calculation.

Figure 3

The electronic band structure for AFM spin ordering obtained with the PBE functional including spin-orbit coupling. Calculations are performed in a $(2\sqrt{3}\times \sqrt{3})$ unit cell with two Sn atoms. The red (mostly spin-up) and blue (mostly spin-down) bands are related to different superpositions of spin-wave functions.

Figure 4

Side view of Sn (pink spheres) on Si(111) (yellow spheres). Atomic displacements for selected phonon modes with $q=0$ of Sn/Si(111) ($2\sqrt{3}\times \sqrt{3}$) are indicated by the arrows. Calculations are performed with PBE for the AFM state. The frequencies (energies) of the modes shown in (a), (b), (c), (d), (e), and (f) at $q=0$ are 1.77 THz (7.35 meV), 1.70 THz (7.04 meV), 1.62 THz (6.71 meV), 1.60 THz (6.65 meV), 0.77 THz and 0.76 THz (both 3.16 meV), respectively. (a) and (c) are the torsion and the wagging mode, respectively.

Figure 5

The electronic band structure for AFM spin ordering obtained with the HSE06 functional including spin-orbit coupling. Calculations are performed in a $(2\sqrt{3}\times \sqrt{3})$ unit cell with two Sn atoms. The red and blue bands are related to (mostly) spin-up and spin-down wave functions.

Figure 6

Energy as function of displacement along the wagging mode at $q=0$ for $(2\sqrt{3}\times \sqrt{3})$ with the AFM ordering of the electron spins taken into account. The displacement is given in units of the oscillator length.

Figure 7

Electronic band structure of Sn/Si(111) ($2\sqrt{3}\times \sqrt{3}$) calculated with the HSE functional for displaced structures. The displacements for (a) the wagging mode and (b) the torsional mode are taken from Fig. 4. By comparing to the band dispersion of the undistorted structure, Fig. 5, the electron-phonon coupling parameter $\lambda$ can be extracted.