Simulation of two-photon photoemission from the bulk $sp$-bands of Ag(111)

Theoretical and experimental studies on the two-photon photoemission excited by femtosecond laser pulses from the $sp$-band of Ag(111) in normal direction are reported. At low temperatures, the two-photon photoemission spectrum is found to consist mainly of nonresonant direct transitions from the lower to the upper $sp$-band across the $L$-projected band gap involving the upper $sp$-band as virtual intermediate states. This direct photoemission signal is calculated from the Ag band structure using optical Bloch equations and dipole moments derived from the nearly free electron approximation. For finite temperatures, a minor additional temperature-dependent, continuous contribution to the two-photon photoemission spectrum is experimentally determined. This background due to the secondary electrons is theoretically modeled combining the Debye model and Fermi Liquid Theory to account for electron relaxation effects during the two-photon photoemission process.

Figure 1

The $sp$-band dispersions of silver along the $\Gamma \text{−}L$ direction showing the relevant bulk optical transitions in 1PP and 2PP. The ordinate is normalized so that $k=1$ corresponds to the electron wave number at the Brillouin zone edge, where the band gap amounts to $4.16\mathrm{eV}$. The Shockley surface state is located $\sim 60\mathrm{meV}$ below the Fermi energy.

Figure 2

The 1PP and 2PP spectra of Ag(111) in normal emission geometry taken, respectively, with 6 and $3.1\mathrm{eV}$ photon energy for the same sample. The peaks at $\sim 1.2\mathrm{eV}$ binding energy correspond to direct optical transitions from the lower to the upper $sp$-bands (see Fig. 1), while the peak near zero binding energy is the Shockley surface state.

Figure 3

The scheme for computation of the coherent 2PP signal showing the band dispersion $E\left(k_{\perp }\right)$ in direction normal to the Ag(111) surface. The first optical excitation step (photon energy $h{\nu }_1$) excites nonresonantly the state $E_m\left(k_m\right)$ of the upper $sp$-band with identical momentum to that of the initial state $E_i\left(k_i\right)$. ${\Delta }_1$ and ${\Delta }_2$ denote the detuning energies for the first transition into the virtual intermediate state and the second transition from the final state, respectively. Absorption of the second photon (photon energy $h{\nu }_2=h{\nu }_1$) induces transitions into the final state $E_f\left(k_f\right)$ above the vacuum level $E_{\mathit{vac}}$ which corresponds to the detected electron. The electron yield $Y(E_f,E_i)$ is obtained by solving the optical Bloch equations for the electron levels $E_i\left(k_i\right)$, $E_m(k_m=k_i)$, and $E_f\left(k_f\right)$. The total electron spectrum is obtained by integration over the lower band.

Figure 4

The 2PP spectra of Ag(111) measured at different temperatures from $100\text{to}500\mathrm{K}$. Increasing the sample temperature enhances the contribution of a continuous background caused by resonant absorption for $k\ne k_{\Gamma -L}$ followed by scattering of photoelectrons into the $\Gamma \text{−}L$ direction through phonon-assisted Drude absorption [phonon momentum $q$ (see inset)]. For a quantitative estimation of the background the temperature dependence is analyzed at the positions marked by arrows.

Figure 5

Analysis of the temperature-dependent background contribution due to phonon-assisted Drude absorption for the 2PP electron signal from Ag(111). The traces A-F in (a) represent the electron signal at the corresponding individual binding energies marked in Fig. 4. The solid lines are fits to the data assuming the Debye model. The dashed lines indicate the extrapolation of the almost linear behavior for $T⩾{\Theta }_D$. (b) The 2PP spectrum of Ag(111) including the estimated phonon-induced contribution (dashed line) for $250\mathrm{K}$. The latter is obtained by fitting the energy dependence deduced from Fermi Liquid Theory to the background values taken from the fits in (a) (open circles).

Figure 6

(a) Comparison of the experimental data (circles) and the simulation (solid line) for $T=100\mathrm{K}$. The Shockley surface state is added in the shape of a Gaussian function to complete the picture and is not the subject of the simulation. (b)–(d) The individual bulk contributions the 2PP spectrum in (a) from the coherent, incoherent absorption and final-state scattering processes. The right-hand scale in (b) refers to the escape function (dashed line) and provides the escape probability for an electron to be transmitted through the surface.

Figure 7

(a) The solid line corresponds to the simulated 2PP electron yield $Y(E_f,E_i)$ for particular $E_i=E_i^{\prime }$ which fulfills $E_f-E_i^{\prime }=2h\nu$ (compare Fig. 3). The FWHM of the peak is $\sim 180\mathrm{meV}$. The six dashed curves display $Y(E_f,E_i)$ for $E_i=E_i^{\prime }\pm 50\mathrm{meV}$, $E_i^{\prime }\pm 100\mathrm{meV}$, and $E_i^{\prime }\pm 150\mathrm{meV}$ for the purpose of analyzing the effect of the ultimate summation of over all $Y(E_f,E_i)$ for all $E_i$ to obtain the final 2PP yield $Y\left(E_f\right)$. (b) The 2PP spectrum for the same conditions as $Y(E_i^{\prime },E_f)$ in (a) only with the dipole element for the second transition step kept constant (FWHM $\sim 300\mathrm{meV}$). The comparison of (a) and (b) proves the narrowing of the $sp$-transition peak to be induced by momentum conservation rules leading to a strong energy dependence of the dipole element for the second transition step.