Band structure effects in surface second harmonic generation: The case of Cu(001)

We have performed a study of simultaneous nonlinear optical second harmonic generation (SHG) and angle-resolved nonlinear photoemission at the Cu(001) surface excited by fundamental photon energies near 3 eV. At these excitation energies, we identify a dominant contribution to the SHG signal that is determined by a two-photon resonance between the $d$ bands and the $n=1$ image-potential state at the Cu(001) surface. The near-resonant behavior of SHG and nonlinear photoemission is studied via Cs adsorption, which allows to systematically lower the work function and thus the reference vacuum level of the image potential. Comparison of the angle-resolved photoemission signal, arising from a restricted region of electronic momentum space, with the simultaneous optical SHG signal allows to identify an additional contribution of electronic states with nonzero surface-parallel momentum to the SHG signal enhancements. A simple model accounting for the observed behavior is developed and the implications for the quantitative understanding of SHG are discussed.

Figure 1

Multiphoton spectra of the Cu(001) surface, before (gray symbols) and after (black symbols) the deposition of 0.1 monolayer of Cs. The spectra were measured under normal-emission geometry, with 3.14 eV photon energy.

Figure 2

Normalized total 3PPE yield in normal-emission geometry (open diamonds), normalized $p_{\text{in}}\text{−}{p}_{\text{out}}$ SHG (black circles), and normalized $s_{\text{in}}\text{−}{p}_{\text{out}}$ SHG (gray circles) as a function of the Cs-induced change in work function $\Delta E_V$. The excitation energy was 2.99 eV (top left), 3.03 eV (top right), 3.09 eV (bottom left), and 3.14 eV (bottom right).

Figure 3

(Color online) Schematic representation of a section of the surface BZ of a clean and cesiated Cu(001). Background panel: surface-projected bulk band structure (shaded area) along the $\overline{\Gamma X}$ direction. The IP state on the clean (cesiated) surface is depicted as the solid black (orange) line. Forefront panels: Cu(001) band structure as a function of perpendicular momentum, after Burdick (Ref. 36) for $k_{\parallel }=0$ (right) and $k_{\parallel }=0.35{\text{Å}}^{-1}$ (center). Possible optically induced transitions at $\omega \left(2\omega \right)$ frequency are represented by the red (blue) vertical lines. 3PPE is schematically indicated by stacking a black vertical arrow on top of two $1\omega$ transitions.

Figure 4

(Color online) Top panel: energy diagram of the electronic states considered for the model calculation of ${\chi }_{ijm}^{\left(2\right)}$. The Cs-induced shift of the IP-state energy is pictorially represented. Vertical lines represent light-induced electronic transitions for different values of parallel momentum. Bottom panel: calculated dependence of ${\chi }_{ijm}^{\left(2\right)}$ as a function of $\Delta E_V$ performed for $k$-space integration over $k_{x\left(y\right)}∊\left[-0.5,0.5\right]{\text{Å}}^{-1}$ (solid black line) and restricted to $\overline{\Gamma }$ (dashed black line). Here $k_{x\left(y\right)}$ are the two components of the surface parallele momentum $k_{\parallel }$. Gray line: calculated value of ${\chi }_{ijm}^{\left(2\right)}$ upon inclusion of a phenomenological adsorbate-induced broadening of the IP state.