Understanding intensities of angle-resolved photoemission with circularly polarized radiation from a Cu(111) surface state

The Cu(111) Shockley surface state has been widely measured by photoemission spectroscopy as a prototypical spectroscopy experiment to show parabolic dispersion and energy resolution. Intensity asymmetries have been observed but never interpreted for lack of systematic evidence and theoretical background. Here we report an extended study of the angle-resolved photoemission intensity of the Cu(111) Shockley surface state as measured in different experimental geometries, variable synchrotron light polarization, and at different photon energies in the 18–70 eV range. Strong asymmetries appear with respect to the $\Gamma$ point. We provide an interpretation of the observed intensity asymmetry based on the symmetry of the face-centered-cubic lattice and by means of density-functional theory calculations: we conclude that the origin of the asymmetry is connected with the contribution of the $p$-to-$d$ photoemission channel to the total photoemission intensity, opening an unexpected view on final-state matrix element effects.

Figure 1

(Color online) Sketch of the experimental geometries. The photon beam (violet or dark gray arrow), the electron momentum (red or light gray arrow), and the surface normal are kept in the same scattering plane (fine green line). The thinly and the coarsely dashed lines represent the crystal high-symmetry planes, respectively: the finely dashed ones represent mirror planes and coarsely dashed ones represent nonmirror planes of the Cu(111) surface. In (a) a mirror plane coincides with the scattering plane (photoemission measures bands along the $\Gamma \text{−}M$ direction) so that the dipole selection rules for the matrix elements are valid. This is not the case for (b) since no mirror plane is parallel to the scattering plane (ARPES measures taken along the $\Gamma \text{−}K$ direction). Geometry (a) is changed to (b) by rotating the crystal about the surface normal by $30°$.

Figure 2

(Color online) Angle-resolved photoemission maps of the Cu(111) surface state measured with $h\nu =18\text{eV}$, and (a) $p$ and (b) $s$ polarizations plotted in a color scale ranging from blue (lowest intensity), white, yellow, red, and black (highest intensity). The intensity ratio between the $s$-to-$p$ polarization data is a function of momentum (or binding energy) but it assumes a maximum value of 6% that can be addressed to incomplete $s$-polarized radiation.

Figure 3

(Color online) Selection of experimental angle-resolved photoemission maps of the Cu(111) Shockley surface state at different photon energies and polarizations. As noticed the PID follow a complex behavior due to matrix element effects. This kind of intensity distribution is not expected on the basis of a free-electron final-state approximation.

Figure 4

(Color online) The ARPES map taken with the $\Gamma \text{−}K$ direction coincident with the scattering plane for the two circular polarizations. No spectral weight transfer is observed as expected since the mirror plane containing $\Gamma \text{−}M$ is perpendicular to the scattering plane forbidding a left-to-right asymmetry in the PID.

Figure 5

(Color online) ARPES maps of the Cu(111) surface state measured with linear $p$ polarization at three photon energies ($h\nu =62$, 70, and 75 eV) and three angles: $95°$ (left panel), $90°$ (central panel), and $88°$(right panel), yielding always a nonsymmetric spectral weight, with more intensity on the right side of the mirror plane (positive value of ${\mathbf{k}}_{\parallel }$). The intensity at the sides of the surface state comes from the bulk-derived $sp$ bands that approach by means of $k$-perpendicular dispersion of the surface states at high photon energy.

Figure 6

(Color online) Angle-resolved photoemission maps of the Cu(111) surface state calculated at the same photon energies as the experimental data of Fig. 3 for right circular polarized ($C^{+}$, left side) and left circular polarized lights ($C^{-}$, right side). Strong variation in the PID are observed in the calculated spectra albeit the correspondence to the experiment is only qualitative as explained in the text

Figure 7

(Color online) Angle-resolved photoemission maps of the Cu(111) surface state calculated at $h\nu =50$ (first and second row), 55 (third row), and 65 eV (fourth row) for right circular polarized ($C^{+}$, left side) and left circular polarized lights ($C^{-}$, right side). In the first two panels from the top, the photoemission maps are calculated suppressing all the $d$ bands in the initial states (truncated basis set). This is also the reason for the difference in the binding energy and effective mass with respect to the calculations including all the possible electron bands (truncated basis set), as in the other six photoemission maps. In the calculations done with the full basis set the centers of gravity of the PID progressively separate from each other when the photon energy is increased while those with the truncated basis set has no asymmetry observed.

Figure 8

(Color online) Valence-band data measured along the $\Gamma \text{−}M$ direction using two photon energies (18 eV in the top part and 31 eV in the bottom part), and $C^{+}$ (left part) and $C^{-}$ (right part) circular polarizations. The PID is affected by strong matrix elements which are photon, energy, and angle dependent.