Theory of circular dichroism in angle-resolved resonant photoemission from magnetic surfaces

A theoretical method is presented for angle-resolved photoemission at the transition metal $L$-edge resonance. It combines atomic multiplet calculations for the second-order resonant photoemission amplitude on the core-level site and a single scattering calculation of the photoelectron final state. The theory is applied to a magnetized Ni(111) surface excited with circularly polarized x rays at the Ni $L_{2,3}$-edge resonance with a focus on the circular dichroism (CD) signal. Good agreement with available experimental data is achieved. It is shown that the CD pattern is composed of a slowly varying magnetic signal induced by the atomic resonant process and a signal of fast angular modulations that are due to the interference of primary and scattered waves, known as the Daimon effect. The two types of CD signals are found to be nearly additive. At the Ni $L_2$-edge resonance, the angular dependence of the magnetic CD is well described by a simple expression known from x-ray magnetic CD. At the $L_3$ edge, however, the angular dependence is more complex and shows a pronounced final state multiplet dependence. With the present theory, it becomes possible to extract element- and site-selective magnetic information of surfaces from the CD in angle-resolved resonant photoemission data.

Figure 1

Cluster model used in the calculation for the Ni(111) surface. Atoms located near the central axis below the red circle are taken as electron emitters. As in the experiment in Ref. [16], light incidence and electron emission directions are fixed at a relative angle of ${55}^{\circ }$ while the sample is rotated.

Figure 2

RPE spectrum at the Ni $L_2$-edge resonance for a Ni atom with the ($3d^9$) ground state calculated with atomic multiplet theory. The total RPE intensity (thick red lines) and the contribution of the direct photoemission process (thin blue lines) are shown as line spectra and Gaussian broadened spectra with a FWHM of 2 eV.

Figure 3

Diffraction patterns and corresponding circular dichroism (CD) for RPE from Ni(111) at the Ni $L_2$ edge. The patterns are stereographic projections in the polar angle range $\theta <{70}^{\circ }$. The magnetization axis is taken as $\left[\overline{1}10\right]$ and corresponds to the horizontal ($x$) axis of the plots. (a) Experimental diffraction pattern obtained with circularly polarized light (positive helicity). (b) Corresponding calculated pattern. (c) Angular plot of the function $\mathbf{L}·\mathbf{m}$, i.e., the empirical model for CD used in [16]. (d) Experimental CD pattern (asymmetry function). Two of the three first-nearest-neighbor focusing peaks are surrounded by dashed ellipses. The observed CD sign change at these peaks is typical for the Daimon effect. (e) Corresponding calculated CD pattern. (f) Theoretical CD pattern for a nonmagnetic ground state. In (d)–(f) the maximum anisotropy (corresponding to red and blue colors) is indicated below the plot. The experimental patterns in (a) and (d) are taken from Ref. [16] and have been rotated to match our reference frame.

Figure 4

Theoretical CD patterns of $L_2$-edge RPE from Ni(111) in RPE for each ($d^8$) final state term. Top row: atomic model; bottom row: cluster model. The binding energy of the five multiplet terms decreases from left to right; see Fig. 2 for the exact energy positions. The pattern “total” corresponds to the energy integrated photoemission intensity. The maximum anisotropy values [Eq. (4)] are indicated in percent under each pattern. The full hemisphere, $\theta <{90}^{\circ }$, is shown.

Figure 5

CD patterns of $L_2$-edge RPE decomposed into resonant and direct process, i.e., computed with only the first or only the second term on the right-hand side of Eq. (2). Top row: resonant process in only the atomic model. Middle row: resonant process in only the cluster model. Bottom row: direct process in only the cluster model.

Figure 6

CD patterns of $L_2$-edge RPE: scattering contribution vs nonmagnetic signal. Top row: extra-atomic contribution to CD patterns of the magnetized Ni(111) surface. Before calculating the anisotropy, the single-atom emission intensity was subtracted from the cluster intensity. Bottom row: CD patterns of the nonmagnetized Ni(111) surface (cluster calculation without any subtraction).

Figure 7

Calculated CD patterns of RPE at the $L_3$ edge. Top row: atomic model. Bottom row: cluster model.