Energy filtering of the photoelectrons is

Energy filtering of the photoelectrons is accomplished by a combination of two hemispherical deflection analyzers (HDAs). Each hemisphere has a mean radius of r0=150mm and was modified from a commercially available MG 262 spectrometer (PHOIBOS 150, Specs GmbH). Electrons that pass the entrance plane of the first analyzer are deflected in the spherically symmetric 1/r potential, and have the largest energy dispersion after a deflection of 180°. The image obtained in the exit plane of the first HDA is energy dispersed and subject to the α2 aberration [23]. An effective refocusing of the electron trajectories was described in Ref. [35] by using an electrostatic lens to couple the trajectories to the entrance of the second HDA, such that an effective 360° deflection path is realized. The same principle was also used in previous work [29] and is described in detail in Refs. [36,37]. In short, the solution for a 360° deflection in the spherical 1/r potential is a well-known problem in classical mechanics and leads to closed trajectories (Kepler ellipses). By this symmetry, electron trajectories are refocused in the exit plane of the second HDA to the same spatial and angular coordinates as was the starting point in the entrance plane of the first HDA, transmitting the full image information.
Fig. 1b shows the electron optical principle of the momentum microscope imaging column with simulated electron trajectories between the sample and the entrance plane of the first HDA at a pass energy of 30eV, consisting of three major parts: the cathode lens, the first retarding stage and the second retarding stage. Simulations were carried out using the SIMION [38] software. For correct modeling of the cathode lens a sufficiently fine computational mesh has to be chosen in the region between sample and anode [39]. Here, we find converging results for mesh densities larger than 200points/mm. Fig. 1b shows trajectories for electrons emitted with a kinetic energy of 16eV. This corresponds to the typical maximum photoelectron energy using the He–I line of a gas discharge source. Trajectories with different color start at the sample surface in a lateral distance of , 0, and from the optical axis.
Retarding of electrons from the anode potential to the pass energy of the HDA takes place in several steps. The first momentum image is formed at an energy of about 1200eV in the focal plane of the objective lens, followed by two decelerating lens groups. The first retarding stage is located between the momentum image and the spatial image. The position of the spatial image is kept fixed, such that a movable aperture can be used to select the analyzed area. Finally, the second retarding stage serves three functions: (i) deceleration to the pass energy of the analyzer. (ii) Selection of momentum image or spatial PEEM image. (iii) Variation of the magnification factor (i.e. the field-of-view) for a fixed retarding ratio for PEEM or momentum imaging.
Momentum images are recorded, when a spatial image is placed in the entrance plane of the analyzer. Then, the maximum analyzed sample area is confined by the analyzer slit, and depends on the total real-space magnification, M, of the intermediate image in the entrance plane. As the analyzer transmits electrons in a limited angular interval , a direct relation between the momentum field-of-view and the total magnification M can be given. With rotational symmetry, Liouville\’s theorem requires , where d0 and d are the image height at the sample and at the analyzer entrance, respectively. On the right side, the length of the electron momentum vector at the analyzer entrance is . The magnification then is given byIn Fig. 1b the total magnification, consisting of the magnification of the objective lens (M) and the first (M1) and second (M2) retarding stage, is , with M2=1. For changing the image diameter, M2 can be varied between 0.30 and 3.0, under practical conditions. For instance, the measurements discussed in Figs. 3 and 5 correspond to M2=1.0 and M2=0.30, respectively.