Supplementary Figures

Transcription

1 Supplementary Figures Supplementary Figure S1: Calculated band structure for slabs of (a) 14 blocks EuRh2Si2/Eu, (b) 10 blocks SrRh2Si2/Sr, (c) 8 blocks YbRh2Si2/Si, and (d) 14 blocks EuRh2Si2/Si slab; a block consists hereby of Si-Rh-Si-Sr/Eu/Yb layers and corresponds to the half of a conventional unit cell.

2 Supplementary Figure S2: Determination of the cleavage plane: (a) interlayer distances as a function of the doubled c-lattice constant; increasing strain pulls apart mainly Eu and Si-Rh-Si structures, whereas the distances inside the Si-Rh-Si block remain approximately constant. If the strain becomes too strong (az~28 A), the crystal breaks into two parts revealing one Eu and one Si terminated surface (the purple symbols depict the maximum distance between next-neighbour Eu-Si surface layers); the inset magnifies the small region of elastic deformation; (b) a slab geometry for az < 28 A; (c) a slab for az > 28 A, corresponding to the data marked with arrows in (a)

3 Supplementary Note 1: Angle-resolved photoemission experiments were performed at the Swiss Light Source, Paul Scherrer Institut, Villigen, Switzerland. Surface sensitive ultraviolet (UVARPES) spectra were taken at the Surface-Interface Spectroscopy (SIS) X09LA instrument, while for probing the truly bulk derived electronic structure respective experiments were performed in the soft x-ray regime (SXARPES) at the novel advanced resonant spectroscopy (ADRESS) station. The UVARPES spectra were taken using a VG-Scienta R4000 electron analyzer, and SXARPES spectra were acquired with a Phoibos 150 (Specs GmbH) machine. The overall energy resolution was set to about 15 mev at the SIS instrument and 80 mev at the ADRESS instrument, respectively. For both types of experiment, high quality single-crystalline samples of EuRh 2 Si 2 were mounted on a low-temperature goniometric manipulator (CARVING) with three angular degrees of freedom and cleaved in situ in ultra-high vacuum at a base pressure better than 5 x mbar and a temperature of T = 10 K. Immediately after cleaving the aforementioned experiments were performed keeping the samples at the same temperature. The time scheduled for studying each sample was about 4h including crystal alignment. The results were found to be fully reproducible in several experimental runs, using more than ten different samples. During experiment aging processes were never observed. Similarly to YbRh 2 Si 2, EuRh 2 Si 2 crystals cleave usually between layers of rare-earth (RE) and silicon atoms, leaving behind either europium- or silicon-terminated surfaces. The cleaved surface may be considered as a mosaic of alternating terminations which size and shape of individual domains are unique for each cleave. This causes certain difficulties taking Fermi surface maps, since it is necessary to keep the light-spot always on the same domain when the manipulator is operated. An Eu termination may easily be identified by an intense Eu 4f surface signal which is separated from the bulk emission by a surface core-level shift of about 1 ev and is found between 1.1 ev to 1.8 ev binding energy. For a Si-terminated surface a respective feature is missing (compare Figs. 1a) and 1b)). In order to emphasize the 4f-character of the studied electronic structure, we chose a photon energy of 120 ev where the atomic photoionization cross-section of Eu 4f states is large, while signals from Rh 4d states are strongly reduced by cross-section effects.

4 As it becomes obvious from a comparison of the experimental data in Fig. 1a) and the model calculation in Fig. 1c), predominantly 4f character is observed at the chosen photon energy. However, the spectral contributions from states of other angular momentum character are weak but not completely zero in the experimental data. This explains why the linear dispersive band remains visible up to high binding energies what is not reproduced in the simulated spectra where only 4f contributions are plotted.

5 Supplementary Note 2: Band structure calculations have been carried out with the FPLO code [24] which is a density functional theory code based on the full-potential local orbital method. Unless otherwise stated, the local density approximation (LDA) in the formulation of Perdew-Wang and a scalarrelativistic approximation have been used. Since LDA fails to describe highly-correlated electrons, the 4f basis has been fixed to an unpolarized configuration of seven f electrons, which is in accordance with the divalent behavior of europium ions in the compound [25]. Furthermore, the overlap of neighboring 4f orbitals has been neglected. In the following, this treatment will be referred to as frozen core approximation. LDA+U calculations (U~ 8 ev, J~ 1 ev, chosen according to [26]) lead to occupation numbers close to seven f electrons supporting the former limit. Substituting europium by strontium and keeping the lattice parameters constant yields a band structure qualitatively similar to the one of the frozen core approximation (Supplementary Fig. S1). However, it is not possible to distinguish between effects due to the frozen core approximation (i.e. the limited flexibility of the basis set, which may influence the surface states) and the difference in size between strontium and europium. Surface effects were simulated within a slab geometry, where the translational symmetry is broken artificially in one direction in order to obtain a surface configuration similar to that in the experiment. Since the single crystalline samples where cleaved perpendicular to the c-axis, three different surface terminations are basically possible: Eu, Si and Rh (neglecting terraces and other defect structures). To determine the most probable cleavage plane, a slab of two conventional unit cells has been gradually stretched along the c-axis. Subsequently, the forces acting onto each atom have been minimized ( relaxation of the atomic positions ). The atomic distances as a function of the slab s length (twice the c-axis parameter) are shown in Supplementary Fig. S2. Up to an increase of the c-axis parameter of 25%, the atomic positions scale with the elongation of the c-axis. Thereby, the distance between Eu and Si increases proportional to the increase of the c-axis while the distance between Si and Rh stays roughly constant indicating that the Si-Rh- Si block is tighter bound than the Si-Eu-Si one. When the potential energy stored in the structural elongation (comparable to a spring s energy) becomes larger than the surface energy, the slab breaks into two parts characterized by a europium and a silicon terminated surface.

6 Same results have been found for different slabs, but one has to admit that only the mostfavorable terminations are found by this method. In addition, the computer experiment reveals that the top-most surface layers undergo an inward relaxation and the distances of RE-Si and Si- Rh layers at the surface are, thus, smaller than their respective distances in the bulk. However, this result can solely be viewed qualitatively, since for a quantitative study it has to be extrapolated to a semi-infinite surface. The finding of only two different surface terminations is supported by photoemission spectra as well as scanning tunneling microscopy experiments [27, 28]. The linear dispersive surface state appears for both surface terminations but is less pronounced if the outermost layer is formed by rare-earth or Sr atoms (Supplementary Fig. S1a and b). Whether this is a limitation of the calculation (i.e. slab approximation, frozen core approximation) or an intrinsic effect has not been explored yet. The perturbation induced by substituting the rareearths by Sr seems to have a small effect on the surface states [compare Supplementary Fig. S1a)+b) and Figs. S1c)+d)], although in case of YbRh 2 Si 2, which is a mixed-valent compound, the number of core electrons (and therefore the effective potential ) as well as the valence (influences the total electron density) differ from the ones of the Eu compound.

7 Supplementary Note 3: Our simplified model to simulate the photoemission spectra is based on the assumption that the coupling between the localized 4f states and the valence band near the Fermi level (mainly Rh 4d, Si 3s and Si 3p derived) is weak and, thus, we can calculate the photoemission signal for the two species separately introducing the interaction afterwards within a simple hybridization model. The latter is similar to the solution the periodic Anderson model within the Gutzwiller approximation [29]. For the atomic limit of the 4f emission the results of Gerken et. al. [30] (configuration interaction calculations) and for the valence band the results of the aforementioned density functional theory calculations are used. The latter is appropriate, since we are solely interested in a qualitative description (neglecting renormalization ) and the ground state of the parent compound reveals metallic character. The resulting hybridization Hamiltonian looks like and the hybridization matrix element V ij is chosen proportional to the overlap of atomic 4f and 4d wavefunctions originating from the Eu and Rh sites, respectively. Supplementary References: 25. B. Chevalier et. al., J. Phys. C: Solid State Phys. 19, (1986) 26. R. Gumeniuk et. al., Phys. Rev. B 82, (2010). 27. G. Kaindl et. al., Phys. Rev. B 51, (1995); W.D. Schneider et. al., Phys. Rev. B 28, (1983). 28. S. Ernst et. al., Nature, 474, (2011). 29. T. M. Rice and K. Ueda, Phys. Rev. Lett. 55, (1985). 30. F. Gerken, Journal of Physics F: Metal Physics 13, (1983).