A combined experimental and theoretical study of Rashba-split surface states on the ( 3 × 3 ) ?> Pb/Ag(111) R 30 ° ?> surface

We report on a combined low-temperature scanning tunneling spectroscopy (STS), angle-resolved photoemission spectroscopy (ARPES), and density functional theory (DFT) investigation of the ( 3 × 3 ) ?> R30°Pb/Ag(111) surface alloy which provides a giant Rashba-type spin splitting. With STS we observed spectroscopic features that are assigned to two hole-like Rashba-split bands in the unoccupied energy range. By means of STS and quantum interference mapping we determine the band onsets, splitting strengths, and dispersions for both bands. The unambiguous assignment of scattering vectors is achieved by comparison to ARPES measurements. While intra-band scattering is found for both Rashba bands, inter-band scattering is only observed in the occupied energy range. Spin- and orbitally-resolved band structures were obtained by DFT calculations. Considering the scattering between states of different spin- and orbital character, the apparent deviation between experimentally observed scattering events and the theoretically predicted spin polarization could be resolved.

Interband spin-orbit coupling between anti-parallel spin states in Pb quantum well states Bartosz Slomski, Gabriel Landolt, Stefan Muff et al.

Introduction
It is well known that the degeneracy of electronic states can be lifted by breaking the inversion symmetry of crystal lattices intrinsically (e.g. Dresselhaus effect) [1] or by introducing surfaces or interfaces (Rashba-Bychkov effect) [2]. For a Rashba-split two-dimensional electron gas spin degeneracy is lifted and the dispersion can be written as where * m is the effective mass and k 2 0 is the momentum splitting. The energy splitting between the band onset and the crossing of the two inner branches at k = 0 is called the Rashba energy The first successful observation of a Rashba-split surface state was reported by La Shell et al [3] for Au(111), which exhibits a Rashba energy Δ = E 2.1 R meV only. Recently, so-called giant splittings have been found for × ( 3 3) surface alloys of heavy post-transition metals with noble metal fcc(111) surfaces [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. In these systems the splitting can reach Δ ≈ E 200 meV R . In particular, the electronic structure of the × ( 3 3) Bi/Ag(111)°R30 surface has been studied intensively [5-7, 9, 14-17]. This surface alloy features two downwards dispersing surface states within the L-projected bulk band gap, an occupied s p , z -like band and an (mostly) unoccupied p p , x y -derived band [7]. The band structure has been observed by angle-resolved photo emission spectroscopy (ARPES) [5,6,9,16], scanning tunneling spectroscopy (STS) [5], and quasi-particle inference mapping (QPI) [15,17]. These experimental results consistently show that the two surface state bands hybridize below the Fermi level [16,17]. This hybridization goes along with a rather complex spin-polarization pattern, which was theoretically predicted by density functional theory (DFT) [7] and experimentally verified by spin-resolved ARPES [9] and QPI [17]. QPI maps observed by tunneling into empty sample states, however, were interpreted as evidence for conventional Rashba behavior in the p p , x y -derived band [15,17], in apparent disagreement to theoretical results [7]. In contrast, the × ( 3 3) Pb/Ag(111) surface, which structurally forms the same alloy as Bi/Ag(111) [19], has been studied to a far lesser extent. Since lead (Pb) has one p electron less than bismuth, the bands shift upwards with respect to the Fermi level. Indeed, DFT calculations predict two spin-split states above the Fermi level [7]. As a result the bands relevant for the surface Rashba effect are mostly unoccupied and thereby largely unaccessible by ARPES [4,5,9,20].
The theoretically determined positions of these Rashba states was found to critically depend on the vertical relaxation of the Pb atoms relative to the Ag atoms, Δ z (see figure 1(a)). Two distinct scenarios have been considered in the DFT calculations of [7], as schematically represented in figures 1(b) and (c). In the first case at the self-consistent relaxation value of Δ = 0.97 z Å ( figure 1(b)), the band dispersion qualitatively resembles the dispersion of Bi/Ag (111) with a hybridization of both bands close to the onset of the lower Rashba band at about 1 eV above the Fermi level. However, this band position disagrees with ARPES measurements [4], which finds an extrapolated binding onset which differs by ≈ 400 meV from the DFT result. This offset is indicated by the blue shaded area in figure 1(b).
In order to improve the agreement between the experimentally observed band positions and the calculated surface electronic structure the relaxation value was tuned to Δ = 0.67 z Å [7]. The resulting band structure is shown in figure 1(c). It agrees well with ARPES data [4] and the characteristically shaped van-Hove-like singularity which appears in STS spectra at 654 meV above the Fermi level [5].
Obviously, the smaller relaxation of the Pb atoms results in a larger inter-band spacing and the absence of any hybridization between the upper and the lower band. Since both models result in a very similar dispersion below the Fermi level, it is virtually impossible to verify one of the two models by ARPES measurements. In the case of figure 1(b) the branches observed below E F would belong to two different surface states with a strong spin splitting, in the case of figure 1(c) both branches would originate from the same Rashba-split surface state with a smaller splitting. Experimentally even smaller outward relaxations of Δ = ± (0.46 0.06) z Å have been reported [21], pointing towards the model presented in figure 1(c).
In this report we study the × ( 3 3) Pb/Ag°R30 surface to solve some of the remaining questions. By means of scanning tunneling microscopy and derived spectroscopic methods we obtain insight into the electronic structure of occupied and empty states. Our STS results show two asymmetric peaks which are the fingerprints of the DOS of two Rashba-split surface states [5]. With this finding we can confirm the second predicted Rashba state and thus solve the unclear band onset situation in the unoccupied energy range. In the second step we used QPI mapping to observe spatial oscillations which originate from coherent elastic scattering between  [7]). In (b) a hybridization between both surface states takes place. Red and blue represents the inplane spin polarization. For smaller relaxation the lower (s p , z -derived) Rashba band is shifting downwards (c), thereby increasing the energy separation between the two bands. See text for details. two momentum eigenstates. By comparison with ARPES data the obtained energy dispersion and anisotropies of QPI are correlated to intra-or inter-band scattering vectors between specific bands. We extract effective masses and show that hexagonal warping exists for both bands. These experimental data are compared to DFT calculations of the ×°R ( 3 3)Pb/Ag 30 system. An excellent agreement, especially regarding the binding energy of the s p , z -orbitals which was chronically overestimated so far, is found for a reduced outward relaxation of the Pb atom (Δ = 0.67 z Å). It is found that the scattering pattern observed in QPI can only be explained if-beyond the spin angular momentum-also the orbital momentum of the involved electronic states is considered.

STM setup
The STM experiments have been performed in a two-chamber UHV system equipped with a low-temperature STM working at 5 K. The system consists of separate chambers for sample preparation and surface analysis (base pressure ⩽ × − p 1 10 10 mbar). STM tips were electrochemically etched from a polycrystalline tungsten wire. The Ag(111) single-crystalline substrate was cleaned by cycles of + Ar -sputtering (E = 500 eV) and subsequent annealing ( ≈ T 700 K). For preparation of the ×°R ( 3 3 )Pb/Ag(111) 30 surface alloy a third of a Pb monolayer was evaporated from a home-built Knudsen cell onto the Ag(111) substrate held at ≈ T 520 K. STS spectra and dI/dU maps for QPI measurements have been taken by means of lock-in technique (ν = 789 kHz).

ARPES setup
The ARPES data were collected with a hemispherical electron analyzer (Scienta R4000) and a monochromated He discharge lamp (MB Scientific) operating at an excitation energy of 21.22 eV (He Iα). The energy and angular resolutions were 7 meV and 0.3°, respectively. ARPES measurements were performed at a base pressure < × − p 2 10 10 mbar and a temperature of 25 K. The single-crystalline Ag(111) substrate was prepared by + Ar -sputtering (E = 500 eV) and annealing ( ≈ T 900 K). Pb was evaporated from a commercial Knudsen cell at a rate of approximately 0.03 ML min −1 . Before each evaporation procedure the Ag(111) substrate was mildly annealed and subsequently allowed to cool down during evaporation. The formation of the surface alloy was confirmed by LEED.

DFT
The calculation of the surface electronic structure is based on DFT employing the full-potential linearized augmented plane-wave method as implemented in the FLEUR code 5 . The surface was simulated in a thin film geometry with a nine-layer Ag(111) film terminated on one end by the × ( 3 3) Pb/Ag(111)°R30 surface alloy reconstruction. For further specifications of the calculation, we refer the reader to [7,13].  [22]. The wetting layer domains shows a characteristic moiré pattern which can also be recognized in the atomically resolved STM image of figure 2(b). It is preferentially found at step edges. Figure 2(c) shows an atomically resolved image of the ×°R ( 3 3 ) Pb/Ag(111) 30 surface alloy with a few defects. The visible modulation originates from QPI and will be discussed in more detail below. figure 2(d) displays the corresponding LEED pattern. It consists of an outer hexagon that belongs to the underlying Ag(111) substrate and an inner hexagon which is rotated by°30 and reflects the ×°R ( 3 3) 30 symmetry of the surface alloy. figure 3 shows an averaged tunneling spectrum of the ×°R ( 3 3 ) Pb/Ag(111) 30 surface alloy. We can recognize two asymmetric peaks, the shape of which closely resembles the singularities described in [5]. The DOS of downwards dispersing Rashba-split bands is described by describes the splitting strength. We have fitted the peaks following the algorithm described in [5]. Obviously, both peaks are very well described by the fits and can clearly be assigned to the s p , z -and the p p , x y -derived Rashba states as schematically represented in figure 1(c) (Δ = 0.67 z ) and described in [7]. For the p p , meV, i.e. slightly further away from the Fermi level than reported previously [5].
We would like to emphasize that in contrast to Bi/Ag(111), where we observed an obvious difference between the measured data and the fit for the occupied surface state due to a distinct shoulder [17], the agreement of the fit is very good for on Pb/Ag(111) at both peak positions. As has been pointed out in [17] the shoulder is ascribed to the opening of a hybridization gap which -as a result of the flat energy dispersion around the gap-leads to an enhanced DOS. Based on these observations we do not find any hint for hybridization of the two Rashba-split surface states on Pb/Ag(111), thereby favoring the model of figure 1(c).
To obtain the energy dispersion we measured two series of QPI maps by recording the spatial variation of the dI/dU signal in the bias range between −600 meV (occupied states) and 1600 meV (empty states) with an increment of 50 meV. Figure 4 Figures 4(b) and (c) show the interference patterns measured at this location at = − U 163 mV and = − U 263 mV, respectively. Standing waves parallel to step edges and around defects are clearly visible. The corresponding FT QPI maps are shown in the insets. One can recognize a hexagonally shaped frame with cusps pointing into ΓK directions. Close inspection of the inset figure 4(c) reveals a second hexagon which is slightly larger and much less pronounced. The latter hexagon is rotated by°30 with respect to the former, consequently pointing into Γ M directions. The QPI maps for determining the electronic properties of the p p , x y -derived state have been measured on the same sample but at a different, second area, due to a tip change. The data obtained at the two positions have some energetic overlap confirming compliance of the observed wave vectors. The constant-current STM topograph of this area is shown in figure 5(a). Again, one can recognize atomically flat terraces which are separated by two step edges. The pattern observed in figure 5(b) is not as strong and pronounced as the interference patterns in figures 4(b) or (c). We speculate that the weaker interference pattern is due to the reduced lifetime of electronic states which are energetically further away from the Fermi level. As we discuss later, there are different scattering processes responsible for the two energy regions. This might also affect the appearance of the patterns. In the corresponding 2D FFT (inset of figure 5(b)) we again observe a hexagonally shaped frame with cusps pointing into the Γ M directions. The detailed analysis of the quasiparticle interference maps recorded over a wide range of bias voltages results in the extracted scattering vector q(E) displayed in figure 6(a). Clearly, we observe an energy dispersion of q(E) for both Rashba bands. Scattering within the p p ,   figure 6) starts at about 1500 meV. The parabolic evolution of scattering vectors confirms the dominant role of intra-band scattering in this energy range ( = q k 2 in this case). The dispersion was fitted with a cosine since it yields the best agreement to the experimentally observed scattering vectors. From this fit we obtain the band onset of the p p , meV. This value is in good agreement with our STS data presented in figure 3, thereby confirming the model given in figure 1(c). We obtain effective masses of = − ± * The analysis of the QPI maps taken in the energy range of the s p , z -derived bands turned out to be not as straightforward. Surprisingly, we observe two and at some bias voltages even three scattering vectors, which are marked D 2 , D 3 and D 4 in figure 6(a). The shorter scattering vector D 2 corresponds well to intra-band scattering events within the s p , z -branches. Fitting D 2 with a parabolic function leads to = ± E (708 22) 2 meV, which is in a good agreement with our STS data presented in Hence, for the p p , x y -band and the s p , z -band the lateral atomic bonding appears to be dominated by Pb-Pb and Pb-Ag nearest neighbour interaction, respectively.
With all parameters being determined we can exclude the band model schematically represented in figure 1(b). Instead, our data are consistent with the smaller relaxation model sketched in figure 1(c), where no hybridization between the two Rashba bands takes part.

Theoretical results: DFT
In figure 7(a) we show the calculated spin-resolved band structure of the Pb-derived states around the Fermi level (E F ) for a Pb relaxation of Δ = Å 0.67 z . It is clearly seen that the s p , z states with a band onset at about 0.5 eV show a strong Rashba-type spin polarization (as well as the p z states at 2.5 eV), while the m 1/2 -bands, the band onset of which is at 1.5 eV, exhibit a more complex spin texture. In particular around the Γ -point the polarization decreases rapidly and even exhibits a reversal of the spin direction at some k-points . As the m 1/2 -bands show no direct crossing with the s p , z -bands (in contrast to the BiAg 2 case and the PbAg 2 with stronger relaxation), for larger k-values the spin direction reverses again and increases to larger momentum values.
To shed light on the origin of this loss of spin polarization near the Γ -point, we show the spin-and orbital-resolved band structure in figure 7(b) focusing on the p x and p y states. Apparently, near the Brillouin-zone the states consist of a mixture of p x with one spinorientation and p y states of the opposite spin orientation. It is well known from hole-states in semiconductors that spin-orbit coupling at small momenta can lead to the formation of orbitalmoment-carrying states ( = m 1/2 j ) [23], while Rashba-type spin-orbit coupling dominates at larger k-values and forces the spin direction in-plane. Additionally, crystal-field effects decouple the p x and p y states according to the momentum direction.
Although an apparent decrease of spin polarization for the m 1/2 states at small k-values is clearly visible in figure 7, we cannot directly identify the origin of this reduction. In principle, also an increased out-of-plane component of the spin moment may lead to the same behavior. We checked for this possibility and found no significant spin component along the surface normal, but rather a mixture of strongly in-plane polarized p x and p y orbitals with opposite spin polarization. We have to note here that, although the total spin polarization is dramatically reduced near the center of the Brillouin-zone, the scattering from k to −k, i.e. between time-reversal partners, is still prohibited by time reversal symmetry [24].
Finally, we can turn to the question why the intra-band scattering-as indicated by the arrow in figure 7(b)-is allowed, despite the fact that initial and final states are dominantly of opposite spin character. Since the total momentum, J, has to be conserved in the scattering process, the associated transfer of spin angular momentum has to be compensated by an equivalent transfer of orbital momentum, L. Loosely speaking we can say that, if Δ Δ Δ = + = J L S 1/2 0, a°180 reversal of spin S needs to be compensated by a°90 rotation New J. Phys. 16 (2014) 045017 L El-Kareh et al of the orbital moment, i.e. from a p x to a p y orbital or vice versa. This is exactly the process allowed by the spin-flip (off-diagonal) part of the spin-orbit coupling 6 [25]. Indeed, the scattering process indicated by the black arrow in figure 7(b) connects p x to p y orbitals of opposite spin and is, therefore, allowed. A forward scattering process, as could be imagined in the PbAg 2 surface alloy between different bands of the same spin, can be excluded since it involves scattering from a p x to a p y orbital without a spin-flip. The interpretation of scattering events in terms of spin polarization of the involved bands cannot be drawn from a plot of the spin polarization of the states (like figure 7(a)) alone, since it is not directly possible to draw conclusions about allowed or forbidden quasi-particle scattering events. Only in a simple case, i.e. when the involved states do not carry orbital moments (s p d , , z z 2), the Rashba picture of spin polarization holds and selection rules can be directly inferred. In the case of the Bi(110) surface analyzed in this way [26] this was meaningful, since the surface states of this surface are of p z character. In the present system, however, such assignments are misleading, since-in addition to the spin-also the orbital character has to be taken into account. In the same spirit the STM data of [17] can be reconciled with the calculations presented in [7], as we checked by explicit calculations. The Pb relaxation is fixed at Δ = Å 0.67 z . Red and blue colors indicate the spin polarization with respect to a spin-quantization axis that is perpendicular to the k-vector and surface normal. (b) Spin-and orbital-resolved band structure of the PbAg 2 surface alloy. Open and full circles show contributions from p x and p y orbitals, respectively. The size of the symbols in (a) and (b) indicates the degree of spin polarization. 6 The spin-orbit coupling Hamiltonian can be written as ξ ξ = · = + + ξ + − − + H l s l s l s l s ( )