Abstract
Gold surfaces host special electronic states that have been understood as a prototype of Shockley surface states. These surface states are commonly employed to benchmark the capability of angleresolved photoemission spectroscopy (ARPES) and scanning tunnelling spectroscopy. Here we show that these Shockley surface states can be reinterpreted as topologically derived surface states (TDSSs) of a topological insulator (TI), a recently discovered quantum state. Based on band structure calculations, the Z_{2}type invariants of gold can be welldefined to characterize a TI. Further, our ARPES measurement validates TDSSs by detecting the dispersion of unoccupied surface states. The same TDSSs are also recognized on surfaces of other wellknown noble metals (for example, silver, copper, platinum and palladium), which shines a new light on these longknown surface states.
Introduction
The history of surface states (SSs)^{1,2,3,4,5,6,7,8,9,10} can be traced back to 1932 when Tamm^{11} predicted the existence of special electronic states near the crystal boundary. Soon Shockley^{3} found that SSs, usually called Shockley SSs later, emerge in an inverted energy gap owing to band crossing, for which the symmetry of the bulk band structure was found to be crucial^{12}. The SSs on (111)oriented surfaces of noble metals (for example, Au, Ag and Cu) have been known as typical Shockleytype SSs (for example, refs 13, 14, 15), wherein SSs appear inside an inverted energy gap of s and p bands at the centre of the surface Brillouin zone (BZ). As a result of inversion symmetry breaking on the surface, these SSs exhibit Rashbatype^{16} spin splitting with spinmomentum locking at the Fermi surface^{2,15,17,18}, which is essential for spintronic devices^{19}. These SSs have been used to design quantum corrals^{20,21,22,23,24} as well as artificial Dirac fermions^{25} and benchmark the capability of angleresolved photoemission spectroscopy (ARPES)^{4,5} and scanning tunnelling spectroscopy (STS)^{6}. As another intrinsic SS, the topological SS (TSS) has recently attracted great research interest in the condensed matter physics community^{9,10}. Metallic TSSs inside the bulk energy gap are induced by the topology of the inherent bulk band structure, which can be understood as an inversion between the conduction and valence bands that have opposite parities^{26,27}. TSSs have been predicted and observed in many compounds^{28}, such as HgTe (refs 26, 29) and Bi_{2}Se_{3} (refs 30, 31), wherein spin and momentum are locked up and form spin texture in the Diracconelike band structure. From a naive viewpoint of the band inversion, Shockley SSs and TSSs are not fully exclusive of each other. TSSs of Bi_{2}Se_{3}, for example, have been described in a generalized Shockley model that includes spinorbit coupling (SOC) by Pershoguba et al.^{32} This inspires us to pose the opposite question: Can some Shockley SSs be understood as TSSs?
In this work, we revisit SSs on Au, Ag and Cu (111) surfaces by ab initio band structure calculations and ARPES performed on the Au(111) surface with a momentum microscope^{33} for the detection of the complete angular distribution of the photoemitted electrons as function of their kinetic energy. We find that these famous Shockley SSs are also TSSs, which originate from the inverted bulk band structure. The Rashbasplitlike energy dispersion can be regarded as a strongly distorted Dirac cone. Although noble metals do not exhibit an energy gap, a Z_{2} topological invariant ν_{0}=1 can be well defined owing to the existence of a direct energy gap above the Fermi energy. Thus, the existence of TSSs is generalized from the insulators to common metals. Besides providing a new understanding of noble metal SSs, finding topological states on late transition metals will provoke interesting questions on the role of topological effects in surfacerelated processes, such as adsorption and catalysis.
Results
Comparison of different types of SSs
Shockley states on gold surface have been commonly modelled as a nearlyfreeelectron (NFE) model^{34}. They exhibit quadratic energy dispersion with the spinsplit that is explained by the Rashba effect, as illustrated in Fig. 1. The NFE model fits well to the band structure near the surface BZ centre , which are the occupied states detected by ARPES. However, this model neglects information of the whole BZ, that is, the topologcial properties of the band structure. Actually, recent inverse photoemission measurement^{35} on the unoccupied states already indicated the discrepancy between NFE model and SS dispersions in the region far away from . We should note that the original Shockley’s theory^{3} implies some information of topology. It predicts SSs inside the inverted energy gap without considering SOC for gold (Fig. 1b), in which the bulk is gapless at the s–p bandcrossing point and SSs are spin degenerate. By introducing SOC, we find two simple consequences in band structure: The bulk opens a direct energy gap at the bandcrossing point and SSs become clearly TSSs. Although they can still be fitted by a NFE model only near , TSSs disperse from the valence to conduction bands inside the direct gap across the whole BZ, which is verified by our following ARPES measurement on unoccupied states. Compared with TSSs commonly observed with a simple Diracconelike dispersion (see Fig. 1d), the surface band bending pulls the Dirac point below the Fermi energy, strongly deforming the shape of the Dirac cone. However, this can not remove the topological nature of gold SSs. We note that one signature of the Dirac cone deformation is the crossing of two spin channels of TSSs in highenergy region away from the point (see Fig. 1c), as revealed by our following calculations.
Bulk band structure
Noble metals share the same facecentredcubic (FCC) lattice structure. We take Au as an example. As shown in Fig. 2a, the primitive unit cell includes a single Au atom as the inversion centre of the lattice. Therefore, the parity of the Bloch wave function is consistent with that of the corresponding atomic orbital: ‘+’ for the Aus and d orbitals and ‘−’ for the Aup orbital. The bulk band structure that includes the SOC effect is shown in Fig. 2c. The Fermi energy crosses the middle of a band wherein s (blue colour) and p (red colour) states hybridize together, wherein the d bands are fully occupied and below the Fermi energy. Because the band inversion only involves sp rather than d states, we project the band structure to Ausp states using Wannier function method for simplicity. Above the halffilled band, a direct energy gap exists in the whole BZ owing to SOC, as indicated by the grey shadow in Fig. 2c, though the indirect gap is still zero. As we will see, this direct energy gap determines the topology of SSs. For the sake of simplicity, we call bands below and above the gap valence and conduction bands, respectively. To illustrate the band inversion clearly, we show the energy dispersion along Γ−X′−L′−Γ lines, wherein X′(L′) is equivalent to X(L). Because of relativistic contraction of Au6s orbitals (ref. 36, and references therein), the s band is lower in energy than the d and p bands at the Γ point. In contrast, the s band energy is even higher than that of p bands at X′(X) and L′(L) points. Thus, one can find that s and p bands preserve the normal order at the Γ point but get inverted at other timereversal invariant momenta (TRIM) X′(X) and L′(L) points. Between the Γ and X′(L′) points, the s and p bands cross each other. The direct energy gap opens at the crossing point because of band anticrossing caused by SOC. The Z_{2} topological invariant ν_{0} of a topological insulator (TI) can be calculated by the product of the parity eigenvalues of all valence bands at all TRIM^{27}. If the parity product is , then ν_{0}=1 represents a TI; otherwise, ν_{0}=0 represents a trivial insulator. In the FCC BZ, eight TRIM include one Γ point, three X points and four L points. Because s and d states are always ‘+’ in parity while only p states are ‘−’, the parity product at a given k point is determined by the number of p states. Therefore, the parity product of the Γ point is ‘+’ since only sd states appear in the valence bands. However, the parity products at X and L are ‘−’, for there is one p state as the top valence band for both X and L points. Thus, the total parity product for eight TRIM is ‘−’, that is, the Z_{2} invariant ν_{0}=1, showing the topologically nontrivial feature. One can see that the topology of the band structure is caused by the s–p band inversion above the Fermi energy, which is related to the relativistic contraction of the Au6s state.
Topological SSs
The nontrivial Z_{2} topological number guarantees the existence of TSSs on the boundary. On the Au(111) surface, the s–p inversion gap (also called Lgap in the literature) remains at the point of the surface BZ while it is reduced to zero at and points. Near the point, a pair of TSSs exist inside the s–p inversion gap with the Dirac point lying below the Fermi energy, as shown in Fig. 3a. Although the topological energy gap is above the Fermi energy, the local surface potential pulls the Dirac cone below the Fermi energy (see Supplementary Figs 1 and 2). As shown in Fig. 3a, the Dirac cone is strongly distorted, with the lefthand spin texture in the upper cone being similar to known TIs^{37,38}. Due to the Dirac cone deformation, two spin channels of TSSs cross each other between the and points at energy ∼3 eV above the Fermi energy. We note that the spinchannel crossing is consistent with previous calculations in ref. 35. Analysis of orbital components reveals that TSSs are mainly composed by sp orbitals. The same energy dispersion has been previous observed using ARPES and revealed in ab initio calculations^{2,17,15,39} this dispersion was interpreted as a Rashbatype split of spderived Shockley SSs. However, in a local region (for example, near the point) in the BZ, one cannot distinguish TSSs from trivial Rashba SSs. These Rashba states are equivalent to TSSs if the Rashbasplit bands are regarded as a strongly distorted Dirac cone wherein the lower cone was pushed above the Dirac point. A similar dispersion of TSSs as a dramatically deformed Dirac cone was recently observed on the surface of HgTe (ref. 40).
Twophotonphotoemission ARPES
To prove the topological origin of these SSs, we follow a twofold strategy. First of all, we use twophotonphotoemission (2PPE) ARPES to measure the energy dispersion of the SSs of the Au(111) surface below and above the Fermi energy. By combining an optical parametric oscillator (OPO) laser system with a modern momentum microscope^{33}, we are able to map the dispersion of the SSs far away from the point, and confirm experimentally the strong deviation from the dispersion of a trivial Rashba SSs. Second, we demonstrate theoretically that these SSs are adiabatically connected to TSSs of a real TI. We first describe the ARPES results. To detect the electronic structure of the Au(111) surface both below and above the Fermi energy (E_{F}), we have performed 2PPE ARPES with an OPO laser system and a momentum microscope^{33}. This photoelectron analyser detects the complete angular distribution of the photoelectrons for a selected kinetic energy (E_{kin}), as exemplarily shown in the insets of Fig. 3b for two selected values of E_{kin}. Varying E_{kin} allows to record a threedimensional data set of the ARPES intensity as function of electron momentum parallel to the surface. In this way, energy distribution curves for all high symmetry directions are recorded simultaneously (see Methods for further informations). The OPO laser system is used as excitation source for 2PPE. By varying the photon energy (between 4.13 and 4.43 eV) and the light polarization (between s and p), we can easily assign the features in the ARPES spectra to either occupied or unoccupied electronic states (that is, states below or above E_{F}) and determine their surface or bulkrelated character. Crucially, using photon energies above 4 eV gives us access to the still unexplored region of the Brilloiun zone far away from the point, where we expect a strong deviation of the dispersion of the Au(111) SSs from trivial Rashba SSs. By a careful analysis of the 2PPE ARPES data (see Supplementary Figs 3 and 4) we can identify three dominant contributions to the spectra: the occupied and unoccupied part of the SSs, an unoccupied image potential resonance and bulk states. The dispersion of the SSs and of the image potential resonance are plotted in Fig. 3b with black and blue circles, respectively. Due to the perfectly spherical shape of both states in momentum space, their dispersions and hence the corresponding data points shown in Fig. 3b are identical for both high symmetry directions. From previous ARPES^{15} and STS^{41} studies it is wellknown that the occupied Shockley SSs can be described by a quasifree electron parabola with an effective mass m_{eff} in the range of 0.25m_{e} (ref. 15) to 0.37m_{e} (ref. 42). The significantly smaller effective mass compared with the free electron mass (m_{e}) is due to an intrinsic coupling of the SSs with bulk states^{43}. Our analysis reveals an effective mass of 0.28m_{e} (grey dashed line in Fig. 3b), in good agreement with previous studies. The unoccupied image potential resonance also follows a freeelectronlike behaviour as it is wellknown from literature^{42}. The unoccupied part of the SSs, on the other hand, shows a clear deviation from the trivial freeelectronlike behaviour. In particular, in the energy range from 2.0 to 3.0 eV above E_{F}, the unoccupied SSs are found at larger k_{}values than expected for a freeelectronlike behaviour with m_{eff}=0.30m_{e}. A similar deviation in this intermediate state range was already reported previously for the Shockley SS of Au(111)^{35} and Cu(111)^{43}. It was explained as the result of hybridization of the SS with the bulk bands of the noble metal. The strength of the hybridization, that is, the deviation of the experimental dispersion from the freeelectronlike behaviour, increases as the SSs approach the Lband edge. Crucially, for intermediate state energies >3.0 eV the dispersion of the SSs in our data changes again and the SSs disperse faster to larger momentum values with increasing energy above E_{F}. To our knowledge, such a strong deviation from the freeelectronlike behaviour was not observed yet for SSs on fcc(111) noble metal surfaces. This observation can be explained by assuming that the SSs disperse into the bulk bands for intermediate state energies >3.0 eV to connect the valence and conduction bands. This behaviour, together with the dispersion of the SSs, is in full agreement with the calculations in Fig. 3a, pointing to the topological nature of the SSs. More details can be found in Supplementary Note 1.
Adiabatic evolution into a TI
To conclusively demonstrate the topological nature of the SSs we now turn to the ab initio calculations. Here we increase the strength of SOC artificially and realize an indirect energy gap in bulk Au, for example, when the SOC strength is 350% of the normal one. The motivation for this approach is that topological nature of the SSs has been long neglected plausibly due to the lack of an energy gap in gold. Thus, we design here a real gap to demonstrate these SSs derived from topology. In the bulk, the band structures with 100% SOC strength and 350% SOC strength are adiabatically connected to each other, exhibiting the same topology with ν_{0}=1. On the surface, an energy gap opens in the band structure for the 350% SOC case (Fig. 2d). Inside this gap, a pair of gapless SSs appears, with one branch merging into conduction bands and the other branch into valence bands, as shown in Fig. 3c. The existence of a single Fermi surface^{44} between and points provides unambiguous evidence of TSSs. Because SSs of the 100% SOC case are adiabatically connected to TSSs of the 350% SOC case, we can conclude that the normal SSs on the Au(111) surface are also TSSs.
Other noble metals Ag, Cu, Pt and Pd
Ag and Cu exhibit bulk and surface band structures that are very similar to those of Au (see Supplementary Fig. 5) and equivalent in topology. Therefore, we conclude that Au, Ag and Cu are all topological metals wherein TSSs exist on the surface (see Supplementary Fig. 2). We note that the spin splitting observed on Cu(111) is surprising larger^{18} than expected from the Rashba effect based on the weak SOC of Cu (ref. 5), indicating the topological origin of SSs. TSSs should also exist on other facets of these noble metals (including Au) owing to the nontrivial Z_{2} index of the bulk. This is consistent with SSs, for example, on (110) and (001) surfaces, reported in the literature (for example, refs 45, 46, and references therein). The topological energy gap above the Fermi energy is similar to the case of a recently discovered oxide TI, BaBiO_{3} (ref. 47), and to another topological metal, Sb. Bulk Sb is a semimetal with a direct energy gap and a nontrivial band structure^{27}. As a consequence, TSSs have been observed on special facets such as the Sb (111) and (110) surfaces in experiments^{48,49,50,51}. After we clarify the topological feature of Au, Ag and Cu, we further generalize the same idea to other FCC noble metals—Pt and Pd. These two metals show very similar bulk band structure to Au (Supplementary Fig. 6). The main difference is that their Fermi energies are lower than that of Au because Pt and Pd have one fewer valence electron than Au. Therefore, TSSs also exist on both Pt and Pd (111) surfaces, as shown in Fig. 4. Compared with those of the Au surface, these TSSs shift downward towards the bulk bands in energy but still lie above the Fermi energy for both Pt and Pd surfaces. For the Pt (111) surface, the Dirac point of TSSs is found to slightly merge into the bulk bands and forms a surface resonance. In the literature, these empty SSs have been observed and also interpreted as Shockley SSs with Rashbasplitting in photoemission and STS for both Pt^{52,53,54} and Pd^{55}. We note that the positions of TSSs in these reports^{52,53,55} are consistent with our results. For example, STS revealed the unoccupied SSs of the Pt (111) surface at with strong SOC splitting above the Fermi energy^{53}. Here we call these SSs on noble metal surfaces topologically derived SSs (TDSSs), to distinguish them from TSSs on a real insulator.
Discussions
Both TDSSs and Shockley SSs are ingap states and originate from the inversion of two bulk bands with different symmetries. The TI requires an odd number of band inversions at TRIM in the BZ, and TDSSs are protected by timereversal symmetry (TRS). In contrast, Shockley states also need band inversions, but they are not limited to finite positions of the zone and to a finite number. Hence, Shockley states may exist more commonly than TDSSs and lack the protection by TRS. The robustness of TSSs, which refers to their existence inside the inverted energy gap in the energy spectra, is protected by the band topology of the bulk. Such topological protection was indeed observed for TDSSs of noble metals in previous experiments. These states remain robust, for example, under the adsorption of alkali metals^{56,57,58,59}, guest noble metals^{54}, rare gases^{60,61}, and CO and oxygen^{56,62,63} and even against surface reconstruction^{2,64}. In contrast to trivial SSs such as dangling bond states, they usually shift in energy rather than get eliminated by adsorbates or reconstruction^{52}. However, the topological protection does not necessarily mean the robustness against any types of electron scattering, for example, by surface defects. Because the Dirac cone is heavily deformed, with both upper and lower cones, as well as bulk bands crossing the Fermi energy, TDSSs can be backscattered on the metal surface, which is different from those of a TI with an energy gap^{9,10}. This weakness allows versatile manipulation of these SSs by quantum confinement (for example, in quantum corrals). Therefore, we conclude that TDSSs on noble metal surfaces are more stable than trivial SSs in energy spectra due to topological protection. In addition, the Dirac point is expected to open an energy gap by breaking TRS, for example, by depositing magnetic impurities on the surface, as previously observed in magnetic TIs^{65,66}.
Methods
Ab initio calculations
The ab initio calculations have been performed within the framework of density functional theory with the generalized gradient approximation^{67}. We employed the Vienna ab initio simulation package with a plane wave basis^{68}. The core electrons were represented by the projectoraugmentedwave potential. The bulk band structure of Au was projected to Ausp orbitals in Fig. 2c using Wannier functions^{69}. The surface band structures were calculated on a slab model that includes thirtythree atomic layers using density functional theory.
ARPES
The ARPES experiments were performed in an ultrahigh vacuum setup with a base pressure of 10^{−11} mBar, equipped with a momentum microscope^{33} for the detection of the complete angular distribution of the photoelectrons as a function of their kinetic energy. The Au(111) surface was prepared in situ by repeated cycles of Ar^{+} ion sputtering at 2 kV and annealing to 570 K of a mechanically polished (111) crystal. For the 2PPE experiments, we used a commercial OPO laser system (Inspire OPO, Spectra Physics) with a pulse width of 150 fs. The laser was focused on the sample with a spot size of a few micrometres at an incidence angle of 65° with respect to the sample surface. The photon energy was tuned between 4.13 and 4.43 eV, while the light polarization was changed between s and p using a λ/2 plate.
Additional information
How to cite this article: Yan, B. et al. Topological states on the gold surface. Nat. Commun. 6:10167 doi: 10.1038/ncomms10167 (2015).
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Acknowledgements
We thank Prof S.S.P. Parkin at IBM Almaden Research Center San Jose and Prof S.C. Zhang at Stanford University for fruitful discussions. B.Y. and C.F. acknowledge financial support from the ERC Advanced Grant (291472) and computing time at HLRN Berlin/Hannover (Germany).
Author information
Affiliations
Max Planck Institute for Chemical Physics of Solids, 01187 Dresden, Germany
 Binghai Yan
 & Claudia Felser
Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany
 Binghai Yan
School of Physical Science and Technology, ShanghaiTech University, Shanghai 200031, China
 Binghai Yan
Department of Physics and Research Center OPTIMAS, University of Kaiserslautern, 67653 Kaiserslautern, Germany
 Benjamin Stadtmüller
 , Norman Haag
 , Sebastian Jakobs
 , Johannes Seidel
 , Dominik Jungkenn
 , Mirko Cinchetti
 & Martin Aeschlimann
I. Physikalisches Institut, GeorgAugustUniversität Göttingen, 37077 Göttingen, Germany
 Stefan Mathias
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Contributions
B.Y. conceived the project. B.Y. and C.F. performed the calculations. B.S., N.H., S.J., J.S., D.J., S.M., M.C. and M.A. performed the ARPES experiment. All authors analysed the results and wrote the manuscript.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to Binghai Yan.
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Supplementary Information
Supplementary Figures 16, Supplementary Note 1 and Supplementary References.
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