Abstract
Bismuthchalchogenides are model examples of threedimensional topological insulators. Their ideal bulktruncated surface hosts a single spinhelical surface state, which is the simplest possible surface electronic structure allowed by their nontrivial Z_{2} topology. However, real surfaces of such compounds, even if kept in ultrahigh vacuum, rapidly develop a much more complex electronic structure whose origin and properties have proved controversial. Here we demonstrate that a conceptually simple model, implementing a semiconductorlike band bending in a parameterfree tightbinding supercell calculation, can quantitatively explain the entire measured hierarchy of electronic states. In combination with circular dichroism in angleresolved photoemission experiments, we further uncover a rich threedimensional spin texture of this surface electronic system, resulting from the nontrivial topology of the bulk band structure. Moreover, our study sheds new light on the surfacebulk connectivity in topological insulators, and reveals how this is modified by quantum confinement.
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Introduction
Topological insulators are an exotic state of quantum matter, guaranteed to have metallic edge or surface states due to an inverted ordering of their bulk electronic bands^{1}. The corresponding topological invariants^{2} dictate that there must be an odd number of such states intersecting the Fermi level between each pair of surface timereversal invariant momenta. In the most widely investigated bismuthchalchogenide family of topological insulators, there is just one of these socalled topological surface states (TSSs) creating a single Dirac cone around the Brillouin zone centre^{3,4,5,6}. These materials are therefore widely regarded ideal templates to realize the predicted exotic phenomena and applications of this TSS. However, exposure to even the minute amount of residual gas in an ultrahigh vacuum chamber induces a drastic reconstruction of the surface electronic structure. The Dirac point shifts to higher binding energies, indicating more electronrich surfaces^{7,8,9,10}. More importantly, additional pairs of twodimensional, almost parabolic states emerge in the vicinity of the bulk conduction band^{7}, which develop large Rashbatype splittings^{8}, whereas new ladders of Mshaped states are created in the original bulk valence bands^{7,11}. Given their propensity for formation, these must be considered part of the intrinsic electronic structure of the surface of any realistic topological insulator, for example when exposed to air or interfaced to another material. However, their existence is not predicted by idealized theoretical models of the bulktruncated surface^{3}, and has proved controversial. Recent proposals attribute their formation to a variety of electronic or structural modifications of the crystal host^{7,8,9,11,12,13,14,15,16,17,18,19,20,21,22}, but to date no single model has been able to simultaneously reproduce the number and binding energies of the experimentally observed electronic states as well as the magnitude of their measured Rashbatype splittings.
In the following, we show that this can be achieved taking account of only a single electronic phenomenon—a nearsurface electrostatic potential variation. Such intrinsic electric fields are familiar from surface spacecharge regions in conventional semiconductors^{23,24}, and are also formed at interfaces where they can be modulated using applied gate voltages. Thus, our findings are not only relevant to understand the free surfaces of generic topological insulators, but also their use in transistorstyle or other thinfilm device architectures, which may ultimately provide one of the most promising routes towards spintronics. Our study elucidates the crucial interplay of quantum confinement with topological order for such systems, and uncovers a unique spin texture of the resulting electronic states. We consider Bi_{2}Se_{3} as a model example, but our computational approach can be freely adapted to virtually all topological insulators, such as other emerging Bichalcogenides or the halfHeusler alloys^{25}, where we would expect a similarly striking role of band bending at free surfaces, interfaces, or in devices.
Results
Bandbendingdriven quantum confinement
We implement such a band bending scenario (Fig. 1a) within a 120 quintuplelayer tightbinding supercell calculation (see Methods). Without incorporating any band bending, our supercell model yields a single TSS spanning from the bulk valence to conduction bands (Supplementary Fig. S1), in good agreement with both firstprinciples slab calculations^{3} and our experimental measurements of the pristine surface (Supplementary Fig. S2). Additionally including an electrostatic potential variation, we find a hierarchy of electronic states emerges that is in excellent agreement with our experimental angleresolved photoemission (ARPES) measurements of a doped topological insulator surface (Fig. 1b). The effective mass is slightly overestimated in our model, which stems from a small overestimation of this quantity in our bulk band structure calculations. Nonetheless, all other features are well reproduced. This agreement is remarkable, given that we do not adjust any parameters in our calculation. All tightbinding matrix elements are obtained from an ab initio calculation of the bulk electronic structure. The magnitude of the potential change is fixed by the experimentally measured shift of the Dirac point between a pristinecleaved surface and the ntype doped one (see Supplementary Fig. S2), and its functional form follows from a solution of Poisson's equation^{26}.
All of the new electronic states observed experimentally arise simply due to the perturbative effect of this electrostatic potential in the system, which causes a triangularlike downward bending of the bulk electronic bands close to the surface by almost 250 meV (Fig. 2b). Combined with the large potential step at the material/vacuum interface, this creates a nearsurface quantum well for electrons. The bulk conduction band states are consequently restructured into ladders of multiple subband states^{7,27,28}. Our layerresolved calculations (Fig. 2) show that the lowest such subband, lying deep within the potential well, is localized in the topmost few quintuple layers of the structure: The electrons that populate this subband are free to move parallel to the surface, but are strongly confined perpendicular to it, forming a twodimensional electron gas (2DEG). This state exhibits a strong Rashba spin–orbit splitting, whose calculated magnitude reproduces our experimental observations. Higherlying states of the subband ladder are less strongly bound within the surface quantum well and contribute significant weight as far as 10–15 quintuple layers below the sample surface. The smaller spin splitting observed experimentally for these higherlying states is again reproduced by our model calculations, confirming that the Rashba splitting is driven by the potential gradient of the confining electrostatic potential^{8}. Similar to the conduction band 2DEG, a ladder of subbands is also observed in the valence band near the zone centre. These holelike states become quantum confined between the surface and the upper edge of a projected bulk band gap^{3,11}, as shown schematically in Fig. 1a. The lowest of these, as for the topological and lower conductionband quantum well states, is strongly confined within the first 3–4 quintuple layers below the surface. We also find a small spin splitting of these valence band quantum well states, which is not resolved experimentally. Its location, around the top of their Mshaped dispersions, indicates a subtle interplay of spin–orbit interactions between the valence and conduction band subbands^{29}. Thus, a hierarchy of electronic states in both energy and spatial extent is created, spanning from the twodimensional limit of the TSS and lowest subbands of the conduction and valence bands to the threedimensional continuum (Fig. 1a).
Threedimensional spin texture
Intriguingly, we find an anticorrelation between this electronic dimensionality, and that of the states' spin texture. This is revealed experimentally via an asymmetry in the matrix element for photoemission when the system is excited by right and left circularly polarized light, respectively (Fig. 3a–c). Such circular dichroism has previously been suggested to probe the spin of the isolated TSS, although the microscopic origin of the dichroism has proved controversial^{30,31,32,33,34}. Recently, Park et al.^{31} proposed the existence of an unquenched local orbital angular momentum (OAM) of the TSS. They found that this has the same chiral structure as the spin angular momentum (SAM), permitting the well known spinmomentum locking of the TSS, but opposite helicity due to the strong spin–orbit coupling. They proposed that the observed dichroism in ARPES measurements of the TSS results due to this orbital texture. Such circular dichroism, therefore, still reveals the spin texture of the system (up to an arbitrary sign), but indirectly via the orbital sector. We therefore employ this to extract the relative spin texture of the entire lowenergy electronic structure of the surfaces of topological insulators. We note that, for the relative changes in dichroism between the bands that we study here we can neglect finalstate effects, as transitions from all of these closely spaced initial states will go into the same (broad) final state.
Along the Γ–M mirror direction, symmetry requires that the spin, and therefore also the OAM, must lie entirely within the surface plane^{35}. Our measurements of circular dichroism extracted along this direction (Fig. 1b) are therefore representative of the inplane spin–orbital texture. The dichroism changes sign upon moving inwards from the outermost Fermi surface sheet of the topological state to the subsequent Fermi surfaces of the lowest Rashbasplit subband (E_{0}) of the 2DEG. This is entirely consistent with the results of spinresolved photoemission for these states^{8}. More extensive analysis (Supplementary Fig. S3) indicates that this trend continues for the next 2DEG subband. Given the lefthanded helicity of the TSS^{9}, these measurements therefore indicate an alternating left–right–left–right–left helicity for the spin of consecutive Fermi surface sheets of the TSS and first (E_{0}) and second (E_{1}) quantized conduction subbands.
Away from this direction, dichroism of the inner E_{0} and E_{1} 2DEG Fermi surface sheets vary sinusoidally with angle, φ, around the Fermi surface (shown in Supplementary Fig. S3). This reveals a model Rashbalike^{36} helical spin texture of these states, with the spinpolarized entirely in the surface plane at all energies. However, for the more spatially localized TSS and outermost 2DEG state, the dichroism additionally develops a strong sin(3φ) component, which is maximal along Γ–K and zero along Γ–M (Fig. 1c and Supplementary Fig. S3). This reflects a strong outofplane spin polarization close to the Fermi level, which is correlated with hexagonal warping of the electronic states^{35,36,37,38,39}, again switches sign between neighbouring states, and decreases with increasing binding energy (Supplementary Fig. S4).
These findings are fully supported by our tightbinding supercell calculations (Fig. 3d; Supplementary Fig. S5), where we find that the OAM and SAM exhibit an alternating helical structure for consecutive Fermi surface sheets, both also developing increasing outofplane components with hexagonal warping. The local OAM, therefore, similar to the spin texture, does lie purely in the surface plane at kpoints along Γ–M, whereas away from this highsymmetry direction it gradually attains a net outofplane component, which is maximal along Γ–K. However, the OAM helicity of each band is opposite to that of its SAM. This not only confirms that the TSS possesses intricately related OAM and SAM structures, but also reveals that the nontrivial topology drives the TSS and 2DEG states to have a novel alternating helical OAM texture. Thus, not only does the band bending promote the creation of additional Fermi surface sheets, it also leads to complex band and bindingenergydependent threedimensional orbital and spin textures of the resulting electronic system, as summarized for the spin in Fig. 3e. Together, this will cause significant complications for the interpretation of spindependent transport experiments, and interband processes will weaken the topological protection against backscattering commonly assumed for the TSS of these compounds. Nonetheless, we note that our resolution limited linewidths of full width at half maximum indicate that the intrinsic scattering remains relatively weak for all Fermi surface sheets.
Surfacebulk connectivity
The complex spin texture of these states also raises questions over their interplay. For any system with timereversal symmetry, the electronic states must be spindegenerate at timereversal invariant momenta, even though a breaking of inversion symmetry by the confining potential and surface allows a lifting of the spin degeneracy at arbitrary kpoints. The hexagonal Brillouin zone of the Bi_{2}Se_{3} surface contains timereversal invariant kpoints at the zone centre (Γ point) and at the sidecentre (M) point. Figure 4a shows the electronic structure calculated along the entire Γ–M direction. At the Γ point, each Rashbasplit pair of 2DEG states clearly become degenerate. For the TSS, this initially appears to be complicated by the presence of bulk states, with the Dirac point looking to be buried deep in the bulk valence band. However, locally, the TSS is still situated within a projected band gap of the (bent) bulk bands, and so remains a welldefined surface state (unlike the 2DEG states that are located within, and indeed derived from, the projected bulklike bands). Our layerprojected calculations (Fig. 2) show that the Kramer's degeneracy of the TSS is preserved in the presence of the bandbending potential, contrary to previous suggestions of a band gap opening in the surface spectrum at the Dirac point^{13,17}.
At the Mpoint, our calculations reveal that the TSS connects to the outer branch of the lower 2DEG state. Assuming a spindown state for the TSS, the pairing 2DEG state has no other choice but to be spinup. Its spindown partner must then become connected with the outer branch of the second subband to satisfy TR symmetry for these states, thereby dictating the overall up–down–up–down–up spin configuration discussed above. Fundamentally, this pairexchange follows from the nontrivial Z_{2} topology of the bulk band structure. This requires that the TSS must connect from the bulk valence to conduction bands. Now, however, the original bulk bands have become quantized into twodimensional subband states in the vicinity of the surface, and our calculations indicate that it is to these subbands which the topological state connects.
If the band bending is reduced (Fig. 4d), the subband states will become splitoff less far below the bulk conduction bands, while their Rashba spin splitting will simultaneously decrease. This will consequently reduce the relative energy separation of the multiple band connectivities at the Mpoint. Extending this continuously to the limit of no band bending, the subbands evolve into the lower bulk conduction band states, while the zoneedge connectivities merge into a single degeneracy at the Mpoint between the topological state and the bulk electronic continuum. This clarifies how the surfacebulk connectivity occurs in the idealized case of topological insulators with a pristine surface and no band bending. In the more realistic case where band bending is present, however, the surfacebulk connectivity is mediated via the quantumwell states. This requires the outer and inner branches of the spin–split subbands to diverge from each other at large kvectors, in stark contrast to conventional Rashba systems^{41} in which the two spin channels merge again at zoneedge timereversal invariant momenta (Fig. 4c). In topologically trivial surface systems where giant Rashba effects have been observed, the spin splitting depends sensitively on details of the nearsurface atomic coordinates relative to the surface potential step^{42}. In contrast, because of the subband connectivity, the magnitude of the Rashba splitting in a topological insulator 2DEG is directly linked to the subband separation. The possibility to control the 2DEG spin splitting electrostatically via the band bending is therefore intrinsic to topological insulator 2DEGs, rendering these systems ideal candidates for applications in spintronics.
Hexagonal warping of the electronic states can additionally induce accidental degeneracies along highsymmetry lines^{43,44}, as seen in Fig. 4. However, as these are not protected by any fundamental symmetry, introduction of perturbations in the system can shift, or even completely remove, such accidental degeneracies. Experimentally, the inner branch of the lower 2DEG state and outer branch of the upper 2DEG state can be seen to approach each other at low binding energies (Fig. 4d), consistent with their proximity to the accidental crossing predicted in our calculations. However, contrary to previous reports^{13,17}, our highresolution measurements reveal that the occupied bands remain nondegenerate and exhibit rather parallel dispersion close to the Fermi level, which could be indicative of an avoided crossing.
Discussion
The electronic structure that we consider here represents the generic surface electronic structure of topological insulators under realistic conditions, such as exposure to atmosphere^{12,21}, or in desired devices such as the (spin)transistor^{8}. We have shown that this entire surface electronic system can be accurately described, considering only a single modification of the pristine crystal—a nearsurface electrostatic potential variation. The quantitative agreement of our parameterfree tightbinding supercell calculations to our photoemission measurements demonstrates that such a band bending alone is sufficient to induce the entire experimentally measured complex electronic structure of topological insulator surfaces. We stress that our approach does not require any additional modifications beyond an electrostatic potential, such as a huge increased van der Waals gap size^{15,16,18,20}, intercalation^{45}, other artificial detachments of a finite number of crystal layers^{14}, or the formation of new TSSs^{13}.
Despite its conceptual simplicity, the band bending has dramatic consequences for the surface electronic structure. Driven by an emergent role of quantum size effects, it leads to a rich surface electronic system simultaneously comprising both electron and holelike subbands of varying spatial confinement and with intertwined threedimensional spin textures. We have shown how these subbands mediate connectivity of the bulk bands and the TSS, one of the defining properties of topological insulators, through a series of band degeneracies required by timereversal symmetry, and further identify possible accidental degeneracies, which we do not observe experimentally. Understanding whether interactions not included in our theoretical treatment have a role in avoiding such accidental degeneracies will be crucial to unravel surfacebulk coupling in these materials, and the role of interactions in limiting mobilities of spinpolarized surface transport in topological insulators.
Methods
Theory
Relativistic electronic structure calculations were carried out within the context of density functional theory using the Perdew–Burke–Ernzerhof exchangecorrelation functional^{46} and the augmented plane wave plus atomic orbitals method as implemented in the WIEN2K program^{47}. For all the atoms, the muffintin radius R_{MT} was set to 2.5 Bohr and the maximum modulus of reciprocal vectors K_{max} was chosen such that R_{MT}K_{max}=7.0. The primitive cell was considered to be hexagonal (space group: ) with lattice parameters and atomic positions taken from experiment^{48}. The corresponding Brillouin zone was sampled using a 10 × 10 × 3 kmesh. To simulate the effect of band bending, a 120 qunituplelayer tightbinding supercell Hamiltonian was constructed by downfolding the augmented plane wave plus atomic orbitals Hamiltonian using maximally localized Wannier functions^{49,50,51}. We chose valence p orbitals of Bi and Se as the projection centres of the Wannier functions. The bending potential was obtained by solving the coupled Poisson–Schrödinger equation^{26}, assuming a static dielectric constant of 70 (ref. 8). The resulting potential was then added to the onsite terms of the tightbinding supercell Hamiltonain. The (quintuplelayerprojected) surface band structure was eventually obtained by diagonalizing the supercell Hamiltonian (and projecting the resulting eigenstates onto the maximally localized Wannier functions of respective surface quintuple layers).
Experiment
Bi_{2}Se_{3} crystals were prepared by reacting highpurity elements (5 N) of Bi and Se in sealed, evacuated quartz tubes at 850 °C for 2 days, followed by cooling at 2–3 °C h^{−1} to 650 °C. The crystals were annealed at this temperature for 7 days before quenching to room temperature. ARPES measurements were performed at 10 K using circularly polarized light with photon energies of 20–30 eV and a Scienta R4000 hemispherical analyser at the SIS beamline of the Swiss Light Source. Surface doping was achieved by depositing potassium at room temperature on freshly cleaved samples from a properly outgassed SAES K getter source. The sample was then immediately transferred to the lowtemperature manipulator for ARPES measurements.
Additional information
How to cite this article: Bahramy, M.S. et al. Emergent quantum confinement at topological insulator surfaces. Nat. Commun. 3:1159 doi: 10.1038/ncomms2162 (2012).
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Acknowledgements
We are grateful for support from the UK EPSRC, the ERC and the Japan Society for the Promotion of Science (JSPS) through the ‘Funding Program for WorldLeading Innovative R&D on Science and Technology (FIRST Program)’, initiated by the council for Science and Technology Policy (CSTP). This work was also supported by GrantinAids for Scientific Research (No. 24224009) from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan, and Strategic International Cooperative Program (Joint Research Type) from Japan Science and Technology Agency.
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MSB developed the tightbinding model with assistance from RA, and performed the tightbinding model calculations. PDCK, ADLT and FB performed the ARPES measurements. PDCK analysed the ARPES data and performed the Poisson–Schrödinger calculations. JC, MS and LP maintained the ARPES endstation at the Swiss Light Source and assisted with the ARPES measurements. PH contributed preliminary measurements and discussions. GB grew the samples. PDCK, MSB and FB wrote the paper with input from all coauthors. All authors discussed the interpretation of the results and commented on the final manuscript. PDCK, MSB, NN and FB were responsible for overall project planning, infrastructure and direction.
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Bahramy, M., King, P., de la Torre, A. et al. Emergent quantum confinement at topological insulator surfaces. Nat Commun 3, 1159 (2012). https://doi.org/10.1038/ncomms2162
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DOI: https://doi.org/10.1038/ncomms2162
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