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# Crossover of the three-dimensional topological insulator Bi2Se3 to the two-dimensional limit

## Abstract

A topological insulator1,2,3,4,5,6,7,8,9 is a new state of quantum matter that is characterized by a finite energy gap in the bulk and gapless modes flowing along the boundaries that are robust against disorder scattering. The topological protection of the surface state could be useful for both low-power electronics10 and error-tolerant quantum computing11,12. For a thin slab of three-dimensional topological insulator, the boundary modes from the opposite surfaces may be coupled by quantum tunnelling, so that a small, thickness-dependent gap is opened up13,14,15. Here we report such results from angle-resolved photoemission spectroscopy on Bi2Se3 films of various thicknesses grown by molecular beam epitaxy. The energy gap opening is clearly seen when the thickness is below six quintuple layers. The gapped surface states also exhibit sizeable Rashba-type spin–orbit splitting because of the substrate-induced potential difference between the two surfaces. The tunable gap and the spin–orbit coupling make these topological thin films ideal for electronic and spintronic device applications.

## Main

Bi2Se3, along with the congeneric Bi2Te3 and Sb2Te3, is theoretically predicted16,17 and experimentally observed17,18,19 to be a three-dimensional (3D) topological insulator with a single Dirac cone. For possible applications of the topological insulators, planar devices would be most promising. Thin films of Bi2Se3 have been grown by molecular beam epitaxy20 (MBE) and by the vapour–liquid–solid method in nanoribbon form21. Finite-size effects have to be considered for a topological insulator film of only several nanometres. For both 2D (ref. 22) and 3D (refs 13, 14, 15) topological insulators, the finite-size effects were studied theoretically within the effective models of refs 3, 16, and by first-principles calculations16. Owing to the coupling between the surface states, a finite energy gap is predicted to open in the 2D limit of the 3D topological insulators. Furthermore, the 2D energy gap is predicted to oscillate between the ordinary insulator gap and the quantum spin Hall (QSH) gap as a function of thickness. In this work, by combining MBE, angle-resolved photoemission spectroscopy (ARPES) and model calculations, we have systematically studied the thickness-dependent evolution of the band and spin structure of 3D topological insulator Bi2Se3 film with a thickness from one quintuple layer (QL) to 50 QL.

Figure 1a and b show the reflection high electron energy diffraction (RHEED) pattern and a scanning tunnelling microscope (STM) image of a typical MBE-grown Bi2Se3 film with a thickness of 50 QL, respectively. The sharp 1×1 streaks and atomically flat morphology demonstrate the high crystal quality of the film. Figure 1c shows the band map of the film. The bands show similar structure and dispersion to bulk Bi2Se3 (ref. 17). Two bands with nearly linear dispersion cross each other at the point, forming a Dirac cone. They are the gapless surface states of Bi2Se3. The band map shown in Fig. 1c does not exhibit the conduction band like in the bulk samples17. The Dirac point is located at 0.12 eV below the Fermi level. The observation indicates that our MBE-grown Bi2Se3 samples are a nearly intrinsic topological insulator with a single Dirac point and contain fewer impurities, compared with the bulk samples17,23.

By delicate control of the growth condition, we were able to achieve QL-by-QL growth of the Bi2Se3 thin films. Figure 1d shows a series of normal emission spectra of Bi2Se3 films from 1 to 6 QL with a step of 0.3 QL. The peaks indicated by broken vertical bars are quantum well states (QWSs) of the valence band, which will be addressed later. With increasing thickness, a stepwise change in the QWS peaks is clearly observed in Fig. 1d, which indicates a QL-by-QL growth mode24. It enables us to investigate the thickness-dependent band structures of the Bi2Se3 films with an accuracy of 1 QL.

Figure 2a–e shows a series of ARPES band maps of Bi2Se3 films from 1 to 6 QL (except for 4 QL) measured at room temperature. All of the spectra are taken along the direction. The spectra of 1 QL (Fig. 2a) exhibit a nearly parabolic band dispersion towards the Fermi level along k//. At 2 QL (Fig. 2b), the band moves down by 40 meV and another band appears below it. As shown later, the two bands finally evolve into the gapless surface states in thicker films. We refer to them hereafter as the upper and lower surface states, respectively, although the distinction between the surface and bulk states is blurred in films this thin. At 3 QL (Fig. 2c), the gap between the upper and lower surface states decreases. Interestingly, the upper surface states are split into two branches. The splitting is more prominent at larger wavevectors but disappears at the point as shown in the energy distribution curves (EDCs) in Fig. 2f. This is a Rashba-type splitting, in which the two sub-bands with different spins shift along the k// axis in opposite directions and degenerate at the point. For thicker films at 5 and 6 QL (Fig. 2d,e), the gap between the two surface states decreases further. Meanwhile, the outer two branches of the split upper surface state fade out gradually. At 6 QL, the gap disappears and so do the outer branches of the upper surface states. The inner branches of the upper surface states connect the lower surface states, resulting in a Dirac cone (Fig. 1c).

The Dirac point at 6 QL is located at −0.26 eV, lower by 0.14 eV than that of the thick (50 QL) Bi2Se3 films shown in Fig. 1c. It suggests that the film is electron doped. Above 6 QL, the Dirac point moves towards the Fermi level with increasing thickness until 20 QL where the saturation value is reached. Figure 3a plots the thickness dependence of the Dirac point position. For comparison, the measured mid-gap energies of the surface states and the spin-degenerate points of the Rashba-split upper surface states are also shown. As STM experiments show little difference in surface defect density between thick and thin films, the thickness-dependent electron doping may come from band bending resulting from charge transfer from the SiC substrate.

Besides the surface states, above 2 QL, there are other features either above the upper surface states or below the lower surface states. With increasing thickness, these bands move up or down towards the surface states, and more bands emerge. In Fig. 3b we summarize the energies of these bands at the point with respect to the mid-gap energies of the surface states. The mid-gap energies are used as a reference to remove the influence of chemical potential change. The energy evolution shows a typical QWS behaviour. Therefore, these bands can be attributed to QWSs of conduction and valence bands.

When the thickness of a film is reduced to only several nanometres, overlapping between the surface-state wavefunctions from the two surfaces of the film becomes non-negligible, and hybridization between them has to be taken into account. For a freestanding and symmetric Bi2Se3 film, the spin-polarized surface states at one surface will be mixed up with the components of opposite spin from the other surface when the thickness is small enough. It will lead to a hybridization gap at the Dirac point to avoid crossing of bands with the same quantum numbers. The observed gap opening below 6 QL can preliminarily be attributed to this effect. The band and spin structures of a 3D topological insulator such as Bi2Se3 near the Fermi level are well described by the four-band effective model16. In the ultrathin limit15, the 3D topological insulator model of ref. 11 reduces to the Bernevig–Hughes–Zhang model3 of the 2D QSH insulator, and the in-plane dispersions of the surface states can be written as15:

The term including the Fermi velocity vF is the Dirac component of the bands, and Δ represents the gap size. The definition of the parameters D and B can be found in ref. 15. For a 1 QL film, only one band can be observed. The absence of the lower surface states may be due to the bonding with the substrate. The ARPES spectra of 2 QL can be well fitted by equation (1). The pink dashed lines in Fig. 2b depict the fitted curves with the parameters listed in Table 1. However, starting from 3 QL, the surface states exhibit Rashba-type splitting. The splitting cannot be obtained with equation (1), and neither by previous first-principles calculation16. Both the effective model and the first-principles calculation are based on a free-standing symmetric topological insulator film, whereas the existence of substrate breaks the structural inversion symmetry, which can result in a Rashba-type spin splitting in the ultrathin-film regime.

To model the substrate effect, we invoke an effective electrical potential V (z) along the film normal direction. From this modified model, we can obtain the dispersions25:

The term represents the effect of the potential V (z). Each spin-degenerate dispersion in equation (1) is now split into two sub-bands of opposite spins (σ=±1) that shift away from each other along the k// axis. If , equation (2) degenerates into equation (1), which corresponds to the 2 QL case discussed above. The splitting of the upper surface states is very well reproduced in the fitting with equation (2) (see the dashed lines in Fig. 2c–e). When Δ is zero, the dispersions obtained from equation (2) evolve into two sets of Dirac cones centred at the point. The two Dirac points are shifted in energy by . In the ARPES data, at 6 QL only one Dirac cone was observed. The outer branches of the Rashba-split upper surface states, which are expected to evolve into part of the other Dirac cone, are nearly invisible. In Fig. 3c, the calculated weights in real space of the surface states at 5 QL are shown with the colours of the lines. It is clear that near critical thickness the outer and inner branches of the Rashba-split surface states localize at different surfaces of the film. The two Dirac cones obtained from equation (2) correspond to the two sets of gapless surface states from the surface and interface sides of the film, respectively. The absence of one Dirac cone in the ARPES spectra is due to its localization at the interface side, which is beyond the detection depth of ARPES. In this context, the observed Rashba splitting can also be understood as a result of the hybridization of the two sets of gapless surface states with different Dirac points from the two surfaces of the film.

The breaking of structural inversion symmetry of the system can originate from the band bending induced by the substrate, as suggested in Fig. 3a, or alternatively from different environments of the two surfaces. In the former case, when the film is much thinner than the thickness of the band-bending region, that is, the Debye length, which is determined by the dielectric constant and carrier density of the material, the potential variation along the z direction of the film will be negligible. So the Rashba effect is weaker in thinner films. In the latter case, the structural asymmetry is more significant in thinner films, and thus the Rashba effect is stronger. Our observation of reduced Rashba splitting with decreasing film thickness suggests that the former case dominates in Bi2Se2 films. In an epitaxial film, band bending can be modified by the surface photovoltage (SPV) effect, which takes place in a p–n or Schottky junction in which the photon-excited electrons and holes are driven in opposite directions by a space charge layer26,27. As a result, the band bending is reduced and even completely removed. The SPV effect is more prominent at low temperature because of less electron–hole recombination. The epitaxial Bi2Se3 films show a strong SPV effect at 150 K, which is confirmed by the 0.06 eV shift of the Fermi edge compared with the room-temperature spectra26. In Fig. 3e, we show EDCs of ARPES spectra of 3 QL Bi2Se3 at 150 K. Rashba splitting cannot be distinguished any more, whereas other features, for example the gap size, are basically similar to the room-temperature data. The thickness dependence of energies of the Dirac point (6 QL) and mid-gap position (<6 QL) measured at low temperature is also shown in Fig. 3a. The variation of chemical potential with thickness is largely reduced, suggesting a nearly flat band. The quench of the Rashba splitting by the SPV effect demonstrates that the splitting is mainly contributed by substrate-induced band bending.

The above conclusion implies that the Rashba splitting can be controlled by modifying band bending, which can be easily realized by applying a gate voltage. It is of great significance for spintronic devices, especially the Datta–Das spin field-effect transistor28, in that the spin structure can be tuned electrically. In Table 1, the fitted Rashba parameter αR values are listed. The Bi2Se3 films (3–6 QL) show at least one order of magnitude larger Rashba splitting than semiconductor heterostructures29, and the splitting can be removed by changing the band bending by 0.3 eV. Therefore, ultrathin films of topological insulators provide a new kind of spintronic materials that may realize high-efficiency electrical spin manipulation.

A thickness-dependent oscillatory transition with a periodicity of 3 QL between QSH and ordinary insulator phases was predicted theoretically14. The oscillation mostly refers to the sign of the gap, but could also lead to an oscillation in the magnitude of the gap size13,14,15. In Fig. 3d, we show the measured gap size as a function of thickness, which is more or less monotonic. According to refs 15, 25, if Δ and B in equation (2) have the same sign, the film is in the QSH phase, otherwise, it is a trivial insulator. From the fitted parameters listed in Table 1, we can see that from 2 to 5 QL, Δ and B have the same sign. Therefore, it illustrates that the films are in the QSH regime. However, because band dispersion can be influenced by other non-universal factors, further experiments by STM and electrical transport measurement with and without external magnetic field, similar to those carried out in ref. 4, are needed to clarify this issue.

## Methods

All experiments were carried out in an ultrahigh-vacuum system (Omicron), equipped with MBE, an STM and ARPES. The base pressure of the system is 1.5×10−10 torr. In the ARPES measurement, photoelectrons are excited by an unpolarized He-Iα light (21.21 eV), and collected by a Scienta SES-2002 analyser (15 meV). Bi2Se3 films were grown under Se-rich conditions on a double-layer-graphene-terminated 6H-SiC (0001) substrate at 220 °C by MBE. Bi (99.9999%) and Se (99.999%) were both evaporated from standard Knudsen cells. The growth rate was calibrated by a real-time RHEED intensity oscillation measured on the (00) diffraction. The Se4(Se2)/Bi flux ratio was between 10 and 15, which leads to a growth rate of 0.3 QL min−1 when the Bi source temperature was set at 550 °C.

## Change history

• ### 12 August 2010

On the first page of the PDF and printed versions of this Letter originally published, the full list of authors and their affiliations should have been included. This has now been corrected on the PDF version of the Letter.

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## Acknowledgements

This work was supported by the National Science Foundation and Ministry of Science and Technology of China and RGC of Hong Kong, China under grant No. HKU 7037/08P. X-L.Q., S-C.Z. and Q.N. are supported by the Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, under contracts DE-AC02-76SF00515 and DE-FG03-02ER45958.

## Author information

Authors

### Contributions

K.H., S-Q.S., X-C.M. and Q-K.X. conceived and designed the experiments. Y.Z., K.H. and C-Z.C. carried out MBE growth and ARPES measurements. C-L.S. carried out substrate preparation and STM observation. L-L.W., X.C., J-F.J., X-C.M. and Q-K.X. assisted in the experiments. W-Y.S. and S-Q.S. carried out the theoretical analyses and data interpretations. Q.N., Z.F., X.D., X-L.Q. and S-C.Z. assisted in the theoretical analyses. Y.Z., K.H., S-Q.S., S-C.Z. and Q-K.X. prepared the manuscript.

### Corresponding authors

Correspondence to Ke He or Qi-Kun Xue.

## Ethics declarations

### Competing interests

The authors declare no competing financial interests.

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Zhang, Y., He, K., Chang, CZ. et al. Crossover of the three-dimensional topological insulator Bi2Se3 to the two-dimensional limit. Nature Phys 6, 584–588 (2010). https://doi.org/10.1038/nphys1689

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• DOI: https://doi.org/10.1038/nphys1689

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