Abstract
Based on an inverted bulk band order, antimony thin films presumably could become topological insulators if quantum confinement effect opens up a gap in the bulk bands. Coupling between topological surface states (TSS) from opposite surfaces, however, tends to degrade or even destroy their novel characters. Here the evolution and coupling of TSS on Sb(111) thin films from 30 bilayers down to 4 bilayers was investigated using insitu Fouriertransform scanning tunneling spectroscopy and density functional theory computations. On a 30bilayer sample, quasiparticle interference patterns are generated by the scattering of TSS from the top surface only. As the thickness decreases, intersurface coupling degrades spin polarisation of TSS and opens up new wavevectordependent scattering channels, resulting in spin degenerate states in most part of the surface Brillouin zone, whereas the TSS near the zone centre exhibit little intersurface coupling, so they remain spinpolarised without opening a gap at the Dirac point.
Introduction
Recent experimental^{1,2,3,4,5} and theoretical^{6,7,8} works have demonstrated that topological surface states (TSS) support massless spinpolarised Dirac fermions due to a strong spinorbit coupling effect. They are robust and immune to backscattering by nonmagnetic defects because of protection by timereversal symmetry^{9}. Unlike the Dirac fermions in graphene with pseudospin texture, the real helical spin polarization TSS exhibit has been widely studied by surfacesensitive experiments such as angleresolved photoemission spectroscopy (ARPES), scanning tunneling microscopy (STM) and scanning tunneling spectroscopy (STS) as summarized in recent review articles^{10,11}. TSS are attractive not only in fundamental condensed matter physics for realising novel entities such as dyons^{12}, imaging magnetic monopoles^{13} and Majorana fermions^{12,14}, but also in practical applications in spintronics and errortolerant quantum computing. Motivated by such perspective, special attention has been focused on thin films hosting TSS suitable for device applications. Therefore, threedimensional (3D) topological insulator (TI) thin films approaching 2D limit have been explored. Theoretical calculations predict that an energy gap can be opened at the Dirac point by intersurface coupling^{15}. Some of these films may exhibit quantum spin Hall effect as that observed in HgTe quantum wells^{16,17}. Experimental studies on Bi_{2}Se_{3}^{18,19} and Bi_{2}Te_{3}^{20} confirm the existence of a thicknessdependent bandgap. However, the detail of intersurface coupling effect, especially the intermediate state between weak and strong coupling of surface states (SS) from two surfaces, still needs to be interrogated. From topological nontrivial to trivial phases, interesting transformation must take place progressively as the overlap of electron wave functions on two surfaces increases with reducing film thickness.
Here we investigate the intersurface coupling issue in antimony (Sb), which has a rhombohedral crystal structure and can be considered as a stacking of (111) bilayers (BL, 1 BL = 3.75 Å). There are several reasons to study Sb(111) films. Firstly, as the “parent” of the firstgeneration 3D TI Bi_{1−x}Sb_{x}^{21}, although bulk Sb is semimetal due to its negative indirect bandgap, its band order is inverted at the L point of the Brillouin zone. Sb(111) has been confirmed to possess TSS, leading to the absence of 180° backscattering and exotic transmission through atomic steps^{22}. Next, the strongly distorted TSS Dirac cone on Sb(111) yields various scattering channels in QPI patterns, from which TSS dispersion and spin information can be extracted. Measuring the patterns at different film thickness helps us understand how the spin texture evolves as intersurface coupling varies. Thirdly, a thin film with a large surfacetovolume ratio can make surface effect more observable. Lastly, compared with wellstudied compound TIs such as Bi_{2}Se_{3}, Sb thin films provide a singleelement simple system for demonstrating topological properties without much influence of residual bulk carriers from selfdoping states^{2,4,23} or spatial fluctuations of charge and potential^{24}. Therefore, by using Fouriertransform scanning tunneling spectroscopy (FTSTS), we measure the quasiparticle interference (QPI) patterns of TSS on thick and thin Sb(111) films. Combining with density functional theory (DFT) calculations, we aim to identify the scattering features in the patterns generated by the intra and intersurface couplings of TSS, and to examine the dependence of the couplings on film thickness and wavevector k.
Results
We grew Sb films on Si(111)√3 × √3:Biβ surface in an ultrahigh vacuum (UHV) chamber^{25,26,27,28}. Shown in Figure 1a is a STM image of an Sb(111) film of thickness ~30 BL. Steps here are all 3.75 Å corresponding to 1 BL of Sb(111). The atomic resolution image in the inset exhibits the hexagonal lattice on the (111) surface. In Fig. 1b, a typical differential tunneling conductance dI/dV_{b} spectrum, where V_{b} is sample bias voltage and I the tunneling current, is displayed to reveal the local density of states (LDOS) of electrons in certain energy range. Unlike Bi_{2}Se_{3} and Bi_{2}Te_{3}, Sb(111) has a strongly distorted TSS Dirac cone, with its lower part bowing up rapidly near point of the first surface Brillouin zone (SBZ) before bending down to merge with bulk valence bands, as shown schematically in Fig. 1b inset. Several extraordinary features of Sb(111) show up starting from the Dirac point E_{D} at E ≈ −240 meV (with the Fermi energy E_{F} at E = 0), which have also been confirmed by previous ARPES measurements^{1,29}. The specific dispersion of TSS on Sb(111) allows us to determine E_{D} being at the appearance E of QPI patterns^{22,30}.
For TSS, it is generally accepted that, without magnetic field or magnetic impurities, scattering between states must obey spin conservation. That is, the scattering intensity P from a state of wavevector k and spin S to another state of and depends on , the angle between S and , as^{5,30} STS can detect the amplitude modulation of LDOS due to the scattering of momentum transfer vector ^{5,31}. After Fourier transform, the realspace modulations can be visualised as a distribution of q in the kspace, offering us information of the SS involved in scatterings and even their spin alignment at a given E. Hence, the dI/dV_{b} maps on 30BL Sb(111), which can be regarded as the bulk surface, were taken under different V_{b}. Above E ≈ −240 meV, 6fold symmetric patterns with dominating intensity along the direction can be identified in the FTSTS power spectra. Fig. 2a shows the coexistence of atomic structure and QPI patterns, while Fig. 2b gives its Fourier transform, which also contains both the contributions from atomic lattice (the outer 6 reciprocal lattice points) and QPI pattern near at the centre. The hexagon marks the first SBZ.
The QPI patterns are our main concern. Starting at E ≈ −40 meV, a set of spots mainly along and appear. We use the pattern at E = 20 meV in Fig. 2c as an example to investigate the origins of these spots marked with scattering vectors q_{A}, q_{B} and q_{C}. From the schematic constant energy contour (CEC) in Fig. 2d with the spin texture^{23}, one can find that q_{A} along represents the scattering between neighbouring hole pockets, and q_{B}_{ }is that between the hole edge and the electron pocket edge across . The q_{A}_{ }scattering has a spin misalignment of 60°, giving a spin probability factor of ¾, while the spins are nearly parallel for the q_{B}, giving a probability of 1. These two scattering channels are both observed along in Fig. 2c with the q_{A}_{ }intensity weaker than that of q_{B}. The q_{C} scattering along between the nextnearestneighbor hole pockets with = 120°, has a probability of ¼, yielding an even weaker intensity. The q_{D} scattering between two hole pockets across , however, is totally forbidden due to antiparallel spins of two relevant states.
In Fig. 3a, a series of FTSTS patterns for E from −40 meV to 150 meV are displayed. Based on the magnitude of q plotted as a function of E in Fig. 3b, the dispersions of q_{A} and q_{B} are approximately linear with slopes of 1.18 eVÅ and 1.09 eVÅ, respectively. These values are close to those reported in previous STS^{22} and ARPES studies^{1}. Similar dispersions have also been obtained from our QPI patterns taken near step edges, confirming that scattering only takes place between TSS with similar spin directions. Moreover, the observed QPI patterns can be compared with DFT computational data. A QPI pattern at certain E can be simulated based on the CEC and spin texture from DFT computations as well as the spin factor given by equation (1). We replot the experimental QPI patterns together with the simulation patterns for E = 5 meV and 80 meV in Fig. 3c–f. Comparing the corresponding patterns, one can find a few common features. Firstly, the dominating scattering vectors q_{B} along appear as bright spots in all cases. Secondly, the spots centred at q_{B} spread more in the direction transverse to at 80 meV than at 5 meV. Thirdly, the relative weak q_{C} spots along become more visible at 80 meV than at 5 meV. The agreement between experimental and computational results on these key features confirms that the QPI patterns acquired on such a “thick” Sb(111) film are generated by spinpolarized TSS only on the top surface.
When Sb(111) film is thin enough, however, the TSS on the top and bottom surfaces will couple eventually. It is normally believed that a gap will open up at the Dirac point and the Dirac cone will turn into Dirac hyperbola when the coupling is significant^{15,18}. In order to gain insight into how the thickness affects the band structure, and furthermore, the coupling of SS from two surfaces, we examined a 9BL Sb(111) film. Interestingly, according to DFT calculations, at this thickness, the coupling of states at is not strong enough to open a gap yet, but the intersurface coupling of states elsewhere in kspace is already significant as revealed by FTSTS mapping. Shown in Fig. 4a is the QPI pattern on 9BL Sb(111) at E = 20 meV. Compared with the 30BL sample, a few changes are obvious in the pattern. Firstly, besides intensity mostly along in the 30 BL case, here comparable intensity along can also be seen. Secondly, instead of discrete q for 30BL Sb, 12 lobes of continuous intensity with clear cutoff vectors labeled as q_{E}_{ }and q_{F} along and , respectively, are observable on 9BL Sb, indicating new scattering channels are effective.
Discussion
Now we discuss the origins of these new scattering channels on 9BL Sb. Notice the scale difference between Fig. 2c and Fig. 4a. The new intensity lobes in Fig. 4a have q lengths longer than those in Fig. 2c. Specifically, the lengths of the cutoff vectors q_{E}_{ }and q_{F} (both shown as dashline arrows) are 0.63 Å^{−1} and 0.54 Å^{−1}, respectively, whereas q_{C} = 0.25 Å^{−1} in Fig. 2c. The q_{E}/q_{F}_{ }length ratio takes a unique value 1.17 in the measured energy range. For comparison, the calculated CEC at E_{F }is shown in Fig. 4b, with superficial spin directions labeled as grey arrows assuming the upper surface separated far from the lower one. The new scattering channels are also marked in Fig. 4b, with q_{E} between the outer edges of opposite hole pockets along , and q_{F} between the outer edges of nextnearestneighbor hole pockets. Such assignments yield a , in agreement with the measured ratio. These scatterings, especially q_{E}, however, seem to violate spin conservation based on the superficial spin texture.
Different from thick film case, the two surfaces of 9BL film are separated not far from each other, so a SS is no long confined to one surface. Now, it is possible for an electron in state k on one surface to scatter into a state with the same spin on the opposite surface. In another viewpoint, for a thick Sb(111) film, there is one TSS Dirac cone on each surface. With timereversal and inversion symmetries, a pair of SS of a particular k are degenerate in energy but with opposite spin. They do not couple noticeably with each other since their wave functions overlap little. In a thin Sb(111) film, the coupling becomes strong if wave function overlap is significant, yielding mixing states with their spin only partially polarized or even totally degenerate. This opens up new scattering channels without violating spin conservation. Therefore, quite different from the 30BL film where the patterns mainly originate from the socalled “intrasurface scattering”, here the QPI patterns include “intersurface scattering”. On the other hand, the intensities in inner hexagonal zone in Fig. 4a corresponding to q_{A}_{ }and q_{B} scatterings marked in Fig. 4b are stronger than those of q_{E} and q_{F}. This part overlaps closely with the QPI patterns on the 30BL sample, demonstrating significant contribution from intrasurface scattering. This twopart pattern clearly shows the coexistence of intra and intersurface scattering events.
Based on DFT computation results of CEC and spin texture near E_{F}, we simulated the QPI pattern as shown in Fig. 4c, which is in good agreement with our experimental pattern in Fig. 4a. First of all, the characteristic outer lobes of comparable intensities along and are reproduced in the simulation pattern. Secondly, the core part of the simulated pattern has stronger intensity along , which is consistent with out observation that “intrasurface scattering” is still nontrivial. It is also worth noting that the experimental patterns were taken at 4.2 K with a 5 mV peaktopeak modulation added to V_{b}, which results in an energy resolution ΔE ≈ 4.5 meV^{32}. If this energy window is considered, the dots and arcs in the simulation pattern should be smeared out to continuous features similar to those in Fig. 4a.
Besides new features in QPI patterns, following the procedure described by Bian et al.^{33,34}, the spin separation of SS can be defined and evaluated for our 9 BL Sb(111) film. In Fig. 4d, the electronic states in the 9 BL film obtained with DFT computation are plotted along . The SS bands remain cross each other at , i.e., the Dirac point is intact. Due to quantum confinement effect, an indirect bandgap of ~0.3 eV can be observed for the bulk bands in Fig. 4d. Simple estimation based on bulk band parameters^{35} gives a bandgap ~0.53 eV, whereas Zhang et al. get a value of 0.25 eV in their computations^{36}. The calculated spin separation as a function of k along for the SS bands is plotted in Fig. 4e, where the limiting values ±1 represent full spin polarization whereas 0 represents total spin degenerate. At , the states are fully spinpolarized, but the spin separation decreases quickly as k moves from towards , and it vanishes at k = 0.34 Å^{−1}. This k point is closer to than the total spin degenerate point of 20BL Sb(111)^{33}, indicating that a stronger intersurface coupling reduces spin polarization of SS.
On the other hand, the variation of spin separation shown in Fig. 4e implies a nonuniform degree of intersurface coupling, which should depend on the SS penetration depth. The calculated realspace distributions of SS in the film normal direction are plotted in Fig. 4f for six k points of the lower SS band along . It is obvious that around the SS are well localized near the surface, so the intersurface coupling is weak. In contrast, as k approaches , the SS become more spread over the thickness, so the coupling is stronger.
To correlate the above computational results with the observed QPI patterns, we overlay the CEC at E_{F }on the spin separation plot in Fig. 4e. The three points marked as A, B and C correspond to the intersections of E_{F} with the SS bands. The states near point B with k ≈ 0.12 Å^{−1} are involved in the q_{B} scatterings in Fig. 2c and d. These states have spin separation of about ±0.9, and they are localized near surface based on Fig. 4f, so they largely maintain the character of TSS. The states at point C with k ≈ 0.34 Å^{−1} are basically spin degenerate and spread over the whole thickness of 9 BL film. According to our DFT computations, the states at this k value in the 30 BL film exhibit SS character with a penetration depth λ ~ 5.7 BL (for comparison, λ ~ 1 BL at k = 0.12 Å^{−1}). In the 9 BL film, with such a λ value, the states originated at two surfaces can couple strongly due to significant overlap in the interior, losing TSS character. These states can scatter with any other states at the CEC of the hole pockets without violating spin conservation, resulting in the outer lobes of continuous intensity terminating at q_{E}_{ }and q_{F} in Fig. 4a. The wave functions for the states with k ≥ 0.34 Å^{−1} in Fig. 4f have quite large magnitude both at the surface and in the film centre. Since the magnitude does not decay when moving from film interior to the surface, these states should be considered as surface resonance states instead of quantum well states (QWSs)^{37}. QWSs derived from the bulk states, with their wave functions decaying significantly from film interior to the surface, are readily observable with ARPES^{33}, but they seem to contribute weakly as diffused background intensity in FT STS QPI patterns. We explain this in terms of different surface sensitivity in these two methods. The photoelectrons detected in ARPES come from the top ~1–3 atomic layers of the sample whereas STS detects electronic states at ~3–10 Å above the top atomic layer.
Along , the solid and dotline arrows in Fig. 4a denote the end points of scattering vector q_{B} due to intrasurface coupling and q_{E} due to intersurface coupling, respectively. The ratio of intensities at these points, I_{E}/I_{B}, in FTSTS patterns is a quantitative measure of the relative strength of intra and intersurface coupling. In order to analyze the thickness dependence of intersurface coupling, we took FTSTS patterns at E = 20 meV for Sb(111) films with thickness in 4–15 BL range. In Fig. 5, the FTSTS patterns for 15, 12, 6 and 5 BL samples are displayed. The measured I_{E}/I_{B} as a function of film thickness is plotted in Fig. 5e. It shows that I_{E} is almost absent in 15 BL sample, and it increases as film thickness reduces from 15 to 5 BL. This monotonic trend unambiguously confirms the evolutionary process of diminishing spin polarization from thick to ultrathin films.
The intersurface coupling of TSS in Bi_{2}Se_{3} thin films has been studied both theoretically and experimentally^{15,18}. There, the most remarkable effect is the opening of a bandgap at for the TSS. Comparing to Sb(111), the TSS on Bi_{2}Se_{3}(111) form a fairly ideal Dirac cone with rather weak warping in the bulk bandgap^{6,8}, so they can be considered as true surface states. Sb(111) has a highly warped Dirac cone. The SS not far from penetrate quite deep into the interior, and they almost behave as surface resonance states. In 9BL Sb(111), the intersurface coupling of these states makes them locate more near the film centre than at the surfaces. The spin splitting of SS due to the Rashba effect, which depends on the overall potential gradient ∇V the electrons experience, is greatly reduced. To the extreme, for those states with k ≥ 0.5 Å^{−1} in Fig. 4f with wave functions totally symmetric about the film centre, the overall ∇V = 0, so the spin polarisation is completely lost. Therefore, our results demonstrate that SS spin polarization diminishes not only with decreasing film thickness, but even more dramatically with k moving from to .
Although not an ideal TI, Sb(111) ultrathin films allow us to investigate the extraordinary scatterings between SS with nearly parallel spin directions, while in other TIs (e.g. Bi_{2}Se_{3}) possessing an ideal Dirac cone it is hard to get similar features exclusively between SS. The “distorted” Dirac cone and dramatic kdependent penetration depth of SS in Sb thin films offer us an interesting quasi2D system to help interpret the detailed evolution of coupling and hybridization of TSS. The dramatic change in QPI patterns with film thickness indicates that the intra and intersurface coupling of TSS dominates in the thick and thin film cases, respectively. The intersurface coupling of TSS in a “thin” film shows strong kdependence. The tuneability of relative contributions of intra and intersurface scatterings by changing thickness can be further explored for tailoring the surface energetic and transport properties for potential applications.
Methods
The experiments are carried out in a UHV LTSTM system. The base pressure is better than 1 × 10^{−10} Torr. Clean Si(111)7 × 7 surface is prepared by degassing at 500°C overnight, a briefing annealing at 850°C and finally flashing at 1200°C for ~1 min. The Si(111)√3 × √3:Biβ surface serving as the substrate for Sb film growth in this work is prepared with 2ML Bi deposition on Si(111)7 × 7 at room temperature followed by annealing at ~450°C for 15 min. Bi and Sb deposition fluxes are generated from Ta boats loaded with high purity (99.999%) source materials. Prior to the deposition, the evaporators are degassed at appropriate temperature for a few hours in order to remove contaminations. Bi deposition flux is calibrated by measuring the coverage and thickness of Bi(110) films on Si(111). 1 ML of Bi(110) represents 9.3 × 10^{14} atom/cm^{2}. Sb is deposited on Si(111)√3 × √3:Biβ at 100°C to form (111)oriented thin films with thickness from 4 BL to >30 BL. All STM images are acquired at 77 K or 4.2 K. The dI/dV_{b} spectra are acquired at 4.2 K using a lockin amplifier with a modulation voltage at a frequency of 700 Hz and a peaktopeak amplitude of 5 mV. The dI/dV_{b} mapping for FTSTS is normally taken in a square area of edge length ~25–40 nm, without any atomic step and other defect in the area.
Firstprinciples DFTbased electronic structure calculations are performed using the VASP package^{38} with a plane wave basis and a 5 × 5 kpoint sampling of the Brillouin zone. In all calculations, generalized gradient approximation (GGA) in PerdewBurkeErnzerhof (PBE) format^{39} and spinorbital coupling are included. The neighbouring Sb(111) film slabs are separated with a 10Å vacuum region along the [111] direction.
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Acknowledgements
We acknowledge the financial support from Singapore MOE AcRF grant (T208B1110), A*STAR SERC (Project No. 122PSF0017), and NUS RGF grants R398000008112 and R144000310112.
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Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542
 Guanggeng Yao
 , Ziyu Luo
 , Feng Pan
 , Wentao Xu
 , Yuan Ping Feng
 & Xuesen Wang
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Contributions
G.Y., Y.P.F. and W.X. carried out the experiments; G.Y., Z.L., Y.P.F., W.X. and X.S.W. analyzed the data; Z.L. and Y.P.F. carried out the theoretical calculations. Y.P.F. and X.S.W. assisted in the theoretical analysis. G.Y. and X.S.W. prepared the manuscript.
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The authors declare no competing financial interests.
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Correspondence to Xuesen Wang.
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