Abstract
Manipulating the spins of the topological surface states represents an essential step towards exploring the exotic quantum states emerging from the time reversal symmetry breaking via magnetic doping or external magnetic fields. The latter case relies on the Zeeman effect and thereby we need to estimate the gfactor of the topological surface state precisely. Here, we report the direct observations of the Zeeman effect at the surfaces of Bi_{2}Se_{3} and Sb_{2}Te_{2}Se by spectroscopicimaging scanning tunnelling microscopy. The Zeeman shift of the zero mode Landau level is identified unambiguously by appropriately excluding the extrinsic effects arising from the nonlinearity in the band dispersion of the topological surface state and the spatially varying potential. Surprisingly, the gfactors of the topological surface states in Bi_{2}Se_{3} and Sb_{2}Te_{2}Se are very different (+18 and −6, respectively). Such remarkable material dependence opens up a new route to control the spins of the topological surface states.
Introduction
The helical Dirac fermions on the surfaces of topological insulators (TIs) host novel relativistic quantum phenomena in solids^{1,2}. When the time reversal symmetry (TRS) of the topological surface state (TSS) is broken, a gap opens at the Dirac point. This brings about novel topological excitations, such as the magnetoelectric effect^{3,4}, the quantum anomalous Hall effect^{3,5} and the magnetic monopole effect^{6}. Magnetic doping has proven to be an effective way to break the TRS via magnetic exchange interactions^{7,8,9}, thereby enabling the experimental observation of the quantum anomalous Hall state^{10}. However, magnetic dopants may introduce charge inhomogeneity^{11} and weaken the spin–orbit coupling of TI compounds^{12}.
The Zeeman effect, the coupling of spins with a magnetic field, offers an alternative way to break the TRS of the TSS without causing any of the above issues. The opened gap Δ is fully tunable by a perpendicular magnetic field B, that is, Δ=g_{s}μ_{B}B, where g_{s} is the electron gfactor of the TSS and μ_{B} is the Bohr magneton^{13}. Consequently, to manipulate the TSS via the Zeeman effect, it is crucial to know its g_{s}.
Under B, Landau levels (LLs) are formed as a result of the cyclotron motion of the electrons. Inclusion of the Zeeman effect can lead to an energetic shift of the LLs as the corresponding TSS lack spin degeneracy (Fig. 1a). This is in stark contrast to the Zeeman splitting of the LLs observed in graphene^{14} and conventional twodimensional (2D) electron systems^{15}. Such a Zeeman effect is most pronounced for the zeroth LL (LL_{0}), and decreases dramatically with increasing Landau index n (Supplementary Note 1).
Understanding the Zeeman shift behaviour and determining the gfactor of the TSS have entailed intense research investigations primarily by the means of quantum oscillation measurements^{13,16,17}. However, the reported data are still largely controversial. The inconsistency is due to the fact that the host materials are inevitably doped by defects. This prevents the lower LLs, which exhibit prominent Zeeman shifts, from reaching the Fermi level and contributing to quantum oscillations in the accessible B (ref. 16). Furthermore, the sign of g_{s} cannot be determined from the Zeeman shift of the nonzero LLs^{17}. A recent tunnelling spectroscopy study appears to be successfully probing the LL_{0} of the TSS formed at an interface with a conventional semiconductor^{18}. However, the method used depends on specific details of bandbending in such heterostructures, and therefore cannot be readily applied to other TI compounds. Spectroscopic imaging scanning tunnelling microscopy (SISTM) can access electronic states in a wide energy range with high spatial and energy resolutions. Thus, it can be used to study any LL, regardless of the doping level of the TI compounds.
Using this technique, we have developed a methodology to unambiguously observe the Zeeman shift of LL_{0}. This allows us to determine the gfactors of the TSSs in Bi_{2}Se_{3} and Sb_{2}Te_{2}Se precisely, which turn out to be very different.
Results
Modelling the Zeeman effect of the TSS
In principle, the Zeeman shift of the LL_{0} energy (E_{0}) of the TSS is linear with B, and its slope determines g_{s}. In practice, however, more factors are involved, hindering its direct observation. On one hand, a finite curvature is superimposed on the linear dispersion of the TSS in actual compounds (Fig. 1b). This induces an extra Blinear change in E_{0} that is irrelevant to the Zeeman effect. On the other hand, there exists spatial potential variations in the TSS originating from the inhomogenously distributed charged defects^{19}. This introduces an extrinsic B dependence of LL energies as the spatial extension of the LL wave functions shrinks with increasing B (Fig. 1c).
To quantify the effects of nonlinear dispersions and the potential variations, we consider the following model Hamiltonian^{17}.
Here, σ are the Pauli matrices and II are the canonical momenta. The first term denotes the nonlinearity of band dispersions, with m* being the effective mass relative to that of the free electron (m_{e}). The second term represents the helical Dirac component of the TSS, with v being the electron velocity. The third term is the Zeeman term and the last term represents the potential variation. The energy of the nth LL, E_{n}, has been given in ref. 20, albeit without contribution from the last term of equation (1).
To include the effect of this term, we consider a 2D parabolic potential model to approximate the shape and location of the potential extremes, where E_{D} is the Diracpoint energy. At the potential extreme, the Bdependence of E_{0} can be calculated using a firstorder approximation^{21} (Supplementary Note 1). Including the effects of nonlinear dispersions and potential variations, the E_{0}(B) is accordingly given as
Note that m* renormalizes g_{s}, and the potential variation introduces an additional 1/B dependence term. To determine the intrinsic g_{s}, we have developed an analysis scheme for our SISTM data, which corrects these extrinsic factors, m* and α_{x}+α_{y}, and applied it to two different TI materials, Bi_{2}Se_{3} and Sb_{2}Te_{2}Se.
SISTM study of the Zeeman effect of the TSS
As the first step, we evaluate m* using momentumresolved LL spectroscopy^{22}. Figure 2a,b display the LL spectra of Bi_{2}Se_{3} and Sb_{2}Te_{2}Se (Supplementary Note 2, Supplementary Figs 1and 2), respectively, as measured at a fixed location at various B. In contrast to the electrondoped Bi_{2}Se_{3}, Sb_{2}Te_{2}Se is holedoped and thus its Dirac point is in the empty state. For each material, the corresponding E_{n} exhibits a quasilinear scaling relation with a scaling variable (nB)^{1/2} (Fig. 2c), which represents the energy–momentum dispersion of the TSS^{22}. The potential effect on E_{n} is more significant for the LLs with small n exposed to high B than those with large n exposed to low B (ref. 21). In this regard, the observed scaling with (nB)^{1/2} demonstrates that such potential effect has negligible influence on it at the measured location. For both compounds, a finite curvature is evidently seen in the dispersions indicating that m* is finite, as also demonstrated by angleresolved photoemission spectroscopy measurements and band calculations^{7,23}. Remarkably, both compounds show a very similar band curvature, despite their different constituent elements. We thus expect m* to be nearly the same for both the compounds. To evaluate this, m* is extracted from the scaling function, while ignoring the potential variations. For each compound, we have performed three measurements at different locations/samples. On the basis of these measurements, m* is found to be 0.12±0.03 for Bi_{2}Se_{3} and 0.13±0.02 for Sb_{2}Te_{2}Se with negligible inhomogeneity.
Next, we assess the impact of potential variations on E_{0}(B). The spatial variation of E_{0} represents the potential landscape, although it is smeared out by the magnetic length l_{B} (refs 19, 21). We start with Bi_{2}Se_{3} and map out the potential landscape by performing a spectroscopic imaging of E_{0} at a high B of 11 T, where l_{B}∼7.7 nm (Fig. 3a,d). We focus on areas around the potential extremes (Fig. 3a,d) and fit their shapes with the 2D parabolic potential model introduced above (Supplementary Fig. 3). After positioning the tip at the fitted potential centre (Fig. 3a,d, cross point), the B dependence of the LL_{0} peak is measured (Fig. 3b,e). The measured E_{0}(B) is plotted in Fig. 3c,f (black symbols). Intriguingly, E_{0}(B) exhibits ∼1/B behaviours at both the potential minimum (α_{x}>0,α_{y}>0) and maximum (α_{x}<0,α_{y}<0), but shifts towards opposite directions. This is exactly expected from equation (2) and directly highlights the influence of potential variations on E_{0}(B).
To eliminate the potential effect, we estimate the last term of equation (2) using the fitted values of α_{x} and α_{y}. After subtracting the estimated potential effect, we obtain a nearly B independent E_{0} in the high B region at both the potential minimum and maximum (red symbols in Fig. 3c,f). This validates our methodology and indicates that the effect of m* and the Zeeman effect contribute oppositely with an accidental cancellation. By further subtracting the contribution from m*, we obtain the genuine Zeeman shift (Fig. 3c,f, blue symbols). Subsequently, g_{s} in Bi_{2}Se_{3} is determined to be +18±4. (The error in g_{s} is the propagation of uncertainty evaluated from the errors of m* and , which are extracted above and measured at different potential extremes, respectively.) This differs significantly from its corresponding bulk value +32 as indicated by magnetotransport and nuclear magnetic resonance measurements^{24,25}.
The same methodology is applied to Sb_{2}Te_{2}Se (Fig. 4; Supplementary Figs 4 and 5). Distinct from Bi_{2}Se_{3}, the LL_{0} state measured at the potential minimum (Fig. 4b) exhibits a nonmonotonic B dependence (Fig. 4c) and increases with B at high fields. Given that the effective masses of both compounds are almost the same, g_{s} in Sb_{2}Te_{2}Se must differ significantly from that of Bi_{2}Se_{3}. Indeed, we evaluate g_{s} in Sb_{2}Te_{2}Se to be −6±2. While both systems share very similar band dispersions in their TSSs, their gfactors turn out to be strikingly different in both size and sign.
Discussion
We elucidate the above results under the framework of k·p theory. For narrow gap semiconductors, the conduction bands are coupled to the spin–orbitsplit valence bands through a secondorder perturbative term. Such a coupling substantially enhances the orbital sector of the gfactor. As a result, the total gfactor of such electrons/holes can significantly deviate from that of the free electrons in both magnitude and sign^{26}. In the case of TIs, the strong atomic spin–orbit coupling creates a symmetryinverted band gap. Thus, the atomic orbital characters of the wave function undergo a strong variation at and in the vicinity of the inverted band edges^{27}. Such a variation manifests itself in the orbital character of TSS around the Dirac point as well^{28}. As Sb_{2}Te_{2}Se and Bi_{2}Se_{3} comprise different elements, their wave functions must have different orbital characters. Such a difference further implies that the gfactors of the two compounds should be different too. Note that a considerable energy dependence of the gfactor is expected, since the orbital character of the wave function changes notably in these systems. Our measurement is merely around the Dirac point, which is directly relevant to the gap opening of the TSS via B. In this regard, the gfactor of the TSS may be different from that of the bulk since they are measured at different energies. Further theoretical investigations regarding those factors are needed to develop a general theory that describes the gfactor.
Considering the significant material dependence of g_{s}, we envision an interesting possibility to tailor the gfactor of the TSS by controlling the chemical composition of the chalcogenide TI materials in the form of solid solutions^{29,30}. This provides a new knob in manipulating the TSS for its spinrelated applications.
Methods
Experiment description
The experiments were performed with a modified commercial UNISOKU low temperature STM at 4.4 or 1.5 K. Magnetic fields up to 12 T can be applied perpendicularly to the sample surface. Sb_{2}Te_{2}Se and Bi_{2}Se_{3} crystals grown by a modified Bridgman technique were cleaved in situ under ultrahigh vacuum conditions at∼77 K. After cleaving, the crystals were transferred quickly to the lowtemperature STM for subsequent measurements. Two Bi_{2}Se_{3} and three Sb_{2}Te_{2}Se samples were measured. A tungsten tip was used as the STM probe, which had been cleaned and characterized with a fieldion microscope. The tunnelling spectra were obtained by lockin detection of the tunnelling current with a modulation voltage at 617.3 Hz feeding into the sample bias. The tip was grounded to provide the reference voltage.
Evaluation of the finite curvature of LL scaling
We evaluated m*, which characterizes the finite curvature, using the scaling analysis of the LL energies shown in Fig. 2c. Since only the Zeeman shift of the LL_{0} is prominent, Supplementary equation (10) (Supplementary Note 1) can be approximated by . This indicates that the scaling of E_{n} with (nB)^{1/2} still applies even in the case of finite m*. Regarding E_{0}, its energy at 3 T was used for the scaling analysis, because its shift is negligible at low B. By fitting the lowenergy part of the scaling relation, the m* value can be obtained. It must be noted that the lowenergy fitting considerably deteriorated when highenergy data were also included. Since the effect of m* on E_{0} is determined by the electronic states around the Dirac point, we only fitted the lowenergy parts.
Fitting the potential extremes with 2D parabolic potential model
First, two sectional lines across the potential extreme were drawn and fitted with a onedimensional parabolic potential to estimate its shape and location. The obtained parameters were subsequently input as the initial guess of the 2D parabolic fitting. The 2D fitting provided the shape and location of the fitted potential, whose equipotential lines were superimposed on the potential map (Supplementary Figs 3a,b and 4a,b). The generated fitting error was small, demonstrating that the 2D parabolic potential model fitted the measured potential well. This conclusion was further augmented by evaluating two sectional lines extracted along the major eclipse axes of both the fitted potential and the measured E_{0} map (Supplementary Figs 3c–f and 4c–f). Since the 2D parabolic potential merely applies to potential extremes, it cannot be used at low B, where the LL_{0} state spatially expands beyond the potential extremes. (The effect of the potential variations at low B is discussed in Supplementary Note 3, Supplementary Figs 6 and 7). Therefore, for the fitting of Figs 3c,f and 4c,f, we did not include the data below the critical B value, where the potentialcorrected E_{0} begins to deviate from the Blinear behaviour. The size of the LL_{0} state (2l_{B}) at the critical B (Supplementary Figs 3a,b and 4a,b, dashed circles) has a good correspondence with the fitted area, which substantiates our model.
Additional information
How to cite this article: Fu, Y.S. et al. Observation of Zeeman effect in topological surface state with distinct material dependence. Nat. Commun. 7:10829 doi: 10.1038/ncomms10829 (2016).
References
 1
Hasan, M. Z. & Kane, C. L. Topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).
 2
Qi, X. L. & Zhang, S. C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 105–1110 (2010).
 3
Qi, X. L., Hughes, T. L. & Zhang, S. C. Topological field theory of timereversal invariant insulators. Phys. Rev. B 78, 195424 (2008).
 4
Essin, A. M., Moore, J. E. & Vanderbilt, D. Magnetoelectric polarizability and axion electrodynamics in crystalline insulators. Phys. Rev. Lett. 102, 146805 (2009).
 5
Yu, R. et al. Quantized anomalous Hall effect in magnetic topological insulators. Science 329, 61–64 (2010).
 6
Qi, X. L., Li, R., Zang, J. & Zhang, S. C. Inducing a magnetic monopole with topological surface states. Science 323, 1184–1187 (2009).
 7
Chen, Y. L. et al. Massive Dirac fermion on the surface of a magnetically doped topological insulator. Science 329, 659–662 (2010).
 8
Checkelsky, J. G., Ye, J. T., Onose, Y., Iwasa, Y. & Tokura, Y. Diracfermionmediated ferromagnetism in a topological insulator. Nat. Phys. 8, 729–733 (2012).
 9
Xu, S. Y. et al. Hedgehog spin texture and Berry’s phase tuning in a magnetic topological insulator. Nat. Phys. 8, 616–622 (2012).
 10
Chang, C. Z. et al. Experimental observation of the quantum anomalous Hall effect in a magnetic topological insulator. Science 340, 167–170 (2013).
 11
Zhang, D. M. et al. Interplay between ferromagnetism, surface states, and quantum corrections in a magnetically doped topological insulator. Phys. Rev. B 86, 205127 (2012).
 12
Zhang, J. S. et al. Topologydriven magnetic quantum phase transition in topological insulators. Science 339, 123905 (2013).
 13
Analytis, J. G. et al. Twodimensional surface state in the quantum limit of a topological insulator. Nat. Phys. 6, 960–964 (2010).
 14
Song, Y. J. et al. Highresolution tunnelling spectroscopy of a graphene quartet. Nature 467, 185–189 (2010).
 15
Hashimoto et al. Quantum Hall transition in real space: from localized to extended states. Phys. Rev. Lett. 101, 256802 (2008).
 16
Xiong, J. et al. Highfield Shubnikov–de Haas oscillations in the topological insulator Bi2Te2Se. Phys. Rev. B 86, 045314 (2012).
 17
Taskin, A. A. & Ando, Y. Berry phase of nonideal Dirac fermions in topological insulators. Phys. Rev. B 84, 035301 (2011).
 18
Yoshimi, R. et al. Dirac electron states formed at the heterointerface between a topological insulator and a conventional semiconductor. Nat. Mater. 13, 253–257 (2014).
 19
Fu, Y. S. et al. Imaging the twocomponent nature of Dirac–Landau levels in the topological surface state of Bi2Se3 . Nat. Phys. 10, 815–819 (2014).
 20
Reynoso, A., Usaj, G., Sánchez, M. J. & Balseiro, C. A. Theory of edge states in systems with Rashba spinorbit coupling. Phys. Rev. B 70, 235344 (2004).
 21
Okada, Y. et al. Visualizing Landau levels of Dirac electrons in a onedimensional potential. Phys. Rev. Lett. 109, 166407 (2012).
 22
Hanaguri, T., Igarashi, K., Kawamura, M., Takagi, H. & Sasagawa, T. Momentumresolved Landaulevel spectroscopy of Dirac surface state in Bi2Se3 . Phys. Rev. B 82, 081305 (R) (2010).
 23
Lin, H. et al. An isolated Dirac cone on the surface of ternary tetradymitelike topological insulators. New J. Phys. 13, 095005 (2011).
 24
Köhler, H. & Wöchner, E. The gfactor of the conduction electrons in Bi2Se3 . Phys. Stat. Sol. (b) 67, 675 (1975).
 25
Mukhopadhyay, S. et al. Hyperfine coupling and spin polarization in the bulk of the topological insulator Bi2Se3 . Phys. Rev. B 91, 081105 (R) (2015).
 26
Roth, L. M., Lax, B. & Zwerdling, S. Theory of optical magnetoabsorption effects in semiconductors. Phys. Rev. 114, 90–104 (1959).
 27
Liu, C. X. et al. Model Hamiltonian for topological insulators. Phys. Rev. B 82, 045122 (2010).
 28
Cao, Y. et al. Mapping the orbital wavefunction of the surface states in threedimensional topological insulators. Nat. Phys. 9, 499–504 (2013).
 29
Arakane, T. et al. Tunable Dirac cone in the topological insulator Bi2xSbxTe3ySey . Nat. Commun. 3, 636 (2012).
 30
Zhang, J. S. et al. Band structure engineering in Bi2xSbxTe3 topological insulators. Nat. Commun. 2, 574 (2011).
Acknowledgements
We thank Y. Tokura, X.C. Xie, J.H. Gao, P. Zhang, Z.G. Wang, Y.Y. Wang and X. Liu for helpful discussions. This work is funded by GrantinAid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology of Japan, as well as the National Science Foundation of China.
Author information
Affiliations
Contributions
Y.S.F. and T.H. performed the experiments and analysed the data. K.I. and T.S. grew single crystals of Sb_{2}Te_{2}Se and Bi_{2}Se_{3}. M.K. and M.S.B. contributed to the theoretical understanding. T.H. supervised the project. Y.S.F. and T.H. wrote the manuscript with contributions from M.K. and M.S.B.
Corresponding authors
Correspondence to YingShuang Fu or T. Hanaguri.
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Supplementary information
Supplementary Information
Supplementary Figures 17, Supplementary Notes 13 and Supplementary References (PDF 1185 kb)
Rights and permissions
This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/
About this article
Cite this article
Fu, Y., Hanaguri, T., Igarashi, K. et al. Observation of Zeeman effect in topological surface state with distinct material dependence. Nat Commun 7, 10829 (2016) doi:10.1038/ncomms10829
Received
Accepted
Published
DOI
Further reading

Conditions for fully gapped topological superconductivity in topological insulator nanowires
SciPost Physics (2019)

Perfect Andreev reflection due to the Klein paradox in a topological superconducting state
Nature (2019)

Tuning InsulatorSemimetal Transitions in 3D Topological Insulator thin Films by Intersurface Hybridization and InPlane Magnetic Fields
Physical Review Letters (2019)

Quantum‐Hall to Insulator Transition in Ultra‐Low‐Carrier‐Density Topological Insulator Films and a Hidden Phase of the Zeroth Landau Level
Advanced Materials (2019)

Quasiparticle interference of Fermi arc states in the typeII Weyl semimetal candidate WTe2
Physical Review B (2018)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.