Landau Quantization of Topological Surface States in Bi 2Se3

Selected for a Viewpoint in Physics week ending PRL 105, 076801 (2010) PHYSICAL REVIEW LETTERS 13 AUGUST 2010

Landau Quantization of Topological Surface States in Bi2Se3

Peng Cheng,1 Canli Song,1 Tong Zhang,1,2 Yanyi Zhang,1 Yilin Wang,2 Jin-Feng Jia,1 Jing Wang,1 Yayu Wang,1 Bang-Fen Zhu,1 Xi Chen,1,* Xucun Ma,2,† Ke He,2 Lili Wang,2 Xi Dai,2 Zhong Fang,2 Xincheng Xie,2 Xiao-Liang Qi,3,4 Chao-Xing Liu,4,5 Shou-Cheng Zhang,4 and Qi-Kun Xue1,2 1Department of Physics, Tsinghua University, Beijing 100084, China 2Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China 3Microsoft Research, Station Q, University of California, Santa Barbara, California 93106, USA 4Department of Physics, Stanford University, Stanford California 94305, USA 5Physikalisches Institut, Universita¨tWu¨rzburg, D-97074 Wu¨rzburg, Germany (Received 25 May 2010; published 9 August 2010)

We report the direct observation of Landau quantization in Bi2Se3 thin ﬁlms by using a low-temperature scanning tunneling microscope. In particular, we discovered the zeroth Landau level, which is predicted to give rise to the half-quantized Hall effect for the topological surface states. The existence of the discrete Landau levels (LLs) and the suppression of LLs by surface impurities strongly support the 2D nature of the topological states. These observations may eventually lead to the realization of quantum Hall effect in topological insulators.

DOI: 10.1103/PhysRevLett.105.076801 PACS numbers: 73.20.r, 68.37.Ef, 71.70.Di, 72.25.Dc

The recent theoretical prediction and experimental real- 0:03 =cm and thickness of 0.1 mm. The graphene was ization [1–14] of topological insulators (TI) have generated grown on SiC by thermal desorption of silicon at 1300 C intense interest in this new state of quantum matter. The after the sample had been heated to 850 C in Si flux for surface states of a three-dimensional (3D) TI such as several cycles to form Si-rich 3 3 reconstruction. High Bi2Se3 [12], Bi2Te3 [13], and Sb2Te3 [14] consist of a purity Bi (99.999%) and Se (99.999%) were thermally single massless Dirac cone [9]. Crossing of the two surface evaporated from Knudsen cells. The temperatures of the state branches with opposite spins in the materials is fully Bi source, the Se source, and the substrate are 455, 170, protected by the time-reversal (TR) symmetry at the Dirac and 230 C, respectively. Figure 1(a) shows a typical STM points, which cannot be destroyed by any TR invariant image of the atomically flat Bi2Se3 film with a thickness of perturbation. Recent advances in thin-film growth [15,16] 50 quintuple layers (QL) grown by MBE. The steps on the have permitted this unique two-dimensional electron sys- surface are preferentially oriented along the close-packing tem (2DES) to be probed by scanning tunneling micros- directions and have the height (0.95 nm) of a quintuple copy (STM) and spectroscopy (STS) [17]. The intriguing layer. The atomically resolved STM image [Fig. 1(b)] TR symmetry protected topological states were revealed in exhibits the hexagonal lattice structure of the Se- STM experiments where the backscattering induced by terminated (111) surface of Bi2Se3 [21–23]. The STM nonmagnetic impurities is forbidden [17–19]. Here we images reveal a very small density of defects (approxi- report the Landau quantization of the topological surface mately 1 per 50 nm2) on the surface. Most of the defects states in Bi2Se3 in magnetic field by using STM and STS. [Fig. 1(c)] are either clover-shaped protrusions [21–23]or The direct observation of the discrete Landau levels (LLs) triangular depressions, which can be assigned to the sub- and the suppression of LLs by surface impurities strongly stitutional Bi defects at Se sites or the Se vacancies, support the 2D nature of the topological states. respectively. Furthermore, the dependence of LLs on magnetic field is In STS, the differential tunneling conductance dI=dV demonstrated to be consistent with the cone structure of the measures the local density of states (LDOS) of electrons surface states. The formation of LLs also implies the high at energy eV, where e is the charge on an electron. dI=dV mobility of the 2DES, which has been predicted to lead to Figure 1(d) shows the spectrum on the Bi2Se3 topological magnetoelectric effect of the TI [7,20]. surface at zero magnetic field. The Dirac point of the The experiments were conducted at 4.2 K in a Unisoku topological states corresponds to the minimum (indicated ultrahigh vacuum low-temperature STM system equipped by an arrow) of the spectrum and is about 200 meV below with molecular beam epitaxy (MBE) for film growth. As the Fermi level. However, the Fermi level determined by shown in previous works [15,16], the MBE films of TI have angle-resolved photoemission spectroscopy (ARPES) [16] lower carrier density than those prepared by the self-flux on 50 QL film is only 120 meV above the Dirac point [24]. technique. We epitaxially grew the single crystalline The discrepancy is due to the electrostatic induction by H ð Þ Bi2Se3 films on graphitized 6 SiC 0001 substrate, the electric field between the STM tip and sample, which which is nitrogen doped (n type) with a resistivity of has been observed in 2D systems with low electron

0031-9007=10=105(7)=076801(4) 076801-1 Ó 2010 The American Physical Society week ending PRL 105, 076801 (2010) PHYSICAL REVIEW LETTERS 13 AUGUST 2010

state on the Dirac cone without a tip are considerably different. To take into account the electrostatic effect, we consider the simplest approximation which models the tunneling junction as a parallel plate capacitor. Knowing the energies of the Dirac points with (ED ¼200 meV) 0 and without (ED ¼120 meV) a tip, a straightforward calculation leads to sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E E0 eV E ¼ eV þ E0 þ D D : D 1 1 0 0 (1) ED ED ED

The calibration by the above equation is needed to reliably interpret the STM data on topological insulators. When magnetic field is applied to a 2DES, the energy spectrum of the system is quantized into LLs as shown in previous experiments on various 2DESs [25–30]. For topological insulators, the high quality of the MBE films ensures the observation of LLs. The magnetic field dependence of tunneling conductance in Fig. 2 clearly reveals the development of well-defined LLs (the series of peaks) in Bi2Se3 with increasing field. The spectra demonstrate a FIG. 1 (color). Bi2Se3 film prepared by MBE. (a) The STM direct measurement of the Landau quantization of the image (200 nm 200 nm) of the Bi2Se3ð111Þ film. Imaging conditions: V ¼ 4:0V, I ¼ 0:1nA. The bias voltage is applied topological surface states. Under strong magnetic field, to the sample. (b) The atomic-resolution image of the more than ten unequally spaced LLs are explicitly resolved above the Dirac point (for example, see the black curve for Bi2Se3ð111Þ surface (1 mV, 0.2 nA). Each spot in the image corresponds to a Se atom. Selenium atom spacing is about a magnetic field of 11 T in Fig. 2). Very likely, the absence ˚ 4.14 A. (c) The defects on the surface of Bi2Se3. The left and of LLs below the Dirac point results from the overlapping the right images (both taken at 0:3V, 0.1 nA) show the of the surface states with the bulk valence band [12]. substitutional Bi defect and the Se vacancy, respectively. dI=dV ð Þ (d) spectrum taken on the Bi2Se3 111 surface. Set point: V ¼ 0:2V, I ¼ 0:12 nA. The Dirac point is indicated by an arrow. The single Dirac cone band structure of Bi2Se3 is sche- matically shown in the inset. (e) The energy band shifts appreci- ably in the presence of STM tip. The red curve depicts the electron density n as a function of the literal distance r from the apex of tip.

density [25]. The MBE film of Bi2Se3 with high quality is a bulk insulator. The surface states of the film are fully re- sponsible for screening and susceptible to external electric field due to the low density of states. The density of the induced charges on the surface in the presence of a tip is V=d V d roughly given by 0 , where and are the sample bias voltage and the distance between tip and sample, respectively. With V 100 mV and d 1nm, the induced density of electrons is typically in the order of 1012 cm2, which is comparable to the intrinsic carrier density on the surface without a tip. With negative bias voltage as shown in Fig. 1(e), the field effect moves the Dirac cone (and also the Dirac point) downwards to accommodate the induced FIG. 2 (color). Landau quantization of the topological states. charges, leading to a bias-dependent Dirac point. The film is 50 QL thick. The differential tunneling spectra were If the electrostatic effect is negligible, a quantum state acquired for various magnetic fields from 0 to 11 T. The field is can be accessed by STM at bias voltage V ¼ E=e, where E perpendicular to the surface. The set point of STS is 0.1 V and is the energy of the state with respect to the Fermi level. 0.12 nA. The bias modulation was 2 mV (rms) at 913 Hz. The eV E However, in the case of Bi2Se3, and the energy of a curves are offset vertically for clarity.

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We further confirm the 2D nature of the quantized states spectrum. The energy of such LL is independent of the by evaporating Ag atoms onto the surface of Bi2Se3 magnetic field [31]. In the lowest order approximation, the [Fig. 3]. Presumably, the LLs are sensitive to the defect topological surface states of Bi2Se3 can be modeled by the scattering if the impurities are located within the 2DES. massless Dirac equation. Unique to the massless Dirac Compared with the undoped samples, the Dirac point shifts fermions, the energy of the nth level LLn has a square- downwards in energy because of the electron transfer root dependence on magnetic field B: from the Ag atoms to the substrate. At low defect density pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 [Fig. 3(a)], no explicit change in the tunneling spectrum En ¼ ED þ sgnðnÞvF 2eB@jnj;n¼ 0; 1; 2; ...; has been observed. The defect scattering comes into effect (2) [Fig. 3(b)] when the distance between impuritiespffiffiffiffiffiffiffiffiffiffiffi in the 2DES is comparable to the magnetic length lB @=eB where vF is the Fermi velocity. Here we neglect the effect 10 nm at 11 T, where @ is Planck’s constant h divided by of the Zeeman splitting which only leads to a correction of 2. Finally, at even higher Ag atom coverage [Fig. 3(c)], about 1 meV to the Landau level spectrum for the magnetic the Landau quantization is completely suppressed as ex- field range (B 11 T) we used. The number of electrons pected. The presence of distinct peaks and the suppression per unit area on a Landau level is proportional to the of LLs by surface impurities strongly indicate that the magnetic field and given by nL ¼ eB=2@. Notably, the peaks in Fig. 2 is due to Landau quantitation of the topo- Dirac fermion model shows that the energy of LL0 is logical surface states within the bulk gap of Bi2Se3. independent of B. We therefore identify the peaks at the The Landau quantization of the topological states of Dirac point (indicated by the vertical dotted line at Bi2Se3 has some common features with that of graphene 200 mV in Fig. 2)asLL0. As predicted by the field [27–29]. Unlike the Landau levels of a 2DES with para- theory of topological insulators, the existence of such bolic dispersion, the peaks in the spectra in Fig. 2 are not zero-mode level is crucial for the topological magneto- equally spaced. In addition, there is always one peak electric effect defining the TI [7,20]. Leaving the n ¼ 0 residing at the Dirac point (200 mV) in each dI=dV level empty or completely filling it gives rise to the quantum Hall effect with Hall conductance 1=2 or þ1=2 in units of e2=h. The Dirac fermion nature of the electrons is further revealed by plotting the energyp offfiffiffiffiffiffiffi LLs, which has been calibrated by Eq. (1), versus nB [Fig. 4]. The peak positions of LLs are determined by fitting the differential conductance in Fig. 2 with multiple Gaussians. In Fig. 4, all the data collapse to a singlepffiffiffiffiffiffiffi line, suggesting that the energy of LLs is scaled as nB. Fitting the data to a

FIG. 3 (color). Suppression of LLs by defects. The three panels show the STM images and the corresponding STS spectra of LLs at different Ag impurity densities, respectively. An individual Ag atom is imaged as a triangular spot. The imaging conditions are 0.5 V and 0.04 nA. The LLs do not change at low density of impurities (0.0002 ML coverage) (a), but are partially suppressed (b) as the average distance between Ag impurities FIG. 4 (color). The fitting to the LL energies for magnetic becomes 10 nm (0.0006 ML coverage). (c) The LLs are fully fields from 8 to 11 T. Different colors correspond to the LL suppressed at even higher impurity coverage. energies at different magnetic field.

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5 straight line gives vF ¼ 3:4 10 m=s, which is com- [9] H. J. Zhang et al., Nature Phys. 5, 438 (2009). pared to 5 105 m=s by ARPES measurement [12]. The [10] M. Ko¨nig et al., Science 318, 766 (2007). difference is probably due to the nonlinearity of the band [11] D. Hsieh et al., Nature (London) 452, 970 (2008). [12] Y. Xia et al., Nature Phys. 5, 398 (2009). dispersion. The value of vF depends on the position of the Fermi level, which varies in the samples prepared by [13] Y.L. Chen et al., Science 325, 178 (2009). [14] D. Hsieh et al., Phys. Rev. Lett. 103, 146401 (2009). different methods [12,16]. [15] Y.Y. Li et al., Adv. Mater. (to be published). The observation of Landau quantization may eventually [16] Y. Zhang et al., Nature Phys. 6, 584 (2010). lead to the realization of quantum Hall effect in topologi- [17] T. Zhang et al., Phys. Rev. Lett. 103, 266803 (2009). cal insulators. Further efforts are required to design a [18] P. Roushan et al., Nature (London) 460, 1106 (2009). feasible procedure to fabricate the multiterminal devices [19] Z. Alpichshev et al., Phys. Rev. Lett. 104, 016401 based on topological insulators for transport measure- (2010). ments. The STM topographic images were processed using [20] X.-L. Qi, R. D. Li, J. D. Zang, and S.-C. Zhang, Science WSXM [32]. 323, 1184 (2009). The work is supported by NSFC and the National Basic [21] Y.S. Hor et al., Phys. Rev. B 79, 195208 (2009). Research Program of China. [22] S. Urazhdin et al., Phys. Rev. B 66, 161306 (2002). et al. 69 Note added.—Recently, we became aware of related [23] S. Urazhdin , Phys. Rev. B , 085313 (2004). [24] For comparison, the Dirac point of bulk crystal is 0.3 eV work by T. Hanaguri et al. [33]. below the Fermi level due to the higher carrier density. [25] M. Morgenstern et al., J. Electron Spectrosc. Relat. Phenom. 109, 127 (2000). [26] T. Matsui et al., Phys. Rev. Lett. 94, 226403 (2005). *[email protected] 3 † [27] G. H. Li and E. Y. Andrei, Nature Phys. , 623 (2007). [email protected] [28] G. H. Li, A. Luican, and E. Y. Andrei, Phys. Rev. Lett. 102, [1] X.-L. Qi and S.-C. Zhang, Phys. Today 63, No. 1, 33 176804 (2009). (2010). [29] D. L. Miller et al., Science 324, 924 (2009). [2] J. E. Moore, Nature (London) 464, 194 (2010). [30] S. Becker, M. Liebmann, T. Mashoff, M. Pratzer, and M. [3] M. Z. Hasan and C. L. Kane, arXiv:1002.389. Morgenstern, Phys. Rev. B 81, 155308 (2010). [4] L. Fu, C. L. Kane, and E. J. Mele, Phys. Rev. Lett. 98, [31] The energies of these n ¼ 0 peaks slightly shift with 106803 (2007). magnetic ﬁeld as a result of the redistribution of electron [5] L. Fu and C. L. Kane, Phys. Rev. B 76, 045302 (2007). density of states due to the formation of LLs. [6] J. E. Moore and L. Balents, Phys. Rev. B 75, 121306 [32] www.nanotec.es (2007). [33] T. Hanaguri, K. Igarashi, M. Kawamura, H. Takagi, and [7] X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Phys. Rev. B 78, T. Sasagawa, arXiv:1003.0100 [Phys. Rev. B (to be pub- 195424 (2008). lished)]. [8] R. Roy, Phys. Rev. B 79, 195321 (2009).

076801-4 Physics 3, 66 (2010)

Viewpoint Quantized topological surface states promise a quantum Hall state in topological insulators

Jacob W. Linder Department of Physics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway Published August 9, 2010

The ﬁrst experimental observation of Landau levels in the surface states of a three-dimensional topological insulator conﬁrms the levels obey unconventional quantization rules. Subject Areas: Semiconductor Physics, Mesoscopics, Quantum Mechanics

A Viewpoint on: Landau Quantization of Topological Surface States in Bi2Se3 Peng Cheng, Canli Song, Tong Zhang, Yanyi Zhang, Yilin Wang, Jin-Feng Jia, Jing Wang, Yayu Wang, Bang-Fen Zhu, Xi Chen, Xucun Ma, Ke He, Lili Wang, Xi Dai, Zhong Fang, Xincheng Xie, Xiao-Liang Qi, Chao-Xing Liu, Shou-Cheng Zhang and Qi-Kun Xue Phys. Rev. Lett. 105, 076801 (2010) – Published August 9, 2010

Momentum-resolved Landau-level spectroscopy of Dirac surface state in Bi2Se3 T. Hanaguri, K. Igarashi, M. Kawamura, H. Takagi and T. Sasagawa Phys. Rev. B 82, 081305 (2010) – Published August 9, 2010

The prediction [1] and experimental discovery [2, 3] are formed on its surface (or edges in a two-dimensional of a class of materials known as topological insulators is material). These electronic states reside on a massless a major recent event in the condensed matter physics Dirac cone, a relativistic energy-momentum dispersion community. Why do two- and three-dimensional in reciprocal space. Dirac cones also appear in graphene, topological insulators (such as HgTe/CdTe [2] and but there are essential differences between topological Bi2Se3 [3], respectively) attract so much interest? Think- insulators and graphene. In a topological insulator, as ing practically, these materials open a rich vista of pos- opposed to the case for graphene, the surface states dis- sible applications and devices based on the unique in- play a remarkable robustness against any type of disor- terplay between spin and charge. More fundamentally, der or perturbation that is invariant under time-reversal there is much to enjoy from a physics point of view, in- symmetry, even though the surface states in both of cluding the aesthetic spin-resolved Fermi surface topol- these systems are Dirac fermions. The reason for this is ogy [3], the possibility of hosting Majorana fermions the seemingly trivial fact that the Brillouin zone in topo- (a fermion that is its own antiparticle) in a solid-state logical insulators features a single Dirac cone, thus an system [4], and the intrinsic quantum spin Hall effect, odd number, whereas in graphene the number of Dirac which can be thought of as two copies of the quan- cones is even. This can be probed experimentally, e.g., tum Hall effect for spin-up and spin-down electrons [5]. via photoemission spectroscopy [3] to reveal how many Now, an exciting new addition to the above list comes Dirac cones exist in the Brillouin zone. In an environ- from two teams that are reporting the ﬁrst experimen- ment with a single Dirac cone, Kramers’ theorem of de- tal observation of quantized topological surface states generacy guarantees that the energy bands of the sur- forming Landau levels in the presence of a magnetic face states in topological insulators must cross at the ﬁeld. The two papers - one appearing in Physical Review time-reversal invariant Γ point at k = 0. As a result, no Letters by Peng Cheng and colleagues at Tsinghua Uni- disorder or perturbation that respects this time-reversal versity in China, and collaborators in the US, the other, invariance can induce a gap in the surface-state spec- appearing as a Rapid Communication in Physical Review trum. B, by Tetsuo Hanaguri at Japan’s RIKEN Advanced Sci- ence Institute in Wako and scientists at the Tokyo Insti- Let us extend this reasoning to a three-dimensional tute of Technology - pave the way for seeing a quantum topological insulator, in which the surface can be Hall effect in topological insulators. thought of as a two-dimensional metal, with carriers that will feature different spins depending on their di- An ideal topological insulator is electrically insulat- rection of propagation. In such a two-dimensional elec- ing in its bulk, but has conducting electronic states that tron system, the much-celebrated phenomenon known DOI: 10.1103/Physics.3.66 c 2010 American Physical Society URL: http://link.aps.org/doi/10.1103/Physics.3.66 Physics 3, 66 (2010)

as the quantum Hall effect (QHE) is known to occur under appropriate circumstances. The QHE [8] was origi- nally discovered in the context of two-dimensional electron gases at low temperatures, and under the appli- cation of a strong magnetic field perpendicular to the sample. In such a scenario, quantized orbits of motion are formed and the electrons in the sample can only oc- cupy energy states belonging to quantized energy eigen- values. More recently, the QHE was experimentally demonstrated to occur in graphene [9], but with a twist compared to the two-dimensional electron gas case. The special properties of the Dirac fermions in graphene cause the quantized conductance to form its character- istic plateaus at half-integer values of e2/h [10], in contrast to the integer values of e2/h observed in conventional two-dimensional electron gases. To understand this, it is instructive to consider the analytical expression for the Landau energy levels for the linear dispersion, q En = EF + sgn(n)vF 2eBh¯ |n|, with n = 0, ±1, ±2, ... . (1) Here, EF is the Fermi level, n is an integer, vF is the FIG. 1: (a) Illustration of the conventional integer quantum Fermi velocity, e is the unit electric charge, B is the Hall effect found, e.g., in 2D semiconductor systems. (b) Illus- field strength, andh ¯ is Planck’s reduced constant. The tration of the unconventional half-integer quantum Hall effect most remarkable aspect of the above equation is the found, e.g., in a graphene layer. (c) The experimentally ob- fact that there exists a Landau level that is completely served Landau quantization of the topological surface states independent of the magnetic field strength, namely, at in Bi2Se3 for several magnetic field strengths. Curves are offset vertically for clarity. Note in particular the peak occurring n = 0, E0 = EF. Compare this with the case of a conventional two-dimensional electron gas system with a at the Fermi level (around −200 mV sample bias), which is in- parabolic dispersion (nonzero mass of the charge carri- dependent of the field strength. This serves as a hallmark of the unconventional quantization form shown in (b). (d) STM ers), where such a zeroth Landau level is not permitted. picture of a Bi2Se3 surface in the presence of triangular-shaped How, then, is this related to the half-integer quantum defects (bright white spots). By selecting one point in such a Hall effect? The key observation here is that the con- map and plotting the dI/dV curves as a function of bias volt- ductance plateaus, serving as the hallmark of the QHE, age, one is able to probe the density of states similarly to (c). occur whenever the chemical potential falls between two (Illustration: (a),(b) Carin Cain, adapted from Novoselov et al., Landau levels. For a conventional QHE system, there Nature Phys. 2, 177 (2006); (c) Cheng et al.[6]; (d) Hanaguri et is no zeroth Landau level, and thus a plateau appears al.[7]) at σxy = 0. In contrast, if a zeroth Landau level exists, a plateau should exist right before and right after the Fermi level. When particle-hole symmetry holds in trons, as shown in Fig. 1(d), where the density of states the system, the conductance is an odd function of en- and the triangular defects in Bi2Se3 are shown. ergy, measured relative to the Fermi level. It therefore In their work, Cheng et al.[6] demonstrate an undis- follows that the first conductance quantization plateaus putable signature of Landau levels forming at the sur- 2 must occur precisely at values ±(1/2)gse /h, thus being face of the topological insulator Bi2Se3 under the influ- 2 separated by gse /h as they should. Both of the above ence of a strong external magnetic field, namely, a well- scenarios are illustrated in Fig. 1(a) and (b). defined series of peaks in the conductance spectrum as Since the surface states in topological insulators are a function of the bias between the STM/STS tip and the Dirac fermions, it follows from the above discussion surface of the sample [see Fig. 1(c)]. Most interestingly, that 3D topological insulators should feature such a a peak is consistently observed right at the Fermi level unique zero-energy Landau level, just like graphene, and does not change with increasing field strength. This and thus display the unconventional quantum Hall ef- is precisely the famous zeroth Landau level. Cheng et fect. But do they? The papers by Cheng et al.[6] and al. then proceed to confirm further the two-dimensional Hanaguri et al.[7] answer this question in the affirma- nature of the Landau states in three-dimensional topo- tive. Both research teams utilized a scanning tunnel- logical insulators by adding impurity silver atoms di- ing microscopy/spectroscopy (STM/STS) technique to rectly on the Bi2Se3 surface by means of evaporation. probe the current-voltage characteristics on the surface With increasing concentration of impurities, the Dirac of high-quality Bi2Se3 samples. The STM provides in- point shifts in energy due to the electron transfer be- formation about the density of states of the surface elec- tween silver atoms and the substrate and, more impor-

DOI: 10.1103/Physics.3.66 c 2010 American Physical Society URL: http://link.aps.org/doi/10.1103/Physics.3.66 Physics 3, 66 (2010)

tantly, the Landau quantization eventually disappears. is weak, an appreciable mass gap will only occur at high This does not mean that the surface states disappear, field strengths. But the profoundly different response of only the quantization. the topological surface states with respect to a Zeeman- Hanaguri et al.[7] also verify the presence of Landau field, because of the single Dirac cone compared to a levels using similar experimental techniques to those conventional parabolic dispersion, suggests that more used in Ref. [6]. But in addition to this, they also de- exciting effects than a mere spin-splitting of the Landau velop of a clever strategy to probe explicitly the energy- levels should occur. Such questions and more remain to momentum dispersion of the topological surface states. be explored in this developing field of research, in ad- As quasiparticle interference modulations are absent in dition to a direct experimental observation of the quan- the Dirac cone on a topological insulator, due to the tized conductance plateaus pertaining to the quantum above-mentioned robustness of these states, this con- Hall effect. ventional tool for probing the k-space structure of electronic states is rendered ineffective. Instead, Hanaguri et al. derive the E−k dispersion directly from the Landau-level spectroscopy by making use of the fact that the Landau orbits are quantized in k space, and also References employing a Bohr-Sommerfeld quantization condition p which results in the formula kn = (2e|n|B/¯h). In this [1] B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Science 314, 1757 way, a set of energy levels En and belonging momenta (2006). [2] M. König et al., Science 318, 766 (2007). kn can be obtained once n and B are specified, since En p [3] D. Hsieh et al., Nature 452, 970 (2008); Y. Xia et al., Nature Phys. 5, scales with |n|B. 398 (2009). These experiments have established that the surfaces [4] N. Read and D. Green, Phys. Rev. B 61,10267 (2000); D. A. Ivanov, of topological insulators support Landau levels with an Phys. Rev. Lett. 86, 268 (2001); S. Das Sarma et al., Phys. Rev. B 73, unconventional (half-integer) quantization. The next 220502 (2006). [5] B. A. Bernevig and S.-C. Zhang, Phys. Rev. Lett. 96, 106802 (2006). step to actually seeing the resulting quantum Hall effect [6] P. Cheng et al., Phys. Rev. Lett. 105, 076801 (2010). will be transport measurements. However, important [7] T. Hanaguri, K. Igarashi, M. Kawamura, H. Takagi, and T. questions remain to be answered. For instance, what is Sasagawa, Phys. Rev. B 82, 081305 (2010). the role of a possible Zeeman splitting with regard to [8] T. Ando, Y. Matsumoto, and Y. Uemura, J. Phys. Soc. Jpn 39, 279 these unconventional Landau levels? It is known that (1975); K. Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45, 494 (1980). an out-of-plane exchange field will induce a mass gap [9] Y. Zhang, Y.-W. Tan, H. L. Stormer, and P. Kim, Nature 438, 201 in the surface-state spectrum of the topological insula- (2005). tors, thus destroying the surface states in a limited en- [10] V. P. Gusynin and S. G. Sharapov, Phys. Rev. Lett. 95, 146801 ergy regime. If the Zeeman coupling to the external field (2005).

About the Author Jacob W. Linder

Jacob Linder obtained his Ph.D. in physics from the Norwegian University of Science and Technology (NTNU) in 2009. In 2010, he received awards from the Royal Norwegian So- ciety of Sciences and Letters and from ExxonMobil for his work on quantum transport and proximity effects in unconventional superconducting hybrid systems. He is now an Associate Professor at NTNU, Norway. His research addresses a variety of topics related to proximity effects and quantum transport in hybrid structures, magnetization dynamics and spin-transfer torque, and properties of unconventional superconductivity.

DOI: 10.1103/Physics.3.66 c 2010 American Physical Society URL: http://link.aps.org/doi/10.1103/Physics.3.66