Unresolved spectroscopic and transport targets in the Landau quantization of the topological surface Dirac cone in Bi2Se3

The 2010 scanning-tunneling observation of Landau levels of the topological surface states of Bi2Se3, including the field-independent zeroth level LL0, left several experimentally important issues open: realizing the quantum Hall effect of these states in multiterminal devices with an insulating bulk, including the half-quantized Hall contribution of a single Dirac cone tied to filling or emptying LL0; resolving Landau levels on the hole side below the Dirac point, where the surface states overlap the bulk valence band; quantitatively accounting for the role of band-dispersion nonlinearity in velocities extracted from Landau-level spectra; and characterizing the Zeeman coupling and field-induced shifts of the n=0 peak. Later literature has closed part of this agenda (Zeeman g-factor of the Bi2Se3 surface, nonlinear-dispersion accounting, integer quantum Hall effect in Bi2Se3 thin films, and half-quantized Hall responses in magnetically engineered topological-insulator heterostructures), but two precise targets remain unrealized: direct observation of hole-side (n<0) surface Landau levels in Bi2Se3-family spectroscopy with a quantitative account of the mechanisms suppressing them, and direct transport observation of the half-integer quantum Hall sequence of a single gapless Dirac cone in a non-magnetic topological-insulator device.

Background

A three-dimensional topological insulator (TI) is a material with an insulating bulk and conducting surfaces that are protected by time-reversal symmetry. The (111) surface of Bi2Se3, a prototypical TI, hosts a single Dirac cone: an energy-momentum dispersion $E(k)=E_D+\hbar v_F k$ that is linear (massless) around the Dirac point $E_D$, with a spin texture that forbids $180^\circ$ backscattering. Angle-resolved photoemission spectroscopy (ARPES) maps this dispersion in momentum space, while scanning tunneling microscopy and spectroscopy (STM/STS) measure the local density of states versus energy; in STM the differential conductance $dI/dV$ at bias $V$ tracks the density of states at energy $eV$.

In a perpendicular magnetic field $B$, a two-dimensional electron system forms discrete Landau levels (LLs). For massless Dirac fermions the levels are $E_n=E_D+\mathrm{sgn}(n)\,v_F\sqrt{2e\hbar B|n|}$ with $n=0,\pm1,\pm2,\dots$: they are unequally spaced, they disperse as $\sqrt{|n|B}$, and the zeroth level $n=0$ sits at the Dirac point independent of $B$. Because the degeneracy of each level is $eB/h$ per unit area times a degeneracy factor $Z$ ($Z=1$ for the single spin-helical cone of a 3D TI, unlike $Z=4$ in graphene), filling or emptying LL0 shifts the Hall conductance by half a conductance quantum, $\sigma_{xy}=\pm e^2/2h$ — the 'half-quantized' Hall response of a single Dirac cone, which underlies the predicted topological magneto-electric effect of TIs (the linear coupling between electric and magnetic polarization at a TI surface whose Hall conductance is half-quantized).

Cheng et al. (Phys. Rev. Lett. 105, 076801 (2010)) observed this Landau quantization by STM/STS on molecular-beam-epitaxy-grown Bi2Se3 thin films: more than ten LLs above the Dirac point at fields up to 11 T, a field-independent LL0 peak near the Dirac point, degeneracy $Z=0.99$ from the field dependence of the level filling, and suppression of the Landau quantization by Ag impurities. The experiment deliberately neglected Zeeman splitting (a correction of several meV at these fields), attributed the absence of LLs below the Dirac point to the involvement of the bulk valence band, noted that small-index levels deviate from the linear $\sqrt{|n|B}$ fit because of tip-induced electrostatic effects and quadratic terms in the band Hamiltonian, and concluded that 'further efforts are required to design a feasible procedure to fabricate the multi-terminal devices based on topological insulators for transport measurements' in order to see the associated quantum Hall effect. A companion experiment (Hanaguri et al., Phys. Rev. B 82, 081305(R) (2010)) confirmed the LLs on bulk Bi2Se3 crystals, derived the surface dispersion $E(k)$ from the LL energies via a Bohr-Sommerfeld quantization analysis (a semiclassical relation between the Landau orbit area in momentum space and $B$), and also found no LLs with $n<0$. The contemporaneous APS Physics Viewpoint (Linder, Physics 3, 66 (2010)) framed the open agenda as: transport measurements toward the quantum Hall effect, and the role of Zeeman splitting in these unconventional Landau levels.

Problem Statement

For the topological surface Dirac cone of Bi2Se3, whose Landau levels $E_n=E_D+\mathrm{sgn}(n)\,v_F\sqrt{2e\hbar B|n|}$ — with $E_D$ the Dirac-point energy, $v_F$ the Fermi velocity, $B$ the perpendicular magnetic field, and level index $n=0,\pm1,\pm2,\dots$ — and whose field-independent zeroth level LL0 were first observed by STM/STS in 2010, the following issues, acknowledged in that work, remain open in whole or in part.

(1) Transport quantum Hall effect: design and fabricate multiterminal, bulk-insulating topological-insulator devices in which the quantum Hall effect of the surface states can be observed, and in particular establish the half-quantized Hall contribution $\pm e^2/2h$ of a single gapless Dirac cone that is predicted to appear when LL0 is emptied or filled — a direct transport observation of the half-integer Hall plateau sequence from LL0 in a non-magnetic (ungapped) surface is still lacking.

(2) Hole-side Landau levels: observe and characterize the Landau levels with $n<0$ below the Dirac point, where the surface states overlap the bulk valence band, and quantitatively disentangle the mechanisms that suppress them — surface-bulk spectral overlap, the smaller Fermi velocity below the Dirac point, quasiparticle damping, and the electrostatic potential of the STM tip.

(3) Dispersion nonlinearity and velocity: quantitatively account for how the nonlinear (deviation-from-linear) Dirac dispersion and tip-induced charging modify the LL spectrum, so that velocities extracted from Landau-level spectra in STM are quantitatively consistent with ARPES band measurements on the same material. The qualitative role of nonlinearity is established, but a quantitative reconciliation of spectroscopic and photoemission velocities on the same samples is not yet complete.

(4) Small corrections to the level spectrum: characterize the Zeeman coupling of the surface states — the splitting of levels by the coupling of the electron spin magnetic moment to the magnetic field, quantified by a dimensionless g-factor — including its effect on LL0 and higher levels, and the field-dependent shifts of the $n=0$ peak, together with the many-body and tip-induced screening contributions to the level positions and widths. The surface g-factor of Bi2Se3 has since been measured and the $n=0$ shifts largely explained, but the interaction- and tip-induced contributions to level positions, widths, and lifetimes remain to be characterized.

The two targets that remain unrealized are items (1) and (2); items (3) and (4) describe parts of the 2010 agenda that later literature has already closed, together with the residual sub-questions that remain.

References

1. P. Cheng, C. Song, T. Zhang, Y. Zhang, Y. Wang, J.-F. Jia, J. Wang, Y. Wang, B.-F. Zhu, X. Chen, X. Ma, K. He, L. Wang, X. Dai, Z. Fang, X. C. Xie, X.-L. Qi, C.-X. Liu, S.-C. Zhang, Q.-K. Xue, 'Landau Quantization of Topological Surface States in Bi2Se3', Phys. Rev. Lett. 105, 076801 (2010). DOI: 10.1103/PhysRevLett.105.076801; preprint arXiv:1001.3220. https://doi.org/10.1103/PhysRevLett.105.076801
2. Jacob Linder, 'Quantized topological surface states promise a quantum Hall state in topological insulators', Physics 3, 66 (2010). DOI: 10.1103/Physics.3.66. https://doi.org/10.1103/Physics.3.66
3. T. Hanaguri, K. Igarashi, M. Kawamura, H. Takagi, T. Sasagawa, 'Momentum-resolved Landau-level spectroscopy of Dirac surface state in Bi2Se3', Phys. Rev. B 82, 081305(R) (2010). DOI: 10.1103/PhysRevB.82.081305; arXiv:1003.0100. https://doi.org/10.1103/PhysRevB.82.081305
4. P. Schwab, M. Dzierzawa, 'Landau levels in a topological insulator', Phys. Rev. B 85, 155403 (2012). DOI: 10.1103/PhysRevB.85.155403; arXiv:1107.0827. https://doi.org/10.1103/PhysRevB.85.155403
5. Y. Jiang, Y. Wang, M. Chen, Z. Li, C. Song, K. He, L. Wang, X. Chen, X. Ma, Q.-K. Xue, 'Landau Quantization and the Thickness Limit of Topological Insulator Thin Films of Sb2Te3', Phys. Rev. Lett. 108, 016401 (2012). DOI: 10.1103/PhysRevLett.108.016401; arXiv:1111.1485. https://doi.org/10.1103/PhysRevLett.108.016401
6. Cui-Zu Chang, Jinsong Zhang, Xiao Feng, Jie Shen, Zuocheng Zhang, Minghua Guo, Kang Li, Yunbo Ou, Pang Wei, Li-Li Wang, Zhong-Qing Ji, Yang Feng, Shuaihua Ji, Xi Chen, Jinfeng Jia, Xi Dai, Zhong Fang, Shou-Cheng Zhang, Ke He, Yayu Wang, Li Lu, Xu-Cun Ma, Qi-Kun Xue, 'Experimental Observation of the Quantum Anomalous Hall Effect in a Magnetic Topological Insulator', Science 340, 167-170 (2013). DOI: 10.1126/science.1234414. https://doi.org/10.1126/science.1234414
7. Ying-Shuang Fu, M. Kawamura, K. Igarashi, H. Takagi, T. Hanaguri, T. Sasagawa, 'Imaging the two-component nature of Dirac-Landau levels in the topological surface state of Bi2Se3', Nature Physics 10, 815-819 (2014). DOI: 10.1038/nphys3084; arXiv:1408.0873. https://doi.org/10.1038/nphys3084
8. N. Koirala, M. Brahlek, M. Salehi, L. Wu, J. Dai, J. Waugh, T. Nummy, Myung-Geun Han, J. Moon, Yimei Zhu, D. Dessau, Weida Wu, N. Peter Armitage, Seongshik Oh, 'Record Surface State Mobility and Quantum Hall Effect in Topological Insulator Thin Films via Interface Engineering', Nano Letters 15, 8245-8249 (2015). DOI: 10.1021/acs.nanolett.5b03770; arXiv:1511.08939. https://doi.org/10.1021/acs.nanolett.5b03770
9. Ying-Shuang Fu, T. Hanaguri, K. Igarashi, M. Kawamura, M. S. Bahramy, T. Sasagawa, 'Observation of Zeeman effect in topological surface state with distinct material dependence', Nature Communications 7, 10829 (2016). DOI: 10.1038/ncomms10829; arXiv:1601.01740. https://doi.org/10.1038/ncomms10829
10. A. Miyata, Z. Yang, A. Surrente, O. Drachenko, D. K. Maude, O. Portugall, L. B. Duffy, T. Hesjedal, P. Plochocka, R. J. Nicholas, 'Ultrahigh magnetic field spectroscopy reveals the band structure of the 3D topological insulator Bi2Se3', Phys. Rev. B 96, 121111(R) (2017). DOI: 10.1103/PhysRevB.96.121111; arXiv:1706.00781. https://doi.org/10.1103/PhysRevB.96.121111
11. Humian Zhou, Hailong Li, Dong-Hui Xu, Chui-Zhen Chen, Qing-Feng Sun, X. C. Xie, 'Transport Theory of Half-Quantized Hall Conductance in a Semimagnetic Topological Insulator', Phys. Rev. Lett. 129, 096601 (2022). DOI: 10.1103/PhysRevLett.129.096601. https://doi.org/10.1103/PhysRevLett.129.096601
12. M. Mogi, Y. Okamura, M. Kawamura, R. Yoshimi, K. Yasuda, A. Tsukazaki, K. S. Takahashi, T. Morimoto, N. Nagaosa, M. Kawasaki, Y. Takahashi, Y. Tokura, 'Experimental signature of the parity anomaly in a semi-magnetic topological insulator', Nature Physics 18, 390-394 (2022). DOI: 10.1038/s41567-021-01490-y; arXiv:2105.04127. https://doi.org/10.1038/s41567-021-01490-y
13. Pengliang Leng, Yingcai Qian, Xiangyu Cao, Nesta Benno Joseph, Yuda Zhang, Ayan Banerjee, Zihan Li, Fengshuo Liu, Zehao Jia, Linfeng Ai, Yong Zhang, Xiaoyi Xie, Shengbing Guo, Chuanying Xi, Li Pi, Jinglei Zhang, Awadhesh Narayan, Faxian Xiu, 'Nondegenerate Integer Quantum Hall Effect from Topological Surface States in Ag2Te Nanoplates', Nano Letters 23, 9026-9033 (2023). DOI: 10.1021/acs.nanolett.3c02703. https://doi.org/10.1021/acs.nanolett.3c02703
14. Huan-Wen Wang, Bo Fu, Shun-Qing Shen, 'Signature of parity anomaly: Crossover from one half to integer quantized Hall conductance in a finite magnetic field', Phys. Rev. B 109, 075113 (2024). DOI: 10.1103/PhysRevB.109.075113; arXiv:2308.04718. https://doi.org/10.1103/PhysRevB.109.075113
15. Deyi Zhuo, Bomin Zhang, Humian Zhou, Han Tay, Xiaoda Liu, Zhiyuan Xi, Chui-Zhen Chen, Cui-Zu Chang, 'Half-Quantized Chiral Edge Current in a C = 1/2 Parity Anomaly State', Phys. Rev. Lett. 136, 016601 (2026). DOI: 10.1103/vxcb-rwbl; arXiv:2509.15525. https://doi.org/10.1103/vxcb-rwbl