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Cited by 699Topological Materials and PhenomenaAdvanced Condensed Matter PhysicsMagnetic properties of thin films

Three-Dimensional Topological Insulators

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Hasan, M. Zahid, Moore, Joel E.·2011·Annual Review of Condensed Matter Physics
DOI 10.1146/annurev-conmatphys-062910-140432
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Abstract

Topological insulators in three dimensions are nonmagnetic insulators that possess metallic surface states (SSs) as a consequence of the nontrivial topology of electronic wavefunctions in the bulk of the material. They are the first known examples of topological order in bulk solids. We review the basic phenomena and experimental history, starting with the observation of topological insulator behavior in Bi x Sb 1− x by angle and spin-resolved photoemission spectroscopy (spin-ARPES) and continuing through measurements on other materials and by other probes. A self-contained introduction to the single-particle theory is then given, followed by the many-particle definition of a topological insulator as a material with quantized magnetoelectric polarizability. The last section reviews recent work on strongly correlated topological insulators and new effects that arise from the proximity effect between a topological insulator and a superconductor. Although this article is not intended to be a comprehensive review of what is already a rather large field, we hope that it serves as a useful introduction, summary of recent progress, and guideline to future directions.

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Written from the paper’s full text.

TL;DR

This review summarizes the experimental discovery and theoretical understanding of three-dimensional topological insulators: nonmagnetic bulk insulators with metallic, spin-momentum-locked surface states protected by time-reversal symmetry. It covers materials such as Bi₁₋ₓSbₓ and Bi₂Se₃, the Z₂ topological classification, magnetoelectric polarizability, and recent connections to superconductivity and Majorana fermions.

Key findings

  • The first experimentally discovered 3D topological insulator was Bi₁₋ₓSbₓ; ARPES on Bi₀.₉Sb₀.₁ showed five surface-state crossings between Γ and M, an odd number consistent with Z₂ = −1, and spin-ARPES revealed a nontrivial Berry’s phase of π and spin-momentum locking.
  • Bi₂Se₃ was identified as a nearly ideal single-Dirac-cone topological insulator with a large bulk band gap of approximately 0.3 eV (equivalent to 3600 K), while Bi₂Te₃ has a smaller gap of about 0.15 eV and stronger trigonal warping.
  • Magnetic Fe deposition on Bi₂Se₃ opens a gap at the Dirac point, whereas nonmagnetic disorder from NO₂ or alkali atoms (K or Na) leaves the Dirac node intact, consistent with time-reversal protection; STM/STS plus spin-ARPES on Bi₀.₉₂Sb₀.₀₈ showed absence of elastic backscattering despite strong atomic-scale disorder.
  • The review connects 3D topological insulators to quantized magnetoelectric polarizability with θ = π, and discusses recent developments including superconductivity in CuₓBi₂Se₃ with Tc = 3.8 K, possible Majorana fermions via proximity to an s-wave superconductor, and half-Heusler candidates such as LuPtBi, YPtBi, and LaPtBi with Z₂ = −1 parent states.

Methods

  • This is a review article, not a primary experimental report; it synthesizes experimental history, single-particle theory, many-particle definitions, and recent developments.
  • Experimental evidence reviewed comes mainly from angle-resolved photoemission spectroscopy (ARPES), spin-resolved ARPES, scanning tunneling spectroscopy (STM/STS), transport measurements, and first-principles/tight-binding calculations across materials including Bi₁₋ₓSbₓ, Bi₂Se₃, Bi₂Te₃, Sb₂Te₃, CuₓBi₂Se₃, and half-Heuslers.
  • Theoretical treatment uses topological invariants, Z₂ classification, Chern-Simons forms, axion electrodynamics, and the orbital magnetoelectric polarizability.

Limitations

  • The paper states it is not a comprehensive review, has length limitations, omits some areas, and says little about other topological single-particle problems such as quasiparticle Hamiltonians of topological superconductors.
  • The paper states that current materials have residual bulk conduction from impurity states even when the band structure has a gap, limiting techniques that do not distinguish bulk from surface states; it also notes that tuning the surface Fermi level to the Dirac point needs to be demonstrated in transport geometry, and that half-Heuslers are not insulating in their native state and require substrates, thin films, or quantum wells to open a gap.
  • Evident from the text: many discussed phenomena, such as Majorana fermions, strongly correlated topological insulators, and some proposed device applications, are theoretical proposals or early-stage directions rather than established experimental results.

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Crossref · 10.1146/annurev-conmatphys-062910-140432

Cites

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Claims the paper makes

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  • Findingcitation
    § 1 INTRODUCTION · p. 3

    There are topological phases in fairly ordinary materials that occur in three dimensions and zero magnetic field.

    “The current excitement in this field results from the experimental discovery that there are topological phases in fairly ordinary materials that occur in three dimensions and zero magnetic field (1, 2, 3).”
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  • Findingcitation
    § 2 BASIC PHENOMENA AND EXPERIMENTS · p. 9

    Despite strong atomic scale disorder and random potential, elastic-backscattering between states of opposite momentum and opposite spin is absent.

    “The results show that despite strong atomic scale disorder and random potential, elastic-backscattering between states of opposite momentum and opposite spin is absent.”
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  • Findingnone given
    § 2 BASIC PHENOMENA AND EXPERIMENTS · p. 6

    Once the band at T drops below the valence band at L, at x ≈ 7%, the system evolves into a direct-gap insulator with massive Dirac-like bulk bands.

    “Once the band at T drops below the valence band at L, at x ≈ 7%, the system evolves into a direct-gap insulator with a massive Dirac-like bulk bands.”
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  • Findingown data
    § 2 BASIC PHENOMENA AND EXPERIMENTS · p. 8

    The observation of five surface state crossings (an odd rather than an even number) between Γ and M, confirmed by the observation of the Kramers degenerate point, confirms the topologically non-trivial nature of the material.

    “The observation of five surface state crossings (an odd rather than an even number) between ¯ Γ and ¯ M , confirmed by the observation of the Kramers degenerate point confirms the topologically non-trivial nature of the material.”
    Quote in the paperSubstantial claim
  • Findingnone given
    § 2 BASIC PHENOMENA AND EXPERIMENTS · p. 8

    In pure bismuth, two surface bands emerge from the bulk band continuum near Γ to form a central electron pocket and an adjacent hole lobe.

    “In pure bismuth (Bi), two surface bands emerge from the bulk band continuum near ¯ Γ to form a central electron pocket and an adjacent hole lobe.”
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    Findingown data
    § 2 BASIC PHENOMENA AND EXPERIMENTS · p. 8

    Spin-ARPES measurements on Sb showed that its surface carries a geometrical (Berry's) phase and chirality property unlike the conventional spin-orbit metals such as gold, which has zero net Berry's phase and no net chirality.

    “Spin-ARPES measurements on Sb showed that its surface carries a geometrical (Berry's) phase and chirality property unlike the conventional spin-orbit metals such as gold, which has zero net Berry's phase and no net chirality.”
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    Findingnone given
    § 1 INTRODUCTION · p. 3

    The Hall conductance in the integer quantum Hall effect can be quantized to an integer multiple of e^2/h with an accuracy of one part per billion.

    “the first example was the Hall (i.e., transverse) conductance in the IQHE, which can be quantized to an integer multiple of e 2 /h with an accuracy of one part per billion.”
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    Findingnone given
    § 2 BASIC PHENOMENA AND EXPERIMENTS · p. 5

    The first three-dimensional topological insulator to be experimentally discovered was the semiconducting alloy Bi_{1-x}Sb_x, whose unusual surface bands were mapped in an angle-resolved photoemission experiment.

    “The first 3D topological insulator to be experimentally discovered was the semiconducting alloy Bi 1 -x Sb x , whose unusual surface bands were mapped in an angleresolved photoemission (ARPES) experiment.”
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    Findingreasoning
    § 2 BASIC PHENOMENA AND EXPERIMENTS · p. 6

    As x is further increased, the gap re-opens with inverted symmetry ordering, leading to a change in sign of Δ at each of the three equivalent L points in the Brillouin zone.

    “As x is further increased this gap re-opens with inverted symmetry ordering, which leads to a change in sign of ∆ at each of the three equivalent L points in the BZ.”
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    Findingnone given
    § 1 INTRODUCTION · p. 4

    Topological insulators in several materials families have been discovered to date.

    “Topological insulators in several materials families have been discovered to date”
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    Findingcitation
    § 2 BASIC PHENOMENA AND EXPERIMENTS · p. 8

    Hsieh et al. measured the topological numbers for Bi_{1-x}Sb_x and provided an identification of its spin texture.

    “Hsieh et.al. (19) measured the topological numbers for Bi 1 -x Sb x and provided an identification of its spin texture, which heretofore was unmeasured despite its surface states (SSs) having been observed.”
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    Findingreasoning
    § 2 BASIC PHENOMENA AND EXPERIMENTS · p. 6

    For concentrations greater than x > 7%, there is no overlap between the valence band at T and the conduction band at L, and the material becomes an inverted-band insulator.

    “For concentrations greater than x > 7% there is no overlap between the valence band at T and the conduction band at L, and the material becomes an inverted-band insulator.”
    Quote in the paperSubstantial claim

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    Findingcitation
    § 2 BASIC PHENOMENA AND EXPERIMENTS · p. 5

    The surface of Bi_{1-x}Sb_x is a new form of two-dimensional metal where the electron's spin and linear momentum are locked one to one and the two-dimensional Fermi surface carries a nontrivial Berry's phase of π.

    “Its surface is a new form of 2D metal where electron's spin and linear momentum are locked one to one and the two dimensional Fermi surface carries a nontrivial Berry's phase of π (6).”
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    Findingcitation
    § 2 BASIC PHENOMENA AND EXPERIMENTS · p. 8

    The measured spin texture reveals the existence of a nonzero geometrical quantum phase, a Berry's phase of π and the handedness or chiral properties.

    “The measured spin texture (Fig.2) reveals the existence of a nonzero geometrical quantum phase, a Berry's phase of π and the handedness or chiral properties.”
    Quote in the paperSubstantial claim

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    Findingnone given
    § 2 BASIC PHENOMENA AND EXPERIMENTS · p. 9

    Spin-ARPES experiments have established that these surface states have a chiral spin structure and carry a non-trivial Berry's phase, which makes them distinct from ordinary surface states.

    “Spin-ARPES experiments have established that these surface states have a chiral spin structure and carries a non-trivial Berry's phase, which makes them distinct from ordinary surface states.”
    Quote in the paperSubstantial claim

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    Findingnone given
    § 2 BASIC PHENOMENA AND EXPERIMENTS · p. 14

    Although a large majority of the well-known Heuslers such as TiNiSn and LuNiBi are rather topologically trivial, the distorted LnPtSb-type (such as LnPtBi or LnPdBi, Ln=fn lanthanides) compounds belonging to the half-Heusler subclass harbour Z2=-1 topological insulator parent states.

    “It has been recently found that although a large majority of the well-known Heuslers such as TiNiSn and LuNiBi are rather topologically trivial, the distorted LnPtSb-type (such as LnPtBi or LnPdBi, Ln=fn lanthanides) compounds belonging to the half-Heusler subclass harbour Z 2 =-1 topological insulator parent states, where Z 2 is the band parity product index.”
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    Findingcitation
    § 2 BASIC PHENOMENA AND EXPERIMENTS · p. 13

    The surface Fermi level of a topological insulator does not have any particular reason to sit at the Dirac point, but by a combination of surface and bulk chemical modification, tuning to the Dirac point in Bi2Se3 was demonstrated by Hsieh et al.

    “The surface Fermi level of a topological insulator does not have any particular reason to sit at the Dirac point, but by a combination of surface and bulk chemical modification, tuning to the Dirac point in Bi 2 Se 3 was demonstrated by Hsieh et.al. (23).”
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    Findingnone given
    § 2 BASIC PHENOMENA AND EXPERIMENTS · p. 11

    Bi2Se3 has a large band gap of approximately 0.3 eV (equivalent to 3600 Kelvin).

    “Finally, and perhaps most important for applications, Bi 2 Se 3 has a large band gap of approximately 0.3 eV (equivalent to 3600 Kelvin).”
    Quote in the paperSubstantial claim

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    Findingcitation
    § 2 BASIC PHENOMENA AND EXPERIMENTS · p. 11

    Zhang et al. also provided a simple tightbinding model to capture the single-Dirac-cone observed in Bi2Te3 and made new predictions about Bi2Se3 and Sb2Te3.

    “Zhang et.al., also provided a simple tightbinding model to capture the single-Dirac-cone observed in Bi 2 Te 3 (28) and made new predictions about Bi 2 Se 3 and Sb 2 Te 3 .”
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    Findingcitation
    § 2 BASIC PHENOMENA AND EXPERIMENTS · p. 14

    Measurements of electronic band structure in Cu_xBi2Se3 (x=0 to 12%) by Wray et al. suggest that superconductivity occurs with electrons in a bulk relativistic kinematic regime and an unconventional doping mechanism causes the nontrivial topological surface character of the undoped compound to be preserved at the Fermi level of the superconducting compound, where Cooper pairing occurs.

    “Measurements of electronic band structure in Cu x Bi 2 Se 3 (x=0 to 12%) by Wray et.al., (40) suggest that superconductivity occurs with electrons in a bulk relativistic kinematic regime and we identify that an unconventional doping mechanism causes the nontrivial topological surface character of the undoped compound to be preserved at the Fermi level of the superconducting compound, where Cooper”
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    Findingcitation
    § 2 BASIC PHENOMENA AND EXPERIMENTS · p. 13

    This was achieved by passivating the Se vacancies with Ca in the bulk to place the Fermi level within the bulk bandgap and then hole doping the surface with NO2 to keep the chemical potential at the Dirac point.

    “This was achieved by passivating the Se vacancies with Ca in the bulk to place the Fermi level within the bulk bandgap and then hole doping the surface with NO 2 to keep the chemical potential at the Dirac point.”
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    Findingnone given
    § 2 BASIC PHENOMENA AND EXPERIMENTS · p. 12

    All the key properties of topological states have been demonstrated for Bi2Se3 which has the most simple Dirac cone surface spectrum and the largest bandgap (0.3 eV).

    “All the key properties of topological states have been demonstrated for Bi 2 Se 3 which has the most simple Dirac cone surface spectrum and the largest bandgap (0.3 eV).”
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    Premisenone given
    § Joel E. Moore · p. 1

    Three-dimensional topological insulators are nonmagnetic insulators that possess metallic surface states as a consequence of the nontrivial topology of electronic wavefunctions in the bulk of the material.

    “Topological insulators in three dimensions are nonmagnetic insulators that possess metallic surface states as a consequence of the nontrivial topology of electronic wavefunctions in the bulk of the material.”
    Quote in the paperSubstantial claim

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    Interpretationnone given
    § 2 BASIC PHENOMENA AND EXPERIMENTS · p. 5

    The surface states of a weak three-dimensional topological insulator are not robust against disorder.

    “The surface states of a WTI are not robust against disorder.”
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    Premisenone given
    § 1 INTRODUCTION · p. 3

    The general term 'topological insulators' was coined to indicate that both the two-dimensional and three-dimensional phases are topological in the same sense as the integer quantum Hall effect, with topologically protected edge or surface states that result from spin-orbit coupling rather than a magnetic field.

    “The general term 'topological insulators' was coined to indicate that both the 2D and 3D phases are topological in the same sense as the IQHE, with topologically protected edge or surface states that result from spin-orbit coupling rather than a magnetic field.”
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    Premisereasoning
    § 1 INTRODUCTION · p. 3

    Topological order can lead to measurable physical properties with extremely precise quantization as a result of the insensitivity of topology to local perturbations such as defects and thermal fluctuations.

    “A related property is that the topological order can lead to measurable physical properties with extremely precise quantization as a result of the insensitivity of topology to local perturbations such as defects and thermal fluctuations”
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    Premisenone given
    § 2 BASIC PHENOMENA AND EXPERIMENTS · p. 8

    The spin polarization is not a conserved quantity in a spin-orbit material.

    “In fact, the spin polarization is not a conserved quantity in a spin-orbit material.”
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    Interpretationnone given
    § 2 BASIC PHENOMENA AND EXPERIMENTS · p. 5

    A layered three-dimensional version of the quantum spin Hall insulator Hg(Cd)Te would be a weak three-dimensional topological insulator with invariant structure (ν0=0,111).

    “a layered three dimensional version of the quantum spin Hall insulator Hg(Cd)Te would be a 'weak 3D topological insulator (WTI)' with invariant structure of ( ν o =0,111).”
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    § 2 BASIC PHENOMENA AND EXPERIMENTS · p. 7

    Such 'topological metal' surface states cannot be realized in a purely two-dimensional electron gas system such as the one realized at the interface of GaAs/GaAlAs or on the surface of Si.

    “Such 'topological metal' surface states cannot be realized in a purely 2D electron gas system such as the one realized at the interface of GaAs/GaAlAs or on the surface of Si.”
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    Premisecitation
    § 1 INTRODUCTION · p. 3

    Simpler forms of these invariants in inversion-symmetric materials were found independently and used to predict the first topological insulator to be discovered, Bi_x Sb_{1-x}.

    “Simpler forms of these invariants in inversion-symmetric materials were found independently (6) and used to predict the first topological insulator to be discovered, Bi x Sb 1 -x (7).”
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    Premisereasoning
    § 2 BASIC PHENOMENA AND EXPERIMENTS · p. 6

    The dispersion relation E(k) = ±√((v·k)^2 + Δ^2) ≈ v·k is highly linear owing to the combination of an unusually large band velocity v and a small gap Δ (such that |Δ/|v|| ≈ 5×10^-3 Å^-1) and has been used to explain various peculiar properties of bismuth.

    “The resulting dispersion relation, E ( ⃗ k ) = ± √ ( ⃗ v · ⃗ k ) 2 +∆ 2 ≈ ⃗ v · ⃗ k , is highly linear owing to the combination of an unusually large band velocity ⃗ v and a small gap ∆ (such that | ∆ / | ⃗ v || ≈ 5 × 10 -3 ˚ A -1 ) and has been used to explain various peculiar properties of bismuth.”
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    Premisenone given
    § 1 INTRODUCTION · p. 3

    Topological phases are typically insulating or gapped in bulk but possess gapless edge or surface states as a result of topology.

    “One distinguishing feature of topological phases is that they are typically insulating or 'gapped' in bulk but possess gapless edge or surface states as a result of the topology.”
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    Interpretationreasoning
    § 2 BASIC PHENOMENA AND EXPERIMENTS · p. 8

    These two bands result from the spin-splitting of a surface state and are thus singly degenerate.

    “These two bands result from the spin-splitting of a surface state and are thus singly degenerate.”
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    Interpretationreasoning
    § 2 BASIC PHENOMENA AND EXPERIMENTS · p. 7

    When there are an odd number of crossings, at least one surface state must remain gapless, which makes it nontrivial.

    “When there are an odd number of crossings, however, at least one surface state must remain gapless, which makes it nontrivial.”
    Quote in the paperSubstantial claim

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    Premisenone given
    § 2 BASIC PHENOMENA AND EXPERIMENTS · p. 6

    Bismuth is a semimetal with strong spin-orbit interactions.

    “Bismuth is a semimetal with strong spin-orbit interactions.”
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    Premisenone given
    § 2 BASIC PHENOMENA AND EXPERIMENTS · p. 10

    Topological phenomena in materials are usually very fragile and impossible to create without liquid helium temperatures and high magnetic fields.

    “Topological phenomena in materials are usually very fragile and impossible to create without liquid helium temperatures and high magnetic fields.”
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    Interpretationreasoning
    § 2 BASIC PHENOMENA AND EXPERIMENTS · p. 9

    The chiral nature of these states protects the spin of the carriers, which is ultimately a consequence of time-reversal symmetry.

    “These observations demonstrate that the chiral nature of these states protects the spin of the carriers which is ultimately a consequence of time-reversal symmetry.”
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    Premisenone given
    § 1 INTRODUCTION · p. 4

    The method used to discover all three-dimensional topological insulators known so far is angle-resolved photoemission spectroscopy (ARPES), which probes directly the unique metallic surface state and consequently the topological invariants.

    “The method used to discover all 3D topological insulators known so far is angleresolved photoemission spectroscopy (ARPES), which probes directly the unique metallic surface state and consequently the topological invariants.”
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    Premisecitation
    § 2 BASIC PHENOMENA AND EXPERIMENTS · p. 6

    The valence and conduction bands at L are derived from antisymmetric (La) and symmetric (Ls) p-type orbitals, respectively, and the effective low-energy Hamiltonian at this point is described by the (3+1)-dimensional relativistic Dirac equation.

    “The valence and conduction bands at L are derived from antisymmetric (L a ) and symmetric (L s ) p -type orbitals, respectively, and the effective low-energy Hamiltonian at this point is described by the (3+1)-dimensional relativistic Dirac equation (18).”
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    § Joel E. Moore · p. 1

    Three-dimensional topological insulators are the first known examples of topological order in bulk solids.

    “They are the first known examples of topological order in bulk solids.”
    • Also stated in: § 1 INTRODUCTION · p. 2 · plain
      “This review discusses recent experimental and theoretical progress on three-dimensional topological insulators, the first topological phase of bulk solids.”
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    Premisenone given
    § 1 INTRODUCTION · p. 3

    Topological order until recently was only observed in two-dimensional electron gases under extreme conditions of low temperature and high magnetic field.

    “However, topological order until recently was only observed in two-dimensional electron gases under extreme conditions of low temperature and high magnetic field.”
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    Interpretationnone given
    § 1 INTRODUCTION · p. 4

    Theoretical work has connected topological insulators to many other areas of condensed matter physics.

    “and theoretical work has connected topological insulators to many other areas of condensed matter physics.”
    Quote in the paperNot a substantial claim

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    Interpretationreasoning
    § 2 BASIC PHENOMENA AND EXPERIMENTS · p. 11

    Bi2Se3 is stoichiometric and hence can be prepared in principle at higher purity.

    “Second, Bi 2 Se 3 is stoichiometric (i.e., a pure compound rather than an alloy like Bi 1 -x Sb x ) and hence can be prepared in principle at higher purity.”
    Quote in the paperSubstantial claim

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    Premisecitation
    § 3 INDEPENDENT-ELECTRON THEORY OF 3D TOPOLOGICAL INSULATORS · p. 16

    The Berry phase is a physically significant phase that accumulates as an eigenstate evolves under a Hamiltonian whose parameters change adiabatically.

    “The idea of the Berry phase (49) is that, as an eigenstate evolves under a Hamiltonian whose parameters change adiabatically, there is a physically significant phase that accumulates.”
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    Interpretationcitation
    § 3.1 Time-reversal invariance in Fermi systems · p. 23

    If there are two Kramers pairs of edge modes, a right-mover can back-scatter to the left-mover that is not its time-reversal conjugate, and this process will in general eliminate these two Kramers pairs from the low-energy theory.

    “However, if we have two Kramers pairs of edge modes, then a right-mover can back-scatter to the left-mover that is not its time-reversal conjugate (62,63). This process will, in general, eliminate these two Kramers pairs from the low-energy theory.”
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    Interpretationreasoning
    § 3 INDEPENDENT-ELECTRON THEORY OF 3D TOPOLOGICAL INSULATORS · p. 18

    To obtain a fully gauge-invariant quantity, one needs to consider a two-dimensional crystal rather than a one-dimensional one, and integrating the Berry curvature rather than the Berry connection gives a well-defined gauge-invariant quantity.

    “The above calculation suggests that, to obtain a fully gauge-invariant quantity, we need to consider a two-dimensional crystal rather than a one-dimensional one. Then integrating the Berry curvature, rather than the Berry connection, has to give a well-defined gauge-invariant quantity.”
    Quote in the paperSubstantial claim

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    Interpretationnone given
    § 2 BASIC PHENOMENA AND EXPERIMENTS · p. 11

    In its simplest possible form, each momentum along the surface has only a single spin state at the Fermi level, and the spin direction rotates as the momentum moves around the Fermi surface ensuring a non-trivial Berry's phase.

    “in its simplest possible form, each momentum along the surface has only a single spin state at the Fermi level, and the spin direction rotates as the momentum moves around the Fermi surface ensuring a non-trivial Berry's phase.”
    Quote in the paperNot a substantial claim

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    Interpretationreasoning
    § 2 BASIC PHENOMENA AND EXPERIMENTS · p. 12

    Topological surface states are protected by time reversal symmetry which implies that the surface state Dirac node would be robust in the presence of non-magnetic disorder but open a gap in the presence of time-reversal breaking perturbation such as magnetic impurities.

    “Topological surface states are protected by time reversal symmetry which implies that the surface state Dirac node would be robust in the presence of non-magnetic disorder but open a gap in the presence of time-reversal breaking perturbation of the system such as the effect of magnetic impurities and the associated disorder.”
    Quote in the paperSubstantial claim

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    Interpretationreasoning
    § 2 BASIC PHENOMENA AND EXPERIMENTS · p. 14

    These experimental observations provide important clues for developing a theory of superconductivity in doped Bi-based topological insulators.

    “These experimental observations provide important clues for developing a theory of superconductivity in doped Bi-based topological insulators.”
    Quote in the paperNot a substantial claim

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    Interpretationnone given
    § 2 BASIC PHENOMENA AND EXPERIMENTS · p. 14

    Heuslers are not insulating in their native state but growing them on substrates making thinfilms or quantum-well structures provide a way to open a band-gap turning them into real insulators.

    “Heuslers are not insulating in their native state but growing them on substrates making thinfilms or quantum-well structures provide a way to open a band-gap turning them into real insulators.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisenone given
    § 2 BASIC PHENOMENA AND EXPERIMENTS · p. 13

    In graphene, the chemistry of carbon atoms naturally locates the Fermi level at the Dirac point where the density of states vanishes.

    “In graphene, the chemistry of carbon atoms naturally locates the Fermi level at the Dirac point - the point at which the two cones intersect - where the density of states vanishes.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation
    § 3 INDEPENDENT-ELECTRON THEORY OF 3D TOPOLOGICAL INSULATORS · p. 20

    An elegant way to generalize the topological invariant definition physically while keeping the same mathematical structure was developed by Niu, Thouless, and Wu.

    “An elegant way to generalize the definition physically, while keeping the same mathematical structure, was developed by Niu, Thouless, and Wu (55).”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [55] source

      Not checked in the free run.

  • May combine two claims.

    Interpretationreasoning
    § 2 BASIC PHENOMENA AND EXPERIMENTS · p. 11

    In its high purity form, the large band gap of Bi2Se3 indicates that topological behavior can be seen at room temperature and greatly increases the potential for applications.

    “In its high purity form, the large band gap of Bi 2 Se 3 indicates that topological behavior can be seen at room temperature and greatly increases the potential for applications (Fig.7).”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Premisenone given
    § 3 INDEPENDENT-ELECTRON THEORY OF 3D TOPOLOGICAL INSULATORS · p. 19

    The TKNN integer or Chern number, expressed as an integral over the Berry flux, is connected to the IQHE and also changes in polarization.

    “This topological quantity (the 'TKNN integer' or 'Chern number', expressed as an integral over the Berry flux, which is the curl of the Berry connection A j = i 〈 u j |∇ k u j 〉 ) is connected to the IQHE and also changes in polarization.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Interpretationreasoning
    § 2 BASIC PHENOMENA AND EXPERIMENTS · p. 13

    This makes graphene's density of carriers highly tunable by an applied electrical field and enables applications of graphene to both basic science and microelectronics.

    “This makes its density of carriers highly tunable by an applied electrical field and enables applications of graphene to both basic science and microelectronics.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationnone given
    § 3 INDEPENDENT-ELECTRON THEORY OF 3D TOPOLOGICAL INSULATORS · p. 19

    The one-dimensional polarization example is related to the two-dimensional integer quantum Hall effect.

    “Indeed, the one-dimensional polarization example is related to the two-dimensional integer quantum Hall effect, which historically came first.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning
    § 3 INDEPENDENT-ELECTRON THEORY OF 3D TOPOLOGICAL INSULATORS · p. 18

    Redefinition of the wavefunctions shifts the Berry phase, and this corresponds to changing the polarization by a multiple of the polarization quantum, which in one dimension is the electron charge.

    “So redefinition of the wavefunctions shifts the Berry phase, and this corresponds to changing the polarization by a multiple of the 'polarization quantum', which in one dimension is just the electron charge.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Premisenone given
    § 2 BASIC PHENOMENA AND EXPERIMENTS · p. 13

    Most of the interesting theoretical proposals that utilize topological insulator surfaces require the chemical potential to lie at the degenerate Dirac node similar to the case in graphene.

    “Most of the interesting theoretical proposals that utilize topological insulator surfaces require the chemical potential to lie at the degenerate Dirac node similar to the case in graphene.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning
    § 3 INDEPENDENT-ELECTRON THEORY OF 3D TOPOLOGICAL INSULATORS · p. 18

    The Berry phase is not gauge-invariant, but any fractional part it had in units of a is gauge-invariant.

    “The Berry phase is not gauge-invariant, but any fractional part it had in units of a is gauge-invariant.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Premisecitation
    § 3.1 Time-reversal invariance in Fermi systems · p. 23

    Kane and Mele showed that starting from the 2D time-reversal-invariant crystal, there are two topological classes; instead of an integer-valued topological invariant, there is a Z2 invariant with two values, even and odd.

    “Kane and Mele showed that starting from the 2D T -invariant crystal, there are two topological classes; instead of an integer-valued topological invariant, there is a ' Z 2 invariant with two values, 'even' and 'odd'.”
    Quote in the paperSubstantial claim0 of 5 sources checked
    • cites [57] source

      Not checked in the free run.

    • cites [64] source

      Not checked in the free run.

    • cites [65] source

      Not checked in the free run.

    • cites [66] source

      Not checked in the free run.

    • cites [4] source

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisenone given
    § 3 INDEPENDENT-ELECTRON THEORY OF 3D TOPOLOGICAL INSULATORS · p. 16

    In crystalline solids, the electrical polarization, the anomalous Hall effect, and one part of the magnetoelectric polarizability all derive from non-topological Berry phases of the Bloch electron states.

    “In crystalline solids, the electrical polarization, the anomalous Hall effect, and one part of the magnetoelectric polarizability all derive from 'non-topological' Berry phases of the Bloch electron states.”
    Quote in the paperNot a substantial claim

  • Classified as probably not a substantial claim.

    Premisenone given
    § 3 INDEPENDENT-ELECTRON THEORY OF 3D TOPOLOGICAL INSULATORS · p. 17

    Bloch's theorem states that for electrons in a periodic potential, the electronic eigenstates can be labeled by a crystal momentum k and written in a form with a lattice-periodic function u.

    “Bloch's theorem for electrons in a periodic potential states that, given a potential invariant under a set of lattice vectors R , V ( r + R ) = V ( r ), the electronic eigenstates can be labeled by a 'crystal momentum' k and written in the form”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S1 (asserted). May combine two claims.

    Interpretationreasoning
    § 3.1 Time-reversal invariance in Fermi systems · p. 23

    A system with an even number of Kramers pairs will, under time-reversal-invariant backscattering, localize in pairs down to zero Kramers pairs, while a system with an odd number of Kramers pairs will wind up with a single stable Kramers pair.

    “The general belief based on this argument is that a system with an even number of Kramers pairs will, under time-reversal-invariant backscattering, localize in pairs down to zero Kramers pairs, while a system with an odd number of Kramers pairs will wind up with a single stable Kramers pair.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisenone given
    § 2 BASIC PHENOMENA AND EXPERIMENTS · p. 12

    Bi2Te3 is a commonly used thermoelectric material in the crucial engineering regime near room temperature.

    “Bi 2 Te 3 , is already well known to materials scientists working on thermoelectricity: it is a commonly used thermoelectric material in the crucial engineering regime near room temperature.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Premisenone given
    § 5 RECENT DEVELOPMENTS: STRONG CORRELATIONS AND MAJORANA FERMIONS · p. 33

    The 2D p+ip superconductor has been understood for some time to support Majorana fermions.

    “that has been understood for some time to support Majorana fermions.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Premisenone given
    § List of Figures · p. 39

    Bi0.9Sb0.1 features 5 surface state Fermi crossings where each band is nondegenerate.

    “Bi 0 . 9 Sb 0 . 1 features 5 such Fermi crossings where each band is nondegenerate.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationcitation
    § 3.2 3D topological insulators · p. 26

    Another way to make a 3D topological insulator is to stack 2D topological insulators, but the edge of such a system is not very stable because two odd edges combine to make an even edge, which is unstable in the presence of time-reversal-invariant backscattering; such a stacked system is called a weak topological insulator.

    “Another way to make a 3D topological insulator is to stack 2D topological insulators, but considering the edge of such a system shows that it will not be very stable: since two 'odd' edges combine to make an 'even' edge, which is unstable in the presence of T -invariant backscattering, we call such a stacked system a 'weak topological insulator' (6).”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [6] source

      Not checked in the free run.

  • May combine two claims.

    Interpretationreasoning
    § 4 TOPOLOGICAL INSULATORS AND ORBITAL MAGNETOELECTRIC POLARIZABILITY · p. 29

    At first glance θ = 0 in any time-reversal-invariant system, since θ → -θ under time-reversal; however, because θ is periodic, θ = π also works and is inequivalent to θ = 0.

    “At first glance, θ = 0 in any time-reversal-invariant system, since θ → -θ under time-reversal. However, since θ is periodic, θ = π also works, as -θ and θ are equivalent because of the periodicity, and is inequivalent to θ = 0.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning
    § 4 TOPOLOGICAL INSULATORS AND ORBITAL MAGNETOELECTRIC POLARIZABILITY · p. 31

    The 4D quantum Hall effect is not directly measurable but is mathematically connected to the 3D magnetoelectric polarizability in the same way as 1D polarization and the 2D IQHE are connected.

    “Although this 4D QHE is not directly measurable, it is mathematically connected to the 3D magnetoelectric polarizability in the same way as 1D polarization and the 2D IQHE are connected.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation
    § 5 RECENT DEVELOPMENTS: STRONG CORRELATIONS AND MAJORANA FERMIONS · p. 33

    There are analogous ways to create a Majorana fermion using strong spin-orbit quantum wells rather than topological insulators.

    “There are analogous ways to create a Majorana fermion using strong spin-orbit quantum wells rather than topological insulators (96, 97, 98).”
    Quote in the paperNot a substantial claim0 of 3 sources checked
    • cites [96] source

      Not checked in the free run.

    • cites [97] source

      Not checked in the free run.

    • cites [98] source

      Not checked in the free run.

  • May combine two claims.

    Interpretationcitation
    § 4 TOPOLOGICAL INSULATORS AND ORBITAL MAGNETOELECTRIC POLARIZABILITY · p. 28

    When the axion field θ(x,t) is constant, it plays no role in electrodynamics because θ couples to a total derivative and does not modify the equations of motion; however, the presence of the axion field can have profound consequences at surfaces and interfaces where gradients in θ(x) appear.

    “An essential feature of the above 'axion electrodynamics' (cf. Wilczek, 1987 (78) and references therein) is that, when the axion field θ ( x , t ) is constant, it plays no role in electrodynamics; this follows because θ couples to a total derivative, ǫ αβγδ F αβ F γδ = 2 ǫ αβγδ ∂ α ( A β F γδ ) (here we used that F is closed, i.e., dF = 0), and so does not modify the equations of motion. However,”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [78] source

      Not checked in the free run.

  • May combine two claims.

    Premisecitation
    § 3.2 3D topological insulators · p. 25

    The edge of the three-dimensional IQHE can form a two-dimensional chiral metal, as the chiral modes from each IQHE combine and point in the same direction.

    “The edge of this three-dimensional IQHE is quite interesting: it can form a two-dimensional chiral metal, as the chiral modes from each IQHE (74) combine and point in the same direction. (75,76)”
    Quote in the paperNot a substantial claim0 of 3 sources checked
    • cites [74] source

      Not checked in the free run.

    • cites [75] source

      Not checked in the free run.

    • cites [76] source

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationnone given
    § 4 TOPOLOGICAL INSULATORS AND ORBITAL MAGNETOELECTRIC POLARIZABILITY · p. 32

    Adding a fractional quantum Hall layer involves additional subtleties related to the topological ground state degeneracy.

    “(Adding a fractional quantum Hall layer involves additional subtleties related to the topological ground state degeneracy.)”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning
    § 3.2 3D topological insulators · p. 26

    The four independent invariants in three dimensions can be taken as (x0, y0, z0, x0 x±1), where the first three describe layered weak topological insulators and the last describes the strong topological insulator, typically meant by 3D topological insulator.

    “We can take these four invariants in three dimensions as ( x 0 , y 0 , z 0 , x 0 x ± 1 ), where the first three describe layered 'weak' topological insulators, and the last describes the strong topological insulator (STI), which is the one typically meant by '3D topological insulator'.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationnone given
    § 5 RECENT DEVELOPMENTS: STRONG CORRELATIONS AND MAJORANA FERMIONS · p. 33

    A magnetic vortex core in such a system may also carry ordinary electronic modes.

    “as well as possibly ordinary electronic modes.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationnone given
    § Surface Dirac points are protected by time reversal symme- · p. 42

    If superconducting parity is odd (∆2 state), the material will be a topological superconductor, and new gapless states will appear below TC.

    “If superconducting parity is odd (∆ 2 state), the material will be a 'topological superconductor', and new gapless states will appear below T C .”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation
    § Surface Dirac points are protected by time reversal symme- · p. 51

    For the two sites depicted, the factor νij of equation (8) is νij = -1.

    “For the two sites depicted, the factor ν ij of equation (8) is ν ij = -1.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S4 (based).

    Premisenone given
    § Surface Dirac points are protected by time reversal symme- · p. 41

    Surface Dirac points are protected by time reversal symmetry.

    “Surface Dirac points are protected by time reversal symme- try”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning
    § Surface Dirac points are protected by time reversal symme- · p. 42

    Majorana fermion surface vortices are found at the end of bulk vortex lines and might be adiabatically manipulated for quantum computation if superconducting pairing is even (∆1).

    “Majorana fermion surface vortices are found at the end of bulk vortex lines and might be adiabatically manipulated for quantum computation if superconducting pairing is even (∆ 1 )”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S4 (based). May combine two claims.

    Interpretationnone given
    § Surface Dirac points are protected by time reversal symme- · p. 42

    If the superconducting wavefunction has even parity, the Fermi level will be fully gapped and vortices on the crystal surface will host braidable Majorana fermions.

    “If the superconducting wavefunction has even parity, the Fermi level will be fully gapped and vortices on the crystal surface will host braidable Majorana fermions.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisenone given
    § List of Figures · p. 41

    The single-Dirac-cone per surface topological insulators belong to the (νo=1,000) topological class.

    “The single-Dirac-cone per surface topological insulators belong to the ( ν o =1,000) topological class.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation
    § Surface Dirac points are protected by time reversal symme- · p. 51

    The phases φx,y describe twisted boundary conditions that can be used to give a pumping definition of the Z2 invariant in the disordered case.

    “The phases φ x,y describe twisted boundary conditions that can be used to give a pumping definition of the Z 2 invariant in the disordered case.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [61] source

      Not checked in the free run.

77 of 77

  • The paper relies on this without support.

    The main remaining limitation of current topological insulator materials is that they have some residual conduction in the bulk from impurity states even if the band structure features an energy gap.

    “The main remaining limitation of current materials, especially for experimental techniques that (unlike ARPES) do not distinguish directly between bulk and surface states, is that they have some residual conduction in the bulk from impurity states even if the band-structure features an energy gap.”

  • The paper relies on this without support.

    The integer quantum Hall effect has a transport quantity quantized to remarkable precision: the transverse Hall conductivity is σxy = ne^2/h, where n is integral to 1 part in 10^9.

    “The integer quantum Hall effect has the remarkable property that, even at finite temperature in a disordered material, a transport quantity is quantized to remarkable precision: the transverse (a.k.a. Hall) conductivity is σ xy = ne 2 /h , where n is integral to 1 part in 10 9 .”

  • The authors state this as an assumption.

    The ordinary Chern numbers are zero, so that an A can be defined in the 3D Brillouin zone.

    “We are assuming in even writing the Chern-Simons formula for θ that the ordinary Chern numbers are zero, so that an A can be defined in the 3D Brillouin zone.”
3 of 3

Claims other papers make about this one

Each is checked against this paper’s own text: does it say what the citing paper says it does?

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

Each contradiction between two passages listed here was confirmed by an independent check.

  • Figure 7 does not show the cited Bi2Se3 band-gap result

    Minor

    The text cites Figure 7 for Bi2Se3's large band gap and room-temperature potential, but Figure 7 is described as a plot of band-inversion strengths for half-Heusler compounds.

1 of 1

Cites

Flagged first, then by citations

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Cited by

Sorted by citations

Authors
Hasan, M. Z., & Moore, J. E.
Affiliations
  • Princeton University (US)
  • Lawrence Berkeley National Laboratory (US)
ORCID
Venue
Annual Review of Condensed Matter Physics
Publisher
Annual Reviews
Published
2011-03-01
Volume
2
Issue
1
Pages
55-78
ISSN
  • 1947-5454 (ISSN-L)
  • 1947-5462
DOI
10.1146/annurev-conmatphys-062910-140432
OpenAlex
W2110292721
Semantic Scholar
45d7ef6837d3bed5da82712f6355f2b5119bc19a
Cited by
699
References
24
Citations by year
2026: 28 · 2025: 50 · 2024: 50 · 2023: 63 · 2022: 57 · 2021: 49 · 2020: 39 · 2019: 36 · 2018: 32 · 2017: 36 · 2016: 45 · 2015: 50 · 2014: 47 · 2013: 52 · 2012: 48
FWCI
4.06
Type
journal article
Language
en
Open access
Locations
  • Annual Review of Condensed Matter Physics (publishedVersion)
  • arXiv (Cornell University) (publishedVersion)
  • arXiv (Cornell University)
Topics
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Topological Materials and Phenomena
  • Physical Sciences › Physics and Astronomy › Condensed Matter Physics › Advanced Condensed Matter Physics
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Magnetic properties of thin films
Keywords
three-dimensional topological insulators, metallic surface states, topological order, Bi-Sb alloys, proximity effect

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