Model Hamiltonian for topological insulators
Open accessAbstract
In this paper we give the full microscopic derivation of the model Hamiltonian for the three-dimensional topological insulators in the ${\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$ family of materials (${\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$, ${\mathrm{Bi}}_{2}{\mathrm{Te}}_{3}$ and ${\mathrm{Sb}}_{2}{\mathrm{Te}}_{3}$). We first give a physical picture to understand the electronic structure by analyzing atomic orbitals and applying symmetry principles. Subsequently, we give the full microscopic derivation of the model Hamiltonian introduced by Zhang et al. [Nat. Phys. 5, 438 (2009)] based both on symmetry principles and the $\mathbf{k}\ensuremath{\cdot}\mathbf{p}$ perturbation theory. Two different types of ${k}^{3}$ terms, which break the in-plane full rotation symmetry down to threefold rotation symmetry, are taken into account. An effective Hamiltonian is derived for the topological surface states. Both bulk and surface models are investigated in the presence of an external magnetic field, and the associated Landau level structure is presented. For a more quantitative fitting to the first principle calculations, we also present a model Hamiltonian including eight energy bands.
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TL;DR
This paper gives a full microscopic derivation of the model Hamiltonian for three-dimensional topological insulators in the Bi₂Se₃ family (Bi₂Se₃, Bi₂Te₃, Sb₂Te₃), using symmetry principles and k·p perturbation theory. It derives bulk and surface effective Hamiltonians, computes Landau levels in a magnetic field, and proposes an eight-band extension for more quantitative agreement with ab initio calculations.
Key findings
- The four-band model Hamiltonian for the Bi₂Se₃ family is derived from crystal symmetry and k·p theory, including two types of k³ terms that break in-plane full rotation symmetry down to threefold rotation symmetry.
- The topological surface states have a helical spin texture whose helicity is traced back to the sign of the atomic spin-orbit coupling; the surface Landau levels show approximately √B dependence at low fields.
- The 0th Landau level gap can be as large as 50 meV for a 10 T magnetic field, which the authors suggest makes the topological magneto-electric effect observable at such energy scales.
- The bulk gap decreases significantly with magnetic field—about 160 meV for 5 T—due to the double-hump valence-band dispersion of Bi₂Se₃.
- A new eight-band model Hamiltonian is introduced and fits ab initio dispersions well, whereas the four-band model is valid only for k_x,z < 0.04 Å⁻¹.
Methods
- Symmetry analysis of the rhombohedral D⁵₃d crystal structure, atomic orbital hybridization, crystal-field splitting, and spin-orbit coupling to identify the four low-energy states near the Fermi surface.
- k·p perturbation theory near the Γ point, using ab initio wavefunctions to determine all model parameters directly from momentum matrix elements, so no fitting is required.
- Surface states are solved with open boundary conditions; bulk and surface Landau levels are computed under a z-direction magnetic field using Peierls substitution and effective Zeeman terms, with an eight-band extension derived by including additional states.
Limitations
- The paper states that the four-band model is valid only in the regime k_x,z < 0.04 Å⁻¹, while the valence-band maximum for Bi₂Se₃ and Bi₂Te₃ lies at about k_x ≈ 0.07 Å⁻¹, so discrepancies are expected for quantitative descriptions.
- The paper states that bulk and surface Landau levels are solved separately—bulk for an infinite sample and surface for a semi-infinite sample—and the mixing effect between them is neglected.
- The paper states that the √B dependence of surface Landau levels is only an approximation for low Landau levels and low magnetic fields; at high fields the behavior becomes a combination of linear and square-root contributions.
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- Findingnone given§ V. SURFACE STATES · p. 7
The two-component wave function φ should be an eigenstate of the Pauli matrix τ1.
“It is obvious that the two-component wave function φ should be the eigen-state of Pauli matrix τ 1 .”
Quote in the paperSubstantial claim- The authors state this as an assumption: The ǫ term can be neglected for simplicity in the surface-state derivation.
- Findingcitation§ III. MODEL HAMILTONIAN DERIVED FROM SYMMETRY PRINCIPLES · p. 5
|Λ±,±3/2> should be a combination of the ̃Γ±_4 and ̃Γ±_5 representations.
“According to (A1), it is expected that | Λ ± , ± 3 / 2 〉 should be a combination of ˜ Γ ± 4 and ˜ Γ ± 5 representations.”
Quote in the paperSubstantial claim- Rests on an assumption: Other orbitals such as s-orbitals of Bi and Se will also mix into the states near the Fermi surface.
- Findingreasoning§ II. CRYSTAL STRUCTURE, ATOMIC ORBITALS AND SYMMETRY · p. 3
The states |P1+,α> and |P2-,α> are near the Fermi surface.
“Therefore as shown in Fig.2 (II), the states | P 1 + , α 〉 and | P 2 -, α 〉 are found to be near the Fermi surface”
Quote in the paperSubstantial claim- The authors state this as an assumption: The states |P1+,α> and |P2-,α> are near the Fermi surface and other states can be neglected.
- Findingcitation§ V. SURFACE STATES · p. 8
The effective Hamiltonian of the surface states Ψ is given by Ref. [1] with k± = kx ± iky = k‖ e^{±iθ}.
“With these expressions, the effective Hamiltonian of the surface states Ψ is given by[1]”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [1] source
Not checked in the free run.
- Rests on an assumption: The form of the surface wave function ψ0 remains valid in realistic materials despite material details.
- Findingreasoning§ III. MODEL HAMILTONIAN DERIVED FROM SYMMETRY PRINCIPLES · p. 6
The above Hamiltonian is the same as that presented by Zhang et al. in Ref. [1], up to a transformation.
“The above Hamiltonian is the same to that presented by Zhang et al in the Ref. [1], which can be shown by performing the transformation”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [1] source
Not checked in the free run.
- The authors state this as an assumption: The four states |P1+-,±1/2> and |P2-+,±1/2> are sufficient for the minimum model Hamiltonian; other states are treated as perturbation.
- Findingown data§ I. INTRODUCTION · p. 1
The gap of the 0th Landau level can be as large as 50 meV for a 10 T magnetic field.
“The gap of 0th Landau level can be as large as 50 meV for 10 T magnetic field”
Quote in the paperSubstantial claim - Findingown data§ V. SURFACE STATES · p. 8
The σ matrix in model Hamiltonian (34) is proportional to the real spin.
“This indicates that σ matrix in our model Hamiltonian (34) is proportional to the real spin.”
Quote in the paperSubstantial claim- Rests on an assumption: The form of the surface wave function ψ0 remains valid in realistic materials despite material details.
- Findingreasoning§ V. SURFACE STATES · p. 7
If λ is a solution for φ+, then -λ is the solution for φ-.
“Another important observation is that if λ is a solution for φ + , then -λ is the solution for φ -.”
Quote in the paperSubstantial claim- The authors state this as an assumption: The ǫ term can be neglected for simplicity in the surface-state derivation.
- Findingown data§ IV. MODEL HAMILTONIAN DERIVED FROM THE k · p PERTURBATION THEORY · p. 6
The maximum of the valence band for Bi2Se3 and Bi2Te3 stays away from the Γ point, at about k_x ≈ 0.07 Å^-1.
“However the maximum of the valence band for Bi 2 Se 3 and Bi 2 Te 3 stays away from Γ point, at about k x ≈ 0 . 07 ˚ A -1”
Quote in the paperSubstantial claim- The authors state this as an assumption: The four states |P1+-,±1/2> and |P2-+,±1/2> are sufficient for the minimum model Hamiltonian; other states are treated as perturbation.
- Findingreasoning§ I. INTRODUCTION · p. 1
The surface Landau levels have a square-root magnetic-field dependence, as expected from the Dirac-type dispersion of the surface states.
“The surface Landau levels have √ B fi eld dependence, as is expected from the Dirac-type dispersion of the surface states.”
Quote in the paperSubstantial claim - Findingown data§ VII. NEW MODEL HAMILTONIAN WITH EIGHT BANDS · p. 12
The band dispersion is found to fit well with that of ab initio calculation, as shown in Fig 9.
“the band dispersion is found to fit well with that of ab initio calculation, as shown in Fig 9.”
Quote in the paperSubstantial claim - Findingreasoning§ VI. MAGNETIC FIELD AND LANDAU LEVEL · p. 10
When magnetic field is increased, the bulk gap is decreased by about 160 meV for 5T magnetic field.
“When the magnetic field is increased, the bulk gap is also decreased by a significant amount (about 160meV for 5T magnetic field)”
Quote in the paperNot a substantial claim - Findingreasoning§ V. SURFACE STATES · p. 9
In the limit k → 0, the spin polarization almost lies in the xy plane.
“In the limit k → 0, the spin polarization almost lies in the xy plane”
Quote in the paperNot a substantial claim - Findingown data§ VI. MAGNETIC FIELD AND LANDAU LEVEL · p. 10
The maximum of the valence band is not located at kz=0 point for small magnetic field.
“From Fig 7 (a), we find that the maximum of the valence band is not located at k z = 0 point for small magnetic field”
Quote in the paperSubstantial claim - Findingreasoning§ V. SURFACE STATES · p. 9
The k-cubic term yields z direction spin polarization.
“but also yield z direction spin polarization”
Quote in the paperNot a substantial claim - Findingown data§ Appendix C: Parameters in k · p theory · p. 17
For the |P1+-, ±1/2> band, there is a strong anisotropy which comes from the large contribution of the second order perturbation of the states |P2-, ±3/2> and |P2--, ±1/2>.
“From table (IV), we find that for | P 1 + -, ± 1 2 〉 band, there is a strong anisotropy which comes from the large contribution of the second order perturbation of the states | P 2 -, ± 3 2 〉 and | P 2 --, ± 1 2 〉 .”
Quote in the paperNot a substantial claim- Rests on an assumption: The wave functions for the basis of the model Hamiltonian can be projected into atomic orbitals.
- Findingreasoning§ VII. NEW MODEL HAMILTONIAN WITH EIGHT BANDS · p. 12
In the eight band model, it is the second order coupling gij.
“while in the eight band model, it is the second order coupling g ij”
Quote in the paperNot a substantial claim - Findingreasoning§ V. SURFACE STATES · p. 8
A0 depends on the sign of the atomic SOC.
“which indicates that A 0 depends on the sign of the atomic SOC”
Quote in the paperSubstantial claim - Findingown data§ VIII. CONCLUSIONS · p. 12
The gap of bulk Landau levels decreases when magnetic field increases.
“The gap of bulk Landau levels is shown to decrease when magnetic field increases”
Quote in the paperSubstantial claim - Findingown data§ VI. MAGNETIC FIELD AND LANDAU LEVEL · p. 11
The largest Landau gap for surface states is between the 0th and 1st Landau level, about 50 meV.
“The largest Landau gap for surface states is between the 0th and 1th Landau level, about 50meV”
Quote in the paperSubstantial claim - Findingreasoning§ V. SURFACE STATES · p. 9
The linear term determines the spin texture of the surface states.
“we have shown how the the linear term determines the spin texture of the surface states”
Quote in the paperSubstantial claim - Findingnone given§ VIII. CONCLUSIONS · p. 12
Within the bulk gap, the surface Landau levels appear as discrete peaks for the LDOS.
“Within the bulk gap, the surface Landau levels appears as discrete peaks for the LDOS”
Quote in the paperSubstantial claim - Findingnone given§ VII. NEW MODEL HAMILTONIAN WITH EIGHT BANDS · p. 11
Our model Hamiltonian can capture the salient topological features of the Bi2Se3 family of materials.
“our model Hamiltonian can capture the salient topological features of the Bi 2 Se 3 family of materials”
Quote in the paperSubstantial claim - Findingreasoning§ VI. MAGNETIC FIELD AND LANDAU LEVEL · p. 10
Non-linear behavior appears for high √NBz.
“the non-linear behavior will appear for high √ NB z .”
Quote in the paperNot a substantial claim - Findingown data§ VI. MAGNETIC FIELD AND LANDAU LEVEL · p. 11
The bulk LDOS shows a gap of about 0.3 eV.
“The bulk LDOS shows a gap of about 0.3eV”
Quote in the paperNot a substantial claim - Findingcitation§ V. SURFACE STATES · p. 9
The k-cubic term induces hexagonal warping of the constant energy contours.
“the k-cubic term comes into play, which will not only induce the hexagonal warping of the constant energy contours[35]”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [35] L. Fu, Phys. Rev. Lett. 103 , 266801 (2009). source
Not checked in the free run.
- Findingown data§ VII. NEW MODEL HAMILTONIAN WITH EIGHT BANDS · p. 11
There are strong couplings between the state |P1+-, ±1/2> and the state |P2-, Γ4,5> or |P2--, ±1/2>.
“we find that there are strong couplings between the state | P 1 + -, ± 1 2 〉 and the state | P 2 -, ˜ Γ 4 , 5 〉 or | P 2 --, ± 1 2 〉”
Quote in the paperSubstantial claim - Findingnone given§ VIII. CONCLUSIONS · p. 12
Our model Hamiltonian captures the main low energy physics, such as the inverted band structure and topologically protected surface states.
“Our model Hamiltonian captures the main low energy physics, such as the inverted band structure and topologically protected surface states.”
Quote in the paperSubstantial claim - Findingreasoning§ V. SURFACE STATES · p. 8
The sign of B0 does not depend on the atomic SOC.
“thus the sign of B 0 will not depend on the atomic SOC”
Quote in the paperSubstantial claim - Findingnone given§ V. SURFACE STATES · p. 9
The spin texture of the surface states can also be affected by the quadratic term and cubic term in the effective Hamiltonian (34).
“which can also be affected by the quadratic term and cubic term in the effective Hamiltonian (34)”
Quote in the paperNot a substantial claim - Findingreasoning§ VI. MAGNETIC FIELD AND LANDAU LEVEL · p. 10
Due to the quadratic term C2 k||^2, the square root dependence of the energy level versus magnetic field is only an approximation applicable for low Landau levels and low magnetic field.
“due to the existence of the quadratic term ˜ C 2 k 2 ‖ , the square root dependence of the energy level verse magnetic field is only an approximation applicable for low Landau levels and low magnetic field.”
Quote in the paperSubstantial claim - Findingreasoning§ VII. NEW MODEL HAMILTONIAN WITH EIGHT BANDS · p. 12
In our model Hamiltonian with four bands, the leading term that breaks the in-plane full rotation symmetry down to R3 symmetry is the third order in perturbation.
“in our model Hamiltonian with four bands, the leading term that breaks the inplane full rotation symmetry down to R 3 symmetry is the third order in perturbation”
Quote in the paperNot a substantial claim - Findingown data§ VII. NEW MODEL HAMILTONIAN WITH EIGHT BANDS · p. 11
At the valence band maximum kx≈0.07 Å^-1, these couplings can be as large as the energy gap between these states.
“at the valence band maximum k x ≈ 0 . 07 ˚ A -1 , we find that these couplings can be as large as the energy gap between these states”
Quote in the paperSubstantial claim - Findingreasoning§ Appendix C: Parameters in k · p theory · p. 16
The effective g0 is approximately 1.2.
“the effective g 0 can be easily calculated, which is found to be g 0 ≈ 1 . 2.”
Quote in the paperNot a substantial claim- Rests on an assumption: The wave functions for the basis of the model Hamiltonian can be projected into atomic orbitals.
- Interpretationreasoning§ II. CRYSTAL STRUCTURE, ATOMIC ORBITALS AND SYMMETRY · p. 4
This band inversion is the key signature of the topological insulator phase in the Bi2Se3 family of materials.
“This is the key signature of the topological insulator phase in Bi 2 Se 3 family of materials[1].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [1] source
Not checked in the free run.
- The authors state this as an assumption: Only p-orbitals of Bi and Se need be considered; other orbitals can be neglected.
- Premisecitation§ I. INTRODUCTION · p. 1
Bernevig, Hughes and Zhang constructed the model Hamiltonian for the two-dimensional topological insulator in HgTe quantum wells.
“Bernevig, Hughes and Zhang (BHZ)[5] constructed the model Hamiltonian for the 2D TI in HgTe quantum wells.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [5] source
Not checked in the free run.
- Premisecitation§ I. INTRODUCTION · p. 1
In the thin film limit, the three-dimensional topological insulator model reduces exactly to the two-dimensional topological insulator model by BHZ.
“Interestingly, in the thin film limit, the 3D TI model reduces exactly to the 2D TI model by BHZ[32-34].”
Quote in the paperNot a substantial claim0 of 3 sources checked- cites [32] J. Linder, T. Yokoyama, and A. Sudb?, Phys. Rev. B 80 , 205401 (2009). source
Not checked in the free run.
- cites [33] C. Liu, H. Zhang, B. Yan, X. Qi, T. Frauenheim, X. Dai, Z. Fang, and S. Zhang, Phys. Rev. B 81 , 041307 (2010). source
Not checked in the free run.
- cites [34] H. Lu, W. Shan, W. Yao, Q. Niu, and S. Shen, Phys. Rev. B 81 , 115407 (2010). source
Not checked in the free run.
- Interpretationown data§ IV. MODEL HAMILTONIAN DERIVED FROM THE k · p PERTURBATION THEORY · p. 6
The model Hamiltonian is valid in the regime k_x,z < 0.04 Å^-1.
“It is shown that our model Hamiltonian is valid in the regime k x,z < 0 . 04 ˚ A -1 .”
Quote in the paperSubstantial claim- The authors state this as an assumption: The four states |P1+-,±1/2> and |P2-+,±1/2> are sufficient for the minimum model Hamiltonian; other states are treated as perturbation.
- Interpretationreasoning§ V. SURFACE STATES · p. 8
The sign change of B0/M1 will change the spin basis of the surface states, which is key to determining the helicity of the Dirac Hamiltonian for the topological surface states.
“We emphasize here the sign change of B 0 /M 1 will change the spin basis of the surface states, which will be the key point to determine the helicity of the Dirac Hamiltonian for the topological surface states.”
Quote in the paperSubstantial claim- The authors state this as an assumption: The ǫ term can be neglected for simplicity in the surface-state derivation.
- Interpretationnone given§ II. CRYSTAL STRUCTURE, ATOMIC ORBITALS AND SYMMETRY · p. 3
When the coupling between |Bα(Sα)> and |B'α(S'α)> is taken into account, the bonding and antibonding states split, with the antibonding state having higher energy than the bonding state.
“When the coupling between | B α ( S α ) 〉 and | B ′ α ( S ′ α ) 〉 is taken into account, the bonding and antibonding states are split, with the anti-bonding state having higher energy than the bonding state.”
Quote in the paperSubstantial claim- The authors state this as an assumption: Only p-orbitals of Bi and Se need be considered; other orbitals can be neglected.
- The authors state this as an assumption: The two neighboring quintuple layers are coupled only by van der Waals force, so one can first focus on one quintuple layer.
- Interpretationreasoning§ IV. MODEL HAMILTONIAN DERIVED FROM THE k · p PERTURBATION THEORY · p. 6
There may be discrepancies when using the four-band model Hamiltonian to describe Bi2Se3 quantitatively.
“therefore we need to keep in mind that there may be some discrepancies when we try to use our model Hamiltonian to describe Bi 2 Se 3 quantitatively.”
Quote in the paperSubstantial claim- The authors state this as an assumption: The four states |P1+-,±1/2> and |P2-+,±1/2> are sufficient for the minimum model Hamiltonian; other states are treated as perturbation.
- Interpretationown data§ IV. MODEL HAMILTONIAN DERIVED FROM THE k · p PERTURBATION THEORY · p. 6
The four-band model Hamiltonian captures the salient features of the band dispersion, especially the inverted band structure.
“Since our model Hamiltonian with four bands already captures the salient features of the band dispersion, especially the inverted band structure”
Quote in the paperSubstantial claim- The authors state this as an assumption: The four states |P1+-,±1/2> and |P2-+,±1/2> are sufficient for the minimum model Hamiltonian; other states are treated as perturbation.
- Premisecitation§ I. INTRODUCTION · p. 1
The BHZ model Hamiltonian demonstrates the basic mechanism of topological insulator behavior through band inversion induced by spin-orbit coupling.
“This model Hamiltonian demonstrates the basic mechanism of TI behavior through band inversion induced by spin-orbit coupling (SOC).”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [5] source
Not checked in the free run.
- Interpretationown data§ I. INTRODUCTION · p. 1
The topological magneto-electric effect can be observable at the energy scale of the 0th Landau level gap for a 10 T magnetic field.
“which suggests that the topological magneto-electric effect[36, 37] can be observable at such energy scales.”
Quote in the paperSubstantial claim0 of 2 sources checked - Interpretationreasoning§ II. CRYSTAL STRUCTURE, ATOMIC ORBITALS AND SYMMETRY · p. 4
Because these two pairs of states have opposite parity, their crossing leads to a band inversion similar to the case in HgTe quantum wells.
“Since these two pairs of states have the opposite parity, their crossing leads to a band inversion , similar to the case in the HgTe quantum wells[5].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [5] source
Not checked in the free run.
- The authors state this as an assumption: Only p-orbitals of Bi and Se need be considered; other orbitals can be neglected.
- Premisecitation§ I. INTRODUCTION · p. 1
ARPES measurements show a single Dirac cone with linear dispersion around the Γ point in both Bi2Se3 and Bi2Te3.
“ARPES measurement[10, 11] indeed shows the single Dirac cone with linear dispersion around the Γ point in both Bi 2 Se 3 and Bi 2 Te 3 .”
Quote in the paperNot a substantial claim0 of 2 sources checked - Interpretationreasoning§ VII. NEW MODEL HAMILTONIAN WITH EIGHT BANDS · p. 12
From the symmetry point of view, our model is an extension of the Kane model to a crystal structure with lower symmetry.
“from the symmetry point of view, our model here is nothing but an extension of the Kane model to a crystal structure with lower symmetry.”
Quote in the paperNot a substantial claim - Premisecitation§ VI. MAGNETIC FIELD AND LANDAU LEVEL · p. 9
The orbital effect can be included by Peierls substitution.
“The orbital effect can be included by Peierls substitution[47]”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [47] J. M. Luttinger and W. Kohn, Phys. Rev. 97 , 869 (1955).
Not checked in the free run.
- Rests on an assumption: For bulk states, the magnetic field contributes through orbital and Zeeman effects.
- Interpretationreasoning§ VII. NEW MODEL HAMILTONIAN WITH EIGHT BANDS · p. 11
To describe this material more accurately, we need to further include the states |P2-, Γ4,5> and |P2--, ±1/2> into our model Hamiltonian.
“The strong couplings between these states indicate that if we want to describe this material more accurately, we need to further include the states | P 2 -, ˜ Γ 4 , 5 〉 and | P 2 --, ± 1 2 〉 into our model Hamiltonian.”
Quote in the paperSubstantial claim - Interpretationreasoning§ V. SURFACE STATES · p. 8
The helicity is determined by the relative sign of A0 and B0/M1.
“Therefore the helicity is determined by the relative sign of A 0 and B 0 /M 1 .”
Quote in the paperSubstantial claim- Rests on an assumption: The form of the surface wave function ψ0 remains valid in realistic materials despite material details.
- Interpretationcitation§ V. SURFACE STATES · p. 9
The k-cubic term effect is similar to the situation in BixSb1-x studied by ab initio calculation.
“similar to the situation in Bi x Sb 1 -x studied by ab initio calculation[46]”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [46] H. Zhang, C. Liu, X. Qi, X. Deng, X. Dai, S. Zhang, and Z. Fang, Phys. Rev. B 80 , 085307 (2009).
Not checked in the free run.
- Interpretationreasoning§ VII. NEW MODEL HAMILTONIAN WITH EIGHT BANDS · p. 11
For full quantitative fitting with first principle calculations, we need to expand the basis set.
“for full quantitative fitting with the first principle calculations, we need to expand the basis set”
Quote in the paperSubstantial claim - Interpretationreasoning§ VII. NEW MODEL HAMILTONIAN WITH EIGHT BANDS · p. 12
These type of terms exist because the states |P2-, Γ4,5> themselves break the rotation symmetry according to expressions (13) and (14).
“These type of terms exist because the states | P 2 -, ˜ Γ 4 , 5 〉 themselves break the rotation symmetry according to the expression (13) and (14).”
Quote in the paperSubstantial claim - Interpretationown data§ VI. MAGNETIC FIELD AND LANDAU LEVEL · p. 11
This gap is large enough for the observation of the topological magneto-electric effect.
“which is large enough for the observation of the topological magneto-electric effect[36, 37]”
Quote in the paperNot a substantial claim0 of 2 sources checked - Interpretationown data§ VII. NEW MODEL HAMILTONIAN WITH EIGHT BANDS · p. 12
The eight band model is suitable to serve as the basis of the quantitative study of Bi2Se3 family of materials.
“This demonstrate that the eight band model is suitable to serve as the basis of the quantitative study of Bi 2 Se 3 family of materials.”
Quote in the paperSubstantial claim - Interpretationreasoning§ V. SURFACE STATES · p. 8
The helicity of the spin texture is originally related to the atomic SOC.
“Finally, we conclude that the helicity of the spin texture is originally related to the atomic SOC.”
Quote in the paperSubstantial claim - Interpretationreasoning§ Appendix C: Parameters in k · p theory · p. 17
The effective g factor is the sum of the atomic g factor and the second-order perturbation contribution.
“Therefore finally our effective g factor is the summation of the above two different contributions”
Quote in the paperNot a substantial claim- Rests on an assumption: The wave functions for the basis of the model Hamiltonian can be projected into atomic orbitals.
- Interpretationreasoning§ V. SURFACE STATES · p. 8
Due to the inversion condition M1 M0 < 0, the sign of M1 is already determined by the gap M0.
“Furthermore, due to the inversion condition M 1 M 0 < 0, the sign of M 1 is already determined by the gap M 0 .”
Quote in the paperSubstantial claim- Rests on an assumption: The form of the surface wave function ψ0 remains valid in realistic materials despite material details.
- Interpretationown data§ VI. MAGNETIC FIELD AND LANDAU LEVEL · p. 10
The decrease of the bulk gap is due to the double hump structure for the valence band dispersion of Bi2Se3.
“which is due to the double hump structure for the valence band dispersion of Bi 2 Se 3”
Quote in the paperSubstantial claim - Interpretationreasoning§ VI. MAGNETIC FIELD AND LANDAU LEVEL · p. 10
For high magnetic field, the energy level will be a combination of the linear contribution and square root contribution.
“For high magnetic field, it will be a combination of the linear contribution and square root contribution.”
Quote in the paperSubstantial claim - Interpretationown data§ VIII. CONCLUSIONS · p. 12
A new model Hamiltonian with eight energy bands can describe Bi2Se3 type of materials quantitatively.
“a new model Hamiltonian with eight energy bands can describe Bi 2 Se 3 type of materials quantitatively”
Quote in the paperSubstantial claim - Interpretationreasoning§ V. SURFACE STATES · p. 8
Within the model Hamiltonian, the helicity of the spin texture is given by the relative sign of the coefficients of the two types of linear terms, A0 and B0.
“Consequently, within our model Hamiltonian the helicity of the spin texture is given by the relative sign of the coefficients of two types of linear terms, A 0 and B 0 .”
Quote in the paperSubstantial claim- Rests on an assumption: The form of the surface wave function ψ0 remains valid in realistic materials despite material details.
- Interpretationreasoning§ VII. NEW MODEL HAMILTONIAN WITH EIGHT BANDS · p. 11
Our model Hamiltonian with four bands is not suitable in this regime.
“Therefore it is not surprise that our model Hamiltonian with four bands is not suitable in this regime.”
Quote in the paperSubstantial claim - Interpretationreasoning§ V. SURFACE STATES · p. 8
The helicity of the spin texture is determined by the sign of the coefficient A~ = A0 α1, where α1 is related to the spin basis of the surface states.
“From (34), the helicity of the spin texture is determined by the sign of the coefficient ˜ A = A 0 α 1 , where α 1 ≡ 〈 ψ 0 | τ 1 | ψ 0 〉 is related to the spin basis of the surface states (31).”
Quote in the paperSubstantial claim- Rests on an assumption: The form of the surface wave function ψ0 remains valid in realistic materials despite material details.
- May combine two claims. 28
- Classified as probably not a substantial claim. 9
- The quote only partly states this claim. 1
May combine two claims.
Findingreasoning§ V. SURFACE STATES · p. 7When B0/M1 > 0, Rλ1,2 < 0, while when B0/M1 < 0, Rλ1,2 > 0.
“Furthermore, it is easy to show that when B 0 /M 1 > 0, ℜ λ 1 , 2 < 0, while B 0 /M 1 < 0, ℜ λ 1 , 2 > 0”
Quote in the paperSubstantial claim- The authors state this as an assumption: The ǫ term can be neglected for simplicity in the surface-state derivation.
May combine two claims.
Findingreasoning§ III. MODEL HAMILTONIAN DERIVED FROM SYMMETRY PRINCIPLES · p. 4With spin-orbit coupling, |Λ+,±1/2> with Λ=P1,P2 belong to the ̃Γ+_6 representation while |Λ−,±1/2> belong to the ̃Γ−_6 representation.
“therefore | Λ + , ± 1 2 〉 with Λ = P 1 , P 2 should belong to ˜ Γ + 6 representaion while | Λ -, ± 1 2 〉 should belong to ˜ Γ -6 representation.”
Quote in the paperSubstantial claim- Rests on an assumption: Other orbitals such as s-orbitals of Bi and Se will also mix into the states near the Fermi surface.
Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
Findingnone given§ V. SURFACE STATES · p. 8In the numerical calculation, the coefficients α1 and α3 are treated as fitting parameters to experiment, given by α1 = 0.99 and α3 = -0.15.
“In the following numerical calculation, the coefficient α 1 and α 3 are treated as two fitting parameters to the experiment, given by α 1 = ˜ A exp A 0 = 0 . 99 and α 3 = ˜ C exp -C 0 M 0 = -0 . 15”
Quote in the paperNot a substantial claim- Rests on an assumption: The form of the surface wave function ψ0 remains valid in realistic materials despite material details.
May combine two claims. The quote only partly states this claim.
Findingnone given§ III. MODEL HAMILTONIAN DERIVED FROM SYMMETRY PRINCIPLES · p. 6The model Hamiltonian up to O(k^3) includes H'_0, which keeps the in-plane rotation symmetry along z, while H'_3 breaks the in-plane rotation symmetry down to threefold rotation symmetry.
“H ′ 0 keeps the in-plane rotation symmetry along z direction, while H ′ 3 breaks the in-plane rotation symmetry down to three fold rotation symmetry.”
Quote in the paperSubstantial claim- The authors state this as an assumption: The four states |P1+-,±1/2> and |P2-+,±1/2> are sufficient for the minimum model Hamiltonian; other states are treated as perturbation.
May combine two claims.
Findingcitation§ V. SURFACE STATES · p. 7The open boundary condition ψ(0)=0 together with normalizability leads to the existence condition of surface states, Rλ1,2<0 (c=d=0) or Rλ1,2>0 (a=b=0), which can only be satisfied with the band inversion condition M0 M1 < 0.
“Similar to the Ref. [31], the open boundary condition ψ (0) = 0, together with the normalizability of the wave function in the region z > 0, leads to the existence condition of the surface states, ℜ λ 1 , 2 < 0 ( c = d = 0) or ℜ λ 1 , 2 > 0 ( a = b = 0), which can only be satisfied with the band inversion condition M 0 M 1 < 0.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [31] M. Koenig, H. Buhmann, L. W. Molenkamp, T. Hughes, C.-X. Liu, X.-L. Qi, and S.-C. Zhang, J. Phys. Soc. Japan 77 , 031007 (2008). source
Not checked in the free run.
- The authors state this as an assumption: The ǫ term can be neglected for simplicity in the surface-state derivation.
May combine two claims.
Findingreasoning§ II. CRYSTAL STRUCTURE, ATOMIC ORBITALS AND SYMMETRY · p. 3There is an energy splitting between p_z and p_x,y orbitals for both P1+ and P2- states; |P1+,p_x,y> have higher energy than |P1+,p_z>, while |P2-,p_x,y> have lower energy than |P2-,p_z>.
“Thus there is an energy splitting between p z and p x,y orbitals for both P 1 + and P 2 -states. We find that | P 1 + , p x,y 〉 orbitals have higher energy than | P 1 + , p z 〉 , while | P 2 -, p x,y 〉 orbitals have lower energy than | P 2 -, p z 〉 .”
Quote in the paperSubstantial claim- The authors state this as an assumption: The states |P1+,α> and |P2-,α> are near the Fermi surface and other states can be neglected.
May combine two claims.
Findingreasoning§ III. MODEL HAMILTONIAN DERIVED FROM SYMMETRY PRINCIPLES · p. 4|Λ+(−),p_x> and |Λ+(−),p_y> belong to the ̃Γ+(−)_3 representation; |Λ+,p_z> belongs to ̃Γ+_1; |Λ−,p_z> belongs to ̃Γ−_2.
“According to the above transformation, we find that | Λ +( -) , p x 〉 and | Λ +( -) , p y 〉 belong to the ˜ Γ +( -) 3 representation. | Λ + , p z 〉 belongs to ˜ Γ + 1 representation and | Λ -, p z 〉 belongs to ˜ Γ -2 representation.”
Quote in the paperSubstantial claim- Rests on an assumption: Other orbitals such as s-orbitals of Bi and Se will also mix into the states near the Fermi surface.
May combine two claims.
Findingreasoning§ VI. MAGNETIC FIELD AND LANDAU LEVEL · p. 10The model Hamiltonian can also be derived from k·p theory and its parameters can be related to momentum matrix elements.
“This model Hamiltonian can also be derived from k · p theory and the parameters g 1 z , g 2 z , g 1 p and g 2 p can be related to the matrix elements of the momentum operator p in the k · p theory”
Quote in the paperNot a substantial claimMay combine two claims.
Findingown data§ VI. MAGNETIC FIELD AND LANDAU LEVEL · p. 11Within the bulk gap, only surface LDOS remains and shows clearly the Landau levels as discrete peaks.
“within the bulk gap, only surface LDOS remains and shows clearly the Landau levels as discreted peaks”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S4 (based).
Findingnone given§ VII. NEW MODEL HAMILTONIAN WITH EIGHT BANDS · p. 12In this Hamiltonian, time-reversal symmetry is already satisfied.
“In this Hamiltonian, time-reversal symmetry is already satisfied.”
Quote in the paperNot a substantial claimMay combine two claims.
Findingreasoning§ VII. NEW MODEL HAMILTONIAN WITH EIGHT BANDS · p. 12R2 rotation symmetry yields U35=U*36, V35=-V*36, U58=-U*68 and V58=V*68.
“R 2 rotation symmetry yields that U 35 = U ∗ 36 , V 35 = -V ∗ 36 , U 58 = -U ∗ 68 and V 58 = V ∗ 68 .”
Quote in the paperSubstantial claimMay combine two claims.
Interpretationreasoning§ II. CRYSTAL STRUCTURE, ATOMIC ORBITALS AND SYMMETRY · p. 2The strongest coupling in the system is between Bi layers and Se layers, causing level repulsion so that Bi energy levels are pushed up and Se energy levels are pushed down.
“Since all the Se layers are seperated by Bi layers, the strongest coupling in this system is the coupling between Bi layers and Se layers. Such coupling causes level repulsion, so that the Bi energy levels are pushed up and form new hybridized states | B α 〉 and | B ′ α 〉 while the Se energy levels”
Quote in the paperSubstantial claim- The authors state this as an assumption: Only p-orbitals of Bi and Se need be considered; other orbitals can be neglected.
- The authors state this as an assumption: The two neighboring quintuple layers are coupled only by van der Waals force, so one can first focus on one quintuple layer.
May combine two claims.
Premisecitation§ I. INTRODUCTION · p. 1Topological insulators are fully gapped in the bulk but have gapless edge or surface states that are topologically protected by time reversal symmetry.
“These insulators are fully gapped in the bulk, but have gapless edge or surface states which are topologically protected by the time reversal symmetry.”
Quote in the paperSubstantial claim0 of 3 sources checkedClassified as probably not a substantial claim. May combine two claims.
Premisecitation§ I. INTRODUCTION · p. 1Zhang et al. derived a model Hamiltonian for the three-dimensional topological insulators Bi2Se3, Bi2Te3 and Sb2Te3, and obtained topological surface states consisting of a single Dirac cone.
“Zhang et al [1] derived a model Hamiltonian for the 3D TI Bi 2 Se 3 , Bi 2 Te 3 and Sb 2 Te 3 , and obtained topological surface states consisting of a single Dirac cone.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [1] source
Not checked in the free run.
May combine two claims.
Interpretationreasoning§ V. SURFACE STATES · p. 7After neglecting the ǫ term, the eigen equation has particle-hole symmetry, so a special surface state with E=0 can exist.
“After neglecting the ˜ ǫ term, the eigen equation (27) has the particle hole symmetry, therefore we would like to expect a sepecial surface states with E = 0 can exist.”
Quote in the paperSubstantial claim- The authors state this as an assumption: The ǫ term can be neglected for simplicity in the surface-state derivation.
May combine two claims.
Premisereasoning§ IV. MODEL HAMILTONIAN DERIVED FROM THE k · p PERTURBATION THEORY · p. 6Due to inversion symmetry, all states at the Γ point have definite parity eigenvalues; since momentum p has odd parity, matrix elements of momentum between two states with the same parity always vanish.
“For example, due to the existence of the inversion symmetry, all the states at Γ point have definite parity eigenvalues. Since the momentum p has odd parity, the matrix elements of momentum between two states with the same parity always vanish.”
Quote in the paperSubstantial claim- The authors state this as an assumption: The four states |P1+-,±1/2> and |P2-+,±1/2> are sufficient for the minimum model Hamiltonian; other states are treated as perturbation.
Classified as probably not a substantial claim.
Premisecitation§ I. INTRODUCTION · p. 1New topological insulators with large bulk gaps of about 0.3 eV and single Dirac cone surface states have been theoretically predicted for Bi2Te3, Sb2Te3 and Bi2Se3.
“More recently, new TIs with large bulk gaps ∼ 0 . 3 eV and single Dirac cone surface states have been theoretically predicted for Bi 2 Te 3 [1], Sb 2 Te 3 [1] and Bi 2 Se 3 [1, 10].”
Quote in the paperSubstantial claim0 of 2 sources checkedMay combine two claims.
Premisecitation§ I. INTRODUCTION · p. 1Transport measurements show a gapless edge channel, demonstrating that the HgTe/CdTe quantum well is a two-dimensional topological insulator with the quantum spin Hall effect.
“Transport measurements[6, 7] show the existence of the gapless edge channel, which demonstrates that HgTe/CdTe quantum well is a two-dimensional (2D) TI with quantum spin Hall effect.”
Quote in the paperNot a substantial claim0 of 2 sources checkedMay combine two claims.
Interpretationreasoning§ II. CRYSTAL STRUCTURE, ATOMIC ORBITALS AND SYMMETRY · p. 4|P1+-,±1/2> is pushed down while |P2-+,±1/2> is pushed up, yielding a level crossing between these two pairs of states when spin-orbit coupling is strong enough.
“Consequently, | P 1 + -, ± 1 2 〉 is pushed down while | P 2 -+ , ± 1 2 〉 is pushed up, which yields the level crossing between these two pairs of states, when the SOC is strong enough”
Quote in the paperSubstantial claim- The authors state this as an assumption: Only p-orbitals of Bi and Se need be considered; other orbitals can be neglected.
May combine two claims.
Interpretationreasoning§ III. MODEL HAMILTONIAN DERIVED FROM SYMMETRY PRINCIPLES · p. 5Near the Fermi surface, the conduction and valence bands are determined by the four states |P1+-,±1/2> and |P2-+,±1/2>, belonging to ̃Γ+_6 and ̃Γ−_6; therefore the minimum model Hamiltonian should be written with these four states as the basis.
“As described above, near the Fermi surface the conduction band and valence band are determined by the four states | P 1 + -, ± 1 2 〉 and | P 2 -+ , ± 1 2 〉 , belonging to the ˜ Γ + 6 and ˜ Γ -6 representation. Therefore the minimum model Hamiltonian for Bi 2 Se 3 should be written with these four states as the basis.”
Quote in the paperSubstantial claim- The authors state this as an assumption: The four states |P1+-,±1/2> and |P2-+,±1/2> are sufficient for the minimum model Hamiltonian; other states are treated as perturbation.
May combine two claims.
Interpretationreasoning§ II. CRYSTAL STRUCTURE, ATOMIC ORBITALS AND SYMMETRY · p. 4Spin-orbit coupling couples |Λ,p_z,↑> (|Λ,p_z,↓>) to |Λ,p_+,↓> (|Λ,p_-,↑>) and induces level repulsion between these two states.
“Now as we see, the SOC couples | Λ , p z , ↑〉 ( | Λ , p z , ↓〉 ) to | Λ , p + , ↓〉 ( | Λ , p -, ↑〉 ) so that it induces the level repulsion between these two states.”
Quote in the paperSubstantial claim- The authors state this as an assumption: Only p-orbitals of Bi and Se need be considered; other orbitals can be neglected.
May combine two claims.
Study design and conductreasoning§ IV. MODEL HAMILTONIAN DERIVED FROM THE k · p PERTURBATION THEORY · p. 6The parameters given here are different from those in Refs. [1] and [41], where parameters are determined by fitting to energy dispersion, which has some ambiguity; in the present method, because the matrix elements of momentum are directly calculated from microscopic wave functions, there is no ambiguity.
“We note that the parameters given here are different from those in Ref. [1] and [41], where the parameters are determined by fitting to the energy dispersion, which has some ambiguity. In the present method, since we directly calculate the matrix elements of momentum from microscopic wave functions, there is no ambiguity.”
Quote in the paperSubstantial claim0 of 2 sources checked- cites [1] source
Not checked in the free run.
- cites [41] W. Shan, H. Lu, and S. Shen, Arxiv:cond-mat/1001.0526 (2010).
Not checked in the free run.
- The authors state this as an assumption: The four states |P1+-,±1/2> and |P2-+,±1/2> are sufficient for the minimum model Hamiltonian; other states are treated as perturbation.
May combine two claims.
Interpretationreasoning§ II. CRYSTAL STRUCTURE, ATOMIC ORBITALS AND SYMMETRY · p. 3Before spin-orbit coupling is considered, the conduction band mainly consists of |P1+,p_z> while the valence band is dominated by |P2-,p_z>.
“Consequently, the conduction band mainly consists of | P 1 + , p z 〉 while the valence band is dominated by the | P 2 -, p z 〉 orbital before SOC is considered”
Quote in the paperSubstantial claim- The authors state this as an assumption: The states |P1+,α> and |P2-,α> are near the Fermi surface and other states can be neglected.
Probably not a substantial claim: may not pass S3 (load-bearing).
Premisecitation§ V. SURFACE STATES · p. 8The k^3 terms have also been found in Ref. [35].
“The k 3 terms have also been found in Ref. [35].”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [35] L. Fu, Phys. Rev. Lett. 103 , 266801 (2009). source
Not checked in the free run.
- Rests on an assumption: The form of the surface wave function ψ0 remains valid in realistic materials despite material details.
Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
Premisecitation§ I. INTRODUCTION · p. 1Bi_xSb_{1-x} was suggested to be a three-dimensional topological insulator with topologically non-trivial surface states, which were observed by ARPES.
“Later, Bi x Sb 1 -x was suggested to be a three-dimensional (3D) TI[8] with topologically non-trivial surface states, which were observed by angle-resolved photoemission spectroscopy (ARPES)[9].”
Quote in the paperNot a substantial claim0 of 2 sources checkedMay combine two claims.
Interpretationreasoning§ II. CRYSTAL STRUCTURE, ATOMIC ORBITALS AND SYMMETRY · p. 3The sign of λ_Λ is always positive for Λ=P1+,P2- because the potential is always attractive for atoms.
“The sign of λ Λ is always positive for Λ = P 1 + , P 2 -since the potential is always attractive for atoms.”
Quote in the paperNot a substantial claim- The authors state this as an assumption: Only p-orbitals of Bi and Se need be considered; other orbitals can be neglected.
Classified as probably not a substantial claim.
Study design and conductnone given§ VI. MAGNETIC FIELD AND LANDAU LEVEL · p. 10The k3 term, which breaks the in-plane rotation symmetry, is neglected here.
“The k 3 term, which breaks the in-plane rotation symmetry, is neglected here”
Quote in the paperSubstantial claimMay combine two claims.
Interpretationreasoning§ V. SURFACE STATES · p. 8In the limit k→0, the linear term in Hamiltonian (34) is dominant, and the surface states show linear dispersion with helical spin texture, which has opposite direction for the conduction and valence bands.
“In the limit k → 0, the linear term in Hamiltonian (34) will be dominant, then the surface states show the linear dispersion with helical spin texture, which has the opposite direction for the conduction and valence band”
Quote in the paperSubstantial claim- Rests on an assumption: The form of the surface wave function ψ0 remains valid in realistic materials despite material details.
May combine two claims.
Interpretationreasoning§ V. SURFACE STATES · p. 8The helical spin texture is similar to one of the Fermi surfaces in the usual 2D electron gas with Rashba spin-orbit coupling, which can be understood from inversion symmetry being broken near the surface.
“Such type of spin texture is similar to one of the fermi surfaces in the usual 2D electron gas with Rashba SOC[40, 44], which can be simply understood from the fact that the inversion symmetry is broken near the surface.”
Quote in the paperNot a substantial claim0 of 2 sources checked- cites [40] R. Winkler, Spin-orbit coupling effects in twodimensional electron and hole systems , Springer Tracts in Modern Physics, 2003.
Not checked in the free run.
- cites [44] Y. A. Bychkov and E. I. Rashba, Journal of Physics C Solid State Physics 17 , 6039 (1984).
Not checked in the free run.
- Rests on an assumption: The form of the surface wave function ψ0 remains valid in realistic materials despite material details.
Probably not a substantial claim: may not pass S3 (load-bearing).
Premisecitation§ Appendix C: Parameters in k · p theory · p. 16The relation between effective mass and effective g factor in ordinary semiconductors is known as Roth's formula.
“The relation between the effective mass and effective g factor in the ordinary semiconductors is known as the Roth's formula[48].”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [48] L. M. Roth, B. Lax, and S. Zwerdling, Phys. Rev. 114 , 90 (1959).
Not checked in the free run.
May combine two claims.
Interpretationnone given§ VII. NEW MODEL HAMILTONIAN WITH EIGHT BANDS · p. 12There is a one-to-one correspondence between the basis of the eight band model and the Kane model: |P1+-, ±1/2> corresponds to the electron band (Γ6), |P2-+, ±1/2> and |P2-, Γ4,5> correspond to the light hole and heavy hole bands (Γ8), and |P2--, ±1/2> corresponds to the spin-orbit split-off band (Γ7).
“there is a one-to-one correspondence between the basis of these two models, in which | P 1 + -, ± 1 2 〉 corresponds to the electron band (Γ 6 ), | P 2 -+ , ± 1 2 〉 and | P 2 -, Γ 4 , 5 〉 correspond to the light hole and heavy hole bands (Γ 8 ), respectively, and | P 2 --, ± 1 2 〉 corresponds to the spin-orbit split-off band (Γ 7 ).”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Study design and conductnone given§ VI. MAGNETIC FIELD AND LANDAU LEVEL · p. 9The mixing effect between bulk and surface Landau levels is neglected in this analysis.
“The mixing effect between bulk and surface Landau levels is neglected here.”
Quote in the paperNot a substantial claimMay combine two claims.
Premisecitation§ V. SURFACE STATES · p. 8The helical spin texture has been calculated from ab initio methods and observed in spin-resolved ARPES measurements.
“The helical spin texture has also been calculated from ab initio method[45] and already observed in the pioneering spinresolved APRES measurement[18].”
Quote in the paperNot a substantial claim0 of 2 sources checked- cites [45] W. Zhang, R. Yu, H. Zhang, X. Dai, and Z. Fang, arxiv: cond-mat/1003.5082 (2010).
Not checked in the free run.
- cites [18] source
Not checked in the free run.
- Rests on an assumption: The form of the surface wave function ψ0 remains valid in realistic materials despite material details.
May combine two claims.
Interpretationnone given§ VIII. CONCLUSIONS · p. 12The topological surface states have well defined spin texture, which can be traced back to the sign of the atomic SOC in these materials.
“The topological surface states have well defined spin texture, which can be traced back to the sign of the atomic SOC in these materials.”
Quote in the paperNot a substantial claim
The paper relies on this without support.
Bi2Se3 has a rhombohedral crystal structure with space group D5_3d (R-3m).
“The crystal structure of Bi 2 Se 3 is rhombohedral with the space group D 5 3 d ( R ¯ 3 m ).”
The authors state this as an assumption.
Only p-orbitals of Bi and Se need be considered; other orbitals can be neglected.
“therefore it is natural to consider only p-orbitals of Bi and Se and neglect other orbitals.”
The authors state this as an assumption.
The two neighboring quintuple layers are coupled only by van der Waals force, so one can first focus on one quintuple layer.
“The chemical bonding is very strong within one quintuple layer but the two neighboring quintuple layers are only coupled by the van der Waals force. Therefore it is reasonable for us to first focus on one quintuple layer.”
The authors state this as an assumption.
The states |P1+,α> and |P2-,α> are near the Fermi surface and other states can be neglected.
“hence we focus on | P 1 + , α 〉 and | P 2 -, α 〉 ( α = p x , p y , p z ) and neglect the other states.”
The authors state this as an assumption.
The four states |P1+-,±1/2> and |P2-+,±1/2> are sufficient for the minimum model Hamiltonian; other states are treated as perturbation.
“Therefore in the following we will focus on these four states and regard the other states as the perturbation.”
The authors state this as an assumption.
The ǫ term can be neglected for simplicity in the surface-state derivation.
“we would like to breifly review the derivation for the explicit form of the surface states by neglecting the ˜ ǫ for simplicity[31].”
The paper relies on this without support.
The form of the surface wave function ψ0 remains valid in realistic materials despite material details.
“In the realistic materials, the detail form of ψ 0 will depend on the material detail, such as the boundary condition or the detail parameters, however the form of the wave function (26) remains valid.”
The paper relies on this without support.
Bi_xSb_{1-x} has a small energy gap, alloy disorder and rather complicated surface states.
“However, Bi x Sb 1 -x has a small energy gap, alloy disorder and rather complicated surface states.”
The paper relies on this without support.
Other orbitals such as s-orbitals of Bi and Se will also mix into the states near the Fermi surface.
“However, other orbitals such as s-orbitals of Bi and Se will also mix into these states.”
The paper relies on this without support.
An important physical consequence of non-trivial topology is the existence of topological surface states.
“An important physical consequence of the non-trivial topology is the existence of topological surface states.”
The paper relies on this without support.
The magnetic field effect cannot be incorporated in ab initio calculations.
“since the magnetic field effect cannot be incorporated in ab initio calculations”
The paper relies on this without support.
For a realistic finite sample, bulk Landau levels always coexist with surface Landau levels.
“For the realistic finite sample, the bulk Landau levels will always coexist with the surface Landau levels”
The paper relies on this without support.
For bulk states, the magnetic field contributes through orbital and Zeeman effects.
“For the bulk state, there are two types of contribution from the magnetic field, the orbital effect and the Zeeman effect.”
The paper relies on this without support.
The surface states only exist near k=0.
“the surface states only exist near k = 0”
The paper relies on this without support.
The wave functions for the basis of the model Hamiltonian can be projected into atomic orbitals.
“The wave functions for the basis of our model Hamiltonian have been calculated from ab initio calculation, which can be projected into the atomic orbitals.”
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?
The origin of the topological insulator state in Bi2Se3 is a symmetry inversion at Γ, and electronic states near the Brillouin-zone boundary resemble those of topologically trivial materials.
“This can be understood because the origin of the topological insulator state in Bi 2 Se 3 is a symmetry inversion that occurs at the point 7,18 , and the electronic states close to the Brillouin-zone boundary are similar to those of topologically trivial materials.”
From A topological insulator surface under strong Coulomb, magnetic and disorder perturbations (2011) · cites this paper as [18]
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.
Eight-band basis names the fourth state inconsistently
MinorThe listed eight-band basis uses a P1 state for its fourth entry, but the paper's subsequent description identifies that pair as the P2 states with angular-momentum labels ±1/2. The two descriptions specify different states for the same basis pair.
“| P 2 -+ , 1 2 〉 , | P 1 -+ , -1 2 〉 , | P 2 -, ˜ Γ 4 〉”
“| P 2 -+ , ± 1 2 〉 and | P 2 -, Γ 4 , 5 〉 correspond to the light hole and heavy hole bands (Γ 8 )”
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Self-citationTopological Insulator Materials
Ando, Y. (2013) · Journal of the Physical Society of Japan · doi:10.7566/jpsj.82.102001 · Cited by 1,789 · see citing works
Quantum anomalous Hall effect and related topological electronic states
Weng, H., Yu, R., Hu, X., Dai, X., & Fang, Z. (2015) · Advances in Physics · doi:10.1080/00018732.2015.1068524 · Cited by 506 · see citing works
Self-citationChiral gauge field and axial anomaly in a Weyl semimetal
Liu, C.-X., Ye, P., & Qi, X.-L. (2013) · Physical Review B · doi:10.1103/physrevb.87.235306 · Cited by 325 · see citing works
Self-citationUltrafast photocurrents at the surface of the three-dimensional topological insulator Bi2Se3
Braun, L., Mussler, G., Hruban, A., Konczykowski, M., Schumann, T., Wolf, M., Münzenberg, M., Perfetti, L., & Kampfrath, T. (2016) · Nature Communications · doi:10.1038/ncomms13259 · Cited by 236 · see citing works
Shift charge and spin photocurrents in Dirac surface states of topological insulator
Kim, K. W., Morimoto, T., & Nagaosa, N. (2017) · Physical Review B · doi:10.1103/physrevb.95.035134 · Cited by 79 · see citing works
- Authors
- Liu, C.-X., Qi, X.-L., Zhang, H., Dai, X., Fang, Z., & Zhang, S.-C.
- Affiliations
- University of Würzburg (DE)
- Stanford University (US)
- Chinese Academy of Sciences (CN)
- Venue
- Physical Review B
- Publisher
- American Physical Society (APS)
- Published
- 2010-07-26
- Volume
- 82
- Issue
- 4
- ISSN
- 1098-0121 (ISSN-L)
- 1550-235X
- OpenAlex
- W2044466999
- Semantic Scholar
- de8bf073db347c678fc9c0d9c0df542102663474
- Cited by
- 923
- References
- 8
- Citations by year
- 2026: 30 · 2025: 55 · 2024: 71 · 2023: 77 · 2022: 54 · 2021: 69 · 2020: 54 · 2019: 52 · 2018: 65 · 2017: 50 · 2016: 60 · 2015: 62 · 2014: 77 · 2013: 53 · 2012: 53
- FWCI
- 25.11
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review B (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- Rare & Special e-Zone (The Hong Kong University of Science and Technology) (submittedVersion)
- 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 › Materials Science › Materials Chemistry › 2D Materials and Applications
- Keywords
- topological insulators, model Hamiltonian, Bi2Se3, Bi2Te3, Sb2Te3, three-dimensional topological insulators, k·p perturbation theory, topological surface states, effective Hamiltonian, Landau levels, external magnetic field, electronic structure, atomic orbitals, symmetry principles
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Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
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