Surface state transport in double-gated and magnetized topological insulators with hexagonal warping effects

Alireza Saffarzadeh; Masomeh Arabikhah

REVIEW 4 major objections 5 minor 36 references

Surface state transport in double-gated and magnetized topological insulators with hexagonal warping effects

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read On a topological-insulator surface, a gate voltage alone can flip the magnetoresistance sign, from about -170% (double barrier) to +60% (double well), for out-of-plane magnetization.

desk verdict Solid but narrow transfer-matrix calculation with a wrong time-reversal-symmetry claim that undermines its stated motivation. read the letter →

arxiv 1908.03862 v1 pith:GESCJ34R submitted 2019-08-11 cond-mat.mes-hall cond-mat.mtrl-sci

Masomeh Arabikhah , Alireza Saffarzadeh This is my paper · ORCID

classification cond-mat.mes-hallcond-mat.mtrl-sci

keywords topologicalinsulatorsurfacestatesmagnetoresistancehexagonalwarpingquantumwellbarrierDiracfermionstransfermatrix

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper studies how two gate-controlled and two magnetized regions on the surface of a three-dimensional topological insulator — with the hexagonal-warping distortion of the Fermi contour included — affect transport of Dirac electrons. The central result is that the magnetoresistance ratio $(G_{\rm Pa}-G_{\rm APa})/G_{\rm Pa}$ can be large and its sign can be switched by the gate voltage: about $-170\%$ for a double-barrier structure and about $+60\%$ for a double-well structure when the magnetization is along the $z$-axis. For magnetization along the $x$-axis, the surface spectrum stays gapless and the magnetoresistance remains positive, exceeding $+50\%$ as the separation between the magnetized regions is varied. The authors conclude that surface-state transport can be controlled by the exchange field and gate voltage without breaking time-reversal symmetry, a capability that could make double-gated, magnetized topological insulators useful as tunable magnetoresistive devices.

What carries the argument

The central object is the transfer-matrix scattering formalism applied to the warped surface Hamiltonian $\hat H = v_F(k_x\sigma_y - k_y\sigma_x) + \lambda(k_x^3 - 3k_x k_y^2)\sigma_z + V(x)$, with $V(x)$ encoding the double gate voltage $U_g$ and the magnetic proximity exchange $\Delta\,\mathbf m\cdot\boldsymbol\sigma$ in two regions of width $W$ separated by $d$. The load-bearing step is the mode classification and matching: the dispersion $E(k_x,k_y)=U_g+\sqrt{[\lambda(k_x^3-3k_xk_y^2)+\Delta\cos\theta]^2 + A}$ is a sextic in $k_x$, giving two real and four complex roots below the critical energy $E_c\approx377$ meV and six real roots above; the scattering solution assumes that continuity of $\psi$, $\partial_x\psi$, and $\partial_x^2\psi$ at all four interfaces completely determines the scattering amplitudes. The resulting transmission coefficients feed a conductance integral over incident angles with a prefactor derived from the warping-modified current density, and the magnetoresistance is defined as $\mathrm{MR} = (G_{\rm Pa}-G_{\rm APa})/G_{\rm Pa}\times 100\%$. The warping term itself deforms the Fermi contour into the snowflake shape that opens additional high-energy transport channels.

What would settle it

An independent numerical solution of the same warped Hamiltonian — for example, a tight-binding or finite-difference lattice calculation — for the Fig. 7 parameters ($\Delta=40$ meV, $W=d=4$ nm, $z$-axis magnetization) should reproduce $\mathrm{MR}\approx -170\%$ in the double-barrier configuration; if it gives a substantially different value, the $\partial_x^2$ matching condition would be falsified.

Watch

Extended reading notes

Core claim

The paper's quantitative claim is that a single gate voltage applied in two regions of width $W$ separated by distance $d$ on a Bi$_2$Te$_3$ surface can flip the sign of the magnetoresistance while the spectrum stays gapless for in-plane magnetization. Using a transfer-matrix calculation with boundary conditions that match $\psi$, $\partial_x\psi$, and $\partial_x^2\psi$ at each of the four interfaces, the authors find that for out-of-plane ($z$) magnetization the magnetoresistance ratio oscillates with energy and separation, reaching $\approx -170\%$ for a double quantum barrier and $\approx +60\%$ for a double quantum well. For in-plane ($x$) magnetization, the ratio is fully positive, exceeding $+50\%$ versus the separation distance $d$, and the conductance remains nonzero for all energies because no gap opens at the Dirac point. The out-of-phase oscillations of the conductance in the parallel and antiparallel configurations are interpreted as quantum interference of propagating waves in the inter-magnet region, and the hexagonal-warping term is shown to enhance high-energy conductance once the constant-energy contour becomes snowflake-shaped.

Load-bearing premise

The results stand on the wave-function matching scheme: the paper assumes that requiring continuity of $\psi$, $\partial_x\psi$, and $\partial_x^2\psi$ at each of the four interfaces, together with the six-root classification of the dispersion, yields a complete and correct transfer matrix for all incident angles and energies.

Editorial extensions

If this is right

A gate voltage alone can switch a topological-insulator surface junction from a state with $\mathrm{MR}\approx -170\%$ (double barrier) to one with $\mathrm{MR}\approx +60\%$ (double well) for out-of-plane magnetization, giving a voltage-controlled sign change in the magnetoresistance.
For in-plane ($x$) magnetization, the surface remains gapless and conductive at all Fermi energies, so the device avoids the resistance suppression that a gap would cause; the $\mathrm{MR}$ is positive and can exceed $+50\%$ as the inter-magnet separation changes.
Hexagonal warping materially increases the high-energy conductance because the snowflake Fermi contour provides more transmission channels, meaning the effect is stronger at energies above the critical value $E_c$.
The oscillations of $\mathrm{MR}$ with separation $d$ resemble RKKY-type coupling between the two magnetic regions, suggesting the transport measurement itself can be used to probe magnetic interactions mediated by the helical surface states.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The gate-tunable sign of the magnetoresistance should be generic to any warped surface state with the same cubic symmetry, so the mechanism may transfer to other topological-insulator families (e.g., Bi$_2$Se$_3$-class materials) beyond the Bi$_2$Te$_3$ parameters used here.
Because the large $\mathrm{MR}$ values appear at modest energies ($E\sim150$ meV) and small separations ($d\sim4$ nm), the proposed double-modulated structure may be realizable with existing ferromagnetic-insulator/topological-insulator heterostructure technology; this is an editorial inference, not a claim of the paper.
The $\mathrm{MR}$-versus-$d$ oscillations could be read as a transport analog of RKKY coupling, which suggests a potential way to map the magnetic coupling between two ferromagnetic islands by a simple two-terminal resistance measurement rather than by spin-polarized probes.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

First, the thing to know: this is a solid but narrow transfer-matrix study, and the headline interpretation is wrong. The combination of double gates, double magnetic regions, and hexagonal warping in one calculation is genuinely new relative to earlier single-barrier or no-warping work. The numerics look careful: the dispersion from Eq. (1) is internally consistent, the gap condition for out-of-plane magnetization matches Δ cosθ, and the parameters are standard bulk values for Bi2Te3. I see no signs of fitting or circularity.

The soft spot is not the one the reader flagged. Continuity of ψ, ψ′, ψ″ at the interfaces is correct: the third-order ODE is just a first-order system in those variables, so conditions (15) are right. The actual problem is the paper's repeated claim that an in-plane magnetization along x preserves time-reversal symmetry. It does not. With T = iσyK, T(Δσx)T⁻¹ = −Δσx, so the exchange term breaks TRS for any nonzero Δ. The gapless spectrum in Fig. 5(b) comes from the Dirac point shifting in momentum space, not from TRS. This error is load-bearing: it is the stated reason the system remains conductive for all energies, and the abstract and conclusions sell the device as working without breaking time-reversal symmetry. The gate-tunable MR predictions may survive as model results, but the physical motivation is undercut.

A fair second criticism: the transfer matrix itself is never shown. The conductance formula is given, but the matching step that produces T(E,β) is black-boxed, making the numerics hard to reproduce without the authors' code.

The citation pattern is fine, with due credit to An & Ting and others. I would send this to referees—it is a coherent calculation with plausible predictions—but the TRS claim must be corrected before publication. I hope a referee catches it.

Referee Report

4 major / 5 minor

Summary. The paper investigates ballistic transport of Dirac surface states in a double-gated, double-magnetized topological insulator with hexagonal warping. The authors model two modulated segments of width W separated by a distance d, each subject to a gate potential Ug and a proximity exchange field Δ with arbitrary magnetization direction. They derive the energy dispersion, solve the scattering problem via boundary conditions for a third-order Hamiltonian, and compute transmission, differential conductance, and magnetoresistance for parallel and antiparallel magnetizations along the z and x axes. The central numerical predictions are large magnetoresistance ratios with sign tunable by the gate voltage: about -170% for double barriers and +60% for double wells for z-axis magnetization, and positive MR exceeding +50% as a function of separation for x-axis magnetization. A repeated physical claim is that the x-axis magnetization does not break time-reversal symmetry and therefore the system remains conductive at all energies.

Significance. If the numerical results are correct, the predicted gate-tunable sign of the magnetoresistance and the large MR values are of interest for spintronic devices based on topological insulator surface states. The transport calculation is essentially parameter-free in the sense that vF and λ are literature values for Bi2Te3 and no fitting is performed; this is a strength. However, the paper's central physical interpretation contains a clear error: an in-plane exchange field does break time-reversal symmetry, and the gapless spectrum is due to a momentum-space shift of the Dirac point, not to symmetry protection. In addition, the transfer matrix and the conductance kernel F(kF,β) are asserted without derivation, which limits reproducibility. The qualitative new phenomenon of gate-controlled MR sign is plausible and the correction of these issues is within the scope of a major revision.

major comments (4)

[Abstract; Sec. III (Fig. 5); Sec. IV] The abstract, Sec. III, and Sec. IV repeatedly state that magnetization along the x-axis does not break time-reversal symmetry (for example, 'without breaking time reversal symmetry' in the abstract and 'due to the time reversal symmetry' in the discussion of Fig. 5(b)). This statement is false: for m along x, the exchange term in Eq. (1) is Δσx, and with the time-reversal operator T = iσyK one obtains T(H0 + Δσx)T⁻¹ = H0(-k) - Δσx, which differs from H(-k) = H0(-k) + Δσx for any Δ ≠ 0. The gapless conductance in Fig. 5(b) is a consequence of the momentum-space shift of the Dirac point (Fig. 1(f)), not of TRS preservation. Since this claim is central to the abstract and conclusions, it must be corrected throughout; the numerical results themselves need not change, but all interpretations relying on TRS preservation must be removed or revised.
[Sec. II, Eqs. (10)–(15)] The paper states that the transfer matrix method yields all reflection and transmission amplitudes, but the actual transfer matrix is not presented. For a third-order Hamiltonian there are six roots for kx, and the mode classification for E > Ec (three positive and three negative group velocities) is given only verbally. The completeness and correctness of the matching conditions for arbitrary incident angle and energy is load-bearing for all transmission, conductance, and magnetoresistance results. Please provide the explicit transfer matrix and the mode-selection rule, or include a detailed derivation in an appendix, specifying how evanescent modes and the hole-like branches are handled.
[Sec. III, Eq. (19)] The conductance formula contains an angular kernel F(kF,β) given by Eqs. (20)–(22), but its origin is not derived. The transmission coefficient T is defined as a sum of |ti|² without velocity ratios, so it is unclear how F accounts for the current density and why this combination yields the differential conductance. Please derive F from the current density in Eq. (16) and state the assumptions (low temperature, small bias, etc.), or provide a reference where the full derivation is given.
[Sec. III, Fig. 1(d)] The text reports an 11 meV gap for magnetization along the y axis, while Eq. (6) predicts a zero gap for any in-plane magnetization when warping is absent. Please clarify whether this gap is a warping-induced effect and reconcile it with the later claim that in-plane magnetization leaves the system gapless. If a small gap does exist for certain in-plane orientations, the statements about x-axis magnetization need to be qualified.

minor comments (5)

[Sec. I] The sentence 'When the in-plane magnetizations are aligned in the growth direction' is contradictory because the growth direction is out of plane; it should be replaced with 'out-of-plane' or 'z-axis'.
[Sec. II] There is a typo 'in-plain magnetization' that should read 'in-plane magnetization'.
[Sec. III] The phrase 'the the system' and the word 'hight' should be corrected.
[Sec. II, Eq. (6)] Equation (6) is stated without derivation; it would be helpful to show that it follows from the quadratic equation for k when λ = 0.
[Fig. 1 caption] The caption says green and blue lines correspond to Δ = 0 and Δ = 70 meV, but the figure panels do not clearly indicate which color is used in which panel; please clarify the legend.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the transport and magnetoresistance results are computed from a standard model with literature parameters and explicit matching conditions, not from fitted or self-referential inputs.

full rationale

The paper's derivation chain is self-contained after adopting the standard Fu/An-and-Ting surface Hamiltonian (Eq. 1) with literature values vF = 2.55 eV Å and λ = 250 eV Å^3 taken from experimental Bi2Te3 data [22]. The scattering problem is solved by explicitly stated wavefunctions (Eqs. 7–14), boundary conditions (Eq. 15), current expressions (Eqs. 16–17), and conductance/MR definitions (Eqs. 19, 23). No parameter is fitted to the computed transmission, conductance, or magnetoresistance; the exchange field Δ, gate voltage Ug, widths W, and separation d are input parameters varied in the study. The mode-counting rules are imported from Ref. [23] (An and Ting), an external reference, not from the present authors, and the rules are stated in the text rather than merely invoked. The self-citations [29,30] are standard transfer-matrix method references; the matching conditions are written out explicitly, so the central numerical results do not reduce to those citations. The paper's repeated claim that x-axis magnetization preserves time-reversal symmetry is physically incorrect, since the exchange term Δσx anticommutes with the time-reversal operator T = iσyK and hence breaks TRS, but this is a correctness error, not a circular-reasoning step. The magnetoresistance results are genuine outputs of the model rather than restatements of its inputs.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central calculation rests on the standard effective surface Hamiltonian of Bi2Te3 (with parameters vF and λ taken from experiment), the transfer-matrix boundary conditions appropriate to the third-order warping Hamiltonian, and the standard Landauer-type conductance formula. These are domain assumptions, not ad hoc inventions; the only hand-chosen input is the exchange field Δ, which is varied over a plausible range.

free parameters (1)

exchange field strength Δ = 30-60 meV in figures; 40 meV for MR spectra
The proximity-induced exchange energy is a free model input, not independently measured for a specific TI/FM interface; the qualitative results (gap for z-magnetization, MR oscillations) depend on its value.

assumptions (5)

domain assumption The surface states of a 3D topological insulator are described by the effective Hamiltonian (1) with bulk states ignored and Fermi level tuned to the surface band.
Stated in Sec. II: 'We only focus on the topological surface states... The interaction between bulk states and the surface states can be ignored by tuning the Fermi level via appropriate doping.' This is the standard Fu/An-and-Ting model.
domain assumption A gate voltage and proximity exchange field enter only as a rigid scalar and a Zeeman term in the modulated regions, Eq. (2).
Sec. II, Eq. (2); the model neglects band-structure changes from the gate and assumes a uniform exchange field inside each modulated region.
standard math The scattering states are matched by requiring continuity of ψ, ∂xψ, and ∂x²ψ at each interface, Eq. (15).
Sec. II after Eq. (14); this is a mathematical consequence of the third-order kx Hamiltonian but is an assumption about the correct form of the matching conditions.
domain assumption The six solutions of E(qx,qy)=EF are selected as two real plus four complex below Ec, and six real above Ec, with incident/reflected/transmitted modes assigned as in Ref. [23].
Sec. II around Eqs. (10)-(14); the transfer matrix uses this mode classification to build the scattering states.
standard math The low-temperature differential conductance is obtained from the current density (16) via the Landauer-type integral (19), with the Fermi function difference replaced by a small bias.
Sec. II, Eq. (19); standard ballistic transport formula, applied with the warping-modified current density.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Surface state transport in double-gated and magnetized topological insulators with hexagonal warping effects." pith.science (2026). https://pith.science/paper/GESCJ34R

@misc{pith2026190803862,
  author       = {Pith},
  title        = {Pith review of: Surface state transport in double-gated and magnetized topological insulators with hexagonal warping effects},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GESCJ34R}},
  note         = {Machine review of arXiv:1908.03862}
}

read the original abstract

We explore the scattering of Dirac electrons in a double-gated topological insulator in the presence of magnetic proximity effects and warped surface states. It is found that a magnetic field can shift the Dirac cone in momentum space and deform the constant-energy contour, or opens up a band gap at the Dirac point, depending on the magnetization orientation. The double gate voltage induces quantum wells and/or quantum barriers on the surface of topological insulators, generating surface resonant tunnelling states. It is found that the hexagonal warping effect can increase the electronic transport at high energies when the constant-energy contour exhibits a snowflake shape. The energy-dependent conductances in the parallel and antiparallel magnetic configurations exhibit out-of-phase oscillations due to the quantum interference of propagating waves in the region between the two magnetized segments. Although the conductance spectrum of the double-well structure is higher than that of the double-barrier structure, the magnetoresistance ratio versus the separation distance between the two magnetized barriers exhibits pronounced oscillations due to the resonant tunnelling states. We show that the surface state transport can be controlled by the exchange field and gate voltage without breaking time reversal symmetry, suggesting that the double gated and magnetized topological insulators can be utilized to achieve a large magnetoresistance ratio with a tunable sign.

Figures

Figures reproduced from arXiv: 1908.03862 by the authors.

**Figure 2.** FIG. 2: Transmission probability [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3: Transmission probability [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: FIG. 4: Calculated conductance as a function of gate voltage [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

**Figure 5.** Figure 5: FIG. 5: Calculated conductance as a function of incident en [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 7.** Figure 7: FIG. 7: Calculated magnetoresistance ratio as a function of [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

36 extracted references · 34 canonical work pages

[1]

Hasan, and C

M.Z. Hasan, and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010)

work page 2010
[2]

We shall assume the Fermi level lies in the upper band ( s = +1)

+ ∆ cos θ]2 +A, (4) wheres = ±1 correspond to the upper and lower surface bands, respectively, and A = (∆ sin θ cosϕ −υFky)2 + (∆ sin θ sinϕ +υFkx)2. We shall assume the Fermi level lies in the upper band ( s = +1). Using (kx 3 − 3ky 2kx) = k3 cos 3β, we can rewrite Eq. (4) as ǫ(k) = √ υ2 Fk2 + ∆ 2 sin2θ −B +C, (5) whereB = 2υFk∆ sin θ sin(β −ϕ) and C = (...

work page
[3]

Y. Ando, J. Phys. Soc. Jpn. 82, 102001 (2013)

2013
[4]

Qi, and S.C

X.L. Qi, and S.C. Zhang, Rev. Mod. Phys. 83, 1057 (2011)

work page 2011
[5]

Y. Wang, D. Zhu, Y. Wu, Y. Yang, J. Yu, R. Ra- maswamy, R. Mishra, S. Shi, M. Elyasi, K.-L. Teo, Y. Wu and H. Yang, Nat. Commun. 8, 1364 (2017)

work page 2017
[6]

J. Han, A. Richardella, S. A. Siddiqui, J. Finley, N. Samarth, and L. Liu, Phys. Rev. Lett. 119, 077702 (2017)

work page 2017
[7]

C. H. Li, O. M. J. van ’t Erve, J. T. Robinson, Y. Liu, L. Li and B. T. Jonker, Nat. Nanotech. 9, 218 (2014)

work page 2014
[8]

Yasuda, A

K. Yasuda, A. Tsukazaki, R. Yoshimi, K. Kondou, K.S. Takahashi, Y. Otani, M. Kawasaki, and Y. Tokura, Phys. Rev. Lett. 119, 137204 (2017)

work page 2017

Show all 36 references

[9]

L. Liu, A. Richardella, I. Garate, Y. Zhu, N. Samarth, and C.-T. Chen, Phys. Rev. B 91, 235437 (2015)

2015
[10]

Dankert, J

A. Dankert, J. Geurs, M.V. Kamalakar, S. Charpentier, and S.P. Dash, Nano Lett. 15, 7976 (2015)

2015
[11]

Zhang and F

Y. Zhang and F. Zhai, Appl. Phys. Lett. 96, 172109 (2010)

2010
[12]

Wang, X.W

H.Y. Wang, X.W. Chen, X.Y. Zhou, L.B. Zhang, and G.H. Zhou, Phys. B. 407, 3664 (2012)

2012
[13]

M. Vali, D. Dideban, N. Moezi, Super. Micro. 100, 1256 (2016)

2016
[14]

Song, Y.X

J.T. Song, Y.X. Li, and Q.F. Sun, J. Phys.: Con- dens.Matter 26, 185007 (2014)

2014
[15]

modelled magnetic-proximity-induced magnetoresis- tance in disordered topological/ferromagnetic insulator bilayers by means of Kubo and Boltzmann theories. It was shown that for in-plane magnetizations the magne- toresistance ratio vanishes, while for out-of-plane magne- tizat...
[16]

Zhang, Z.-C

K.-H. Zhang, Z.-C. Wang, Q.-R. Zheng, and G. Su, Phys. Rev. B 86, 174416 (2012)

2012
[17]

Chiba, S

T. Chiba, S. Takahashi, and G. E. W. Bauer, Phys. Rev. B 95, 094428 (2017)

2017
[18]

A. A. Burkov and D. G. Hawthorn, Phys. Rev. Lett. 105, 066802 (2010)

2010
[19]

Y. Ando, T. Hamasaki, T. Kurokawa, K. Ichiba, F. Yang, M. Novak, S. Sasaki, K. Segawa, Y. Ando, and M. Shi- raishi, Nano Lett. 14, 6226 (2014)

2014
[20]

Fu, Phys

L. Fu, Phys. Rev. Lett. 103, 266801 (2009)

2009
[21]

Nomura, S

M. Nomura, S. Souma, A. Takayama, T. Sato, T. Taka- hashi,K. Eto, K. Segawa, and Y. Ando, Phys. Rev. B 89, 045134 (2014)

2014
[22]

C. M. Wang and F. J. Yu, Phys. Rev. B 84, 155440 (2011)

2011
[23]

Li and J

Z. Li and J. P. Carbotte, Phys. Rev. B 87, 155416 (2013)

2013
[24]

Y. L. Chen, J. G. Analytis, J.-H. Chu, Z. K. Liu, S.-K. Mo, X. L. Qi, H. J. Zhang, D. H. Lu, X. Dai, Z. Fang, S. C. Zhang, I. R. Fisher, Z. Hussain, Z.-X. Shen, Science 325, 178 (2009)

2009
[25]

An and C

J. An and C. S. Ting, Phys. Rev. B 86, 165313 (2012)

2012
[26]

H. Li, J. M. Shao, H. B. Zhang, Dao-Xin Yao, and G. W. Yang, J. Appl. Phys. 114, 093703 (2013)

2013
[27]

Akzyanov and A.L

R.S. Akzyanov and A.L. Rakhmanov, Phys. Rev. B 97, 075421 (2018)

2018
[28]

Z.-G. Fu, P. Zhang, M. Chen, Z. Wang, F.-W. Zheng and H.-Q. Lin, Sci. Rep. 4, 5544 (2014)

2014
[29]

Siu, M.B.A

Z.B. Siu, M.B.A. Jalil and S.G. Tan, Sci. Rep. 4, 5062 (2014)

2014
[30]

Z.M. Yu, D.S. Ma, H. Pan, and Y. Yao, Phys. Rev. B 96,125152 (2017)

2017
[31]

Saﬀarzadeh, J

A. Saﬀarzadeh, J. Phys.: Condens. Matter 15 3041 (2003)

2003
[32]

Saﬀarzadeh, M

A. Saﬀarzadeh, M. Bahar, and M. Banihasan, Phys. E 27 462 (2005)

2005
[33]

Katsnelson, K.S

M.I. Katsnelson, K.S. Novoselov, and A. K. Geim, Nat. 9 Phys. 2, 620 (2006)

2006
[34]

Y. Li, Q. Wan, Y. Peng, G. Wang, Z. Qian, G. Zhou, and M. B. A. Jalil, Sci. Rep. 5, 18458 (2015)

2015
[35]

Garate and M

I. Garate and M. Franz, Phys. Rev. B 81, 172408 (2010)

2010
[36]

Biswas and A.V

R.R. Biswas and A.V. Balatsky, Phys. Rev. B 81 233405 (2010)

2010

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Hasan, and C

M.Z. Hasan, and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010)

work page 2010

[2] [2]

We shall assume the Fermi level lies in the upper band ( s = +1)

+ ∆ cos θ]2 +A, (4) wheres = ±1 correspond to the upper and lower surface bands, respectively, and A = (∆ sin θ cosϕ −υFky)2 + (∆ sin θ sinϕ +υFkx)2. We shall assume the Fermi level lies in the upper band ( s = +1). Using (kx 3 − 3ky 2kx) = k3 cos 3β, we can rewrite Eq. (4) as ǫ(k) = √ υ2 Fk2 + ∆ 2 sin2θ −B +C, (5) whereB = 2υFk∆ sin θ sin(β −ϕ) and C = (...

work page

[3] [3]

Y. Ando, J. Phys. Soc. Jpn. 82, 102001 (2013)

2013

[4] [4]

Qi, and S.C

X.L. Qi, and S.C. Zhang, Rev. Mod. Phys. 83, 1057 (2011)

work page 2011

[5] [5]

Y. Wang, D. Zhu, Y. Wu, Y. Yang, J. Yu, R. Ra- maswamy, R. Mishra, S. Shi, M. Elyasi, K.-L. Teo, Y. Wu and H. Yang, Nat. Commun. 8, 1364 (2017)

work page 2017

[6] [6]

J. Han, A. Richardella, S. A. Siddiqui, J. Finley, N. Samarth, and L. Liu, Phys. Rev. Lett. 119, 077702 (2017)

work page 2017

[7] [7]

C. H. Li, O. M. J. van ’t Erve, J. T. Robinson, Y. Liu, L. Li and B. T. Jonker, Nat. Nanotech. 9, 218 (2014)

work page 2014

[8] [8]

Yasuda, A

K. Yasuda, A. Tsukazaki, R. Yoshimi, K. Kondou, K.S. Takahashi, Y. Otani, M. Kawasaki, and Y. Tokura, Phys. Rev. Lett. 119, 137204 (2017)

work page 2017