Magnetic field Characterization of edge currents in quantum spin Hall insulators

Andrew D Greentree; Brant C Gibson; Eitan Dvorquez; Felipe Pinto; Jeronimo R Maze; Qiang Sun; Ricardo C Heitzer; Roberto Rodriguez

REVIEW 2 major objections 4 minor 44 references

Magnetic field Characterization of edge currents in quantum spin Hall insulators

T0 review · 2 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Spin accumulation at the edge of a quantum spin Hall strip produces a magnetic field decaying as $1/d^2$, while the edge current field decays as $1/d$, so their cancellation point exposes the spin accumulation to NV-center magnetometry.

desk verdict Clear presentation of a useful idea, but Eq. (5) is not a legitimate Biot-Savart expression and the quantitative cancellation maps rest on it. read the letter →

arxiv 2411.11701 v1 pith:CMPPJHWR submitted 2024-11-18 cond-mat.mes-hall

Felipe Pinto , Ricardo C Heitzer , Eitan Dvorquez , Roberto Rodriguez , Qiang Sun , Andrew D Greentree , Brant C Gibson , Jeronimo R Maze This is my paper · ORCID

classification cond-mat.mes-hall

keywords quantumspinHalleffecthelicaledgestatesaccumulationmagneticfieldNVcentermagnetometrynanodiamondsensingtopologicalinsulatorlinearresponsetheory

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

Quantum spin Hall insulators conduct through helical edge states whose spin and momentum are locked, so a bias voltage creates both an edge current and a local spin accumulation at each edge. This paper calculates the magnetic field of both contributions near the edge of a two-terminal strip and shows they have different decay laws: the current field falls as $1/d$ while the spin-accumulation field falls as $1/d^2$. At a distance set by the material parameters, the two cancel, producing a null point whose position and angular signature could let a scanning nitrogen-vacancy center detect the spin accumulation, and thus the QSH phase, locally. The calculation predicts that larger gaps and lower Fermi velocities make the spin-accumulation field stronger and closer to the edge, while smaller gaps push the detectable signature beyond 20 nm. The paper concludes that NV-center magnetometry with roughly 10 nm nanodiamonds has the sensitivity to characterize this edge magnetic field with minimal back-action.

What carries the argument

The load-bearing object is the pair of field expressions in Eqs. (4) and (5). The spin-accumulation field is built from the linear-response magnetization $m(q,y)$ (Eq. 3), a Fermi-distribution weighting of the edge-state density $|u_k(y)|^2$ with a delta function encoding the uniform bias along the edge, propagated through the dipole tensor $D_{ij}(r-r')$. The current field is computed by Biot-Savart from the same edge-state density with $I=(e^2/h)V$. The contrasting decay in $d$ arises because the magnetization acts as a surface-bound dipole layer ($1/d^2$) while the current acts as a line current ($1/d$), and the cancellation point where the two magnitudes cross is what the detection strategy targets.

What would settle it

Place a scanning NV center in a nanodiamond within a few tens of nanometers of the edge of a QSH strip biased at roughly 1 mV and map the total field magnitude and angle as a function of distance. If no zero-crossing appears at the predicted gap-dependent position, or if the field profile matches a pure $1/d$ current field with no spin-accumulation contribution at any accessible height, the central claim is refuted. A cleaner check is to vary the band gap by material choice or gating and see whether the cancellation point moves toward the edge as the gap increases, as Eqs. (4) and (5) predict.

Watch

Extended reading notes

Core claim

The paper's central claim is that the magnetic field near the edge of a quantum spin Hall strip carries a separable signature of edge spin accumulation because the two field sources scale differently with distance. Using linear response theory on the helical edge states of a Bernevig-Hughes-Zhang-type model, the magnetization $m(q,y)$ produced by spin-locking is weighted by the edge-state density $|u_k(y)|^2$; its field $B_{\mathrm{spin}}$ decays as $1/d^2$, while the Biot-Savart field $B_{\mathrm{current}}$ from the quantized edge current $I=(e^2/h)V$ decays as $1/d$. Consequently the total field has a cancellation point where $|B_{\mathrm{spin}}|=|B_{\mathrm{current}}|$, and this point moves closer to or farther from the edge with gap and Fermi velocity. The paper argues this spatial signature is measurable by NV centers in sub-10 nm nanodiamonds, whose sensitivity and coherence allow the field profile to be mapped without disturbing the helical edge states.

Load-bearing premise

The calculation assumes the sample is a clean quantum spin Hall conductor at low temperature, with all current carried by edge states and no extra spin-orbit coupling tilting the spins; any bulk leakage or spin tilt would shift or erase the cancellation point that the detection relies on.

Editorial extensions

If this is right

A single scan perpendicular to the edge can locate the exact distance where $|B_{\mathrm{spin}}|=|B_{\mathrm{current}}|$; finding that null identifies the spin-accumulation contribution and distinguishes it from a purely orbital current field.
Materials with larger gap and lower Fermi velocity are the best targets for near-edge detection, since their spin-accumulation field is stronger and the cancellation point sits closer to the boundary.
For small-gap or low-velocity materials, the spin-accumulation signature survives to distances beyond 20 nm, which is within reach of nanodiamond NV magnetometry and relaxes the requirement on probe-sample separation.
Because the sample as a whole has zero net magnetization while each edge carries opposite spin accumulation, the local field measurement can certify the QSH phase without contacting the material.
Under the inequality $\hbar v_F \pi/a \gg eV \gg 2\mu_B|B|$, the auxiliary magnetic field used by the NV measurement leaves the helical edge spectrum essentially unchanged, so the probe is non-invasive.

Reading between the lines

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

Beyond the paper, the same $1/d^2$ versus $1/d$ decomposition suggests that scanning at multiple heights and subtracting scaled profiles could separate the two contributions even when no exact cancellation point occurs, which would extend the method to devices with disorder or finite temperature.
Beyond the paper, materials with Rashba spin-orbit coupling would rotate the spin-accumulation direction out of the assumed plane, so the cancellation point would shift or split; a null-field search could then double as a probe of spin-orbit strength.
Beyond the paper, the field-angle peak just before entering the material indicates that angular-resolved vector magnetometry, not just field magnitude, is the discriminating observable; future NV setups that reconstruct all three field components would make the signature easier to identify.

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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

You should know two things about this manuscript. It gives the cleanest statement I've seen of the idea that the spin-accumulation field of a QSH edge decays as 1/d^2 while the current field decays as 1/d, creating a distance where they cancel, and it maps that distance against NV-center sensitivity. That is genuinely useful for experimental planning. But the quantitative engine of the paper, Eq. (5), is not right as written, and the cancellation distances in Figs. 5-7 are not supported by the printed equations.

The good parts first. The BHZ model, the linear-response magnetization in Eq. (3), and the NV sensitivity estimates are conventional, and the authors say clearly where they are approximating. The 1/d versus 1/d^2 scaling argument is plausible, and if the prefactors are corrected the qualitative conclusion -- that there is a distance range where spin accumulation should be separable -- may well survive. The parameter maps of the cancellation distance (Figs. 5-6) are a new and sensible way to organize the discussion.

The soft spot is large. Eq. (5) writes the current field as an integral over the full Brillouin zone of the edge-state probability |u_q(y)|^2 multiplied by the total current I. But for a single ballistic channel, the non-equilibrium current density at low temperature is (e/h)V |u_{E_F}(y)|^2; the occupied window collapses to the Fermi point, not to the whole BZ. As printed, the integral gives a B_current that scales as 1/a and diverges as a->0. That is unphysical, and it means the relative magnitude of B_current and B_spin -- which sets every cancellation distance in the paper -- is overcounted by roughly 2*pi/a. Without knowing the code, I cannot say what was actually implemented, but the manuscript as written does not reproduce the figures. That is a load-bearing flaw, not a cosmetic one.

Other issues are smaller: the abstract says a larger gap gives a stronger magnetic field, while Sec. 3 and Fig. 5b say the magnitude is largely unaffected; Eq. (4)'s tensor is not derived; no code or data are shipped.

The paper is competently written and the concept is worth keeping. A serious referee should see it, but only with the request that Eq. (5) be re-derived and the numerics recomputed, or that the wavevector window be justified explicitly. I would not cite it in its current form.

Referee Report

2 major / 4 minor

Summary. The paper studies the magnetic field generated near the edge of a two-dimensional quantum spin Hall insulator when a bias voltage drives current through helical edge states. Using the Bernevig-Hughes-Zhang Hamiltonian, the authors derive an edge-state wavefunction, compute a linear-response magnetization due to spin accumulation, and combine it with the Biot-Savart field of the edge current. They find that the spin-accumulation field decays as 1/d^2 and the current field as 1/d, producing a distance-dependent cancellation point. They map this cancellation point as a function of band gap, Fermi velocity, and band curvature, and argue that NV centers in nanodiamonds can detect the spatially varying field, especially for small-gap or low-Fermi-velocity materials where the spin-accumulation signal is claimed to persist beyond 20 nm from the edge.

Significance. If the central expressions are correct, the paper offers a concrete, falsifiable prediction: a measurable spatial signature of helical edge spin accumulation, with the cancellation distance set by the QSH material parameters. The study is not circular: it uses an explicit model, literature parameter ranges, and a standard linear-response/magnetostatic framework, and it does not fit any free parameter to the claimed detection distance. The explicit parameter maps and the comparison with NV sensitivity are useful for designing experiments. The main quantitative conclusions, however, rest on the printed expression for the current-produced magnetic field and on assumptions about an ideal clean, inversion-symmetric QSH device, both of which need careful tightening.

major comments (2)

[Section 2, Eq. (5), and Figs. 5–7] As written, Eq. (5) does not give the magnetic field of a single helical edge channel. For the two-terminal QSH device in the manuscript, the low-temperature bias current has spatial density j_x(y') = (e/h) ∫ dE [f(E) - f(E - eV)] |u_E(y')|^2, which for a narrow bias window reduces to I |u_{k_F}(y')|^2, with I = (e^2/h)V; there is no ∫_{-π/a}^{π/a} dq over all wavevectors. Because Eq. (2) demands ∫dy' |u_q(y')|^2 = 1, the printed q integral contributes a factor 2π/a to the integrated current density, making B_current depend on the lattice cutoff and diverge as a → 0. Unless the numerical code implicitly used a normalized (a/2π)∫dq or an occupied-window sum, the cancellation contours in Fig. 6 and the “distances exceeding 20 nm” statement in Section 5 are not supported by the equations as printed. Please re-derive the current density from the ballistic transport window and recompute the figures, or explicitly state and justify any normalization used in the numerics.
[Section 2 and Section 5] The paper assumes inversion symmetry is preserved and Rashba spin-orbit coupling is negligible, and it assumes all transport is through the helical edge states with conductance e^2/h. These are stated in Section 2 and after Eq. (5). The proposed detection strategy relies on a quantitative cancellation point in the magnetic field angle and magnitude. A Rashba term would rotate the spin-accumulation magnetization or introduce a spin texture, and bulk leakage or disorder would add parasitic current fields; either can shift or partially erase the predicted cancellation. The manuscript should estimate the size of these corrections for the specific material classes it cites, or explicitly restrict the feasibility conclusion to ideal inversion-symmetric clean QSH insulators.

minor comments (4)

[Abstract vs. Section 3, Fig. 5(b)] The abstract states that a larger band gap results in a stronger magnetic field, but Section 3 and Fig. 5(b) state that the magnitude remains largely unaffected while the angle changes. The Section 5 summary correctly emphasizes the angular and detectability changes; the abstract should be reworded to match the body.
[Section 2, Eq. (4)] The kernel D_{iz}(q, y - y', z) is only described verbally as a semi-Fourier transform of ∂_i ∂_j (a^2/|r - r'|). Since Eq. (4) is central to the spin-accumulation field, please write out the explicit form of D_{iz} and the Fourier transform convention, including the handling of the δ(q) factor in Eq. (3).
[Figure captions and text] The units of ℏv_F are used inconsistently: Fig. 2 gives “v_F ℏ = 0.5 nm^{-1} eV,” Fig. 7 and Fig. A1 use “ℏv_F/a = 0.3 eV” and “ℏv_F = 0.03 eV nm,” respectively. Please use a single, dimensionally consistent notation throughout.
[General presentation] There are several typographical issues: “Accomulation” in the Section 4 header, “and and” in the Fig. 5 caption, and duplicate reference [23] identical to [21]. A data availability statement for the numerical maps would also be helpful.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the magnetic-field profiles are derived from the stated BHZ model and linear response with parameters scanned from literature; no fitted input is renamed as a prediction.

full rationale

The derivation chain starts from the continuous BHZ Hamiltonian (Eq. 1), obtains edge-state wavefunctions (Eq. 2), computes the spin-accumulation magnetization via linear response (Eq. 3 and Appendix A), and then obtains magnetic fields via the dipole-integral (Eq. 4) and Biot-Savart (Eq. 5) expressions. The material parameters (M, vF, C, a) are scanned over literature ranges; no parameter is fitted to the claimed cancellation distance or field profiles. The NV sensitivity of about 0.1 microtesla per root hertz is an external benchmark that only sets detectability thresholds, not the predicted physics. The paper's self-citation [29] (Maze et al.) is used for the general capability of NV nanoscale magnetometry, not to justify the QSH model or the cancellation prediction, so it is not load-bearing. The skeptic's concern about Eq. (5) — that the printed q integral over the full Brillouin zone may overcount the single helical channel and introduce a lattice-cutoff dependence — is a correctness or regularization issue about the prefactor of B_current, not a circularity; even if the prefactor were corrected, the derivation would still be an independent calculation from stated assumptions. No step in the paper defines the predicted quantity in terms of itself, nor does it use a fitted parameter as a prediction. The central claims (1/d vs 1/d^2 scaling and parameter-dependent cancellation distances) are self-contained outputs of the model, not inputs disguised as conclusions.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central derivation rests on the BHZ model of QSH edge states, linear response to a voltage bias, and magnetostatic formulas. The model parameters M, vF, C, A, and a are chosen by hand to represent candidate materials and are scanned, so the quantitative field values and detectability distances are parameter-dependent predictions rather than material-specific ab initio results. No invented physical entities are introduced.

free parameters (5)

Band gap parameter M = M = -0.1 eV in main figures; gap 2M scanned from ~0.01 to 0.2 eV
Chosen to model candidate QSH insulators; the central claim about gap dependence is a scan over this parameter, not fixed by data.
Fermi velocity vF = hbar*vF/a from 0.1 to 0.5 eV (e.g., hbar*vF ~ 0.1 eV*nm)
Hand-picked input controlling the edge dispersion slope; key results such as field magnitude and cancellation distance depend on it.
Band curvature C = C = -0.5 eV*nm^-2
Hand-picked model parameter; affects edge-state localization and thus the spin-accumulation field.
Lattice constant a = a = 3.5 Angstrom
Chosen for numerical integrations; sets the Brillouin zone cutoff used in the Fourier transforms.
Band asymmetry A = A = 0
Set to zero in all simulations; a simplifying choice, not fitted.

assumptions (6)

domain assumption The BHZ Hamiltonian (Eq. 1) describes a 2D topological insulator with helical edge states when sign(MC) = 1.
This is the starting model for the edge states and their dispersion; if the real material deviates from this model, the calculated fields change.
domain assumption Inversion symmetry is preserved and Rashba spin-orbit coupling is neglected, so the induced magnetization lies in the plane perpendicular to the sample plane (Section 2).
This sets the orientation of B_spin; if false, the predicted field direction signature changes.
domain assumption The strip is wide enough that the two edges do not interact; only one edge is considered (Section 2).
Finite-size hybridization between Kramers partners or opposite edges would alter edge dispersion and spin accumulation.
domain assumption Linear response in the bias voltage is valid; first-order occupation shifts with chemical potentials mu_R and mu_L are sufficient (Appendix A).
Used for Eq. (3) and Eq. (A.8); higher-order or out-of-equilibrium effects are neglected.
domain assumption The edge transport is ballistic with negligible thermal excitation, so I = (e^2/h)V at low temperature (after Eq. 5).
Bulk leakage or thermal activation would add extra current fields not present in the model.
standard math The magnetic field of the local magnetization is computed via the dipolar tensor D_ij with a semi-Fourier transform over the x-direction (Eq. 4).
Assumes standard magnetostatics; the explicit form of the dimensionless tensor D_iz is not derived in the manuscript.

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Pith. "Pith review of Magnetic field Characterization of edge currents in quantum spin Hall insulators." pith.science (2026). https://pith.science/paper/CMPPJHWR

@misc{pith2026241111701,
  author       = {Pith},
  title        = {Pith review of: Magnetic field Characterization of edge currents in quantum spin Hall insulators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CMPPJHWR}},
  note         = {Machine review of arXiv:2411.11701}
}

read the original abstract

Quantum spin Hall (QSH) insulators are materials with nontrivial topological properties, characterized by helical edge currents. In 2D strips, the application of a bias voltage along the edge generates a magnetization that can be measured using quantum sensors and magnetometry techniques. In this work, we calculate the magnetic field in the vicinity of the edge and explore the potential role of nitrogen-vacancy (NV) centers in diamond as local probes for the characterization of QSH edge states in topological insulators. We characterize the magnetic field near the edges produced by both electron currents and spin accumulation at the edge. We focus on identifying the position from the edge at which the effects of spin accumulation become detectable. We observe that a larger gap between the conduction and valence bands, along with a lower Fermi velocity, results in a stronger magnetic field, with the detectable spin accumulation being more concentrated near the edge. Conversely, a smaller gap results in a slight reduction in the magnetic field magnitude, but the field associated with spin accumulation becomes detectable further from the edge. This work provides insights that could be useful for the characterization of topological materials and the development of novel electro-optical devices.

Figures

Figures reproduced from arXiv: 2411.11701 by the authors.

**Figure 1.** Schematic representation of a topological insulator and an NV center based sensor on a scanning AFM probe. Conductive edge states are depicted in colour along the top and bottom sides of the TI, along with their probability density across the yaxis, highlighting that for a given spin, one edge state becomes more populated when an electric potential is applied. The magnetic field near the edge is illustrated by fiel… view at source ↗

**Figure 2.** (a) Dispersion relation for A = 0, M = −0.1 eV, vF ℏ = 0.5 nm−1 eV, and C = −0.5 nm−2 eV, highlighting the band gap width as 2M and the Fermi velocity as the slope of the dispersion curves. (b) Cross-sectional profiles of the edge states penetrating into the bulk for three values of k within the range − q M C < k < q M C , showing the spatial region where the probability density of the edge state is non-zero in this… view at source ↗

**Figure 3.** At low temperatures, a pair of helical edge states with unequal populations is observed when connected to leads with chemical potentials µL and µR. In this regime, the ballistic transport through the edge states is described by the LandauerB¨uttiker framework interactions . In this work, we assume that inversion symmetry is preserved in the material, neglecting the Rashba spin-orbit interaction . Consequently, the … view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Cross-sectional magnetic field lines generated in a QSH insulator strip, with different colors indicating the field intensity. Panel (a) depicts the magnetic field lines arising from spin accumulation at the edge, panel (b) shows the magnetic field lines generated by t…

**Figure 5.** Figure 5: Magnitude and and angle with respect to the y axis of the total magnetic field as a function of the topological insulator parameters for three spatial points shown in the inset. Calculations were performed at a voltage V = 1 mV and temperature T = 50 K. Panel (a) displ…

**Figure 6.** Figure 6: Ratio between the magnetic field magnitudes due to spin accumulation at the edge and electron current as a function of the topological insulator gap and distance along the y-axis for z = 0 for three different Fermi’s velocities. The dashed lines represent points at whi…

**Figure 7.** Figure 7: Magnetic field magnitude (depicted in orange) and angle with respect to the y axis (shown in cyan) as it traverses perpendicularly through the edge along the z-axis (a) and y-axis (b). The zero mark denotes the position of the edge. In both panels, an inset illustrates…

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Works this paper leans on

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Introduction Topological insulators have gained great relevance in the last decades due to their applications in advanced nano-electronic and spintronic devices . This new class of solid-state materials exhibits a bulk band gap like an ordinary insulator but protected conducting states on their edge or surface, arising from the topology of the material. a...

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Edge Magnetic Field from Spin Accumulation and Electron Current The simplest Hamiltonian representing a QSH insulator can be written in continuous form as H(kx, ky) = A(k)I2 + d(k) · σ 0 0 A(k)I2 + (d(−k) · σ)† , (1) where I2 is a 2 × 2 unit matrix, σ are the Pauli matrices, A(k) = −A(k2 x + k2 y), with A representing the asymmetry between the valence and...

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Summary and Conclusions In this work, we explore the magnetic field generated at the edge of a topological insulator in the QSH phase. Specifically, we examined a two-terminal device with two well-separated boundaries, where opposing magnetizations are induced at each edge by an electric current driven by a potential difference V that is two orders of mag...

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Acknowledgments The authors acknowledge the support from the Asian Office of Aerospace Research and Development (AOARD) F A2386-21-1-4125, Fondecyt Regular No 1221512, and ANID BECAS/DOCTORADO NACIONAL 21201120. Appendix A. Magnetization calculation Consider the magnetization generated at the edge as described by the equation: m(r) = µB ⟨n↑(r)⟩ − ⟨n↓(r)⟩ ...

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Introduction Topological insulators have gained great relevance in the last decades due to their applications in advanced nano-electronic and spintronic devices . This new class of solid-state materials exhibits a bulk band gap like an ordinary insulator but protected conducting states on their edge or surface, arising from the topology of the material. a...

work page Pith review arXiv 2024

[2] [2]

This model is highly versatile and sufficient for our purposes, as it allows us to modify both the band gap and the Fermi velocity

Edge Magnetic Field from Spin Accumulation and Electron Current The simplest Hamiltonian representing a QSH insulator can be written in continuous form as H(kx, ky) = A(k)I2 + d(k) · σ 0 0 A(k)I2 + (d(−k) · σ)† , (1) where I2 is a 2 × 2 unit matrix, σ are the Pauli matrices, A(k) = −A(k2 x + k2 y), with A representing the asymmetry between the valence and...

work page

[3] [3]

The latter being a result of spin-locking of the conducting states

Magnetic Field Dependence on material properties In the quantum spin Hall (QSH) phase, contributions to the magnetic field arise from both the electron current and spin accumulation at the edge. The latter being a result of spin-locking of the conducting states. This behavior is sensitive to the topological insulator’s parameters, temperature, and voltage...

work page

[4] [4]

These sensors must be positioned close enough to the edge to differentiate the effects of spin accumulation

Magnetometry of Edge Spin Accomulation To characterize the magnetic fields generated by the edge state currents, NV centers in nanodiamonds or nanopillars can be employed. These sensors must be positioned close enough to the edge to differentiate the effects of spin accumulation. NV nanodiamonds with diameters smaller than 10 nm and stable photoluminescen...

work page

[5] [5]

Summary and Conclusions In this work, we explore the magnetic field generated at the edge of a topological insulator in the QSH phase. Specifically, we examined a two-terminal device with two well-separated boundaries, where opposing magnetizations are induced at each edge by an electric current driven by a potential difference V that is two orders of mag...

work page

[6] [6]

Appendix A

Acknowledgments The authors acknowledge the support from the Asian Office of Aerospace Research and Development (AOARD) F A2386-21-1-4125, Fondecyt Regular No 1221512, and ANID BECAS/DOCTORADO NACIONAL 21201120. Appendix A. Magnetization calculation Consider the magnetization generated at the edge as described by the equation: m(r) = µB ⟨n↑(r)⟩ − ⟨n↓(r)⟩ ...

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C. Yue, S. Jiang, H. Zhu, L. Chen, Q. Sun, and D. Zhang, ”Device Applications of Synthetic Topological Insulator Nanostructures,” Electronics, vol. 7, p. 225, Oct. 2018

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Kong and Y

D. Kong and Y. Cui, ”Opportunities in chemistry and materials science for topological insulators and their nanostructures,” Nature Chemistry, vol. 3, pp. 845-849, Nov. 2011

work page 2011