Band inversion at critical magnetic fields in a silicene quantum dot

Romera, E.; Calixto, M.

doi:10.1209/0295-5075/111/37006

Cited by 17 publications

(16 citation statements)

References 21 publications

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“…The eigenvalue problem can be easily solved. Using the Landau gauge,

\vec{A} = (0, B x, 0)

, the corresponding eigenvalues and eigenvectors for the K and

K'

points are given by

E_{n, Δ_{z}}^{s, ξ} = {\begin{array}{l} left & sgn (n) \sqrt{| n | ℏ^{2} ω^{2} + Δ_{s, ξ}^{2}}, n \neq 0, \\ left & - ξ Δ_{s ξ}, n = 0, \end{array}

and

{| n 〉}_{s, ξ} = (\begin{matrix} \begin{matrix} - i A_{n, Δ_{z}}^{s, ξ} | | n | - ξ_{+} 〉 \\ B_{n, Δ_{z}}^{s, ξ} | | n | - ξ_{-} 〉 \end{matrix} \end{matrix})

where we denote by

ξ_{\pm} = (1 \pm ξ) / 2

, the Landau level index

n = 0, \pm 1, \pm 2, \dots

, the cyclotron frequency

ω = v_{normalF} \sqrt{2 e B / ℏ}

, the lowest band gap

Δ_{s, ξ} \equiv (Δ_{z} - s ξ Δ_{so}) / 2

and the constants

A_{n, Δ_{z}}^{s, ξ} ...

…”

Section: Low Energy Hamiltonianmentioning

confidence: 99%

“…Topological phases are characterized by topological charges like Chern numbers. Theoretical information measures and uncertainty relations have been used to characterize TPTs . Recently, we have studied the time evolution of electron wave packets in silicene under perpendicular magnetic and electric fields to characterize topological‐band insulator transitions …”

Section: Introductionmentioning

confidence: 99%

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Fidelity as a marker of topological phase transitions in 2D Dirac materials

Bolívar

Cordero

Nagy

et al. 2018

Int J of Quantum Chemistry

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Silicene, as other 2D buckled structures, is a gapped Dirac material with intrinsic spin-orbit coupling whose band structure can be controlled by applying a perpendicular electric field. It presents a topological phase transition from a topological insulator to a band insulator at the charge neutrality point. We present in this article a characterization of this phase transition by using fidelity of Loschmidt echoes when a magnetic and two slightly different electric fields are applied by considering the time evolution of two kinds of wave packets, one with a single Gaussian profile and the other with a double Gaussian profile creating a cat state. We also show that Zitterbewegung, classical, and revival Loschmidt periods diverge close to but not exactly at the charge neutrality point and explain this behavior.Silicene is a two dimensional crystal of silicon, belonging to a group of 2D gapped Dirac materials analogous to graphene but with a relevant intrinsic spin-orbit coupling (as compared to graphene) and a buckled structure. It has been studied theoretically [1,2] and experimentally. [3][4][5][6][7] Other gapped Dirac materials are germanene, stannene, and Pb. [8] In the case of silicene, the low energy electronic properties can be described by a Dirac Hamiltonian with a Fermi velocity v F = 4:2 × 10 5 ms − 1 and a value of the spin-orbit coupling Δ so = 4:2 meV. It has a buckling length l = 0:22Å which allows to control the band structure by applying an electric field, ℰ z = Δ z =l, perpendicular to the silicene monolayer generating a tunable band gap jΔ s, ξ j = jðΔ z − sξΔ so Þ=2j (s and ξ denote spin and valley, respectively). Silicene has a topological phase transition (TPT) [9] from a topological insulator (TI, jΔ z j < Δ so ) to a band insulator (BI, jΔ z j > Δ so ), at a charge neutrality point (CNP) Δ ð0Þ z = sξΔ so , with a gap cancellation between the perpendicular electric field and the spin-orbit coupling, thus exhibiting a semimetal behavior.Topological phases are characterized by topological charges like Chern numbers. Theoretical information measures and uncertainty relations have been used to characterize TPTs. [10][11][12][13] Recently, we have studied the time evolution of electron wave packets in silicene under perpendicular magnetic and electric fields to characterize topological-band insulator transitions. [14]

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“…The eigenvalue problem can be easily solved. Using the Landau gauge,

\vec{A} = (0, B x, 0)

, the corresponding eigenvalues and eigenvectors for the K and

K'

points are given by

E_{n, Δ_{z}}^{s, ξ} = {\begin{array}{l} left & sgn (n) \sqrt{| n | ℏ^{2} ω^{2} + Δ_{s, ξ}^{2}}, n \neq 0, \\ left & - ξ Δ_{s ξ}, n = 0, \end{array}

and

{| n 〉}_{s, ξ} = (\begin{matrix} \begin{matrix} - i A_{n, Δ_{z}}^{s, ξ} | | n | - ξ_{+} 〉 \\ B_{n, Δ_{z}}^{s, ξ} | | n | - ξ_{-} 〉 \end{matrix} \end{matrix})

where we denote by

ξ_{\pm} = (1 \pm ξ) / 2

, the Landau level index

n = 0, \pm 1, \pm 2, \dots

, the cyclotron frequency

ω = v_{normalF} \sqrt{2 e B / ℏ}

, the lowest band gap

Δ_{s, ξ} \equiv (Δ_{z} - s ξ Δ_{so}) / 2

and the constants

A_{n, Δ_{z}}^{s, ξ} ...

…”

Section: Low Energy Hamiltonianmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Fidelity as a marker of topological phase transitions in 2D Dirac materials

Bolívar

Cordero

Nagy

et al. 2018

Int J of Quantum Chemistry

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show abstract

“…There is a TPT: we have the phase TI if | z | < so and the phase BI when | z | > so . The effective gap | sξ | disappears at CNP [18,[24][25][26][35][36][37][38]. The eigenvalue equation of the model is [10,[19][20][21]…”

Section: Low Energy Eigenvalue Problem Of 2d Crystalsmentioning

confidence: 99%

“…in position space and I γ sξ n ≤ 4σ ρ sξ n (36) in momentum space. Combining inequalities (31), (32), (35) and (36) we obtain upper bounds to the FI:…”

Section: Variance and Bounds For The Phase-space Fisher Informationmentioning

confidence: 99%

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Phase-space Fisher information of 2D gapped Dirac materials

2019

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It is shown that phase-space Fisher information marks topological phase transitions in 2D Dirac gapped materials. The phase-space Fisher information determined for electrons and holes exhibit minimum at the charge neutrality points. Topological quantum numbers are defined to identify different phases utilizing combined information measures. Upper and lower bounds are derived for the phase-space Fisher information.

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Rényi–Fisher entropy product as a marker of topological phase transitions

Bolívar

Nagy

Romera

2018

Physica A: Statistical Mechanics and its Applications

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The combined Rényi-Fisher entropy product of electrons plus holes displays a minimum at the charge neutrality points. The Stam-Rényi difference and the Stam-Rényi uncertainty product of the electrons plus holes, show maxima at the charge neutrality points. Topological quantum numbers capable of detecting the topological insulator and the band insulator phases, are defined. Upper and lower bounds for the position and momentum space Rényi-Fisher entropy products are derived.

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Band inversion at critical magnetic fields in a silicene quantum dot

Cited by 17 publications

References 21 publications

Fidelity as a marker of topological phase transitions in 2D Dirac materials

Fidelity as a marker of topological phase transitions in 2D Dirac materials

Phase-space Fisher information of 2D gapped Dirac materials

Rényi–Fisher entropy product as a marker of topological phase transitions

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