Zero-field and time-reserval-symmetry-broken topological phase transitions in graphene

Guassi, Marcos Rafael; Diniz, G. S.; Sandler, Nancy; Qu, Fanyao

doi:10.1103/physrevb.92.075426

Cited by 17 publications

(34 citation statements)

References 49 publications

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“…In order to get the Chern number, let us first calculate the Berry curvature

Ω_{x y}^{n} (K_{x}, K_{y})

of the n ‐th bands using the Kubo formula,

Ω_{x y}^{n} (K_{x}, K_{y}) = - \sum_{n^{'} \neq n} \frac{2 Im 〈 Ψ_{n k} | v_{x} | Ψ_{n^{'} k} 〉 〈 Ψ_{n^{'} k} | v_{y} | Ψ_{n k} 〉}{{(ω_{n^{'}} - ω_{n})}^{2}},

where

ω_{n} = E_{n} / ℏ

with

E_{n}

the energy eigenvalue of the n ‐th band and

v_{x (y)} = ℏ^{- 1} \partial H / \partial K_{x (y)}

is the Fermi velocity operator. With the Berry curvature at hand, then the

scriptC

can be calculated by

C = \frac{1}{2 π} \sum_{n} \int_{BZ} d^{2} k Ω_{x y}^{n} (K_{x}, K_{y}),

where the summation...…”

Section: Theoretical Modelmentioning

confidence: 79%

Topological nature of in‐gap bound states in disordered large‐gap monolayer transition metal dichalcogenides

Villegas‐Lelovsky

Diniz

2016

Physica Rapid Research Ltrs

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We propose a physical model based on disordered (a hole punched inside a material) monolayer transition metal dichalcogenides (TMDs) to demonstrate a large‐gap quantum valley Hall insulator. We find an emergence of bound states lying inside the bulk gap of the TMDs. They are strongly affected by spin–valley coupling, rest‐ and kinetic‐mass terms and the hole size. In addition, in the whole range of the hole size, at least two in‐gap bound states with opposite angular momentum, circulating around the edge of the hole, exist.Their topological insulator (TI) feature is analyzed by the Chern number, characterized by spacial distribution of their probabilities and confirmed by energy dispersion curves (energy vs. angular momentum). It not only sheds light on overcoming low‐temperature operating limitation of existing narrow‐gap TIs, but also opens an opportunity to realize valley‐ and spin‐qubits. (© 2016 WILEY‐VCH Verlag GmbH &Co. KGaA, Weinheim)

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“…In order to get the Chern number, let us first calculate the Berry curvature

Ω_{x y}^{n} (K_{x}, K_{y})

of the n ‐th bands using the Kubo formula,

Ω_{x y}^{n} (K_{x}, K_{y}) = - \sum_{n^{'} \neq n} \frac{2 Im 〈 Ψ_{n k} | v_{x} | Ψ_{n^{'} k} 〉 〈 Ψ_{n^{'} k} | v_{y} | Ψ_{n k} 〉}{{(ω_{n^{'}} - ω_{n})}^{2}},

where

ω_{n} = E_{n} / ℏ

with

E_{n}

the energy eigenvalue of the n ‐th band and

v_{x (y)} = ℏ^{- 1} \partial H / \partial K_{x (y)}

is the Fermi velocity operator. With the Berry curvature at hand, then the

scriptC

can be calculated by

C = \frac{1}{2 π} \sum_{n} \int_{BZ} d^{2} k Ω_{x y}^{n} (K_{x}, K_{y}),

where the summation...…”

Section: Theoretical Modelmentioning

confidence: 79%

Topological nature of in‐gap bound states in disordered large‐gap monolayer transition metal dichalcogenides

Villegas‐Lelovsky

Diniz

2016

Physica Rapid Research Ltrs

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“…Besides, Rashba SOC pushes the valence band up and the conduction band down, reducing the bulk gap. Following Reference [45], we present our results for the ZGNR in Figure 17, which shows the effects of intrinsicand Rashba-SOCs and EX upon the energy band of the ZGNR [49]. Notice in Figure 17(a) that the interplay between intrinsic-and Rashba-SOCs partially lifts the degeneracies of both bulkand edge-state, breaks particle-hole symmetry and pushes the valence band up.…”

Section: Quantum Phase Transitions In Strained Graphene Nanoribbonmentioning

confidence: 89%

“…But it has been found a similar state, called pseudo-Hall quantum spin state or quantum spin Hall state with broken time-reversal symmetry [45] in which was possible to observe the spin polarized current on the edges of nanoribbon. In addition, the exchange field is critical to control the transition between electronic states of quantum anomalous Hall to quantum spin Hall [49].…”

Section: Hamiltonian Gives Energiesmentioning

confidence: 99%

“…Quantum spin Hall and quantum anomalous Hall (QAH) states have topologically protected edge states, where the electron back scattering is forbidden, making these systems good candidates for electronic devices with dissipationless electronic transport [33,35,38,41]. The potential possibility to explore the different Quantum Hall phases in strained graphene has motivated us to study the strain-related physics at zero magnetic field in graphene nanoribbons [49]. If the mirror symmetry about the graphene plane is preserved, then the intrinsic SOC which opens gaps around Dirac points is the only allowed spin-dependent term in the Hamiltonian.…”

Section: Quantum Phase Transitions In Strained Graphene Nanoribbonmentioning

confidence: 99%

“…Thus, strain, EX and SOC can be used as a versatile tool for control of topological phase transitions [32]. These facts motivated us to propose ways in which the spin-orbit coupling, uniaxial mechanical strain and exchange (instead of an external magnetic field) to be used to carry out phase transitions in graphene nanoribbons [49].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Electronic Structure and Topological Quantum Phase Transitions in Strained Graphene Nanoribbons

Qu¹,

Diniz²,

Guassi³

2016

Recent Advances in Graphene Research

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In this chapter, we discuss the new classes of matter, such as the quantum spin Hall (QSH) and quantum anomalous Hall (QAH) states, that have been theoretically predicted and experimentally observed in graphene and beyond graphene systems. We further demonstrate how to manipulate these states using mechanical strain, internal exchange field, and spin-orbit couplings (SOC). Spin-charge transport in strained graphene nanoribbons is also discussed assuming the system in the QAH phase, exploring the prospects of topological devices with dissipationless edge currents. A remarkable zero-field topological quantum phase transition between the time-reversalsymmetry-broken QSH and quantum anomalous Hall states is predicted, which was previously thought to take place only in the presence of external magnetic field. In our proposal, we show as the intrinsic SOC is tuned, how it is possible to two different helicity edge states located in the opposite edges of the graphene nanoribbons exchange their locations. Our results indicate that the strain-induced pseudomagnetic field could be coupled to the spin degrees of freedom through the SOC responsible for the stability of a QSH state. The controllability of this zero-field phase transition with strength and direction of the strain is also explored as additional phase-tuning parameter. Our results present prospect of strain, electric and magnetic manipulation of the QSH, and QAH effect in these novel two-dimensional (2D) materials.

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Strain Engineering: Electromechanical Properties of Graphene

Zhu

2019

Handbook of Graphene

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Zero-field and time-reserval-symmetry-broken topological phase transitions in graphene

Cited by 17 publications

References 49 publications

Topological nature of in‐gap bound states in disordered large‐gap monolayer transition metal dichalcogenides

Topological nature of in‐gap bound states in disordered large‐gap monolayer transition metal dichalcogenides

Electronic Structure and Topological Quantum Phase Transitions in Strained Graphene Nanoribbons

Strain Engineering: Electromechanical Properties of Graphene

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