Effective Hamiltonian of strained graphene

Linnik, T. L.

doi:10.1088/0953-8984/24/20/205302

Cited by 31 publications

(38 citation statements)

References 48 publications

(135 reference statements)

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“…s ¼ AE would label valleys K and K 0 in this case. The anisotropic velocity term parametrized by α would be proportional to the strain, much in the same way as in strained graphene [23]. For TMDCs one obtains a scale ℏα=β ∼ 0.2 Å, using a Fermi velocity of v ≈ 4.5 × 10 5 m=s, a gap β ≈ 1.5 eV, and α ¼ 0.1v.…”

Section: H Y S I C a L R E V I E W L E T T E R Smentioning

confidence: 98%

Quantum Nonlinear Hall Effect Induced by Berry Curvature Dipole in Time-Reversal Invariant Materials

2015

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It is well known that a nonvanishing Hall conductivity requires broken time-reversal symmetry. However, in this work, we demonstrate that Hall-like currents can occur in second-order response to external electric fields in a wide class of time-reversal invariant and inversion breaking materials, at both zero and twice the driving frequency. This nonlinear Hall effect has a quantum origin arising from the dipole moment of the Berry curvature in momentum space, which generates a net anomalous velocity when the system is in a current-carrying state. The nonlinear Hall coefficient is a rank-two pseudotensor, whose form is determined by point group symmetry. We discus optimal conditions to observe this effect and propose candidate two-and three-dimensional materials, including topological crystalline insulators, transition metal dichalcogenides, and Weyl semimetals. DOI: 10.1103/PhysRevLett.115.216806 PACS numbers: 73.43.-f, 03.65.Vf, 72.15.-v, 72.20.My Introduction.-The Hall conductivity of an electron system whose Hamiltonian is invariant under time-reversal symmetry is forced to vanish. Crystals with sufficiently low symmetry can have resistivity tensors which are anisotropic, but Onsager's reciprocity relations [1] force the conductivity to be a symmetric tensor in the presence of time-reversal symmetry. Hence, when the electric field is along its principal axes the current and the electric field are collinear, at least to the first order in electric fields. However, this constraint is only about the linear response and does not necessarily enforce the full current to flow collinearly with the local electric field.In this Letter we study a special type of such nonlinear Hall-like currents. We will demonstrate that metals without inversion symmetry can have a nonlinear Hall-like current arising from the Berry curvature in momentum space. The conventional Hall conductivity can be viewed as the zeroorder moment of the Berry curvature over occupied states, namely, as an integral of the Berry curvature within the metal's Fermi surface. The effect we discuss here is determined by a pseudotensorial quantity that measures a first-order moment of the Berry curvature over the occupied states, and hence we call it the Berry curvature dipole. This nonlinear Hall effect has a quantum origin arising from the anomalous velocity of Bloch electrons generated by the Berry curvature [2], but it is not expected to be quantized.In a time-reversal invariant system, the Berry curvature is odd in momentum space, Ω a ðkÞ ¼ −Ω a ð−kÞ, and hence its integral weighed by the equilibrium Fermi distribution is forced to vanish, because Kramers pair states at k and −k are equally occupied. However, the second-order response is determined by the integral of the Berry curvature evaluated in the nonequilibrium distribution of electrons computed to first order in the electric field. Since the

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Section: H Y S I C a L R E V I E W L E T T E R Smentioning

confidence: 98%

Quantum Nonlinear Hall Effect Induced by Berry Curvature Dipole in Time-Reversal Invariant Materials

2015

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“…δH 21 and δH 22 are obtained from δH 12 and δH 11 respectively by making the replacement v n → − v n . Note the symmetric split of the phonon momentum q among the incoming and outgoing electrons in (12), which in position space implies the symmetric derivative convention used in the last section.…”

Section: Generalized Tight-binding Hamiltonianmentioning

confidence: 99%

“…17 The symmetry approach has been applied to the particular problem of strained graphene, for example, in Refs. [18][19][20][21][22]. A highly detailed symmetry construction has been used in Ref.…”

Section: Introductionmentioning

confidence: 99%

Generalized effective Hamiltonian for graphene under nonuniform strain

et al. 2013

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We use a symmetry approach to construct a systematic derivative expansion of the low-energy effective Hamiltonian modifying the continuum Dirac description of graphene in the presence of nonuniform elastic deformations. We extract all experimentally relevant terms and describe their physical significance. Among them there is a new gap-opening term that describes the Zeeman coupling of the elastic pseudomagnetic field and the pseudospin. We determine the value of the couplings using a generalized tight-binding model.

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“…The resulting term is proportional to the torsion tensor components unlike the final expression in their result, which vanishes for uniform plastic strain. Secondly, electron scattering in graphene sheets by strains [17] and impurities and corrugations [44] has been studied extensively recently. The specific problem of dislocations in graphene was studied using a covariant formalism by de Juan et al [45].…”

Section: Numerical Estimates and Discussionmentioning

confidence: 99%

“…This is the most commonly used technique to estimate the shift in band degeneracies [17]. This method has also been applied to elastic media containing dislocations without a microscopic model [18,19].…”

Section: Introductionmentioning

confidence: 99%

Geometric treatment of conduction electron scattering by crystal lattice strains and dislocations

Viswanathan

Chandrasekar

2014

Journal of Applied Physics

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A theory for conduction electron scattering by inhomogeneous crystal lattice strains is developed, based on the differential geometric treatment of deformations in solids. The resulting fully covariant Schrödinger equation shows that the electrons can be described as moving in a non-Euclidean background space in the continuum limit of the deformed lattice. Unlike previous work, the formalism is applicable to cases involving purely elastic strains as well as discrete and continuous distributions of dislocations -in the latter two cases it clearly demarcates the effects of the dislocation strain field and core and differentiates between elastic and plastic strain contributions respectively. The electrical resistivity due to the strain field of edge dislocations is then evaluated using perturbation theory and the Boltzmann transport equation. The resulting numerical estimate for Cu shows good agreement with experimental values, indicating that the electrical resistivity of edge dislocations is not entirely due to the core, contrary to current models. Possible application to the study of strain effects in constrained quantum systems is also discussed.

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Effective Hamiltonian of strained graphene

Cited by 31 publications

References 48 publications

Quantum Nonlinear Hall Effect Induced by Berry Curvature Dipole in Time-Reversal Invariant Materials

Quantum Nonlinear Hall Effect Induced by Berry Curvature Dipole in Time-Reversal Invariant Materials

Generalized effective Hamiltonian for graphene under nonuniform strain

Geometric treatment of conduction electron scattering by crystal lattice strains and dislocations

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