Analytical models of approximations for wave functions and energy dispersion in zigzag graphene nanoribbons

Moradinasab, Mahdi; Nematian, H.; Pourfath, Mahdi; Fathipour, Morteza; Kosina, H.

doi:10.1063/1.3702429

Cited by 10 publications

(9 citation statements)

References 19 publications

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“…A system consists of N x N y sites, so the eigenvalue problem (4) should have N x N y different eigenvalues and eigenvectors. We have N y values of k y (10), N x /2 values of k x (11) and two different values of the parameter s = ±1. The coefficient c 1 can be found from the following normalization conditions:…”

Section: A Wave Functions For Armchair Nanoribbons and Nanotubesmentioning

confidence: 99%

“…where we have introduced the parameter ϕ ∈ [0; 2π) which depends on the ratio k x /(k y − 2π/3). The possible values of wavenumber k y (10) are obtained from the boundary conditions. The wavenumber k x is either a continuous variable for an infinite nanoribbon or is quantized (11) due to periodic boundary conditions for armchair nanotubes.…”

Section: A Wave Functions For Armchair Nanoribbons and Nanotubesmentioning

confidence: 99%

“…Insights into the electronic properties of graphene and its derivatives can be obtained from exact analytical approaches. There are two basic approaches for describing rectangular structures: based on the division of graphene into two sublattices [6][7][8][9][10][11][12] (equivalently, choosing two atom unit cell) or choosing a unit cell consisting of four atoms [13][14][15][16][17]. The first one is usually applied when describing infinite systems and nanoribbons, the second one is used mostly for finite systems.…”

Section: Introductionmentioning

confidence: 99%

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Wave functions and edge states in rectangular honeycomb lattices revisited: nanoflakes, armchair and zigzag nanoribbons and nanotubes

Talkachov¹,

Babaev²

2022

Preprint

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Properties of bulk and boundaries of materials can, in general, be quite different, both for topological and non-topological reasons. One of the simplest boundary problems to pose is the tight-binding problem of noninteracting electrons on a finite honeycomb lattice. Despite its simplicity, the problem is quite rich and directly related to the physics of graphene. We revisit this long-studied problem and present an analytical derivation of the electron spectrum and wave functions for graphene rectangular derivatives. We provide an exact analytical description of extended and localized states, the transition between them, and a special case of a localized state when the wave function is nonzero only at the edge sites. The later state has zero energy, we discuss its existence in zigzag nanoribbons, zigzag nanotubes with number of sites along a zigzag edge divisible by 4, and rectangular graphene nanoflakes with an odd number of sites along both zigzag and armchair edges.

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Section: A Wave Functions For Armchair Nanoribbons and Nanotubesmentioning

confidence: 99%

Section: A Wave Functions For Armchair Nanoribbons and Nanotubesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Wave functions and edge states in rectangular honeycomb lattices revisited: nanoflakes, armchair and zigzag nanoribbons and nanotubes

Talkachov¹,

Babaev²

2022

Preprint

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“…( 7) from the parity of a ZPNR, namely φ A (N + 1 − m)=±φ B (m) [40]. The reason is that the wavefunction of even-N ZPNRs is either symmetric or antisymmetric which is similar to that in ZGNRs [43]. Specifically, combined with the translational invariance along the x-direction, the wavefuntion of even-N ZPNRs is specified as…”

Section: B Analytical Calculation On Electronic Structurementioning

confidence: 99%

Width Dependent Two-Photon Absorption in Monolayer Black Phosphorus Nanoribbons

et al. 2019

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We analytically study the electronic structures and optical properties of zigzag-edged black phosphorene nanoribbons (ZPNRs) utilizing the tight-binding (TB) Hamiltonian and Kubo formula. By solving the discrete Schordinger equation directly, we obtain the energy spectra and wavefunctions for a N-ZPNR with N number of transverse zigzag atomic chains, and classify the eigenstates according to the lattice symmetry. We then obtain the optical transition selection rule of ZPNRs based on the symmetry analysis and the analytical expressions of the optical transition matrix elements. Under an incident light linearly-polarized along the ribbon, importantly, we find that the optical transition selection rule for the N-ZPNR with even-or odd-N is qualitatively different. In specification, for even-N ZPNRs the inter-(intra-) band selection rule is ∆n =odd (even), since the parity of the wavefunction corresponding to the nth subband in the conduction (valence) band is (−1) n [(−1) (n+1) ] due to the presence of the C 2x symmetry. In contrast, all optical transitions are possible among all subbands due to the absence of the C 2x symmetry. Our findings provide a further understanding on the electronic states and optical properties of ZPNRs, which are useful in the explanation of the optical experiment data on ZPNR samples. *

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“…Edge states were usually studied in 2D lattice electron systems and within the framework of tight-binding models [8,[23][24][25], though the Dirac equation approaches have been also carried out [26,27] (see Ref. [28][29][30] for more mathematical treatment).…”

Section: Introductionmentioning

confidence: 99%

Edge States in 2D Lattices with Hopping Anisotropy and Chebyshev Polynomials

et al. 2014

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Analytic technique based on Chebyshev polynomials is developed for studying two-dimensional lattice ribbons with hopping anisotropy. In particular, the tight-binding models on square and triangle lattice ribbons are investigated with anisotropic nearest neighbouring hoppings. For special values of hopping parameters the square lattice becomes topologically equivalent to a honeycomb one either with zigzag or armchair edges. In those cases as well as for triangle lattices we perform the exact analytic diagonalization of tight-binding Hamiltonians in terms of Chebyshev polynomials. Deep inside the edge state subband the wave functions exhibit exponential spatial damping which turns into power-law damping at edge-bulk transition point. It is shown that strong hopping anisotropy crashes down edge states, and the corresponding critical conditions are found.

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Analytical models of approximations for wave functions and energy dispersion in zigzag graphene nanoribbons

Cited by 10 publications

References 19 publications

Wave functions and edge states in rectangular honeycomb lattices revisited: nanoflakes, armchair and zigzag nanoribbons and nanotubes

Wave functions and edge states in rectangular honeycomb lattices revisited: nanoflakes, armchair and zigzag nanoribbons and nanotubes

Width Dependent Two-Photon Absorption in Monolayer Black Phosphorus Nanoribbons

Edge States in 2D Lattices with Hopping Anisotropy and Chebyshev Polynomials

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