the low **energy** range of <0.5 eV. These massless fermions exhibit various quantum electrodynamics(QED) in the low **energy** range that can be explored. It is therefore of interest to understand and simulate the nature of the bands and understand their interactions along the edges and corners of the first brillouin zone. Various forms of monolayer graphene exist - A Graphene nano-ribbon is a sheet of graphene of narrow width and high aspect ratio existing in 2 different edge orientations. This variation in the band gap of different morphologies is known to affect the On current characteristics for corresponding GNRFETs and other applications.

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In both the Laplace and rectangular barrier potential profiles, the DOS(E) and T(E) for our TBDE **model** are in satisfactory agreement with that calculated from TB- π **model** within about 1.5 eV around the mid-gap. At higher energies, significant deviations in the DOS(E) and T(E) are consistent with the discrepancies we observed in E(k) (as shown in Figure 3b), as discussed earlier. Nonetheless, these deviations are limited to the high- **energy** range that is of little relevance to the electron transport in GNR devices. Therefore, our TBDE approach is expected to be valid and as a practical and efficient alternative to TB-π for studying carrier trans- port involving arbitrary self-consistent electrostatic potentials for device simulations [22,23].

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As for the α parameter, we used the one obtained previously for the non-magnetic atom. We plot the obtained curves in figure 4. From this plot we see that the λ dependence of the eigenvalues is very well described. In particular, notable deviations between the ab initio and **tight**-**binding** curves are only observed in some cases and for large values of λ, although they are never larger than 0.006 au. This is in contrast with the case of the magnetic atom. This indicates that the inclusion of the spin–orbit coupling as a perturbation in **tight**-**binding** models might only be a good approximation for closed-shell (i.e. non-magnetic) systems. This is probably due to the interplay between the spin–orbit coupling and the magnetic moment of the system. Furthermore, these results for the dimer validate the use of the α parameter fitted for the non-magnetic atom in the **tight**-**binding** **model** for the dimer, suggesting that it is transferable to other systems containing bismuth.

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function is obtained from Equation (12) for different val- ues of the real freque cy ω, the wave vector q and chemical potential (Fermi **energy**) μ. Moreover, we as- sume κ = 4. The chemical potential level μ is selected to be relative to Dirac points whose existence is not affected by a variation of the parameter t but are shifted by 3 t ([11]), as shown in Figure 2.

The trends shown in Fig. 2 and the values of our fitted parameters in Table I are in general agreement with previ- ously published work. 8–14 What is unique to this study, how- ever, is that we show that a single TB parameter set can be used to **model** both AGNR and ZGNR systems. Without edge-hopping perturbation, the combination of the required AGNR and ZGNR parameters leads to a minimal, general- ized TB **model** 共set D, Table I兲 that adequately reproduces the band-structure features about the Fermi **energy** and gap trends of the ab initio results—note that the gaps for both the AGNR and ZGNR have been adequately obtained **using** the t 3 term 共 see also Refs. 10 and 14 兲 and the Hubbard U, re-

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1.2. Further remarks. **Tight** **binding** **model**. **Tight** **binding** models are minimalistic quantum mechanics type molecular models used to investigate and pre- dict properties of molecules and materials in condensed phases. In terms of both accuracy and computational cost they are situated between accurate but computa- tionally expensive ab initio methods and fast but limited empirical methods. While **tight** **binding** models are interesting in their own right, they also serve as a convenient toy **model** for more accurate electronic structure models such as (Kohn–Sham) density functional **theory**.

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The honeycomb structure has significant prevalence in modern condensed matter. The first significant theoretical analysis of the carbon layers of graphite in a **tight**-**binding** setting was conducted by P.R. Wallace in 1947[30], who identified an approximately linear **energy** **dispersion** around two distinct points in the Brillouin zone. These points were analysed by D. P. DiVincenzo and E.J. Mele in 1984[31], and the hamiltonian was brought into a form identical to a two-dimensional Dirac equation, replacing spin with a sublattice index. In 2004, A. Geim and K. Novoselov isolated individual layers of hexagonal carbon, known as Graphene. Their experimental confirmation of theoretical results, along with further groundbreaking research[32], resulted in them being awarded the 2010 Nobel Prize in Physics.

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Non-exponential decay in our chain is stipulated by nonlocality of Lindblad operators in equations (12) and (13)). It is interesting and instructive to compare the dynamics of population decay in our chain with dynamics of collective spontaneous emission of dense atomic cloud with dipole–dipole interaction [7, 8]. The continuous limit of master equation correspondent to the **model** considered in [7,8] is given by equation (A.5) in the appendix A. Equation (A.5) looks similar to equation (18) and also manifests an algebraic law of population decay, 1 γt [7]. However, it holds in 3D-space, and describes quite different physical process. The reason for it is the formal equivalence of equation (A.5) to the master equation with nonlocal Lindblad operators. This nonlocality arises from non-conservation of the excitation number and from accounting for quantum states corresponding to two excited atoms and one virtual photon with ‘negative’ **energy** [7].

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The Navier-Stokes equations of the boundary layer are non-linearity and dis- sipation interact with each other summand. Non-linearity causes distortion of the original signal. The dissipation reduces the amplitude of the signal. However, we know that in addition to the above factors for a number of tasks are impor- tant **dispersion** effects, i.e. splitting the signal into individual harmonics. Classic- al equation, which is characterized by the presence and interaction of non- linearity, dissipation and **dispersion**, is the Korteweg-de Vries-Burgers equation. Waveform is changing. If in the equations of motion of the system of Navi- er-Stokes equations we introduce an additional summand with the third deriva- tive, then it will turn into the Korteweg-de Vries-Burgers equation. Usually we suggested that non-linearity and dissipation for large gradients of laminar flows can change flows to turbulent flows. In the Reynolds **model** actually stand fast and slow variables. In the resulting averaged equations establish the connection between pressure and the Reynolds-averaged flow parameters, but have not the answer to the question of the form of the closing ratio. The process of building relationships based or on empirical evidence or written out of the equation for higher order moments, such as turbulent (fluctuating) kinetic **energy**. In these equations include new unknowns, and the process is repeated for the specified circuit scenario. For the inertial flow in the equilibrium case, a well-known law of N.A. Kolmogorov’s **theory** of the dimension is performed. At the heart of all theories are the Navier-Stokes equations. Even if involved the Boltzmann equa- tion in order to obtain from it the equations of turbulent flow, the output me- thod is focused on validation of the **model** Reynolds, built on the basis of the Navier-Stokes equations. It is hardly possible to derive the equation of continui- ty for the tube of flow for turbulent flow. The main directions of current re- search include [12] [13] [14] [15] [16]:

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break invariance under these transformations (as, for ex- ample, the vector potential appears explicitely). Hence, the partition function ˜ Z describes a massive spin-1 the- ory that does not allow easy interpretations. We would like to recast this **theory** in a more suitable form given in terms of a “vector potential” and a “curvature” field, which naturally leads to next sections’ topic. This pro- cess is analogous to the (2+1)-dimensional one where a duality between a selfdual free massive field **theory** and a topologically massive **theory** [36] has been demonstrated. As a final note, it is important to stress that it is not pos- sible to apply the bosonization procedure proposed here to free Dirac fermions, i.e. without the presence of the current-current interactions in Eq. (29). This means that we cannot tune the parameter g 2 to zero without encoun-

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The importance of many-body contributions is evident from the collective eigenvectors of the QHO Hamiltonian that represent the eigenstates of the long-range correla- tion problem in the basis of atomic positions. Although these eigenstates merely constitute the canonical basis of the QHO **model** Hamiltonian and do not correspond to any actual physically observable quantities, their change upon adsorption can help to qualitatively analyze the col- lective mutual polarization and depolarization between different domains of the system. Figure 6 shows two such representative MBD eigenmodes (top and bottom) for an isolated graphene sheet and graphene adsorbed on Ag(111) (left and right). The blue vectors indicate the direction and magnitude of polarization on each localized QHO (depicted as orange spheres). The first eigenmode (a and b in Fig. 6) describes lateral polarization within the graphene sheet and is almost unaffected upon ad- sorption on the metal surface. The second mode (c and d in Fig. 6) represents polarization orthogonal to the graphene plane and is strongly modified due to adsorp- tion. In fact, this mode is fully delocalized over adsorbate and substrate (not shown here) and describes the collec- tive polarization between the subsystems. These visually apparent changes can also be seen in the **energy** of the eigenstates. Whereas the **energy** of the fist eigenmode only decreases by about 1.7 meV upon adsorption, the second mode contributes about 32 meV to the **dispersion** **energy** defined by the sum of eigenenergies. Due to the **energy** shifts of the MBD modes we can in fact pinpoint which polarization modes yield the most important con- tributions to the **dispersion** **energy**.

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Because space is the place in which physics happens, it is natural to expect that topol- ogy would play some role in physical theories. And indeed, many areas of physics, mainly within high-**energy** physics, fundamentally rely, although somewhat implicitly, on the no- tion of topology. Examples are general relativity, theoretical particle physics and string **theory**. It is however only relatively recent that the use of topology has emerged in con- densed matter physics as well. It turns out that one can meaningfully assign topological invariants to physical systems in the same way as one does for topological spaces. The topo- logical invariant of a system defines its phase, and this type of phase of matter is called a topological phase. The invariant can only change if the system undergoes a so-called topological phase transition. The discovery of topological phases was quite revolutionary, and in fact the 2016 Nobel prize in physics has been awarded to Thouless, Haldane and Kosterlitz for their discovery of topological phases of matter [2], [4].

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FIG. 1. Minimal **model** of a non-trivial square root. (a) The bow-tie chain is composed of a sequence of dimers (tri- angles), each supporting two nondegenerate modes (onsite en- ergies ±β and intradimer coupling γ) where one mode couples to the left and the other couples to the right (interdimer cou- pling κ). In the regular case the dimer orientations alternate, resulting in a periodic system with four bands. The depicted orientation defect generates robust states in both of the finite **energy** bands. Here this is illustrated for β = γ = κ = 1, cor- responding to the change of the topological index ˜ ξ (see text); further defect configurations are shown in Figs. 4 and 5. (b) Interpretation of the dimer chain as a non-trivial square root of a two-legged ladder system (a **tight**-**binding** system with β 0 = β 2 + γ 2 + κ 2 , κ 0 = 2βκ, γ 0 = γκ). The parent system has two sites per unit cell, thus only features two bands. After taking the non-trivial square root we obtain a **tight**-**binding** system with four sites per unit cell, which can be unfolded into a linear chain with nearest-neighbor couplings. The bow-tie chain emerges after a Z 2 gauge transformation, which renders

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Density functional **theory** (DFT) has proven to be a valuable tool to investigate and predict MA **energy** (MAE) in various systems. The relativistic effects of valence electrons are often treated **using** various approximations to reduce computational complexity and cost. Instead of directly solving the four-component Dirac equation self-consistently, one usually treats SOC as perturbation and starts first with the two-component scalar-relativistic (SR) Hamiltonian [4], omitting SOC but including all other relativistic effects such as mass-velocity and Darwin terms. SOC can be added directly into the SR Hamiltonian or included in a subsequent step **using** the basis (often a subset of it) of SR wave functions (second variation) [5,6]. Because the charge- and spin-density variations caused by SOC vanish to first order in the SOC strength [7], the magnetic force theorem

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Here we describe how rich topological effects arise when one takes an analogous nontrivial square root on a **tight**- **binding** lattice. **Tight**-**binding** lattices provide an ubiquitous description of electronic bands in crystalline solids, but also extend to atoms and photons in suitably engineered optical and photonic lattices. This includes topological systems in all universality classes, such as the paradigmatic Su-Schrieffer- Heeger **model**, originally proposed for polyacetylene [16], and nontopological variants such as the Rice-Mele **model** for conjugated polymers [17], both of which have been implemented on a wide range of platforms [18–21]. Both models possess two bands in their clean incarnation. The SSH **model** features a chiral symmetry which constraints the Bloch states and allows to define a topological winding number [22]. Defects between regions of different winding number introduce localized, square-normalizable defect states of a fixed chirality that are pinned to the midgap **energy**. The procedure of taking square roots of lattice systems proposed here provides a mechanism to generate a wider class of models, including models with multiple band gaps, where some of the topological properties can be traced back to features of a parent system while others emerge from the square-root operation. Given a suitable parent system with **energy** bands at positive energies, taking the nontrivial square root provides us with a symmetric arrangement of **energy** bands at positive and negative energies. If the original system harboured 2| ν | protected modes around a spectral symmetry point E 2

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A key ingredient in our analysis is a notion of locality of the electronic structure **model**, which was also used in [14] to exhibit locality of the potential **energy**, and which we extend here to other physical quantities, specifically the number of elec- trons. Roughly speaking, the dependence of a local physical property such as the local density of states and hence local physical quantities on the environment decays exponentially fast away from the physical location of interest. Therefore, away from the boundary of the finite domain, the electronic structure behaves as that of the infinite problem. A subtlety arises for the canonical ensemble as the Fermi level for the finite system depends globally on the atom configuration, which would destroy the locality. The key idea to overcome this difficulty is to view the Fermi-level as an independent variable, which together with the nuclei positions, solves the con- straint for the number of electrons together with the force balance equation. The thermodynamic limit can then be viewed as the convergence of the solution to the coupled system as the domain tends to infinity.

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In the density functional **theory** 共DFT兲, the exchange- correlation potential is often approximated by **using** the ex- change correlation present in a homogeneous electron gas 共 LDA 兲 , which has been proven to be very successful for solids even if not all systems are equally well described. Materials with strongly correlated electrons, however, are ex- amples where this mean-field approach most strikingly fails. LDA is, in fact, a one-electron method with an orbitally in- dependent potential, and applying it to a system containing transition metals 共TM兲 or rare earths 共REs兲 with partially filled d or f shells gives results consistent with a metallic electronic structure and itinerant d or f electrons, which is definitely wrong for most RE compounds and several ex- amples of TM systems 共 NiO being the classic example 兲 . Other choices for the exchange correlation such as general- ized gradient 共GGA兲 can also be applied, but as with LDA, this is a mean-field correction for the noninteracting system and so suffers from the same pathology. In the strongly cor- related systems, the d or f electrons are often strongly local- ized, and there is a noticeable **energy** difference between occupied and unoccupied states with strong d or f character,

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In this paper, we simulate charge transport in a gra- phene nanoribbon and a nanoribbon junction **using** a NEGF based on a third nearest-neighbour **tight** **binding** **energy** **dispersion**. For transport studies in nanoribbons and junctions, the formulation of the problem differs from that required for bulk graphene. Third nearest- neighbour interactions introduce additional exchange and overlap integrals significantly modifying the Green ’ s function. Calculation of device characteristics is facili- tated by the inclusion of a Sancho-Rubio [11] iterative scheme, modified by the inclusion of third nearest- neighbour interactions, for the calculation of the self- energies. We find that the conductance is significantly altered compared with that obtained based on the nearest-neighbour **tight** **binding** **dispersion** even in an isolated nanoribbon. Hong et al. [12] observed that the conductance is modified (increased as well as decreased) by the presence of defects within the lattice. Our results show that details of the band structure can significantly modify the observed conductivities when defects are included in the structure.

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Abstract—The **tight** **binding** **model** is one of the strong theoretical techniques to calculate the band gap of the materials. The band structures of some semiconductors and the semiconductors made by their combination is calculated here **using** simulation technique. The basic **theory** of **Tight** **Binding** **Model** is presented since beginning to the final matrix formulation. Band structure of the four semiconductors AlAs, AlP, GaAs and GaSb and their corresponding alloys Al𝐴𝑠 𝑥 𝑃 1−𝑥 & Ga 𝐴𝑠 𝑥 𝑆𝑏 1−𝑥 is

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The phonon **dispersion** relations for carbon nanotubes can be determined by folding that of a graphene layer. In general (n, n) armchair carbon nanotubes yield 4n **energy** subbands by means of 2n conduction and 2n valence bands. Out of these 2n bands, two are non-degenerate and n-1 is doubly degenerate. The degeneracy comes from the two subbands with the same **energy** **dispersion**, but different ν - values. In all zigzag carbon nanotubes have the lowest conduction and the highest valence bands are doubly degenerate where as all armchair carbon nanotubes have band degeneracy between the highest valence and the lowest conduction band. Both armchair as well as zigzag carbon nanotubes bands are symmetric with respect to k=0. An armchair carbon nanotubes bands have two valleys at around k = ±2π/3a points and the zigzag and chiral carbon nanotubes can have at most one valley. The bands in armchair cross the Fermi level at k = ±2π/3a thus they are considered to exhibit metallic nanotube [2]. There is no **energy** gap for few carbon nanotubes. It is valuable to build up an approximate relation that describes the **dispersion** relations in the regions around the Fermi **energy** E F = 0 as

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