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A Green’s function-tight-binding-based approach for T-graphene analysis

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Abstract

The framework of the tight-binding model and Green’s function formalism have been derived to investigate the electronic properties of few-layer T-graphene nanoribbons (TGNRs) with different edges including zigzag, bearded, and armchair TGNR (zTGNR, bTGNR, and aTGNR) and the results are compared with those of monolayers. It was observed that single-layer zTGNRs and bTGNRs with metallic properties retain this characteristic as the number of layers increases. In addition, by virtue of their mirror symmetry, aTGNR monolayers exhibit metallic and semiconducting properties by even and odd widths, respectively. When layers are added, symmetric aTGNRs display metallic behavior, while asymmetric aTGNRs display metallic and semiconducting characteristics, depending on the layer stacking. It is also found that the band structure of symmetric aTGNRs and metallic few-layer asymmetric aTGNRs contain Dirac points which increase with increasing their width and layers.

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Data availability statement

The datasets analyzed during the current study are available from the corresponding author on reasonable request.

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Authors and Affiliations

  1. Department of Physics, Razi University, Kermanshah, Iran

    Hamze Mousavi, Samira Jalilvand & Sara Paikar

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  1. Hamze Mousavi
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  2. Samira Jalilvand
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Correspondence to Samira Jalilvand.

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Appendix: TB Hamiltonian matrix for single-layer of bTGNRs and aTGNRs

Appendix: TB Hamiltonian matrix for single-layer of bTGNRs and aTGNRs

We start by calculating the general form of TB Hamiltonian matrix for bTGNRs monolayer. The lattice structure of bTGNR monolayer is shown in Fig. 1b. According to this figure, the BLUC of this structure includes \(N_{a}=4W_{b}+2\) atoms. Therefore, the TB Hamiltonian of single-layer of bTGNR is generally represented by a \(N_{a}\times N_{a}\) matrix:

$$\begin{aligned} {\varvec{h}}_{pp}({\varvec{k}})=\left( \begin{array}{cccccccccccc} 0 &{} {\varvec{h}}^{\prime }_{1b} &{} 0 &{} 0 &{} \cdots &{} 0 &{} 0 &{} 0 \\ \left( {\varvec{h}}^{\prime }_{1b}\right) ^{\dag } &{} {\varvec{h}}_{0z}({\varvec{k}}) &{} {\varvec{h}}^{\prime }_{z} &{} 0 &{} \cdots &{} 0 &{} 0 &{} 0 \\ 0 &{} \left( {\varvec{h}}^{\prime }_{z}\right) ^{\dag } &{} {\varvec{h}}_{0z}({\varvec{k}}) &{} {\varvec{h}}^{\prime }_{z} &{} \cdots &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} \left( {\varvec{h}}^{\prime }_{z}\right) ^{\dag } &{} {\varvec{h}}_{0z}({\varvec{k}}) &{} \cdots &{} 0 &{} 0 &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} \left( {\varvec{h}}^{\prime }_{z}\right) ^{\dag } &{} {\varvec{h}}_{0z}({\varvec{k}}) &{} {\varvec{h}}^{\prime }_{2b} \\ 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} 0 &{} \left( {\varvec{h}}^{\prime }_{2b}\right) ^{\dag } &{} 0\\ \end{array} \right) , \end{aligned}$$
(18)

in which the first row and column, as well as the last row and column, are due to the bearded edges. In addition, \({\varvec{h}}_{0z}({\varvec{k}})\) and \({\varvec{h}}^{\prime }_{z}\) have been introduced previously in Eqs. (7) and (8). In addition, the sub-matrices \({\varvec{h}}^{\prime }_{1b}\) and \({\varvec{h}}^{\prime }_{2b}\) are determined by

$$\begin{aligned} {\varvec{h}}^{\prime }_{1b}=\left( \begin{array}{cccccccccccc} t_{2} &{} 0 &{} 0 &{} 0\\ \end{array} \right) , \quad {\varvec{h}}^{\prime }_{2b}=\left( \begin{array}{cccccccccccc} 0 \\ 0 \\ t_{2} \\ 0 \\ \end{array} \right) . \end{aligned}$$
(19)

According to Fig. 1c, the BLUC of aTGNR monolayer with even width contains \(N_{a}=4W_{a}\) atoms, therefore the TB Hamiltonian of this structure can be generally written as follows:

$$\begin{aligned} {\varvec{h}}_{pp}({\varvec{k}})=\left( \begin{array}{cccccccccccc} {\varvec{h}}_{0e}({\varvec{k}}) &{} {\varvec{h}}^{\prime }_{e} &{} 0 &{} \cdots &{} 0 &{} 0 \\ \left( {\varvec{h}}^{\prime }_{e}\right) ^{\dag } &{} {\varvec{h}}_{0e}({\varvec{k}}) &{} {\varvec{h}}^{\prime }_{e} &{} \cdots &{} 0 &{} 0 \\ 0 &{} \left( {\varvec{h}}^{\prime }_{e}\right) ^{\dag } &{} {\varvec{h}}_{0e}({\varvec{k}}) &{} \cdots &{} 0 &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} 0 &{} \cdots &{} {\varvec{h}}_{0e}({\varvec{k}}) &{} {\varvec{h}}^{\prime }_{e} \\ 0 &{} 0 &{} 0 &{} \cdots &{} \left( {\varvec{h}}^{\prime }_{e}\right) ^{\dag } &{} {\varvec{h}}_{0e}({\varvec{k}}) \\ \end{array} \right) , \end{aligned}$$
(20)

where the sub-matrix \({\varvec{h}}_{0e}({\varvec{k}})\) is a \(8\times 8\) matrix that is given by

$$\begin{aligned} {\varvec{h}}_{0e}({\varvec{k}})=\left( \begin{array}{cccccccccccc} 0 &{} t_{1} &{} 0 &{} t_{1} &{} 0 &{} 0 &{} 0 &{} t_{2} \\ t_{1} &{} 0 &{} t_{1} &{} 0 &{} t_{2} &{} 0 &{} 0 &{} 0 \\ 0 &{} t_{1} &{} 0 &{} t_{1} &{} 0 &{} t_{2} &{} 0 &{} 0 \\ t_{1} &{} 0 &{} t_{1} &{} 0 &{} 0 &{} 0 &{} t_{2} &{} 0 \\ 0 &{} t_{2} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} g({\varvec{k}})\\ 0 &{} 0 &{} t_{2} &{} 0 &{} 0 &{} 0 &{} g({\varvec{k}}) &{} 0\\ 0 &{} 0 &{} 0 &{} t_{2} &{} 0 &{} g^{*}({\varvec{k}}) &{} 0 &{} 0\\ t_{2} &{} 0 &{} 0 &{} 0 &{} g^{*}({\varvec{k}}) &{} 0 &{} 0 &{} 0\\ \end{array} \right) , \end{aligned}$$
(21)

in which \(g({\varvec{k}})=t_{1}\exp \left( \textrm{i}k_{x}a\right)\) with \(a=2a_{1}+\sqrt{2}a_{2}\). Further, the sub-matrix \({\varvec{h}}^{\prime }_{e}\) is also an 8D matrix, with all elements zero except for \({\varvec{h}}^{\prime }(6,5)=t_{1}\) and \({\varvec{h}}^{\prime }(7,8)=t_{1}\).

As it is observed in Fig. 1d, the BLUC of aTGNR monolayer with odd width includes \(N_{a}=4W_{a}\) atoms. Therefore, the general form of the TB Hamiltonian matrix of this structure for each odd width can be shown as follows:

$$\begin{aligned} {\varvec{h}}_{pp}({\varvec{k}})=\left( \begin{array}{cccccccccccc} {\varvec{h}}_{0o}({\varvec{k}}) &{} {\varvec{h}}^{\prime }_{1o} &{} 0 &{} 0 &{} \cdots &{} 0 &{} 0 \\ {\varvec{h}}^{\prime }_{1o} &{} {\varvec{h}}_{0o}({\varvec{k}}) &{} {\varvec{h}}^{\prime }_{2o} &{} 0 &{} \cdots &{} 0 &{} 0 \\ 0 &{} \left( {\varvec{h}}^{\prime }_{2o}\right) ^{\dag } &{} {\varvec{h}}^{\prime }_{0o}({\varvec{k}}) &{} {\varvec{h}}^{\prime }_{2o} &{} \cdots &{} 0 &{} 0 \\ 0 &{} 0 &{} \left( {\varvec{h}}^{\prime }_{2o}\right) ^{\dag } &{} {\varvec{h}}_{0o}({\varvec{k}}) &{} \cdots &{} 0 &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} {\varvec{h}}_{0o}({\varvec{k}}) &{} {\varvec{h}}^{\prime }_{2o} \\ 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} \left( {\varvec{h}}^{\prime }_{2o}\right) ^{\dag } &{} {\varvec{h}}^{\prime }_{0o}({\varvec{k}}) \\ \end{array} \right) , \end{aligned}$$
(22)

in which the sub-matrices \({\varvec{h}}_{0o}({\varvec{k}})\), \({\varvec{h}}^{\prime }_{0o}({\varvec{k}})\), \({\varvec{h}}^{\prime }_{1o}\), and \({\varvec{h}}^{\prime }_{2o}\) are \(4\times 4\) and equal to, respectively:

$$\begin{aligned} {\varvec{h}}_{0o}({\varvec{k}})= & {} \left( \begin{array}{cccccccccccc} 0 &{} t_{1} &{} t_{2} &{} 0 \\ t_{1} &{} 0 &{} 0 &{} t_{2} \\ t_{2} &{} 0 &{} 0 &{} g^{*}({\varvec{k}}) \\ 0 &{} t_{2} &{} g({\varvec{k}}) &{} 0 \\ \end{array} \right) , \nonumber \\ {\varvec{h}}^{\prime }_{0o}({\varvec{k}})= & {} \left( \begin{array}{cccccccccccc} 0 &{} g^{*}({\varvec{k}}) &{} t_{2} &{} 0 \\ g({\varvec{k}}) &{} 0 &{} 0 &{} t_{2} \\ t_{2} &{} 0 &{} 0 &{} t_{1} \\ 0 &{} t_{2} &{} t_{1} &{} 0 \\ \end{array} \right) , \end{aligned}$$
(23)
$$\begin{aligned} {\varvec{h}}^{\prime }_{1o}= & {} \left( \begin{array}{cccccccccccc} t_{1} &{} 0 &{} 0 &{} 0 \\ 0 &{} t_{1} &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 \\ \end{array} \right) , \quad {\varvec{h}}^{\prime }_{2o}=\left( \begin{array}{cccccccccccc} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 \\ t_{1} &{} 0 &{} 0 &{} 0 \\ 0 &{} t_{1} &{} 0 &{} 0 \\ \end{array} \right) . \end{aligned}$$
(24)

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Mousavi, H., Jalilvand, S. & Paikar, S. A Green’s function-tight-binding-based approach for T-graphene analysis. Appl. Phys. A 129, 195 (2023). https://doi.org/10.1007/s00339-023-06424-4

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