# Semi-Empirical Pseudopotential Method for Graphene and Graphene Nanoribbons

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## Abstract

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## 1. Introduction

## 2. Calculation Methods

#### 2.1. B-Splines

#### 2.2. Kinetic and Overlap Matrix Elements within Planar Basis

#### 2.3. Implementation of the Semi-Empirical Local Pseudopotential for Graphene

**R**denotes a bulk lattice vector, and ${\mathsf{\tau}}_{\sigma}$ denotes the position of different atoms within the bulk unit cell. We note that ${V}_{L}^{\sigma}\left(\mathbf{r}\right)$ consists of a long-range term that decays like ${Z}^{\sigma}/r$ for large r. For charge-neutral systems, there is a counter long-range term in ${V}_{H}\left(\mathbf{r}\right)$ due to the valence charges, and the sum of ${V}_{ion}\left(\mathbf{r}\right)+{V}_{H}\left(\mathbf{r}\right)$ will be short-ranged.

#### 2.4. Fitting of the Non-Local Pseudopotential for Graphene

#### 2.5. Matrix Elements of Local and Nonlocal Pseudopotential

#### 2.6. Nonlocal Corrections in Overlap and Potential

## 3. Results and Discussions

#### 3.1. Band Structure of Graphene

#### 3.2. Band Structure of Armchair Graphene Nanoribbon

#### 3.3. Modification of Pseudopotential for Edge Atoms of Armchair Graphene Nanoribbon

#### 3.4. Comparison with Experiments

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

#### Matrix Elements of Nonlocal Corrections in Overlap and Potential for Graphene

## Appendix B

#### Matrix Elements for the Hamiltonian of Armchair Graphene Nanoribbon

## References

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**Figure 1.**B-spline consists of an exponential-type of a 17-point knot sequence of order $\mathsf{\kappa}=4$. Here, (●) denotes the knot points in the sequence.

**Figure 2.**Fourier transform of effective local pseudopotential ${\stackrel{~}{V}}_{loc}(z,\mathbf{G})$ for reciprocal lattice vectors ($\mathbf{G}$ ) in the first few shells. The magnitudes of $\mathbf{G}$ in subfigures are: (

**a**) ${G}_{}=0,$ (

**b**) ${G}_{}={G}_{1}=$ 1.56 a.u., (

**c**) ${G}_{}={G}_{2}=$ 2.70 a.u., and (

**d**) ${G}_{}={G}_{3}=$ 3.12 a.u. for shells 0, 1, 2, and 3, respectively.

**Figure 3.**The net local pseudopotential in real space ${V}_{loc}\left(\mathsf{\rho},z\right)$ along various lines in the plane with $z=-0.5\Delta z$ (half a grid from the center of the supercell). (

**a**) Along the line at $y=\frac{a}{2\sqrt{3}},$ which passes through a row of C atoms in the graphene sheet. (

**b**) Along two perpendicular lines (along the x-axis and y-axis) both passing through the origin, which is the center of the hexagon cell. (

**c**) The difference of ${V}_{loc}\left(\mathsf{\rho},z\right)$ obtained by DFT and SEP plotted along the x-axis (green) or y-axis (red). The blue curve indicates the average of the red and green curves. (

**d**) The best-fit results of the average $\Delta {V}_{b}\left(\mathsf{\rho},z\right)$.

**Figure 4.**The net local potential of graphene, ${V}_{loc}\left(\mathbf{0},z\right),$ as a function of z obtained by DFT (red curve) and the best-fit result to ${V}_{loc}\left(\mathbf{0},z\right)$ (dashed black curve). Here, $W=3.25a$ is the width of the domain along the z-axis used to define the B-spline basis.

**Figure 5.**Fitting results of $\beta $ functions used in the non-local pseudopotential of C atom. (

**a**) 2${S}_{1}$ state. (

**b**) 2${S}_{2}$ state. (

**c**) 2${P}_{1}$ state. (

**d**) 2${P}_{2}$ state.

**Figure 6.**(

**Left**) Primitive vectors and position of atoms for graphene. Atoms on sublattice A and B are colored blue and yellow, respectively. (

**Right**) Hexagonal 2D Brillouin zone of graphene with main symmetry points. The primitive reciprocal lattice vectors shown are in units of $\frac{4\mathsf{\pi}}{\sqrt{3}\mathrm{a}}$.

**Figure 7.**Band structure of graphene obtained by the present SEP with best-fitted parameters (solid curves). For comparison, the band structure obtained by self-consistent calculation based on DFT with Vanderbilt USPP is also included (dotted curves).

**Figure 8.**(

**a**) Position of atoms of a 9 × 2 AGNR and the 16 × 2 supercell used in the calculation (enclosed within the green rectangular box). The red x marks the origin of the coordinate system to illustrate the inversion symmetry. (

**b**) Rectangular 2D Brillouin zone (BZ) of the 16 × 2 supercell for the AGNRs are indicated with the green box, and the non-equivalent AGNRs reciprocal lattice vectors enclosed within the first BZ of graphene ${\mathbf{g}}_{\mathit{j}}$ $(j=1,\dots ,2M)$ are indicated by black dots. The blue rectangular box indicates the BZ of a 1 × 2 supercell for graphene. The black dots outside the red hexagon (BZ of graphene) can be shifted inside the box by adding a reciprocal lattice vector.

**Figure 9.**The band structure of AGNRs with $\mathit{M}=16$ and $\mathit{N}=9$. (

**a**) Our SEP results without the relaxation of the edge atoms. (

**b**) Our SEP results with edge relaxation but without modifying the pseudopotentials on the edge atoms. (

**c**) DFT results were obtained by using the method described in [12] for the AGNRs with relaxed atomic positions for the edge atoms. (

**d**) Our SEP results include the modification of pseudopotentials on the edge atoms. In SEP results (

**a**,

**b**,

**d**), the bands in blue ($\mathbf{w}\mathbf{i}\mathbf{t}\mathbf{h}{\mathit{\pi}}_{-}\mathbf{s}\mathbf{y}\mathbf{m}\mathbf{m}\mathbf{e}\mathbf{t}\mathbf{r}\mathbf{y}$ ) and red (with ${\mathit{\pi}}_{+}$ symmetry) are derived from $\mathit{\pi}$-bonded states (odd with respect to the z-mirror), while the bands in green are derived from the $\mathit{\sigma}$-bonded states (even with respect to the z-mirror). Here, we do not distinguish the ${\mathit{\sigma}}_{+}$ from ${\mathit{\sigma}}_{-}$ states since the important states are edge states, and they are essentially degenerate.

**Figure 10.**Contour plot of net local potential in one supercell of the AGNR obtained by DFT [11]. Here, $a$ is the lattice constant of graphene, and $L=\sqrt{3}a$ is the the length of supercell along the $y$ -axis.

**Figure 11.**The net local pseudopotential in one supercell of the AGNRs obtained by the current SEP (green line) and by DFT (red line) along the lines with $z\approx 0$ (near the AGNRs plane) and some selected values of $y$. (

**a**) $y=-L/2$ (along the bottom black line going through the centers of ten bonds in Figure 10), (

**b**) $y=-3L/17$ (along the black line going through the centers of nine atoms in Figure 10), (

**c**) $y=0$ (along the middle black line going through the centers of nine bonds in Figure 10), and (

**d**) $y=6L/17$ (along the black line going through the centers of ten atoms in Figure 10).

**Figure 12.**Difference in the local potentials between DFT and the current SEP results (red curves) and fitted by expression (48) (green curves) evaluated at (

**a**) $y=-L/2$, (

**b**) $y=-3L/17$, (

**c**) $y=0,$ and (

**d**) $y=6L/17$, respectively.

**Figure 13.**The net local potential of AGNR, ${V}_{loc}\left({x}_{b},0,z\right),$ as a function of z obtained by DFT (dashed black curve) and the best-fit result to ${V}_{loc}\left({x}_{b},0,z\right)$ (blue curve). Here, $W=3.25a$ is the width of the domain along the z-axis used to define the B-spline basis.

**Figure 14.**The HOMO (blue) and LUMO (red) levels of AGNRs with various numbers of dimer lines (${N}_{d}$) calculated by SEPM.

Exponents | Coefficients | ||||
---|---|---|---|---|---|

${\mathit{\alpha}}_{1}^{}$ | ${\mathit{\alpha}}_{2}^{}$ | ${\mathit{\alpha}}_{3}^{}$ | ${\mathit{C}}_{1}^{}$ | ${\mathit{C}}_{2}^{}$ | ${\mathit{C}}_{3}^{}$ |

0.0396 | 1.4100 | 0.3461 | −0.3682 | −1.7360 | −1.5710 |

**Table 2.**Fitting parameters for the short-range and long-range shape function ${{f}_{S}}_{}\left(z\right)$ and ${{f}_{L}}_{}\left(z\right)$ for the correction terms to ${\stackrel{~}{V}}_{loc}\left(z,\mathbf{G}\right)$ used in this work.

Type | Exponent | $\mathbf{Coefficients}\mathbf{for}{\mathit{f}}_{\mathit{\gamma}}\left(\mathit{z}\right)$ | $\mathbf{Coefficients}\mathbf{for}{\stackrel{~}{\mathit{D}}}_{\mathit{\gamma}}\left(\mathbf{G}\right)$ | |||||
---|---|---|---|---|---|---|---|---|

$\mathit{\gamma}$ | ${\alpha}_{0}^{}$ | ${C}_{1}^{\gamma}$ | ${C}_{2}^{\gamma}$ | ${C}_{3}^{\gamma}$ | ${C}_{4}^{\gamma}$ | ${P}_{1}^{\gamma}$ | ${P}_{2}^{\gamma}$ | ${P}_{3}^{\gamma}$ |

Short Range (S) | 2.07 | 2.0372 | −16.164 | 13.912 | −2.8969 | 0.04494 | −0.00574 | 0.000224 |

Long Range (L) | 2.07 | 2.6251 | −5.6668 | 2.1280 | 1.0239 | −0.1650 | 0.03132 | −0.002615 |

**Table 3.**Fitting parameters for the bond-charge contribution in pseudopotential localized at the center of the hexagon cell as described by Equations (25)–(27).

Exponents in Equations (25) and (28) | Coefficients in Equation (25) | |||||||
---|---|---|---|---|---|---|---|---|

${\alpha}_{b}^{}$ | ${\alpha}_{h1}^{}$ | ${\alpha}_{h2}^{}$ | ${C}_{0}^{b}$ | ${C}_{1}^{b}$ | ${C}_{2}^{b}$ | ${C}_{3}^{b}$ | ${C}_{4}^{b}$ | ${a}_{h}$ |

3.0053 | 0.3601 | 0.0383 | −0.1727 | 1.5253 | −6.4817 | 11.5249 | −5.0681 | 0.6930 |

**Table 4.**Fitting parameters (a.u.) for $\beta $ functions used in the non-local pseudopotential of C atom.

Orbitals | ${\mathit{\alpha}}_{}$ | ${\mathit{R}}_{\mathit{s}}$ | ${\mathit{C}}_{0}$ | ${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ | ${\mathit{C}}_{4}$ |
---|---|---|---|---|---|---|---|

2${S}_{1}$ | 2.747 | 1.3174 | −2.999 | −4.209 | −11.95 | 7.612 | 0 |

2${S}_{2}$ | 2.171 | 1.3174 | 6.206 | −9.434 | −21.03 | 14.21 | 0 |

2${P}_{1}$ | 0.5104 | 1.3174 | −3.941 | −1.411 | 5.485 | −1.939 | 0 |

2${P}_{2}\left(\mathrm{s}\mathrm{e}\mathrm{g}1\right)$ | 1.134 | 0.9228 | −2.492 | 94.68 | −365.4 | 480.9 | −209.8 |

2${P}_{2}\left(\mathrm{s}\mathrm{e}\mathrm{g}2\right)$ | 0.0 | 1.2953 | −0.9771 | −350.7 | 4210 | −5711 | −4377 |

n | n’ | l | ${\mathit{E}}_{\mathit{l}}^{\mathit{n}\mathit{n}\mathbf{\prime}}$ | ${\mathit{q}}_{\mathit{l}}^{\mathit{n}\mathit{n}\mathbf{\prime}}$ |
---|---|---|---|---|

1 | 1 | 0 | 3.490422 | −0.449056 |

1 | 2 | 0 | 0.207297 | 0.344889 |

2 | 2 | 0 | −2.748230 | −0.212785 |

3 | 3 | 1 | 2.474918 | 1.236379 |

3 | 4 | 1 | −5.902130 | −0.938122 |

4 | 4 | 1 | 9.289400 | 0.631727 |

**Table 6.**Fitting parameters ${S}_{a}$, ${S}_{b},$ and ${S}_{c}$ (in a.u.) at four selected values of y.

${\mathit{S}}_{\mathit{a}}$ | ${\mathit{S}}_{\mathit{b}}$ | ${\mathit{S}}_{\mathit{c}}$ | ${\mathit{S}}_{\mathit{d}}$ |
---|---|---|---|

−0.25 | −0.34 | 0.10 | −0.42 |

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Paudel, R.K.; Ren, C.-Y.; Chang, Y.-C.
Semi-Empirical Pseudopotential Method for Graphene and Graphene Nanoribbons. *Nanomaterials* **2023**, *13*, 2066.
https://doi.org/10.3390/nano13142066

**AMA Style**

Paudel RK, Ren C-Y, Chang Y-C.
Semi-Empirical Pseudopotential Method for Graphene and Graphene Nanoribbons. *Nanomaterials*. 2023; 13(14):2066.
https://doi.org/10.3390/nano13142066

**Chicago/Turabian Style**

Paudel, Raj Kumar, Chung-Yuan Ren, and Yia-Chung Chang.
2023. "Semi-Empirical Pseudopotential Method for Graphene and Graphene Nanoribbons" *Nanomaterials* 13, no. 14: 2066.
https://doi.org/10.3390/nano13142066