# Evolution of the Electronic and Optical Properties of Meta-Stable Allotropic Forms of 2D Tellurium for Increasing Number of Layers

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

## 3. Results

#### 3.1. Geometry and Stability

#### 3.2. Electronic Bandstructures

#### 3.3. Optical Absorption

#### 3.4. 2D Exciton Model

#### 3.5. Tellurium Interchain Interaction

- interatomic distances between $\pm 0.7\mathrm{\AA}$ the vdW radii sum fall into the vdW peak, while longer distances should indicate non-interacting atoms;
- distances shorter than the vdW radii sum by more than about $1.3\mathrm{\AA}$ correspond most likely to a chemical bond, and those between $0.7$ to $1.3\mathrm{\AA}$ shorter fall within the so called “vdW gap” (see Figure 10), thus suggesting a special bonding situation that asks for a deeper analysis.

**Figure 10.**Pictorial scheme of a general atom-atom bonding length distribution, as described by Alvarez [54].

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References and Notes

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**Figure 1.**Perspective and side views of the crystal structures of bilayer (2L) $\alpha $-, $\beta $- and $\gamma $-phase of Tellurene.

**Figure 2.**Total energy per atom (rescaled with respect to an isolated Te atom), for increasing number of layers, of the three studied phases. The $\gamma $-phase is the most stable in the ML configuration, while the $\alpha $-phase is preferred for larger layer thicknesses. Note that the ML $\alpha $-phase is unstable.

**Figure 3.**Electronic bandstructures, obtained by using a PBE functional, with the inclusion of SOC, for ML $\gamma $-Te (

**a**) and $\beta $-Te (

**b**). Energy rescaled with respect to the VBM. 2D hexagonal (

**c**,

**left**) and orthorhombic (

**c**,

**right**) BZ, with the high symmetry points used in electronic bandstructure calculations.

**Figure 4.**Electronic bandstructures, obtained by using a PBE functional, with the inclusion of SOC, for 2L $\beta $-Te (

**a**), 3L $\beta $-Te (

**b**) and 4L $\beta $-Te (

**c**). Energy rescaled with respect to the VBM. High-symmetry points related to Figure 3 (orthorhombic BZ).

**Figure 5.**Electronic bandstructures, obtained by using a PBE functional, with the inclusion of SOC, for 2L $\gamma $-Te (

**a**), 3L $\gamma $-Te (

**b**) and 4L $\gamma $-Te (

**c**). Energy rescaled with respect to the VBM (and Fermi energy for 4L). High-symmetry points related to Figure 3 (hexagonal BZ).

**Figure 6.**Electronic bandstructures, obtained by using a PBE functional, with the inclusion of SOC, for 2L $\alpha $-Te (

**a**), 3L $\alpha $-Te (

**b**) and 4L $\alpha $-Te (

**c**). Energy rescaled with respect to the VBM. High-symmetry points related to Figure 3 (orthorhombic BZ). Note that lower values of the gaps were found out of high-symmetry directions and they are reported in Table 2.

**Figure 7.**In-plane optical absorbance of (

**a**) ML $\beta $- and (

**b**) $\gamma $-Te, with the inclusion of SOC. Absorption energy threshold estimated values of 1.02 eV and 0.54 eV, respectively.

**Figure 8.**In-plane optical absorbance comparison between 2L, 3L and 4L $\gamma $- (

**a**), $\beta $- (

**b**) and $\alpha $-Te (

**c**), with the inclusion of SOC. Overall, the absorption energy threshold decreases for increasing number of layers (see Table 2).

**Figure 9.**Numerical solutions of the 2D exciton model, for ${E}_{B}/{R}_{ex}$ (red) and ${r}_{ex}/{a}_{ex}$ (blue), as expressed by Equation (5), showing the two limits discussed. ${E}_{B}$ and ${r}_{ex}$ are the exciton binding energy and radius, respectively; ${R}_{ex}={R}_{H}\frac{\mu}{m}$ is the 3D hydrogenoid Rydberg ${R}_{H}$, renormalised by the ratio between the effective reduced mass $\mu $ and the free electron mass m; ${a}_{ex}={a}_{B}\frac{m}{\mu}$ is the Bohr radius ${a}_{B}$, renormalised by the free electron mass and the effective reduced mass ratio. Results for Te are all from this work. Credits to [37,38,39] for the other results reported (see top left inset): InN (green), GaN and Graphane (grey), BN (orange), AlN (cyan), Plumbene:H (magenta).

**Figure 11.**Interchain binding energy as a function of the interchain distance, with and without the inclusion of the Grimme’s DFT-D2 vdW correction. In both case, the chains possess the same fixed geometry. Red dots correspond to the equilibrium interchain separation of minimum energy for the two cases. Energy rescaled with respect to the relative isolated chains (with and without vdW).

**Figure 12.**Electron localization function (ELF) for two Te helical chains. Vertical 2D plot cutting through a Te−Te bond axis along a chain (

**a**) and between the chains (

**b**).

**Table 1.**Optimised lattice parameters (

**a**

**,**

**b**) and (average) buckling parameter (${\mathit{d}}_{\mathit{z}}$), for increasing number of layers, of the three studied phases. Results from the literature are also reported below each related value. Note that ML $\mathit{\alpha}$-Te is unstable. In the case of $\mathit{\gamma}$-phase, $\mathit{a}=\mathit{b}$.

1L | 2L | 3L | 4L | |||||
---|---|---|---|---|---|---|---|---|

a, b (Å) | ${\mathit{d}}_{\mathit{z}}$ (Å) | a, b (Å) | ${\mathit{d}}_{\mathit{z}}$ (Å) | a, b (Å) | ${\mathit{d}}_{\mathit{z}}$ (Å) | a, b (Å) | ${\mathit{d}}_{\mathit{z}}$ (Å) | |

$\alpha $-phase | ||||||||

This work | - | - | $5.79,4.23$ | $2.10$ | $5.88,4.28$ | $2.08$ | $5.92,4.30$ | $2.11$ |

[30] | $5.80,4.27$ | $5.88,4.31$ | $5.92,4.34$ | |||||

$\beta $-phase | ||||||||

This work | $5.48,4.17$ | $2.17$ | $5.81,4.18$ | $2.08$ | $5.92,4.20$ | $2.04$ | $5.96,4.20$ | $2.03$ |

[14,45] | $5.49,4.17$ | $2.16$ | $5.71,4.13$ | $5.81,4.13$ | $5.85,4.14$ | |||

$\gamma $-phase | ||||||||

This work | $4.15$ | $3.68$ | $4.19$ | $3.71$ | $4.19$ | $3.71$ | $4.20$ | $3.72$ |

[14] | $4.15$ | $3.67$ |

**Table 2.**Calculated DFT electronic bandgaps, for increasing number of layers and with the inclusion of SOC, of the three studied phases. When the bandgap is indirect, we also report the direct bandgap value (in square brackets). Note that ML $\alpha $-Te is unstable and 4L $\gamma $-Te is metallic.

1L | 2L | 3L | 4L | |
---|---|---|---|---|

${\mathit{E}}_{\mathit{G}}$ (eV) | ${\mathit{E}}_{\mathit{G}}$ (eV) | ${\mathit{E}}_{\mathit{G}}$ (eV) | ${\mathit{E}}_{\mathit{G}}$ (eV) | |

$\alpha $-phase | − | $0.67$ $\left[0.86\right]$ | $0.50$ $\left[0.62\right]$ | $0.42$ $\left[0.48\right]$ |

$\beta $-phase | $1.02$ | $0.31$ | $0.051$ | $0.006$ $\left[0.075\right]$ |

$\gamma $-phase | $0.42$ $\left[0.54\right]$ | $0.15$ $\left[0.26\right]$ | $0.026$ $\left[0.19\right]$ | 0 |

**Table 3.**Exciton binding energy (${E}_{B}$) and radius (${r}_{ex}$), as calculated within the Rytova-Keldysh model, compared to reported GW-BSE results in literature (${E}_{B}^{BSE}$). Extrapolated exciton reduced mass renormalised with respect to the free electron mass ($\mu /m$) is also reported.

$\frac{\mathit{\mu}}{\mathit{m}}$ | ${\mathit{E}}_{\mathit{B}}$ (eV) | ${\mathit{r}}_{\mathit{e}\mathit{x}}\left(\mathbf{\AA}\right)$ | ${\mathit{E}}_{\mathit{B}}^{\mathbf{BSE}}$ (eV) | |
---|---|---|---|---|

2L $\alpha $-Te | $0.36$ | $0.26$ | 15 | |

3L $\alpha $-Te | $0.34$ | $0.17$ | 19 | |

4L $\alpha $-Te | $0.26$ | $0.12$ | 25 | |

1L $\beta $-Te | $0.34$ | $0.57$ | 9 | $0.57$ [46] |

2L $\beta $-Te | $0.17$ | $0.15$ | 26 | |

3L $\beta $-Te | $0.09$ | $0.06$ | 57 | |

4L $\beta $-Te | $0.02$ | $0.01$ | 267 | |

1L $\gamma $-Te | $0.10$ | $0.20$ | 27 | $0.15$ [50] |

2L $\gamma $-Te | $0.07$ | $0.07$ | 63 | $0.10$ [50] |

3L $\gamma $-Te | $0.13$ | $0.06$ | 52 | $0.07$ [50] |

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**MDPI and ACS Style**

Grillo, S.; Pulci, O.; Marri, I.
Evolution of the Electronic and Optical Properties of Meta-Stable Allotropic Forms of 2D Tellurium for Increasing Number of Layers. *Nanomaterials* **2022**, *12*, 2503.
https://doi.org/10.3390/nano12142503

**AMA Style**

Grillo S, Pulci O, Marri I.
Evolution of the Electronic and Optical Properties of Meta-Stable Allotropic Forms of 2D Tellurium for Increasing Number of Layers. *Nanomaterials*. 2022; 12(14):2503.
https://doi.org/10.3390/nano12142503

**Chicago/Turabian Style**

Grillo, Simone, Olivia Pulci, and Ivan Marri.
2022. "Evolution of the Electronic and Optical Properties of Meta-Stable Allotropic Forms of 2D Tellurium for Increasing Number of Layers" *Nanomaterials* 12, no. 14: 2503.
https://doi.org/10.3390/nano12142503