# Extended Hückel Semi-Empirical Approach as an Efficient Method for Structural Defects Analysis in 4H-SiC

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## Abstract

**:**

## 1. Introduction

## 2. Simulation Software and Models

## 3. Ideal 4H-SiC

#### 3.1. Energy Band-Gap

#### 3.2. Density of States Function

## 4. Density of States for 4H-SiC with Defects

#### 4.1. Point Defects

#### 4.2. Shockley Stacking Fault Defects

#### 4.3. Threading Edge Dislocations

#### 4.3.1. Bulk Structure

#### 4.3.2. Nanowire Structure

## 5. Summary and Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

DFT | density functional theory |

DOS | density of states |

GGA | generalized gradient approximation |

HSE | Heyd–Scuseria–Ernzerhof hybrid functional approach |

LCAO | linear combination of atomic orbitals |

LCAO-HSE | hybrid-functional method for linear combination of atomic orbitals |

MBJLDA | modified Becke Johnson functional combined with local density approximation |

MetaGGA | meta-generalized gradient approximation |

PBEs | Perdew- Burke- Ernzerhof functionals for solids |

PDOS | projected density of states |

PW | plane wave |

PW-HSE | hybrid-functional calculations with plane-wave basis sets |

SD | screw dislocations |

SEeH | semi-empirical extended Hückel |

SSF | Shockley stacking fault |

## References

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**Figure 1.**${E}_{g}$ obtained from semi-empirical extended Hückel (SEeH) calculations as a function of interstitial channel length. The length is defined as a number of atomic bilayers [22].

**Figure 2.**Density of states (DOS) of ideal 3C- and 6H-SiC obtained from QuantumATK using different models: (a) semi-empirical extended Hückel, (b) LCAO-HSE, and (c) PW-HSE models.

**Figure 3.**Density of states of ideal 4H-SiC obtained from different methods: (a) semi-empirical extended Hückel, (b) LCAO-HSE, (c) PW-HSE, (d) meta-generalized gradient approximation using modified Becke Johnson functional combined with local density approximation (MetaGGA-MBJLDA), MedeA-VASP software, and (e) analytical expression for electron DOS.

**Figure 4.**Atomic structures of ideal 4H-SiC (a) elementary cell, (b) $1\times 1\times 4$ supercell, (c) $10\times 10\times 1$ supercell, and (d) a $10\times 10\times 1$ 4H-SiC nanowire with side walls passivated by hydrogen atoms.

**Figure 5.**Density of states of ideal 4H-SiC for: (a) elementary cell, (b) $1\times 1\times 4$ supercell, (c) $10\times 10\times 1$ supercell, and (d) a $10\times 10\times 1$ 4H-SiC nanowire with side walls passivated by hydrogen atoms.

**Figure 7.**Density of states for 4H-SiC with Shockley stacking faults defects: (a) 4H-SiC_0SSF (reference structure with no defects), (b) 4H-SiC_1SSF, (c) 4H-SiC_2SSF, (d) 4H-SiC_3SSF, and (e) 4H-SiC_4SSF.

**Figure 8.**4H-SiC structure with two threading edge dislocations (TEDs): (

**a**) strain colored atom shells and (

**b**) atomic structure with bonds unveiled.

**Figure 9.**4H-SiC structure after relaxation and DOS plots: (

**A**) relaxed atomic structure and selected atoms around dislocation core and (

**B**) DOS plots.

**Figure 10.**4H-SiC nanowire structure and DOS plots: (

**A**) relaxed atomic structure, (

**B**) selected atoms around the dislocation core, and (

**C**) DOS plots.

Model | Energy Band-Gap ${\mathit{E}}_{\mathit{g}}$ (eV) | ||
---|---|---|---|

3C-SiC | 4H-SiC | 6H-SiC | |

Extended Hückel semi-empirical with Cerda parameters | 2.15 | 3.22 | 2.90 |

PW-HSE ^{1} | 2.27 | 3.23 | 2.95 |

LCAO-HSE ^{2} | 2.26 | 3.19 | 2.94 |

Reference values in the literature | 2.36 | 3.23 | 3.00 |

^{1}PW-HSE: hybrid-functional calculations with plane-wave basis sets.

^{2}LCAO-HSE: hybrid-functional method for linear combination of atomic orbitals.

**Table 2.**List of Shockley stacking faults structures and calculated energy band-gaps. The stacking fault sequence is shown in bold.

Structure | Sequence of Layers | Number of Atoms | ${\mathit{E}}_{\mathit{g}}\left(\mathbf{eV}\right)$ | $\mathbf{\Delta}{\mathit{E}}_{\mathit{g}}\left(\mathbf{eV}\right)$ |
---|---|---|---|---|

4H-SiC_0SSF | ABCB|ABCB|ABCB|ABCB | 32 | 3.25 | 0.0 |

4H-SiC_1SSF | ABCB|ABCB|CABA|C|ABCB|ABCB | 42 | 2.95 | 0.3 |

4H-SiC_2SSF | ABCB|ABCA|BCAC|BC|ABCB|ABCB | 44 | 2.5 | 0.75 |

4H-SiC_3SSF | ABCB|ACAB|CABA|C|ABCB|ABCB | 42 | 2.6 | 0.65 |

4H-SiC_4SSF | ABCB|ACBC|ABCB|ABCB | 32 | 2.7 | 0.55 |

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**MDPI and ACS Style**

Wozny, J.; Kovalchuk, A.; Podgorski, J.; Lisik, Z.
Extended Hückel Semi-Empirical Approach as an Efficient Method for Structural Defects Analysis in 4H-SiC. *Materials* **2021**, *14*, 1247.
https://doi.org/10.3390/ma14051247

**AMA Style**

Wozny J, Kovalchuk A, Podgorski J, Lisik Z.
Extended Hückel Semi-Empirical Approach as an Efficient Method for Structural Defects Analysis in 4H-SiC. *Materials*. 2021; 14(5):1247.
https://doi.org/10.3390/ma14051247

**Chicago/Turabian Style**

Wozny, Janusz, Andrii Kovalchuk, Jacek Podgorski, and Zbigniew Lisik.
2021. "Extended Hückel Semi-Empirical Approach as an Efficient Method for Structural Defects Analysis in 4H-SiC" *Materials* 14, no. 5: 1247.
https://doi.org/10.3390/ma14051247