#
Effective Low-Energy Hamiltonians and Unconventional Landau-Level Spectrum of Monolayer C_{3}N

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## Abstract

**:**

## 1. Introduction

## 2. Band Structure Calculations

## 3. Effective $\mathbf{k}\xb7\mathbf{p}$ Hamiltonians

#### 3.1. $\Gamma $ Point

#### 3.2. **M** Point

## 4. Comments on the Optical Properties

## 5. Landau Levels

#### 5.1. Effective Model at the $\Gamma $ Point

#### 5.2. M Point

## 6. Summary

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DFT | density functional theory |

CB | conduction band |

VB | valence band |

LL | Landau level |

## Appendix A. k·p Model at the Γ Point

${\mathit{D}}_{6\mathit{h}}$ | E | $2{\mathit{C}}_{6}\left(\mathit{z}\right)$ | $2{\mathit{C}}_{3}$ | ${\mathit{C}}_{2}$ | $3{\mathit{C}}_{2}^{{}^{\prime}}$ | $3{\mathit{C}}_{2}^{"}$ | i | $2{\mathit{S}}_{3}$ | $2{\mathit{S}}_{6}$ | ${\mathit{\sigma}}_{\mathit{h}}\left(\mathbf{xy}\right)$ | $3{\mathit{\sigma}}_{\mathit{d}}$ | $3{\mathit{\sigma}}_{\mathit{v}}$ | Linear Functions, Rotations |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${A}_{1g}$ | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | - |

${A}_{2g}$ | +1 | +1 | +1 | +1 | −1 | −1 | +1 | +1 | +1 | +1 | −1 | −1 | ${R}_{z}$ |

${B}_{1g}$ | +1 | −1 | +1 | −1 | +1 | −1 | +1 | −1 | +1 | −1 | +1 | −1 | - |

${B}_{2g}$ | +1 | −1 | +1 | −1 | −1 | +1 | +1 | −1 | +1 | −1 | −1 | +1 | - |

${E}_{1g}$ | +2 | +1 | −1 | −2 | 0 | 0 | +2 | +1 | −1 | −2 | 0 | 0 | $({R}_{x},{R}_{y})$ |

${E}_{2g}$ | +2 | −1 | −1 | +2 | 0 | 0 | +2 | −1 | −1 | +2 | 0 | 0 | − |

${A}_{1u}$ | +1 | +1 | +1 | +1 | +1 | +1 | −1 | −1 | −1 | −1 | −1 | −1 | - |

${A}_{2u}$ | +1 | +1 | +1 | +1 | −1 | −1 | −1 | −1 | −1 | −1 | +1 | +1 | z |

${B}_{1u}$ | +1 | −1 | +1 | −1 | +1 | −1 | −1 | +1 | −1 | +1 | −1 | +1 | - |

${B}_{2u}$ | +1 | −1 | +1 | −1 | +1 | −1 | +1 | −1 | −1 | +1 | +1 | −1 | - |

${E}_{1u}$ | +2 | +1 | −1 | −2 | 0 | 0 | −2 | −1 | +1 | +2 | 0 | 0 | $(x,y)$ |

${E}_{2u}$ | +2 | −1 | −1 | +2 | 0 | 0 | −2 | +1 | +1 | −2 | 0 | 0 | - |

**Table A2.**Irreducible representations for nine bands at the $\Gamma $ point. Note, that VB-1 and VB-2 are degenerate, and they are described by the partners of the two-dimensional ${E}_{1g}$ representation. Similarly, CB and CB+1 are degenerate, and they are described by the partners of the two-dimensional ${E}_{2u}$ representation.

Band | Irreducible Representation |
---|---|

VB−3 | ${B}_{1g}$ |

VB−2 | ${E}_{1g}$ |

VB−1 | ${E}_{1g}$ |

VB | ${A}_{2u}$ |

CB | ${E}_{2u}$ |

CB+1 | ${E}_{2u}$ |

CB+2 | ${A}_{1g}$ |

CB+3 | ${A}_{1g}$ |

CB+4 | ${A}_{2u}$ |

${\mathit{H}}_{\mathbf{k}\xb7\mathbf{p}}$ | CB+4 | CB+3 | CB+2 | VB−1 | VB−2 | VB−3 | VB | CB | CB+1 |
---|---|---|---|---|---|---|---|---|---|

CB+4 | 0 | 0 | 0 | ${\lambda}_{1}{q}_{-}$ | ${\lambda}_{2}{q}_{+}$ | 0 | 0 | 0 | 0 |

CB+3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

CB+2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

VB−1 | ${\lambda}_{1}^{*}{q}_{+}$ | 0 | 0 | 0 | 0 | 0 | ${\lambda}_{3}{q}_{+}$ | 0 | ${\lambda}_{4}{q}_{-}$ |

VB−2 | ${\lambda}_{2}^{*}{q}_{-}$ | 0 | 0 | 0 | 0 | 0 | ${\lambda}_{5}{q}_{-}$ | ${\lambda}_{6}{q}_{+}$ | 0 |

VB−3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ${\lambda}_{7}{q}_{-}$ | ${\lambda}_{8}{q}_{+}$ |

VB | 0 | 0 | 0 | ${\lambda}_{3}^{*}{q}_{-}$ | ${\lambda}_{5}^{*}{q}_{+}$ | 0 | 0 | 0 | 0 |

CB | 0 | 0 | 0 | 0 | ${\lambda}_{6}^{*}{q}_{-}$ | ${\lambda}_{7}^{*}{q}_{+}$ | 0 | 0 | 0 |

CB+1 | 0 | 0 | 0 | ${\lambda}_{4}^{*}{q}_{+}$ | 0 | ${\lambda}_{8}^{*}{q}_{-}$ | 0 | 0 | 0 |

## Appendix B. k·p Model at the M Point

${\mathit{D}}_{2\mathit{h}}$ | E | ${\mathit{C}}_{2}\left(\mathit{z}\right)$ | ${\mathit{C}}_{2}\left(\mathit{y}\right)$ | ${\mathit{C}}_{2}\left(\mathit{x}\right)$ | i | $\mathit{\sigma}\left(\mathbf{xy}\right)$ | $\mathit{\sigma}\left(\mathbf{xz}\right)$ | $\mathit{\sigma}\left(\mathbf{yz}\right)$ | Linear Functions, Rotations |
---|---|---|---|---|---|---|---|---|---|

${A}_{g}$ | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | - |

${B}_{1g}$ | +1 | +1 | −1 | −1 | +1 | +1 | −1 | −1 | ${R}_{x}$ |

${B}_{2g}$ | +1 | −1 | +1 | −1 | +1 | −1 | +1 | −1 | ${R}_{y}$ |

${B}_{3g}$ | +1 | −1 | −1 | +1 | +1 | −1 | −1 | +1 | ${R}_{z}$ |

${A}_{u}$ | +1 | +1 | +1 | +1 | −1 | −1 | −1 | −1 | - |

${B}_{1u}$ | +1 | +1 | −1 | −1 | −1 | −1 | +1 | +1 | x |

${B}_{2u}$ | +1 | −1 | +1 | −1 | −1 | +1 | −1 | +1 | y |

${B}_{3u}$ | +1 | −1 | −1 | +1 | −1 | +1 | +1 | −1 | z |

Band | Irreducible Representation |
---|---|

VB−3 | ${B}_{3u}$ |

VB−2 | ${B}_{1u}$ |

VB−1 | ${B}_{2g}$ |

VB | ${B}_{3g}$ |

CB | ${A}_{1u}$ |

CB+1 | ${A}_{1g}$ |

CB+2 | ${B}_{2u}$ |

CB+3 | ${B}_{1u}$ |

CB+4 | ${B}_{2u}$ |

CB+5 | ${A}_{1g}$ |

CB+6 | ${B}_{3g}$ |

CB+7 | ${B}_{1u}$ |

CB+8 | ${B}_{3g}$ |

${\mathit{H}}_{\mathbf{k}\xb7\mathbf{p}}$ | VB−3 | VB−2 | VB−1 | CB+1 | CB+2 | CB+3 | CB+4 | CB+5 | CB+6 | CB+7 | CB+8 | VB | CB |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

VB−3 | 0 | 0 | 0 | ${\gamma}_{1}{q}_{x}$ | 0 | 0 | 0 | ${\gamma}_{2}{q}_{x}$ | 0 | 0 | 0 | 0 | 0 |

VB−2 | 0 | 0 | ${\gamma}_{3}{q}_{x}$ | 0 | 0 | 0 | 0 | 0 | ${\gamma}_{4}{q}_{y}$ | 0 | ${\gamma}_{5}{q}_{y}$ | ${\gamma}_{6}{q}_{y}$ | 0 |

VB−1 | 0 | ${\gamma}_{3}^{*}{q}_{x}$ | 0 | 0 | 0 | ${\gamma}_{7}{q}_{x}$ | 0 | 0 | 0 | ${\gamma}_{8}{q}_{x}$ | 0 | 0 | ${\gamma}_{9}{q}_{y}$ |

CB+1 | ${\gamma}_{1}^{*}{q}_{x}$ | 0 | 0 | 0 | ${\gamma}_{10}{q}_{y}$ | 0 | ${\gamma}_{11}{q}_{y}$ | 0 | 0 | 0 | 0 | 0 | 0 |

CB+2 | 0 | 0 | 0 | ${\gamma}_{10}^{*}{q}_{y}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

CB+3 | 0 | 0 | ${\gamma}_{7}^{*}{q}_{x}$ | 0 | 0 | 0 | 0 | 0 | ${\gamma}_{12}{q}_{y}$ | 0 | ${\gamma}_{13}{q}_{y}$ | ${\gamma}_{14}{q}_{y}$ | 0 |

CB+4 | 0 | 0 | 0 | ${\gamma}_{11}^{*}{q}_{y}$ | 0 | 0 | 0 | ${\gamma}_{15}{q}_{y}$ | 0 | 0 | 0 | 0 | 0 |

CB+5 | ${\gamma}_{2}^{*}{q}_{x}$ | 0 | 0 | 0 | 0 | 0 | ${\gamma}_{15}^{*}{q}_{y}$ | 0 | 0 | 0 | 0 | 0 | 0 |

CB+6 | 0 | ${\gamma}_{4}^{*}{q}_{y}$ | 0 | 0 | 0 | ${\gamma}_{12}^{*}{q}_{y}$ | 0 | 0 | 0 | ${\gamma}_{16}{q}_{y}$ | 0 | 0 | ${\gamma}_{17}{q}_{x}$ |

CB+7 | 0 | 0 | ${\gamma}_{8}^{*}{q}_{x}$ | 0 | 0 | 0 | 0 | 0 | ${\gamma}_{16}^{*}{q}_{y}$ | 0 | ${\gamma}_{18}{q}_{y}$ | ${\gamma}_{19}{k}_{y}$ | 0 |

CB+8 | 0 | ${\gamma}_{5}^{*}{q}_{y}$ | 0 | 0 | 0 | ${\gamma}_{13}^{*}{q}_{y}$ | 0 | 0 | 0 | ${\gamma}_{18}^{*}{q}_{y}$ | 0 | 0 | ${\gamma}_{20}{q}_{x}$ |

VB | 0 | ${\gamma}_{6}^{*}{q}_{y}$ | 0 | 0 | 0 | ${\gamma}_{14}^{*}{q}_{y}$ | 0 | 0 | 0 | ${\gamma}_{19}^{*}{q}_{y}$ | 0 | 0 | ${\gamma}_{21}{q}_{x}$ |

CB | 0 | 0 | ${\gamma}_{9}^{*}{q}_{y}$ | 0 | 0 | 0 | 0 | 0 | ${\gamma}_{17}^{*}{q}_{x}$ | 0 | ${\gamma}_{20}^{*}{q}_{x}$ | ${\gamma}_{21}^{*}{q}_{x}$ | 0 |

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**Figure 1.**Crystal structure of C${}_{3}$N monolayer. Purple and blue circles refer to carbon and nitrogen atoms, respectively. Dashed black lines show the unit cell. The orange line show the hexagonal unit cell, which can be useful for understanding certain optical properties; see Section 4.

**Figure 2.**(

**a**) DFT band structure calculations for C${}_{3}$N along the $\Gamma -K-M-\Gamma $ line in the BZ. (

**b**) orientation of the BZ and the high-symmetry points $\Gamma $, K, M.

**Figure 3.**Atomic orbital weight of (

**a**) carbon and (

**b**) nitrogen atoms in the energy bands of C${}_{3}$N monolayer.

**Figure 4.**Landau levels in the CB at the $\Gamma $ point of the BZ as a function of the out-of-plane magnetic field ${B}_{z}$. Blue lines show ${E}_{0}^{\Gamma}$ and ${E}_{1}^{\Gamma}$ given below Equation (15) and red lines indicate the first two LLs that can be obtained from the Ansatz in Equation (16). Black lines show the "conventional" LLs, see the text.

**Figure 5.**Landau levels in the CB at the $\Gamma $ point of the BZ as a function of the LL index n for ${B}_{z}=10$ T. Magenta dots indicate LLs which can be obtained from the Ansatz in Equation (16). Black dots show the “conventional” LLs, see the text.

All Directions | M–$\mathbf{\Gamma}$ Line | M–K Line | |
---|---|---|---|

${m}_{vb}^{\Gamma}/{m}_{e}$ | 0.27 | - | - |

${m}_{cb}^{\Gamma}/{m}_{e}$ | 0.73 | - | - |

${m}_{cb+1}^{\Gamma}/{m}_{e}$ | 0.29 | - | - |

${m}_{vb}^{M}/{m}_{e}$ | - | −0.82 | −0.12 |

${m}_{cb}^{M}/{m}_{e}$ | - | −0.87 | 0.10 |

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**MDPI and ACS Style**

Shahbazi, M.; Davoodi, J.; Boochani, A.; Khanjani, H.; Kormányos, A.
Effective Low-Energy Hamiltonians and Unconventional Landau-Level Spectrum of Monolayer C_{3}N. *Nanomaterials* **2022**, *12*, 4375.
https://doi.org/10.3390/nano12244375

**AMA Style**

Shahbazi M, Davoodi J, Boochani A, Khanjani H, Kormányos A.
Effective Low-Energy Hamiltonians and Unconventional Landau-Level Spectrum of Monolayer C_{3}N. *Nanomaterials*. 2022; 12(24):4375.
https://doi.org/10.3390/nano12244375

**Chicago/Turabian Style**

Shahbazi, Mohsen, Jamal Davoodi, Arash Boochani, Hadi Khanjani, and Andor Kormányos.
2022. "Effective Low-Energy Hamiltonians and Unconventional Landau-Level Spectrum of Monolayer C_{3}N" *Nanomaterials* 12, no. 24: 4375.
https://doi.org/10.3390/nano12244375