#
Stress-Tuned Optical Transitions in Layered 1T-MX_{2} (M=Hf, Zr, Sn; X=S, Se) Crystals

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## Abstract

**:**

_{2}; DFT; bulk; band structure; pressure coefficients; transition metal dichalcogenides

## 1. Introduction

## 2. Methods and Materials

## 3. Results

#### 3.1. Theoretical Analysis

#### 3.2. Comparison with Experiment

## 4. Summary

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Sample Availability

## Abbreviations

CB | Conduction band |

CBM | Conduction band minimum |

DFT | Density functional theory |

GGA | Generalized gradient approximations |

mBJ | modified Becke–Johnson |

PBE | Perdew–Burke–Ernzerhof |

SO | Spin-orbit |

TMDs | Transition metal dichalcogenides |

VB | Valence band |

VBM | Valence band maximum |

vdW | van der Waals |

XC | Exchange-correlation |

## Appendix A. Band Gap Dependence on Geometrical Parameters

**Figure A1.**Band structures of ZrSe${}_{2}$ calculated with mBJ on top of geometry obtained with full PBE optimization (

**a**) with vdW D3−BJ correction, (

**b**) without vdW correction, and experimental lattice constants (

**c**) with relaxed atomic positions and (

**d**) with fixed atomic positions.

## Appendix B. Stress vs. Strain Dependencies

**Figure A2.**Stresses (${\sigma}_{1}$ = ${\sigma}_{2}$ for biaxial, ${\sigma}_{3}$ for uniaxial) due to (

**a**) biaxial (${\epsilon}_{1}$ = ${\epsilon}_{2}$) (

**b**) uniaxial (${\epsilon}_{3}$) strain for all system, applied as illustrated in (

**c**,

**d**), respectively.

## Appendix C. Indirect Transition

**Figure A3.**Stress/pressure dependence of the fundamental indirect transition energy for all studied compounds. The values of pressure coefficients from linear fitting in selected pressure ranges are provided.

**Table A1.**Pressure ranges and linear pressure coefficients of fundamental indirect transitions in all studied compounds.

Pressure Range of ${\mathit{a}}_{\mathit{n}}^{\mathit{h}}$ (kbar) | Stress Range of ${\mathit{a}}_{\mathit{n}}^{\mathit{u}}$ (kbar) | Stress Range of ${\mathit{a}}_{\mathit{n}}^{\mathit{b}}$ (kbar) | ||||
---|---|---|---|---|---|---|

I | II | I | II | I | II | |

HfS${}_{2}$ | [0–60] | [60–180] | [0–23.5] | [23.5–60.9] | [0–58.5] | [58.5–239.5] |

HfSe${}_{2}$ | [0–40] | [40–60] | [0–24.5] | [24.5–61.3] | [0-62.9] | |

ZrS${}_{2}$ | [0–60] | [60–120] | [0–23.2] | [23.2–53.01] | [0–53.8] | [53.8–169.0] |

ZrSe${}_{2}$ | [0–20] | [0–25.5] | [25.5–63.5] | [0–27.56] | ||

SnS${}_{2}$ | [0–180] | [0–20.6] | [20.6–29.5] | [0–32.3] | [50.5–169.8] | |

SnSe${}_{2}$ | [0–80] | [0–5.26] | [8.2–25.2] | [0–11.8] | [25.6–129.2] | |

${\mathit{a}}_{\mathit{n}}^{\mathit{h}}$(meV/kbar) | ${\mathit{a}}_{\mathit{n}}^{\mathit{u}}$ (meV/kbar) | ${\mathit{a}}_{\mathit{n}}^{\mathit{b}}$ (meV/kbar) | ||||

n | I | II | I | II | I | II |

HfS${}_{2}$ | −10.66 | −6.75 | −9.40 | −4.63 | −3.78 | −8.33 |

HfSe${}_{2}$ | −12.61 | −9.37 | −8.90 | −4.95 | −10.95 | |

ZrS${}_{2}$ | −10.57 | −7,19 | −9.04 | −3.94 | −4.23 | −8.53 |

ZrSe${}_{2}$ | −13.67 | −8.06 | −3.84 | −8.56 | ||

SnS${}_{2}$ | −8.52 | −11.92 | −22.53 | 3.71 | −4.51 | |

SnSe${}_{2}$ | −7.19 | −13.70 | −32.98 | 2.43 | −7.38 |

## Appendix D. Tables with Pressure Coefficients

Point | v | c | E | $\mathit{\sigma}$ | ${\mathbf{a}}_{\mathbf{I}}^{\mathbf{h}}$ | ${\mathbf{a}}_{\mathbf{II}}^{\mathbf{h}}$ | ${\mathbf{a}}_{\mathbf{III}}^{\mathbf{h}}$ | ${\mathbf{a}}_{\mathbf{I}}^{\mathbf{u}}$ | ${\mathbf{a}}_{\mathbf{II}}^{\mathbf{u}}$ | ${\mathbf{a}}_{\mathbf{III}}^{\mathbf{u}}$ | ${\mathbf{a}}_{\mathbf{I}}^{\mathbf{b}}$ | ${\mathbf{a}}_{\mathbf{II}}^{\mathbf{b}}$ | ${\mathbf{a}}_{\mathbf{III}}^{\mathbf{b}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathbf{HfS}}_{\mathbf{2}}$ | |||||||||||||

L | VB | CB | 2.076 | ‖/ ⊥ | −7.61 | −5.45 | −5.05 | −5.16 | −2.91 | −1.72 | −5.49 | −6.99 | −7.68 |

$\mathrm{\Gamma}$ | VB | CB | 2.448 | ⊥ | −6.64 | −3.56 | −3.78 | −12.83 | −9.51 | −8.00 | 1.08 | -6.29 | −9.26 |

n.$\mathrm{\Gamma}$A | VB | CB | 2.687 | ‖ | −4.19 | −1.71 | −0.96 | −9.16 | −7.96 | −7.03 | −1.71 | −2.07 | −2.98 |

$\mathrm{\Gamma}$ | VB-1 | CB | 2.690 | ‖ | 0.87 | 0.66 | 0.66 | −0.21 | 0.14 | 0.25 | −1.63 | −6.74 | −9.75 |

L | VB-1 | CB | 2.781 | ‖/ ⊥ | −2.63 | −1.26 | −0.94 | 1.06 | 2.63 | 3.50 | −6.89 | −4.89 | −4.98 |

A | VB | CB | 2.833 | ‖ | 0.98 | 0.84 | 0.80 | 2.44 | 2.34 | 2.23 | −1.56 | −1.85 | −2.06 |

$\mathrm{\Gamma}$ | VB-2 | CB+1 | 2.842 | ‖ | −1.54 | −1.06 | −1.06 | −0.26 | 0.33 | 0.54 | −0.73 | 2.32 | 2.52 |

A | VB-1 | CB+1 | 2.955 | ‖ | 0.86 | 0.85 | 0.85 | 2.42 | 2.37 | 2.28 | −1.85 | −1.84 | −2.18 |

M | VB | CB | 3.120 | ‖ | −0.89 | −0.91 | −0.93 | 5.29 | 4.73 | 4.44 | −6.07 | −5.50 | −5.59 |

$\mathrm{\Gamma}$ | VB | CB+2 | 3.134 | ⊥ | −8.26 | −5.23 | −3.57 | −8.88 | −5.68 | −4.29 | −4.98 | −4.13 | −2.76 |

$\mathrm{\Gamma}$ | VB-1 | CB+2 | 3.356 | ‖ | −4.00 | −3.50 | −2.55 | 3.10 | 3.57 | 3.65 | −6.57 | −3.70 | −2.70 |

n.AL | VB-2 | CB+1 | 3.486 | ⊥ | 0.03 | 1.07 | 1.38 | −6.03 | −5.59 | −5.65 | 4.10 | 2.86 | 2.07 |

A | VB | CB+2 | 3.841 | ‖ | 0.84 | 0.44 | 0.32 | 3.97 | 3.43 | 3.16 | −3.82 | −4.60 | −5.40 |

A | VB-1 | CB+2 | 4.082 | ⊥ | 0.74 | 0.37 | 0.25 | 3.97 | 3.45 | 3.18 | −3.92 | −4.67 | −5.47 |

${\mathbf{HfSe}}_{\mathbf{2}}$ | |||||||||||||

$\mathrm{\Gamma}$ | VB | CB | 1.699 | ‖ | −9.04 | −6.57 | −5.79 | −10.77 | −8.70 | −7.94 | −8.15 | −13.52 | −14.72 |

$\mathrm{\Gamma}$ | VB-1 | CB | 1.716 | ‖/ ⊥ | −4.02 | −4.06 | −4.79 | 0.84 | 1.03 | 1.13 | −4.34 | −12.03 | −13.93 |

L | VB | CB | 1.758 | ‖/ ⊥ | −10.18 | −8.11 | −6.97 | −3.32 | −2.02 | −1.26 | −8.63 | −9.98 | −9.97 |

$\mathrm{\Gamma}$ | VB-1 | CB+1 | 1.811 | ‖ | −2.92 | −2.33 | −1.96 | 0.34 | 0.71 | 0.94 | 0.40 | −2.10 | −2.58 |

A | VB | CB | 1.962 | ‖ | 1.32 | 1.02 | 0.79 | 4.02 | 3.63 | 3.57 | −3.06 | −3.65 | −4.49 |

$\mathrm{\Gamma}$ | VB-1 | CB+2 | 2.140 | ⊥ | −5.78 | −4.69 | −3.47 | 3.35 | 3.89 | 4.34 | −3.46 | −2.85 | −2.77 |

$\mathrm{\Gamma}$ | VB | CB+2 | 2.161 | ‖ | −10.81 | −7.20 | −4.47 | −8.26 | −5.84 | −4.72 | −7.27 | −4.35 | −3.57 |

L | VB-1 | CB | 2.183 | ‖/ ⊥ | −9.48 | −7.63 | −6.59 | −1.24 | 1.46 | 3.00 | −5.95 | −5.51 | −5.32 |

$\mathrm{\Gamma}$ | VB-2 | CB | 2.256 | ⊥ | −5.25 | −4.32 | −4.75 | −1.68 | 0.11 | 0.64 | −3.39 | −5.52 | −5.64 |

A | VB-1 | CB | 2.317 | ‖ | 1.35 | 1.05 | 0.81 | 3.93 | 3.52 | 3.47 | −2.93 | −3.52 | −4.36 |

A | VB-1 | CB+1 | 2.318 | ‖ | 1.09 | 0.86 | 0.69 | 3.79 | 3.45 | 3.44 | −2.92 | −3.07 | −2.89 |

$\mathrm{\Gamma}$ | VB-2 | CB+1 | 2.341 | ‖ | −4.15 | −2.60 | −1.92 | −2.18 | −0.21 | 0.45 | 1.34 | 4.41 | 5.71 |

M | VB | CB | 2.476 | ‖ | −0.15 | −0.65 | −0.82 | 6.80 | 5.87 | 5.79 | −7.30 | −7.71 | −7.66 |

$\mathrm{\Gamma}$ | VB-2 | CB+1 | 2.673 | ‖/ ⊥ | −7.02 | −4.96 | −3.41 | 0.83 | 2.97 | 3.85 | −2.52 | 3.66 | 5.52 |

A | VB | CB+2 | 2.728 | ‖ | 1.41 | 0.86 | 0.54 | 5.45 | 4.57 | 4.38 | −5.38 | −7.07 | −7.29 |

A | VB-1 | CB+2 | 3.055 | ‖ | 1.45 | 0.89 | 0.57 | 5.35 | 4.46 | 4.27 | −5.25 | −6.94 | −7.15 |

n.AL | VB-2 | CB | 3.433 | ⊥ | 2.28 | 2.00 | 1.83 | −0.26 | −0.81 | −1.32 | −2.76 | −7.23 | −7.53 |

Point | v | c | E | $\mathit{\sigma}$ | ${\mathbf{a}}_{\mathbf{I}}^{\mathbf{h}}$ | ${\mathbf{a}}_{\mathbf{II}}^{\mathbf{h}}$ | ${\mathbf{a}}_{\mathbf{III}}^{\mathbf{h}}$ | ${\mathbf{a}}_{\mathbf{I}}^{\mathbf{u}}$ | ${\mathbf{a}}_{\mathbf{II}}^{\mathbf{u}}$ | ${\mathbf{a}}_{\mathbf{III}}^{\mathbf{u}}$ | ${\mathbf{a}}_{\mathbf{I}}^{\mathbf{b}}$ | ${\mathbf{a}}_{\mathbf{II}}^{\mathbf{b}}$ | ${\mathbf{a}}_{\mathbf{III}}^{\mathbf{b}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathbf{ZrS}}_{\mathbf{2}}$ | |||||||||||||

$\mathrm{\Gamma}$ | VB | CB+1 | 1.777 | ‖ | −5.41 | −2.53 | −1.69 | −11.51 | −8.27 | −7.22 | −3.73 | −3.09 | −2.86 |

L | VB | CB | 1.949 | ‖/ ⊥ | −7.64 | −5.70 | −5.30 | −5.17 | −2.70 | −1.52 | −1.68 | −1.01 | −0.60 |

$\mathrm{\Gamma}$ | VB-1 | CB+2 | 2.073 | ‖ | −4.41 | −4.44 | −4.10 | 3.65 | 4.20 | 4.69 | 1.18 | 1.57 | 1.86 |

$\mathrm{\Gamma}$ | VB-2 | CB+1 | 2.164 | ‖ | −1.37 | −0.96 | −0.78 | −0.16 | 0.23 | 0.40 | −0.05 | 0.09 | 0.16 |

A | VB | CB | 2.209 | ‖ | 1.36 | 1.11 | 1.13 | 2.96 | 2.67 | 2.63 | 0.96 | 1.00 | 1.04 |

L | VB-1 | CB+1 | 2.259 | ‖ | 0.83 | 1.32 | 1.45 | −2.60 | −2.09 | −2.03 | −0.84 | −0.78 | −0.80 |

A | VB-1 | CB+1 | 2.299 | ‖ | 1.40 | 1.15 | 1.15 | 2.93 | 2.67 | 2.64 | 0.95 | 1.00 | 1.05 |

A | VB-1 | CB+2 | 2.331 | ‖ | 1.32 | 0.64 | 0.48 | 5.45 | 4.53 | 4.34 | 1.76 | 1.69 | 1.72 |

$\mathrm{\Gamma}$ | VB | CB+2 | 2.547 | ⊥ | −8.45 | −5.98 | −5.00 | 5.45 | −4.29 | −2.90 | −2.51 | −1.61 | −1.15 |

M | VB | CB | 2.702 | ‖ | −0.07 | −0.45 | −0.82 | 6.52 | 5.76 | 5.73 | 2.11 | 2.15 | 2.27 |

$\mathrm{\Gamma}$ | VB-1 | CB | 2.834 | ‖ | −1.38 | −1.03 | −1.55 | −0.16 | 0.16 | 0.29 | −0.05 | 0.06 | 0.11 |

$\mathrm{\Gamma}$ | VB-2 | CB+2 | 2.927 | ‖ | −4.41 | −4.42 | −4.08 | 3.58 | 4.21 | 4.72 | 1.16 | 1.57 | 1.87 |

M | VB-1 | CB+1 | 3.366 | ‖ | 5.93 | 3.13 | 3.71 | 2.94 | 0.01 | −1.80 | 0.95 | 0.01 | −0.71 |

n.AL | VB-2 | CB | 3.400 | ⊥ | −2.42 | −0.46 | −0.35 | −6.41 | −4.89 | −4.47 | −2.07 | −1.83 | −1.77 |

A | VB | CB+2 | 3.486 | ‖ | 1.33 | 0.64 | 0.49 | 5.47 | 4.53 | 4.34 | 1.77 | 1.69 | 1.72 |

M | VB-2 | CB | 3.894 | ‖/ ⊥ | −1.71 | −2.81 | −2.88 | −2.25 | −2.04 | −1.70 | −0.73 | −0.76 | −0.67 |

L | VB | CB+1 | 4.272 | ‖ | −0.01 | 1.55 | 1.90 | −9.14 | −7.58 | −7.38 | −2.96 | −2.83 | −2.93 |

${\mathbf{ZrSe}}_{\mathbf{2}}$ | |||||||||||||

$\mathrm{\Gamma}$ | VB | CB+1 | 1.151 | ‖ | −6.76 | −3.50 | −2.46 | −8.29 | −5.75 | −4.92 | −2.92 | −3.48 | 0.60 |

n.$\mathrm{\Gamma}$A | VB | CB | 1.252 | ‖ | −5.31 | −2.36 | −1.46 | −10.08 | −7.87 | −7.06 | −3.62 | −10.30 | −12.88 |

$\mathrm{\Gamma}$ | VB-1 | CB | 1.252 | ‖ | −2.87 | −2.88 | −5.39 | 1.34 | 1.38 | 1.34 | −1.65 | −12.33 | −9.91 |

L | VB | CB | 1.320 | ‖/ ⊥ | −10.03 | −7.75 | −7.10 | −3.25 | −1.86 | −0.98 | −9.22 | −10.43 | −10.32 |

A | VB | CB | 1.460 | ‖ | 1.64 | 1.33 | 0.89 | 4.46 | 3.91 | 3.87 | −2.94 | −3.10 | −3.34 |

A | VB | CB+1 | 1.465 | ‖ | 1.68 | 1.35 | 0.90 | 4.49 | 3.92 | 3.87 | −2.95 | −3.01 | −3.11 |

$\mathrm{\Gamma}$ | VB | CB+2 | 1.513 | ⊥ | −11.36 | −7.40 | −3.97 | −5.66 | −2.54 | −1.15 | −10.02 | −3.77 | 0.50 |

$\mathrm{\Gamma}$ | VB-1 | CB+2 | 1.604 | ‖ | −7.28 | −5.86 | −3.40 | 3.19 | 4.04 | 4.75 | −6.41 | −1.67 | −2.53 |

L | VB-1 | CB | 1.653 | ‖ | −9.60 | −7.59 | −6.93 | −0.65 | 2.10 | 3.65 | −6.73 | −5.97 | −5.78 |

$\mathrm{\Gamma}$ | VB-2 | CB+1 | 1.663 | ‖ | −3.23 | −1.94 | −1.76 | 0.42 | 0.65 | 0.79 | 0.23 | 3.85 | 5.16 |

A | VB-1 | CB+1 | 1.798 | ‖ | 1.82 | 1.47 | 1.02 | 4.37 | 3.79 | 3.72 | −2.70 | −2.77 | −2.88 |

$\mathrm{\Gamma}$ | VB-2 | CB+2 | 2.031 | ‖/ ⊥ | −7.83 | −5.84 | −3.27 | 3.04 | 3.86 | 4.55 | −6.87 | 3.56 | 5.06 |

M | VB | CB | 2.133 | ‖ | 0.49 | −0.12 | −0.51 | 7.90 | 6.75 | 6.63 | −7.60 | −7.90 | −7.85 |

A | VB | CB+2 | 2.379 | ‖ | 1.63 | 0.92 | 0.45 | 7.07 | 5.78 | 5.53 | −6.55 | −8.38 | −9.36 |

A | VB-1 | CB+2 | 2.663 | ‖ | 1.78 | 1.05 | 0.57 | 6.95 | 5.65 | 5.37 | −6.30 | −8.13 | −9.12 |

L | VB-1 | CB+1 | 3.939 | ‖ | 0.52 | 1.86 | 2.21 | −4.54 | −2.68 | −2.17 | 7.12 | 8.78 | 8.82 |

L | VB | CB+2 | 4.107 | ‖ | 1.98 | 3.16 | 3.19 | −7.75 | −7.38 | −7.74 | 6.45 | 4.05 | 2.07 |

M | VB | CB+1 | 4.272 | ‖ | 9.62 | 8.37 | 7.57 | 5.71 | 4.46 | 4.00 | 4.14 | 4.85 | 4.91 |

H | VB-1 | CB | 4.372 | ‖/ ⊥ | 2.58 | −0.12 | 0.03 | −5.76 | −5.51 | −5.96 | 2.37 | 0.99 | 0.32 |

L | VB-1 | CB+2 | 4.413 | ‖ | 2.41 | 3.33 | 3.36 | −5.16 | −3.42 | −3.12 | 8.93 | 8.51 | 6.61 |

K | VB-1 | CB | 4.500 | ‖/ ⊥ | 5.25 | 4.87 | 4.64 | −4.69 | −4.92 | −5.46 | 9.90 | 9.73 | 1.37 |

Point | v | c | E | $\mathit{\sigma}$ | ${\mathbf{a}}_{\mathbf{I}}^{\mathbf{h}}$ | ${\mathbf{a}}_{\mathbf{II}}^{\mathbf{h}}$ | ${\mathbf{a}}_{\mathbf{III}}^{\mathbf{h}}$ | ${\mathbf{a}}_{\mathbf{I}}^{\mathbf{u}}$ | ${\mathbf{a}}_{\mathbf{II}}^{\mathbf{u}}$ | ${\mathbf{a}}_{\mathbf{III}}^{\mathbf{u}}$ | ${\mathbf{a}}_{\mathbf{I}}^{\mathbf{b}}$ | ${\mathbf{a}}_{\mathbf{II}}^{\mathbf{b}}$ | ${\mathbf{a}}_{\mathbf{III}}^{\mathbf{b}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathbf{SnS}}_{\mathbf{2}}$ | |||||||||||||

$\mathrm{\Gamma}$ | VB | CB | 2.595 | ⊥ | −4.73 | 0.22 | 1.28 | −28.59 | −25.78 | −24.78 | 13.08 | 1.03 | −1.09 |

L | VB-2 | CB | 3.786 | ‖/ ⊥ | 2.20 | 4.21 | 8.71 | −4.56 | −1.89 | −0.30 | 4.32 | 1.38 | −2.14 |

H | VB | CB | 3.871 | ‖ | 3.01 | 3.00 | 2.85 | −2.63 | −3.11 | −3.71 | 3.84 | 2.21 | 1.42 |

H | VB-1 | CB | 3.889 | ‖ | 2.89 | 3.01 | 2.85 | −2.48 | −2.99 | −3.60 | 3.78 | 2.16 | 1.38 |

K | VB | Cb | 3.953 | ‖ | 3.87 | 2.52 | 1.65 | 1.38 | 0.96 | 0.55 | 2.64 | 1.52 | 0.96 |

L | VB-1 | CB | 3.960 | ‖ | 11.38 | 9.33 | 3.81 | 14.60 | 18.29 | 20.07 | 1.79 | 1.56 | 1.23 |

A | VB-2 | CB | 4.439 | ‖ | 13.55 | 9.42 | 3.37 | 9.98 | 8.16 | 7.35 | 6.32 | 4.01 | 2.43 |

${\mathbf{SnSe}}_{\mathbf{2}}$ | |||||||||||||

$\mathrm{\Gamma}$ | VB | CB | 1.446 | ‖/ ⊥ | −5.31 | −0.15 | 1.17 | −34.07 | −31.48 | −32.20 | 11.54 | −0.14 | −2.98 |

$\mathrm{\Gamma}$ | VB-2 | CB | 1.959 | ‖ | 3.18 | 8.11 | 8.06 | −6.45 | −5.74 | 2.35 | 4.45 | 0.08 | −2.75 |

H | VB | CB | 2.969 | ‖/ ⊥ | 2.74 | 3.00 | 2.56 | −4.41 | −5.21 | −6.68 | 4.16 | 2.25 | 1.21 |

L | VB-2 | CB | 2.986 | ‖/ ⊥ | 4.16 | 4.95 | 4.96 | 1.36 | 5.21 | 8.35 | 1.66 | −2.95 | −4.30 |

H | VB-1 | CB | 3.056 | ‖ | 2.92 | 3.12 | 2.66 | −3.91 | −4.71 | −6.15 | 4.02 | 2.13 | 1.10 |

K | VB | CB | 3.072 | ‖ | 4.17 | 3.24 | 1.43 | 0.47 | −0.52 | −1.76 | 2.73 | 1.42 | 0.69 |

K | VB-1 | CB | 3.158 | ‖ | 3.56 | 2.52 | 2.16 | −0.38 | −1.65 | −3.34 | 2.75 | 1.42 | 0.67 |

A | VB-2 | CB | 3.620 | ‖ | 16.97 | 12.85 | 5.78 | 14.57 | 12.49 | 11.58 | 5.25 | 2.93 | 0.86 |

A | VB | CB+1 | 3.676 | ‖ | −0.08 | 0.27 | 5.86 | −6.28 | −6.90 | −8.13 | 4.06 | 3.08 | 2.62 |

A | VB-1 | CB+1 | 3.937 | ‖ | 0.01 | 0.33 | 5.92 | −6.45 | −7.13 | −8.36 | 4.29 | 3.29 | 2.83 |

$\mathrm{\Gamma}$ | VB-1 | CB+1 | 3.955 | ‖ | 6.62 | 1.84 | 1.78 | 6.22 | 7.03 | −0.44 | 7.55 | 2.74 | 2.37 |

M | VB-2 | CB | 4.092 | ‖ | 16.77 | 12.96 | 11.78 | 24.12 | 24.44 | 26.34 | −2.00 | −2.70 | −3.50 |

K | VB-2 | CB | 4.445 | ⊥ | −10.08 | −6.11 | −5.83 | −25.72 | −26.06 | −29.05 | 6.28 | 3.92 | 2.70 |

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**Figure 1.**Top and side views of the (

**a**) trigonal prismatic (2H) and (

**b**) octahedral (1T) polytypes of MX${}_{2}$. (

**c**) The first Brillouin zone (BZ) with high-symmetry k-points and lines denoted in blue.

**Figure 2.**The electronic structure of bulk MS${}_{2}$ (M=Hf, Zr, Se) for (

**a**) and MSe${}_{2}$ for (

**b**) high symmetry lines in BZ obtained using mBJ potential on the top of PBE+D3-BJ+SO geometry optimization. The VBM and CBM are denoted in blue circles.

**Figure 3.**The electronic band structure for unstressed (grey lines) and stressed samples under hydrostatic pressure (blue lines), uniaxial stress (green lines), biaxial stress (violet lines), respectively, for (

**a**) HfS${}_{2}$ and (

**b**) SnSe${}_{2}$. The uniaxial and biaxial stresses correspond to $-8\%$ and $-2$% strains, respectively. Zero in energy is rigidly set to the VBM.

**Figure 4.**Electronic band structures with orbital projections for (

**a**) HfS${}_{2}$, (

**b**) SnSe${}_{2}$. The arrows in (

**a**,

**b**) mark the optically active transitions with highest intensities. The color of the arrow reflects the position from which the transition occurs. For instance, the blue arrow denotes the transition from VB-1 to CB, whereas the light green—from VB to CB. (

**c**,

**d**) graphs show energy of direct transition between selected bands (the other pairs of bands are omitted for the sake of clarity), and (

**e**,

**f**) demonstrate the corresponding dipole strengths and polarizations of transitions.

**Figure 5.**The energy transition under the hydrostatic pressure at $\mathrm{\Gamma}$ point for (

**a**) HfS${}_{2}$ (

**c**) SnSe${}_{2}$ bulk structures. Three linear ranges have been selected (I, II, III), different for both systems. Corresponding pressure coefficients for HfS${}_{2}$ and SnSe${}_{2}$, are presented in (

**b**,

**d**), respectively, along the entire k-path.

**Figure 6.**Pressure dependence of absorption spectra measured at room temperature for (

**a**) HfS${}_{2}$ and (

**b**) HfSe${}_{2}$. Inset shows the absorption edge determined from the absorption spectra (solid circles) together with the linear fit (red lines). Pressure dependencies of the absorption edge energy (black points) and calculated VB-CB transition at $\mathrm{\Gamma}$ (green) and fundamental indirect transition $\mathrm{\Gamma}$-L (violet) as a linear fit of points on the same range and derived pressure coefficients (meV/kbar) for (

**c**) HfS${}_{2}$ and (

**d**) HfSe${}_{2}$.

System | ${\mathbf{a}}^{\mathbf{DFT}}$ (Å) | ${\mathbf{c}}^{\mathbf{DFT}}$ (Å) | ${\mathbf{E}}_{\mathbf{g}}^{\mathbf{DFT}}$ (eV) | ${\mathbf{a}}^{\mathbf{exp}}$ (Å) | ${\mathbf{c}}^{\mathbf{exp}}$ (Å) | ${\mathbf{E}}_{\mathbf{g}}^{\mathbf{exp}}$ (eV) |
---|---|---|---|---|---|---|

HfS${}_{2}$ | 3.59 | 5.75 | 1.50 | 3.63 [56] | 5.86 [56] | 1.96 [57], 1.80 [58], 1.87 [59] |

HfSe${}_{2}$ | 3.70 | 6.08 | 0.71 | 3.67 [56] | 6.00 [56] | 1.13 [57], 1.15 [58] |

ZrS${}_{2}$ | 3.63 | 5.72 | 1.12 | 3.66 [57] | 5.82 [57] | 1.68 [57], 1.70 [60], 1.78 [61] |

ZrSe${}_{2}$ | 3.74 | 6.04 | 0.33 | 3.77 [57] | 6.14 [57] | 1.20 [60], 1.10 [62], 1.18 [63] |

SnS${}_{2}$ | 3.67 | 5.80 | 2.14 | 3.65 [64] | 5.90 [64] | 2.88 [65] |

SnSe${}_{2}$ | 3.84 | 6.00 | 1.10 | 3.82 [64] | 6.14 [64] | 1.63 [65] |

**Table 2.**Direct optical transitions for particular high symmetry k-points for all employed compounds. The $n.XY$ represents the nesting bands transition between the X and Y k-points, whereas the v (valence band) and c (conduction band) indicate positions for which the transition occurs. The E, ${P}_{\Vert}^{2}$ and ${P}_{\perp}^{2}$ denote the energy of transition, intensity of in-plane and out-of plane polarization of light (given in $e{V}^{2}$Å${}^{2}$), respectively. The last nine columns indicate the pressure coefficients (given in $meV/kbar$), where the lower index denotes the pressure regions defined in Appendix C, and the upper index indicates the type of pressure: h—hydrostatic, u—uniaxial, and b—biaxial.

Point | v | c | E | ${\mathbf{P}}_{\Vert}^{2}$ | ${\mathbf{P}}_{\perp}^{2}$ | ${\mathbf{a}}_{\mathbf{I}}^{\mathbf{h}}$ | ${\mathbf{a}}_{\mathbf{II}}^{\mathbf{h}}$ | ${\mathbf{a}}_{\mathbf{III}}^{\mathbf{h}}$ | ${\mathbf{a}}_{\mathbf{I}}^{\mathbf{u}}$ | ${\mathbf{a}}_{\mathbf{II}}^{\mathbf{u}}$ | ${\mathbf{a}}_{\mathbf{III}}^{\mathbf{u}}$ | ${\mathbf{a}}_{\mathbf{I}}^{\mathbf{b}}$ | ${\mathbf{a}}_{\mathbf{II}}^{\mathbf{b}}$ | ${\mathbf{a}}_{\mathbf{III}}^{\mathbf{b}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathbf{HfS}}_{\mathbf{2}}$ | ||||||||||||||

L | VB | CB | 2.076 | 21 | 11 | −7.61 | −5.45 | −5.05 | −5.16 | −2.91 | −1.72 | −5.49 | −6.99 | −7.68 |

$\mathrm{\Gamma}$ | VB | CB | 2.448 | 0 | 3 | −6.64 | −3.56 | −3.78 | −12.83 | −9.51 | −8.00 | 1.08 | -6.29 | −9.26 |

n.$\mathrm{\Gamma}$A | VB | CB | 2.687 | 67 | 0 | −4.19 | −1.71 | −0.96 | −9.16 | −7.96 | −7.03 | −1.71 | −2.07 | −2.98 |

$\mathrm{\Gamma}$ | VB-1 | CB | 2.690 | 66 | 0 | 0.87 | 0.66 | 0.66 | −0.21 | 0.14 | 0.25 | −1.63 | −6.74 | −9.75 |

L | VB-1 | CB | 2.781 | 30 | 2 | −2.63 | −1.26 | −0.94 | 1.06 | 2.63 | 3.50 | −6.89 | −4.89 | −4.98 |

${\mathbf{HfSe}}_{\mathbf{2}}$ | ||||||||||||||

$\mathrm{\Gamma}$ | VB | CB | 1.699 | 52 | 0 | −9.04 | −6.57 | −5.79 | −10.77 | −8.70 | −7.94 | −8.15 | −13.52 | −14.72 |

$\mathrm{\Gamma}$ | VB-1 | CB | 1.716 | 5 | 5 | −4.02 | −4.06 | −4.79 | 0.84 | 1.03 | 1.13 | −4.34 | −12.03 | −13.93 |

L | VB | CB | 1.758 | 14 | 14 | −10.18 | −8.11 | −6.97 | −3.32 | −2.02 | −1.26 | −8.63 | −9.98 | −9.97 |

$\mathrm{\Gamma}$ | VB-1 | CB+1 | 1.811 | 22 | 0 | −2.92 | −2.33 | −1.96 | 0.34 | 0.71 | 0.94 | 0.40 | −2.10 | −2.58 |

A | VB | CB | 1.962 | 68 | 0 | 1.32 | 1.02 | 0.79 | 4.02 | 3.63 | 3.57 | −3.06 | −3.65 | −4.49 |

${\mathbf{ZrS}}_{\mathbf{2}}$ | ||||||||||||||

$\mathrm{\Gamma}$ | VB | CB+1 | 1.777 | 4 | 0 | −5.41 | −2.53 | −1.69 | −11.51 | −8.27 | −7.22 | −3.73 | −3.09 | −2.86 |

L | VB | CB | 1.949 | 10 | 20 | −7.64 | −5.70 | −5.30 | −5.17 | −2.70 | −1.52 | −1.68 | −1.01 | −0.60 |

$\mathrm{\Gamma}$ | VB-1 | CB+2 | 2.073 | 11 | 0 | −4.41 | −4.44 | −4.10 | 3.65 | 4.20 | 4.69 | 1.18 | 1.57 | 1.86 |

$\mathrm{\Gamma}$ | VB-2 | CB+1 | 2.164 | 41 | 0 | −1.37 | −0.96 | −0.78 | −0.16 | 0.23 | 0.40 | −0.05 | 0.09 | 0.16 |

A | VB | CB | 2.209 | 61 | 0 | 1.36 | 1.11 | 1.13 | 2.96 | 2.67 | 2.63 | 0.96 | 1.00 | 1.04 |

${\mathbf{ZrSe}}_{\mathbf{2}}$ | ||||||||||||||

$\mathrm{\Gamma}$ | VB | CB+1 | 1.151 | 14 | 0 | −6.76 | −3.50 | −2.46 | −8.29 | −5.75 | −4.92 | −2.92 | −3.48 | 0.60 |

n.$\mathrm{\Gamma}$A | VB | CB | 1.252 | 54 | 0 | −5.31 | −2.36 | −1.46 | −10.08 | −7.87 | −7.06 | −3.62 | −10.30 | −12.88 |

$\mathrm{\Gamma}$ | VB-1 | CB | 1.252 | 53 | 0 | −2.87 | −2.88 | −5.39 | 1.34 | 1.38 | 1.34 | −1.65 | −12.33 | −9.91 |

L | VB | CB | 1.320 | 23 | 6 | −10.03 | −7.75 | −7.10 | −3.25 | −1.86 | −0.98 | −9.22 | −10.43 | −10.32 |

A | VB | CB | 1.460 | 60 | 0 | 1.64 | 1.33 | 0.89 | 4.46 | 3.91 | 3.87 | −2.94 | −3.10 | −3.34 |

${\mathbf{SnS}}_{\mathbf{2}}$ | ||||||||||||||

$\mathrm{\Gamma}$ | VB | CB | 2.595 | 0 | 61 | −4.73 | 0.22 | 1.28 | −28.59 | −25.78 | −24.78 | 13.08 | 1.03 | −1.09 |

L | VB-2 | CB | 3.786 | 3 | 35 | 2.20 | 4.21 | 8.71 | −4.56 | −1.89 | −0.30 | 4.32 | 1.38 | −2.14 |

H | VB | CB | 3.871 | 4 | 0 | 3.01 | 3.00 | 2.85 | −2.63 | −3.11 | −3.71 | 3.84 | 2.21 | 1.42 |

H | VB-1 | CB | 3.889 | 7 | 0 | 2.89 | 3.01 | 2.85 | −2.48 | −2.99 | −3.60 | 3.78 | 2.16 | 1.38 |

K | VB | CB | 3.953 | 20 | 0 | 3.87 | 2.52 | 1.65 | 1.38 | 0.96 | 0.55 | 2.64 | 1.52 | 0.96 |

${\mathbf{SnSe}}_{\mathbf{2}}$ | ||||||||||||||

$\mathrm{\Gamma}$ | VB | CB | 1.446 | 8 | 67 | −5.31 | −0.15 | 1.17 | −34.07 | −31.48 | −32.20 | 11.54 | −0.14 | −2.98 |

$\mathrm{\Gamma}$ | VB-2 | CB | 1.959 | 112 | 0 | 3.18 | 8.11 | 8.06 | −6.45 | −5.74 | 2.35 | 4.45 | 0.08 | −2.75 |

H | VB | CB | 2.969 | 3 | 3 | 2.74 | 3.00 | 2.56 | −4.41 | −5.21 | −6.68 | 4.16 | 2.25 | 1.21 |

L | VB-2 | CB | 2.986 | 14 | 14 | 4.16 | 4.95 | 4.96 | 1.36 | 5.21 | 8.35 | 1.66 | −2.95 | −4.30 |

K | VB | CB | 3.072 | 22 | 0 | 4.17 | 3.24 | 1.43 | 0.47 | −0.52 | −1.76 | 2.73 | 1.42 | 0.69 |

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Rybak, M.; Woźniak, T.; Birowska, M.; Dybała, F.; Segura, A.; Kapcia, K.J.; Scharoch, P.; Kudrawiec, R.
Stress-Tuned Optical Transitions in Layered 1T-MX_{2} (M=Hf, Zr, Sn; X=S, Se) Crystals. *Nanomaterials* **2022**, *12*, 3433.
https://doi.org/10.3390/nano12193433

**AMA Style**

Rybak M, Woźniak T, Birowska M, Dybała F, Segura A, Kapcia KJ, Scharoch P, Kudrawiec R.
Stress-Tuned Optical Transitions in Layered 1T-MX_{2} (M=Hf, Zr, Sn; X=S, Se) Crystals. *Nanomaterials*. 2022; 12(19):3433.
https://doi.org/10.3390/nano12193433

**Chicago/Turabian Style**

Rybak, Miłosz, Tomasz Woźniak, Magdalena Birowska, Filip Dybała, Alfredo Segura, Konrad J. Kapcia, Paweł Scharoch, and Robert Kudrawiec.
2022. "Stress-Tuned Optical Transitions in Layered 1T-MX_{2} (M=Hf, Zr, Sn; X=S, Se) Crystals" *Nanomaterials* 12, no. 19: 3433.
https://doi.org/10.3390/nano12193433