# Kramers Degeneracy and Spin Inversion in a Lateral Quantum Dot

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## Abstract

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## 1. Introduction

## 2. Symmetry of Rashba SOI

## 3. Effective Hamiltonian Model

## 4. The 2D Model

## 5. Summary

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Derivation of Effective Hamiltonian on 2D Square Lattice

**Figure A1.**Connection between bounded system B and electrode C. The coupling operator $\langle j|\widehat{V}|{j}^{\prime}\rangle =\langle {j}^{\prime}|\widehat{V}{|j\rangle}^{*}=\langle ({j}_{x},{j}_{y})|\widehat{V}|({j}_{x}^{\prime},{j}_{y}^{\prime})\rangle $ is nonzero only for ${j}_{x}=0$, ${j}_{y}^{\prime}=1$ and ${j}_{y}={j}_{y}^{\prime}=1$ at the connection to the left electrode and for ${j}_{x}=N$, ${j}_{y}^{\prime}=N+1$ and ${j}_{y}={j}_{y}^{\prime}=1$ at the connection to the right electrode.

#### Appendix A.1. Normalization Constant of the Electrode Eigenfunctions

#### Appendix A.2. Integral Over Zone

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**Figure 1.**Sketch of the 2D device that consists of a circular lateral QD. The effective QD’s confinement includes a constant potential with the Rashba interaction (gray region). Additional thin (0.1d) constant potential shell (dark gray region) controls the coupling between the QD and two electrodes.

**Figure 2.**The probability for the direct transmission $|{S}_{2\uparrow 1\uparrow}{|}^{2}$ as function of the electron energy and the parameter $R\alpha $. Red crosses indicate $(E,R\alpha )$ that correspond to $|{S}_{2\uparrow 1\uparrow}{|}^{2}$ = 0 for: $n=0,j=1/2$, $n=0,j=3/2$, $n=1,j=1/2$.

**Figure 3.**Quantum dot with radius $R=2d$ without any additional potential. The probability for the direct transmission $|{S}_{2\uparrow 1\uparrow}{|}^{2}$ as function of the electron energy and the parameter $R\alpha $ (

**a**). Average effectivity of spin invertor $\int dE|{S}_{2\downarrow 1\uparrow}{|}^{2}/\int dE\left(\right|{S}_{2\downarrow 1\uparrow}{|}^{2}+|{S}_{2\uparrow 1\uparrow}{|}^{2})$ over energy range $2{(d/{a}_{0})}^{2}(1-cos(\pi {a}_{0}/d))<E<2{(d/{a}_{0})}^{2}(1-cos(2\pi {a}_{0}/d))$ (

**b**).

n | j | $\mathit{R}\mathit{\alpha}$ |
---|---|---|

0 | 1/2 | 1.68 |

0 | 3/2 | 1.85 |

1 | 1/2 | 1.57 |

0 | 5/2 | 1.98 |

1 | 3/2 | 1.61 |

2 | 1/2 | 1.59 |

0 | 7/2 | 2.1 |

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**MDPI and ACS Style**

Pichugin, K.; Puente, A.; Nazmitdinov, R.
Kramers Degeneracy and Spin Inversion in a Lateral Quantum Dot. *Symmetry* **2020**, *12*, 2043.
https://doi.org/10.3390/sym12122043

**AMA Style**

Pichugin K, Puente A, Nazmitdinov R.
Kramers Degeneracy and Spin Inversion in a Lateral Quantum Dot. *Symmetry*. 2020; 12(12):2043.
https://doi.org/10.3390/sym12122043

**Chicago/Turabian Style**

Pichugin, Konstantin, Antonio Puente, and Rashid Nazmitdinov.
2020. "Kramers Degeneracy and Spin Inversion in a Lateral Quantum Dot" *Symmetry* 12, no. 12: 2043.
https://doi.org/10.3390/sym12122043