# Current Correlations in a Quantum Dot Ring: A Role of Quantum Interference

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## Abstract

**:**

## 1. Introduction

## 2. Calculations of Currents and Their Correlations in Triangular Quantum Dot System

#### 2.1. Model

#### 2.2. Calculation of Currents

#### 2.3. Calculation of Current Correlations

## 3. Bond Currents and Their Correlations: Driven Circular Current in the Case of $\mathsf{\Phi}=\mathbf{0}$

## 4. Persistent Current and Its Noise: The Case $\mathit{V}=\mathbf{0}$

## 5. Correlation of Persistent and Transport Currents, $\mathsf{\Phi}\ne \mathbf{0}$ and $\mathit{V}\ne \mathbf{0}$

## 6. Summary

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Coupling to Atomic Chain Electrodes: Analytical Results

## References and Note

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**Figure 1.**Model of the triangular system of three quantum dots (3QDs) threaded by the magnetic flux $\mathsf{\Phi}$ and attached to the left (L) and the right (R) electrodes.

**Figure 2.**(Top) Transmission and dimensionless bond conductances: $\mathcal{T}$—black, ${\mathcal{G}}_{12}$—blue, and ${\mathcal{G}}_{13}$—green. (Bottom) Dimensionless spectral function of the shot noise: ${\mathcal{S}}_{tr,tr}^{sh}$—black, ${\mathcal{S}}_{12,12}^{sh}$—blue, ${\mathcal{S}}_{13,13}^{sh}$—green, and $-{\mathcal{S}}_{12,13}^{sh}$—red; calculated as a function of the electron energy E for the equilateral triangle system of 3QDs (with the inter-dot hopping ${t}_{12}={t}_{23}={t}_{31}=-1$, which is taken as unity in this paper) in the linear response limit $V\to 0$. The dot levels are ${\epsilon}_{1}={\epsilon}_{2}=0$ and ${\epsilon}_{3}=-2$, 0, 2, for left, center, and right columns, respectively. The coupling with the electrodes is taken to be ${\mathsf{\Gamma}}_{L}={\mathsf{\Gamma}}_{R}=0.25$. Note that the cross-correlation function ${S}_{12,13}^{sh}$ (red) is plotted negatively to show the zero crossing more clearly.

**Figure 3.**Fano factor as a function of the Fermi energy ${E}_{F}$ for the equilateral triangle 3QDs system (${t}_{12}={t}_{23}={t}_{31}=-1$ and ${\epsilon}_{1}={\epsilon}_{2}={\epsilon}_{3}=0$) (

**a**) for various bias voltages $eV=0.01$, 0.5, 1.0, 1.5, and 2.0, at $T=0$; and (

**b**) for various temperatures ${k}_{B}T=0.001$, 0.1, 0.2, 0.3, and 0.4, for $V\to 0$. The coupling to the electrodes is taken as ${\mathsf{\Gamma}}_{L}={\mathsf{\Gamma}}_{R}=0.25$, and the chemical potentials in the electrodes are ${\mu}_{L}={E}_{F}-eV/2$ and ${\mu}_{R}={E}_{F}+eV/2$.

**Figure 4.**Persistent current ${I}^{\varphi}$ versus the flux $\varphi $ threading the equilateral triangle system of 3QDs (${t}_{12}={t}_{23}={t}_{31}=-1$ and ${\epsilon}_{1}={\epsilon}_{2}={\epsilon}_{3}=0$). The coupling is taken as ${\mathsf{\Gamma}}_{L}={\mathsf{\Gamma}}_{R}=\mathsf{\Gamma}=0.01$ and 0.25; the Fermi energies are ${E}_{F}$= −1.5 (

**black**), −0.75 (

**blue**), 1.5 (

**red**); and $T=0$.

**Figure 5.**Flux dependence of spectral function of the persistent current correlator ${\mathcal{S}}_{\varphi ,\varphi}$ (

**black**) and its components: ${\mathcal{S}}_{12,12}$ (

**blue**), ${\mathcal{S}}_{13,13}$ (

**green**), -${\mathcal{S}}_{12,13}$ (

**red**), and ${\mathcal{S}}_{12,12}^{LL}={({\mathcal{G}}_{12}^{L})}^{2}$ (

**blue-dashed**), ${\mathcal{S}}_{12,12}^{RR}={({\mathcal{G}}_{12}^{R})}^{2}$, (

**blue-dotted**), respectively. We assume strong coupling: ${\mathsf{\Gamma}}_{L}={\mathsf{\Gamma}}_{R}=1.0$, ${E}_{F}=-1.5$, and $T=0$.

**Figure 6.**(

**Top**) Energy dependence of driven conductance ${\mathcal{G}}_{12}$ (blue), ${\mathcal{G}}_{13}$ (

**green**) and transmission $\mathcal{T}$ (

**black**). (

**Bottom**) Shot noise ${\mathcal{S}}_{tr,tr}^{sh}$ (

**black**) with the components: ${\mathcal{S}}_{12,12}^{sh}$ (

**blue**), ${\mathcal{S}}_{13,13}^{sh}$ (

**green**), and $-{\mathcal{S}}_{12,13}^{sh}$ (

**red**) for the considered triangular 3QD system threaded by the flux $\varphi =2\pi /16$ (

**left**) or $\varphi =2\pi /4$ (

**right**); the coupling is ${\mathsf{\Gamma}}_{L}={\mathsf{\Gamma}}_{R}=0.25$, and $T=0$. Note that we plot $-{S}_{12,13}^{sh}$.

**Figure 7.**Fano factor as a function of ${E}_{F}$ for the considered 3QD system threaded by the flux $\varphi =2\pi /16$ and for various bias voltages $eV=0.01$, 0.5, 1.0, 1.5, and 2.0. The coupling is ${\mathsf{\Gamma}}_{L}={\mathsf{\Gamma}}_{R}=0.25$, the chemical potentials are ${\mu}_{L}={E}_{F}-eV/2$ and ${\mu}_{R}={E}_{F}+eV/2$, and $T=0$.

**Figure 8.**Circular current ${I}^{c}={I}^{dr}+{I}^{\varphi}$ (solid curves), its driven component ${I}^{dr}$ (dashed curves), and the net transport current ${I}^{tr}$ (dotted curves) versus $\varphi $ for various bias voltages: $eV=0.01$, 0.5, 1.0, and 1.5. We assume a strong coupling ${\mathsf{\Gamma}}_{L}={\mathsf{\Gamma}}_{R}=1$, the chemical potentials are ${\mu}_{L}={E}_{F}-eV/2$, ${\mu}_{R}={E}_{F}+eV/2$, ${E}_{F}=0.9$, and $T=0$.

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Bułka, B.R.; Łuczak, J.
Current Correlations in a Quantum Dot Ring: A Role of Quantum Interference. *Entropy* **2019**, *21*, 527.
https://doi.org/10.3390/e21050527

**AMA Style**

Bułka BR, Łuczak J.
Current Correlations in a Quantum Dot Ring: A Role of Quantum Interference. *Entropy*. 2019; 21(5):527.
https://doi.org/10.3390/e21050527

**Chicago/Turabian Style**

Bułka, Bogdan R., and Jakub Łuczak.
2019. "Current Correlations in a Quantum Dot Ring: A Role of Quantum Interference" *Entropy* 21, no. 5: 527.
https://doi.org/10.3390/e21050527