# Thermoelectric Inversion in a Resonant Quantum Dot-Cavity System in the Steady-State Regime

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## Abstract

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## 1. Introduction

## 2. Modeling and Formalism

#### Results

## 3. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

QD | Quantum dot |

NEGF | Non-equilibrium Green’s function |

${T}_{L}$ | Temperature of the left lead |

${T}_{R}$ | Temperature of the right lead |

MB | Many-body states |

GS | Ground-state energy |

$\gamma $GS | One-photon replica of the ground-state |

FES | First-excited state energy |

## References

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**Figure 1.**(

**a**) Schematic diagram showing the quantum dot (QD) system (black) connected to the leads or the electron reservoirs, where the temperature of the left electron reservoir (${T}_{L}$) (red) is higher than the temperature of the right electron reservoir (${T}_{R}$) (blue). The cyan zigzags represent the photon field in the cavity (cyan rectangle). (

**b**) The potential of the QD system to be connected diametrically to the left and right leads in the x-direction.

**Figure 2.**Many-body energy of the QD system versus the gate voltage (${\mathrm{V}}_{\mathrm{p}}$), where 0ES (green squares) indicate zero-electron states, 1ES (blue circles) display one-electron states, and 2ES (red circles) refer to two-electron states. The golden line is the chemical potential of the leads where ${\mu}_{L}={\mu}_{R}=\mu =1.2\phantom{\rule{3.33333pt}{0ex}}\mathrm{meV}$. GS indicates the one-electron ground-state, $\gamma $GS is the one-photon replica of the one-electron ground-state, and FES is the one-electron first-excited state. The photon energy $\hslash {\omega}_{\gamma}=1.31$ meV, the electron–photon coupling strength is ${g}_{\gamma}=0.05$ meV, and and the photon field is linearly polarized in the x-direction. The magnetic field is $B=0.1\phantom{\rule{3.33333pt}{0ex}}\mathrm{T}$, and $\hslash {\mathsf{\Omega}}_{0}=2.0\phantom{\rule{3.33333pt}{0ex}}\mathrm{meV}$.

**Figure 3.**Thermoelectric current from the left lead to the QD system (${I}_{L}$) and the thermoelectric current from the QD system to the right lead (${I}_{R}$) (

**a**) and occupation (

**b**) versus the gate voltage ${\mathrm{V}}_{\mathrm{p}}$ for the QD system without the photon cavity. The temperature of the left (right) lead is fixed at ${\mathrm{T}}_{\mathrm{L}}=1.16$ K (${\mathrm{T}}_{\mathrm{R}}=0.58$ K), implying thermal energy of $0.1$ meV ($0.05$ meV). The chemical potential of the leads are fixed at ${\mu}_{\mathrm{L}}={\mu}_{\mathrm{R}}=1.2$ meV. The golden vertical lines indicate the resonance condition for the ground-state (GS) at ${\mathrm{V}}_{\mathrm{p}}=1.95$ mV and the first-excited state (FES) at ${\mathrm{V}}_{\mathrm{p}}=0.271$ mV. The magnetic field is $B=0.1\phantom{\rule{3.33333pt}{0ex}}\mathrm{T}$, and $\hslash {\mathsf{\Omega}}_{0}=2.0\phantom{\rule{3.33333pt}{0ex}}\mathrm{meV}$.

**Figure 4.**Thermoelectric current (${I}_{\mathrm{L}}$) in the case of ${\overline{n}}_{\mathrm{R}}=0$ (

**a**) and 1 (

**b**), and electron occupation (

**c**) for the QD system without (w/o ph) (blue color) and with (w ph) (red color) photon field. The photon energy is $\hslash {\omega}_{\gamma}=1.31$ meV (off-resonance regime), ${g}_{\gamma}=0.05$ meV, and the photon field is polarized in the x-direction. The temperature of the left lead is ${\mathrm{T}}_{\mathrm{L}}=1.16$ K and that of the right lead is ${\mathrm{T}}_{\mathrm{R}}=0.58$ K, which raises the thermal energy of the left lead to $0.1$ meV and of the right lead to $0.05$ meV. The chemical potential of the leads are fixed at ${\mu}_{\mathrm{L}}={\mu}_{\mathrm{R}}=1.2$ meV. The golden vertical lines display the resonance condition for the GS at ${\mathrm{V}}_{\mathrm{p}}=1.95$ mV, the $\gamma $GS at ${\mathrm{V}}_{\mathrm{p}}=0.65$ mV, and the FES at ${\mathrm{V}}_{\mathrm{p}}=0.271$ meV. The weak external magnetic field is $B=0.1\phantom{\rule{3.33333pt}{0ex}}\mathrm{T}$, and $\hslash {\mathsf{\Omega}}_{0}=2.0\phantom{\rule{3.33333pt}{0ex}}\mathrm{meV}$.

**Figure 5.**Diagram showing the photon-activated resonance energy levels and electron transition by the photon-induced processes via $\gamma $GS.

**Figure 6.**Thermoelectric current (${I}_{\mathrm{L}}$) for the QD system coupled to the photon field with x- (red color) and y-polarized (green color) photon fields. The photon energy is $\hslash {\omega}_{\gamma}=1.31$ meV (off-resonance regime), and ${g}_{\gamma}=0.05$ meV, ${\overline{n}}_{\mathrm{R}}=1$. The temperature of the left and the right leads is fixed at ${\mathrm{T}}_{\mathrm{L}}=1.16$ K and ${\mathrm{T}}_{\mathrm{R}}=0.58$ K, respectively. The given temperatures imply that the thermal energy of the left lead is $0.1$ meV and that of the right lead is $0.05$ meV. The chemical potential of the leads are fixed at ${\mu}_{\mathrm{L}}={\mu}_{\mathrm{R}}=1.2$ meV. The golden vertical lines indicate the resonance condition for the GS at ${\mathrm{V}}_{\mathrm{p}}=1.95$ mV, the $\gamma $GS at ${\mathrm{V}}_{\mathrm{p}}=0.65$ mV, and the FES at ${\mathrm{V}}_{\mathrm{p}}=0.271$ meV. The weak external magnetic field is $B=0.1\phantom{\rule{3.33333pt}{0ex}}\mathrm{T}$, and $\hslash {\mathsf{\Omega}}_{0}=2.0\phantom{\rule{3.33333pt}{0ex}}\mathrm{meV}$.

**Figure 7.**Thermoelectric current (${I}_{\mathrm{L}}$) versus gate voltage for the QD system coupled to the photon cavity with ${g}_{\gamma}=0.05$ (red color), $0.10$ (magenta color), and $0.15$ meV (navy blue color). The photon energy is $\hslash {\omega}_{\gamma}=1.31$ meV (off-resonance regime), ${\overline{n}}_{\mathrm{R}}=1$, and the photon field is polarized in the direction of electron motion, the x-direction. The temperatures of the left and right leads are fixed at ${\mathrm{T}}_{\mathrm{L}}=1.16$ K and ${\mathrm{T}}_{\mathrm{R}}=0.58$ K, respectively. The chemical potential of the leads are fixed at ${\mu}_{\mathrm{L}}={\mu}_{\mathrm{R}}=1.2$ meV. The golden vertical lines show the resonance condition for the GS at ${\mathrm{V}}_{\mathrm{p}}=1.95$ mV, the $\gamma $GS at ${\mathrm{V}}_{\mathrm{p}}=0.65$ mV, and the FES at ${\mathrm{V}}_{\mathrm{p}}=0.271$ meV. The weak external magnetic field is $B=0.1\phantom{\rule{3.33333pt}{0ex}}\mathrm{T}$, and $\hslash {\mathsf{\Omega}}_{0}=2.0\phantom{\rule{3.33333pt}{0ex}}\mathrm{meV}$.

**Figure 8.**Many-body (MB) energy of the QD system versus the gate voltage (${\mathrm{V}}_{\mathrm{p}}$), for the x- (

**a**) and y-polarized (

**b**) photon fields. The MB energy spectrum of the $\gamma $GS and FES versus photon energy is plotted for x- (

**c**) and y-polarized (

**d**) photon fields. The 0ES (green squares) are zero-electron states, 1ES (blue circles) are one-electron states, and 2ES (red circles) are two-electron states. The golden line is the chemical potential of the leads where ${\mu}_{L}={\mu}_{R}=\mu =1.2\phantom{\rule{3.33333pt}{0ex}}\mathrm{meV}$. GS indicates the one-electron ground-state, $\gamma $GS is the one-photon replica of the one-electron ground-state, and FES is the one-electron first-excited state. The photon energy $\hslash {\omega}_{\gamma}=1.68$ meV, and ${g}_{\gamma}=0.05$ meV. The magnetic field is $B=0.1\phantom{\rule{3.33333pt}{0ex}}\mathrm{T}$, and $\hslash {\mathsf{\Omega}}_{0}=2.0\phantom{\rule{3.33333pt}{0ex}}\mathrm{meV}$.

**Figure 9.**Thermoelectric current (${I}_{\mathrm{L}}$) for the QD system without (w/o ph) (blue color) and with (w ph) x- (red color) and y-polarized (green color) photon fields. The photon energy is $\hslash {\omega}_{\gamma}=1.68$ meV, ${g}_{\gamma}=0.05$ meV, and ${\overline{n}}_{\mathrm{R}}=1$. The temperatures of the left and right leads are constant and fixed at ${\mathrm{T}}_{\mathrm{L}}=1.16$ K and ${\mathrm{T}}_{\mathrm{R}}=0.58$ K, respectively. The chemical potential of the leads are fixed at ${\mu}_{\mathrm{L}}={\mu}_{\mathrm{R}}=1.2$ meV. The golden vertical lines indicate the resonance condition for the GS at ${\mathrm{V}}_{\mathrm{p}}=1.95$ mV, and the Rabi-splitting states between the $\gamma $GS and the FES at ${\mathrm{V}}_{\mathrm{p}}=0.271$ meV. The weak external magnetic field is $B=0.1\phantom{\rule{3.33333pt}{0ex}}\mathrm{T}$, and $\hslash {\mathsf{\Omega}}_{0}=2.0\phantom{\rule{3.33333pt}{0ex}}\mathrm{meV}$.

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**MDPI and ACS Style**

Abdullah, N.R.; Tang, C.-S.; Manolescu, A.; Gudmundsson, V. Thermoelectric Inversion in a Resonant Quantum Dot-Cavity System in the Steady-State Regime. *Nanomaterials* **2019**, *9*, 741.
https://doi.org/10.3390/nano9050741

**AMA Style**

Abdullah NR, Tang C-S, Manolescu A, Gudmundsson V. Thermoelectric Inversion in a Resonant Quantum Dot-Cavity System in the Steady-State Regime. *Nanomaterials*. 2019; 9(5):741.
https://doi.org/10.3390/nano9050741

**Chicago/Turabian Style**

Abdullah, Nzar Rauf, Chi-Shung Tang, Andrei Manolescu, and Vidar Gudmundsson. 2019. "Thermoelectric Inversion in a Resonant Quantum Dot-Cavity System in the Steady-State Regime" *Nanomaterials* 9, no. 5: 741.
https://doi.org/10.3390/nano9050741