# Enhanced Efficiency at Maximum Power in a Fock–Darwin Model Quantum Dot Engine

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## Abstract

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## 1. Introduction

## 2. Model

## 3. The Endoreversible Otto Cycle

## 4. Results and Discussions

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**a**–

**d**) Contour maps showing constant entropy curves as a function of temperature, T, (in Kelvin) and the external magnetic field B (in Tesla) for different values of the geometric confinement with ${\omega}_{0}=2.67$ THz.

**Figure 2.**Entropy as a function of temperature for magnetic field strengths of $B=0$ T (blue), $B=2$ T (orange), $B=4$ T (magenta), and $B=6$ T (red).

**Figure 3.**Entropy versus external field diagram for the endoreversible Otto Cycle. Note that the system is only in contact with the thermal reservoirs during the isochoric (vertical) strokes. Under the assumptions of endoreversibility, the working substance does not fully thermalize to the temperatures of the hot and cold reservoirs, ${T}_{h}$ and ${T}_{c}$, at points 3 and 1.

**Figure 4.**Total work as a function of (

**a**) the external field and (

**b**) the compression ratio for geometric confinement frequencies of $0.5{\omega}_{0}$ (blue), ${\omega}_{0}$ (orange), $1.5{\omega}_{0}$ (magenta), and $2{\omega}_{0}$ (red). We used ${T}_{c}=13$ K, ${T}_{h}=25$ K, and ${B}_{1}=1$ T. Note the clear decrease in the net work with increasing values of the dot confinement. In particular, for these values a transition to negative net work is observed for the case of $2{\omega}_{0}$ (red curve). At this transition point the cycle switches from behaving as an engine to behaving as a refrigerator.

**Figure 5.**Efficiency as a function of r for geometric confinement frequencies of $0.5{\omega}_{0}$ (blue), ${\omega}_{0}$ (orange), $1.5{\omega}_{0}$ (magenta), and $2{\omega}_{0}$ (red). We used ${T}_{c}=13$ K and ${T}_{h}=25$ K. We observe that the system’s efficiency increases as we increase the geometric confinement.

**Figure 6.**Cycle behavior as a function of the external magnetic field strength and cold bath temperature for a geometric confinement of (

**a**) 0.5${\omega}_{0}$ and (

**b**) $2{\omega}_{0}$. Note the increased size of the refrigerator region as the value of the parabolic trap frequency increases.

**Figure 7.**Efficiency at maximum power as a function of the bath temperature ratio for $0.5{\omega}_{0}$ (blue, short dashed), $1.0{\omega}_{0}$ (green, dot-dashed), $1.5{\omega}_{0}$ (brown, long dashed), and $2.0{\omega}_{0}=$ (cyan, dot-dash-dashed). The Carnot (black, dotted) and Curzon-Ahlborn (red, solid) efficiencies are provided for comparison. We observe that the efficiency at maximum power exceeds CA for lower values of dot confinement frequency at low bath temperature ratios. Parameters are ${T}_{h}$ = 25 K and magnetic field ${B}_{h}=$ 12 T.

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**MDPI and ACS Style**

Peña, F.J.; Myers, N.M.; Órdenes, D.; Albarrán-Arriagada, F.; Vargas, P.
Enhanced Efficiency at Maximum Power in a Fock–Darwin Model Quantum Dot Engine. *Entropy* **2023**, *25*, 518.
https://doi.org/10.3390/e25030518

**AMA Style**

Peña FJ, Myers NM, Órdenes D, Albarrán-Arriagada F, Vargas P.
Enhanced Efficiency at Maximum Power in a Fock–Darwin Model Quantum Dot Engine. *Entropy*. 2023; 25(3):518.
https://doi.org/10.3390/e25030518

**Chicago/Turabian Style**

Peña, Francisco J., Nathan M. Myers, Daniel Órdenes, Francisco Albarrán-Arriagada, and Patricio Vargas.
2023. "Enhanced Efficiency at Maximum Power in a Fock–Darwin Model Quantum Dot Engine" *Entropy* 25, no. 3: 518.
https://doi.org/10.3390/e25030518