# Magnetic Otto Engine for an Electron in a Quantum Dot: Classical and Quantum Approach

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## Abstract

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## 1. Introduction

## 2. Model

## 3. First Law of Thermodynamics and the Quantum and Classical Otto Cycle

- Finding the relation between the magnetic field and the temperature along an isentropic trajectory by solving the differential equation of first order given by$$dS(B,T)={\left(\frac{\partial S}{\partial B}\right)}_{T}dB+{\left(\frac{\partial S}{\partial T}\right)}_{B}dT=0,$$$$\frac{dB}{dT}=-\frac{{C}_{B}}{T{\left(\frac{\partial S}{\partial B}\right)}_{T}},$$
- By matching two points within an isentropic trajectory$$\begin{array}{c}S({T}_{l},{B}_{h})=S({T}_{\mathrm{A}},{B}_{l})\hfill \\ S({T}_{h},{B}_{l})=S({T}_{\mathrm{C}},{B}_{h})\phantom{\rule{0.277778em}{0ex}},\hfill \end{array}$$

## 4. Results and Discussions

#### 4.1. Classical Magnetic Otto Cycle

#### 4.2. Magnetic Quantum Otto Cycle

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Fock–Darwin energy spectrum with $\sigma =-1$ for the first six radial number $n=0,1,\dots ,6$ and for each of them the azimuthal quantum number taking the values between $m=-6,-5,\dots ,5,6$. (

**b**) Fock–Darwin energy spectrum with $\sigma =+1$ for the first six radial number $n=0,1,\dots ,6$ and for each of them the azimuthal quantum number taking the values between $m=-6,-5,\dots ,5,6$. We clearly observe the confinement of the energy levels at high magnetic fields $({\omega}_{c}/2{\omega}_{0}>>1)$.

**Figure 2.**Classical thermodynamic quantities entropy $\left(S\right)$, internal energy $\left(U\right)$ and magnetization $\left(M\right)$ as a function of: external magnetic field (B) (

**a**–

**c**); and temperature (T) (

**d**–

**f**). In (

**a**–

**c**), the colors blue to red represent temperatures from $0.1$ K to 10 K, respectively. For (

**d**–

**f**), the colors blue to red represent lower to higher external magnetic field, from $0.1$ T to 5 T. The value of the parabolic trap is approximately to 1.7 meV. Additionally, we show how the Otto cycle appears in terms of the thermodynamic quantities considered.

**Figure 3.**The magnetic Otto engine represented as an entropy $\left(S\right)$ versus a magnetic field $\left(B\right)$ diagram. The way to perform the cycle is in the form $\mathrm{B}\to \mathrm{A}\to \mathrm{D}\to \mathrm{C}\to \mathrm{B}$.

**Figure 4.**Solution of classical isentropic path. (

**a**) The entropy as a function of magnetic field (horizontal axis) and temperature (vertical axis). The level curves (constant entropy values) highlight three different cases for S: first, red-black curve corresponding to $S=0.05$; secondly, yellow-black curve, corresponding to $S=0.10$ and finally, white-black curve for the case of $S=0.13$. (

**b**) The three constant values for the entropy ($S=0.05,S=0.10,S=0.13$) in a graphic of entropy as a function of B for temperatures from 1 K (blue) up to 10 K (red). Due to the form of the entropy obtained for this system, the solution for $S=0.13$ needs to work with temperatures higher than 10 K for an external magnetic field lower than 3 T (white dots in (

**a**,

**b**)). The value of the parabolic trap corresponds to 1.7 meV.

**Figure 5.**Proposed magnetic Otto cycle showing three different thermodynamic quantities, Entropy (S), Magnetization (M) and Internal Energy (U) ((

**a**–

**c**), respectively) as a function of the external magnetic field and different temperatures from 0.1K (blue) to 10K (red). (

**d**) The total work extracted multiplied by efficiency $\left(\mathcal{W}\eta \right)$; (

**e**) the total work extracted $\left(\mathcal{W}\right)$; and (

**f**) the efficiency $\left(\eta \right)$ for the classical cycle. The black points in (

**d**–

**f**) represent exactly the cycle B → A → D → C → B, presented in (

**a**–

**c**). The value of the parabolic trap corresponds to 1.7 meV. The fixed temperatures are ${T}_{\mathrm{B}}=6.19$ K and ${T}_{\mathrm{D}}=10$ K.

**Figure 6.**Proposed magnetic Otto cycle in three different thermodynamics quantities, Entropy, Magnetization and internal energy ((

**a**–

**c**), respectively) as a function of the external magnetic field and different temperatures from 0.1 K (blue) to 10 K (red). Total work extracted multiplied by efficiency $\left(W\eta \right)$ (

**d**) total work extracted $\left(W\right)$ (

**e**) and efficiency $\left(\eta \right)$ (

**d**) for the cycle. The black point in (

**d**–

**f**) represents the value of 0.02 meV of total work extracted and corresponds exactly to the cycle B → A → D → C → B, shown in (

**a**–

**c**). The value of the parabolic trap correspond to 1.7 meV. The fixed temperatures are ${T}_{\mathrm{B}}=2.69$ K and ${T}_{\mathrm{D}}=5.40$ K.

**Figure 7.**Work, efficiency and work multiply by efficiency (

**a**–

**c**) for different values of ${T}_{\mathrm{D}}$ for ${T}_{\mathrm{B}}=2.69$ fixed. The value of the parabolic trap corresponds to 1.7 meV.

**Figure 8.**(

**a**) Total work extracted for classical (blue line) and quantum version of Otto cycle (red line). The parameters for this case displayed are: ${T}_{\mathrm{D}}=10$ K, ${T}_{\mathrm{B}}=6.19$ K and ${B}_{\mathrm{B}}=4$ T as starting value of the external magnetic field. The value of ${B}_{\mathrm{D}}$ moves from 4 T to $1.99$ T and this variation is reflected in the movement of r in the form of $r=\sqrt{\frac{4}{{B}_{\mathrm{D}}}}$, same parameter as the results shown in Figure 5. (

**b**) Total work extracted (W) presented in Figure 6e versus the values obtaining in the quantum version of the Otto cycle. The parameters for this figure are ${T}_{\mathrm{D}}=5.40\phantom{\rule{4pt}{0ex}}K$, ${T}_{\mathrm{B}}=2.69\phantom{\rule{4pt}{0ex}}K$ and ${B}_{\mathrm{B}}=2.995$ T and ${B}_{\mathrm{D}}$ moves from $2.995$ to $0.250$ T. The parabolic trap is fixed to the value of 1.7 meV and the effective mass value of ${m}^{*}=0.067{m}_{e}$.

**Figure 9.**Total quantum work extracted $\left(W\right)$ per energy level for the case of $\sigma =1$ (

**a**) and for the case of $\sigma =-1$ (

**b**). The lines marked with circles correspond to the sum of all contributions of the energy level for each spin. The parameters used for this figure are the same as the one used in Figure 8b.

**Figure 10.**$\eta \times W$ (

**a**); and total work extracted (

**b**,

**c**) efficiency for the case of $\Delta T={T}_{h}-{T}_{l}=3\phantom{\rule{4pt}{0ex}}K$ for different regions of temperature parameter for classical approach (solid line) and quantum version of the magnetic Otto cycle (dotted line). For all graphics, we use the initial external magnetic field in the value of ${B}_{\mathrm{B}}=3.5$ T and the minimum value of the field, ${B}_{D}$ moves between $3.5$ T and $2.0$ T. Therefore, the r parameter moves between $1\le r\le 1.32$. The parabolic trap is fixed to the value of 1.7 meV and the effective mass value of ${m}^{*}=0.067{m}_{e}$.

**Figure 11.**$\eta \times W$ (

**a**); and total work extracted (

**b**,

**c**) efficiency for the case of $\Delta T={T}_{h}-{T}_{l}=3\phantom{\rule{4pt}{0ex}}K$ for different regions of temperature parameter. For all cases, we use the initial external magnetic field at the value of ${B}_{\mathrm{B}}=5.0$ T and the minimum value of the field, ${B}_{D}$ moves between $5.0$ T and $3.5$ T. Therefore, the r parameter moves between $1\le r\le 1.19$. The parabolic trap is fixed to the value of 1.7 meV and the effective mass value of ${m}^{*}=0.067{m}_{e}$.

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## Share and Cite

**MDPI and ACS Style**

Peña, F.J.; Negrete, O.; Alvarado Barrios, G.; Zambrano, D.; González, A.; Nunez, A.S.; Orellana, P.A.; Vargas, P.
Magnetic Otto Engine for an Electron in a Quantum Dot: Classical and Quantum Approach. *Entropy* **2019**, *21*, 512.
https://doi.org/10.3390/e21050512

**AMA Style**

Peña FJ, Negrete O, Alvarado Barrios G, Zambrano D, González A, Nunez AS, Orellana PA, Vargas P.
Magnetic Otto Engine for an Electron in a Quantum Dot: Classical and Quantum Approach. *Entropy*. 2019; 21(5):512.
https://doi.org/10.3390/e21050512

**Chicago/Turabian Style**

Peña, Francisco J., Oscar Negrete, Gabriel Alvarado Barrios, David Zambrano, Alejandro González, Alvaro S. Nunez, Pedro A. Orellana, and Patricio Vargas.
2019. "Magnetic Otto Engine for an Electron in a Quantum Dot: Classical and Quantum Approach" *Entropy* 21, no. 5: 512.
https://doi.org/10.3390/e21050512