# Magnetocaloric Effect in an Antidot: The Effect of the Aharonov-Bohm Flux and Antidot Radius

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## Abstract

**:**

## 1. Introduction

_{x}Ga

_{1$-x$}As ($x=0.3$). Thus, the electrons in GaAs are confined in a 1-D potential well of length L in the z-direction. Therefore, electrons are trapped in 2D space, where a magnetic field along z-axis can be applied [40]. A natural extension of the work [41] corresponds to the study of the magnetocaloric response for an ensemble of antidots. In simple words, an antidot is a potential hill inaccessible to 2D electrons [42,43,44,45,46,47]. The advances in technology allow these systems to work even below $T=1$ K in temperature [48,49,50,51]. The model used is the one proposed by Bogachek and Landman model [52], that constitutes a combination of repulsive potential ($U\left(r\right)\propto {r}^{-2}$) and attractive potential ($U\left(r\right)\propto {r}^{2}$) leaving the electron confined in a finite region of space. Therefore, we investigated a confined electron in a ring topology in the presence of a uniform external magnetic field and subjected to an Aharonov-Bohm flux in the middle of the ring, as shown in Figure 1. In particular, we show that the Aharonov-Bohm flux can be detected by measuring the magnetocaloric effect.

## 2. Model

#### Magnetocaloric Observables

## 3. Results and Discussion

#### 3.1. Influence of Antidot Radius on the MCE

#### 3.2. The Influence of AB-flux in the MCE for Antidots

#### 3.3. The Role of the Harmonic Trap in the MCE Effect

#### 3.4. The Role of the Spin in the MCE Effect for Antidot

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Pictorial representation of an antidot, with an electron trapped in a ring structure subjected to an uniform magnetic field, plus an Aharonov-Bohm flux in the middle of the ring, depicted as an infinite solenoid producing a magnetic field confined inside it.

**Figure 2.**

**Upper row:**Specific heat, magnetization and entropy for a quantum dot without intrinsic spin for our numerical calculations using the parameters $\alpha =0$ and $a=0$ in Equation (6). The inset images correspond to the exact calculations obtained in the Reference [41] for the same observables. We clearly observe a very good convergence of numerical results.

**Lower row:**Specific heat, magnetization and entropy for the case of antidot with $a=0.5$ and $\alpha =0.5$. The inset images correspond to the exact calculations obtained in the Reference [41] for the case of an electron in a dot with an intrinsic spin. We observe here similar behaviour at low temperatures for the thermodynamics observables displayed. Therefore, for the thermal observables, the inclusion of AB-flux in an antidot shows similar behaviour as a function of temperature as compared to the case of an electron trapped in a quantum dot with intrinsic spin.

**Figure 3.**Entropy as a function of temperature for different values of the a parameter in absence of AB-flux. The range of the external magnetic field is between $0.6\le B\le 5$ in Tesla units. (

**a**) Entropy for the case of Fock-Darwin energy levels (i.e., $\alpha =0,a=0$) which represents an electron trapped in a quantum dot. (

**b**) Antidot entropy with $a=1.5$. (

**c**) Antidot entropy with $a=3.0$. We observe in (

**b**,

**c**), non monotonic behavior of S vs T for some magnetic fields at low temperatures.

**Figure 4.**$-\Delta S$ as a function of temperatures for different values of a parameter in absence of AB-flux. The range of the external magnetic field is between $0.6\le B\le 5$ in units of Tesla. (

**a**)$-\Delta S$ for the case of Fock-Darwin energy levels (i.e., $\alpha =0,a=0$) which represents an electron trapped in a quantum dot. Clearly we always appreciate negative values and absence of crosses for different values of external magnetic field. (

**b**) $-\Delta S$ for and antidot with $a=1.5.$ (

**c**) $-\Delta S$ for an antidot with $a=3.0$. Figure b,c show positive values for $-\Delta S$ at low temperatures, $T<7$ K and then negative values for the entire remaining temperature range.

**Figure 5.**$-\Delta S$ as a function of temperature for large sizes of antidot radii. The (

**a**) panel correspond to the case of $a=5$ and the (

**b**) panel the case of $a=10$. Clearly we see that $-\Delta S<0$ therefore direct MCE ($\Delta T>0$) does not occur for this choice of parameters.

**Figure 6.**MCE effect for electrons in an antidot in absence of AB-flux. $\Delta T$ as a function of temperatures for different values of antidot radii. The (

**a**) panel correspond to a values of $a=0$. The (

**b**) panel corresponds to $a=1.5$ and the (

**c**) panel to $a=3.0$. For all graphics shown here, the initial value of the magnetic field is given by ${B}_{i}=0.6$ T. The quantity $\Delta T(T,B)$ is in units of Kelvin. Here, the horizontal axis represents the initial temperature of the system.

**Figure 7.**$-\Delta S$ as a function of temperature between $0.1$ K to 40 K. In the (

**a**) panel we consider $\alpha =0$ and $a=0.2$ (pure antidot radius effect). In the (

**b**) panel we use $\alpha =0.5$ and $a=0.2$. We observe notorious positive peak close to 4 K for a direct MCE. The positive peak on the right is caused by the switching on of the AB-flux. For these two graphics, the value of the initial field is ${B}_{i}=0.6$ T.

**Figure 8.**MCE effect for a small antidot radius and the effect of AB-flux. In the (

**a**) panel we plot the case in the absence of AB-flux. We observe only the typical inverse response in the MCE effect. The case of $\alpha =0.5$ is presented in the (

**b**) panel, a positive MCE is observed at low temperatures caused by the AB-flux. The quantity $\Delta T(T,B)$ is in units of Kelvin. Here, the horizontal axis represents the initial temperature of the system.

**Figure 9.**Comparative MCE effect for a fixed small antidot radius and different values of the AB-flux. The a parameter is fixed at the value of $a=0.2$. The (

**a**) panel shows the results for the case $\alpha =0.2$, in the (

**b**) panel results for the case $\alpha =0.5$ and the (

**c**) panel, results for $\alpha =0.8$. The quantity $\Delta T(T,B)$ is in units of Kelvin. Here, the horizontal axis represents the initial temperature of the system.

**Figure 10.**MCE effect for electron in an antidot with AB-flux in different direction. In the (

**a**) panel we show the case without AB-flux. The (

**b**,

**c**) panels shows a comparative MCE effect for a positive AB-flux and negative AB-flux respectively. The quantity $\Delta T(T,B)$ is in units of Kelvin. Here, the horizontal axis represents the initial temperature of the system.

**Figure 11.**MCE effect ($\Delta T$) for three different values of harmonic trap frequencies and three different values of AB-flux, with a fixed value of the antidot radius $a=1.2$.

**Upper row:**We display the case of $\alpha =0.6$,

**middle row:**$\alpha =0$ and

**lower row:**$\alpha =-0.6$.

**Left column:**We treat the case of parabolic trap frequency ${\omega}_{d}=2.2\times {10}^{12}$ s ${}^{-1}$, which in terms of energy represent $1.448$ meV.

**Central column**: The case of ${\omega}_{d}=4.4\times {10}^{12}$ s${}^{-1}$, which in terms of energy represent 2.896 meV.

**Right column:**The case of ${\omega}_{d}=8.8\times {10}^{12}$ s${}^{-1}$, which in terms of energy represent 5.792 meV. The inset in each figure shows $\Delta T$ in a larger range of temperature, up to $T=50$ K. In general we observe an enhancement of the positive peak in the MCE for the system with higher frequency. In addition, the differences in the MCE for the cases with positive and negative AB fluxes can be noticed in the system with higher frequency. Therefore, there is a clear way to distinguish an AB flux by measuring the MCE. The quantity $\Delta T(T,B)$ is in units of Kelvin. Here, the horizontal axis represents the initial temperature of the system.

**Figure 12.**The MCE effect for a electron with spin in an antidot. For all graphics displayed in this figure, we use the value of ${\omega}_{d}=8.8\times {10}^{12}$ s${}^{-1}$ and for a para meter the value of $a=1.2$. This case corresponds to the one shown in Figure 11, right column, for a spinless electron. For (

**a**) we select $\alpha =0.6$, for (

**b**) the case of $\alpha =0$ and for (

**c**) $\alpha =-0.6$. The quantity $\Delta T(T,B)$ is in units of Kelvin. Here, the horizontal axis represents the initial temperature of the system.

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**MDPI and ACS Style**

Negrete, O.A.; Peña, F.J.; Vargas, P. Magnetocaloric Effect in an Antidot: The Effect of the Aharonov-Bohm Flux and Antidot Radius. *Entropy* **2018**, *20*, 888.
https://doi.org/10.3390/e20110888

**AMA Style**

Negrete OA, Peña FJ, Vargas P. Magnetocaloric Effect in an Antidot: The Effect of the Aharonov-Bohm Flux and Antidot Radius. *Entropy*. 2018; 20(11):888.
https://doi.org/10.3390/e20110888

**Chicago/Turabian Style**

Negrete, Oscar A., Francisco J. Peña, and Patricio Vargas. 2018. "Magnetocaloric Effect in an Antidot: The Effect of the Aharonov-Bohm Flux and Antidot Radius" *Entropy* 20, no. 11: 888.
https://doi.org/10.3390/e20110888