# Magnetocaloric Effect in Non-Interactive Electron Systems: “The Landau Problem” and Its Extension to Quantum Dots

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}O

_{2}tunable by external electromagnetic fields and that was extended by Chotorlishvili et al. [40] under the implementation of shortcuts to adiabaticity, finding a reasonable output power for the proposed machine. Therefore, the study of the role of degeneracy may be of interest to the MCE community. Besides, nowadays, it is physically possible to confine electrons in two dimensions (2D). For instance, quantum confinement can be achieved in semiconductor heterojunctions, such as GaAs and AlGaAs. At room temperature, the bandgap of GaAs is 1.43 eV, while it is 1.79 eV for AlxGa

_{1−x}As ($x=0.3$). Thus, the electrons in GaAs are confined in a 1D potential well of length L in the z-direction. Therefore, electrons are trapped in 2D space, where a magnetic field along the z-axis can be applied [45].

## 2. Model

#### Magnetocaloric Observables

## 3. Results and Discussion

#### 3.1. Landau Problem: Influence of Energy Degeneracy on the MCE

#### 3.2. MCE for Electrons Trapped in a Quantum Dot

_{e}. This effective mass is associated with a cylindrical quantum dot of GaAs with a typical radius of 20 to 100 nm [54,55]. For the characteristic frequency of the trap ${\omega}_{d}$, we use the value of $1.6407\times {10}^{12}\phantom{\rule{4pt}{0ex}}{\mathrm{s}}^{-1}$, which in terms of energy represents approximately $\hslash {\omega}_{d}\sim 1.07$ meV. The selection of this particular value is to compare the intensity of the trap with the typical energy of intra-band optical transition of the quantum dots. The order of this transition is approximately around ∼1 meV for GaAs quantum dots [54].

#### 3.3. MCE for Electrons with Spin Trapped in a Quantum Dot

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Warburg, E. Magnetische Untersuchungen. Ueber einige Wirkungen der Coërcitivkraft. Ann. Phys. (Leipzig)
**1881**, 249, 141–164. (In German) [Google Scholar] [CrossRef] - Weiss, P.; Piccard, A. Le pheénoméne magnétocalorique. J. Phys. (Paris)
**1917**, 7, 103–109. (In French) [Google Scholar] - Weiss, P.; Piccard, A. Sur un nouveau phénoméne magnétocalorique. Comptes Rendus
**1918**, 166, 352–354. (In French) [Google Scholar] - Debye, P. Einige Bemerkungen zur Magnetisierung bei tiefer Temperatur. Ann. Phys.
**1926**, 81, 1154–1160. (In Germany) [Google Scholar] [CrossRef] - Giauque, W.F.; Macdougall, D.P. The Production of Temperatures below One Degree Absolute by Adiabatic Demagnetization of Gadolinium Sulfate. J. Am. Chem. Soc.
**1935**, 57, 1175–1185. [Google Scholar] [CrossRef] - Brown, G.V. Magnetic heat pumping near room temperature. J. Appl. Phys.
**1976**, 47, 3673–3680. [Google Scholar] [CrossRef] - Pecharsky, V.K.; Gschneidner, K.A., Jr. Giant Magnetocaloric Effect in Gd
_{5}(Si_{2}Ge_{2}). Phys. Rev. Lett.**1997**, 78, 4494–4497. [Google Scholar] [CrossRef] - Pathak, A.K.; Gschneidner, K.A.; Pecharsky, V.K. Negative to positive magnetoresistance and magnetocaloric effect in Pr
_{0.6}Er_{0.4}Al_{2}. J. Alloys Compd.**2015**, 621, 411–414. [Google Scholar] [CrossRef] - Florez, J.M.; Vargas, P.; Garcia, C.; Ross, C.A. Magnetic entropy change plateau in a geometrically frustrated layered system: FeCrAs-like iron-pnictide structure as a magnetocaloric prototype. J. Phys. Condens. Matter
**2013**, 25, 226004. [Google Scholar] [CrossRef] [PubMed] - Hudl, M.; Campanini, D.; Caron, L.; Hoglin, V.; Sahlberg, M.; Nordblad, P.; Rydh, A. Thermodynamics around the first-order ferromagnetic phase transition of Fe
_{2}P single crystals. Phys. Rev. B**2014**, 90, 144432. [Google Scholar] [CrossRef] - Miao, X.F.; Caron, L.; Roy, P.; Dung, N.H.; Zhang, L.; Kockelmann, W.A.; De Grootm, R.A.; Van Dijk, N.H.; Brück, E. Tuning the phase transition in transition-metal-based magnetocaloric compounds. Phys. Rev. B
**2014**, 89, 174429. [Google Scholar] [CrossRef] [Green Version] - Sosin, S.; Prozorova, L.; Smirnov, A.; Golov, A.; Berkutov, I.; Petrenko, O.; Balakrishnan, G.; Zhitomirsky, M.E. Magnetocaloric effect in pyrochlore antiferromagnet Gd
_{2}Ti_{2}O_{7}. Phys. Rev. B**2005**, 71, 2005094413. [Google Scholar] [CrossRef] - Wang, F.; Yuan, F.-Y.; Wang, J.-Z.; Feng, T.-F.; Hu, G.-Q. Conventional and inverse magnetocaloric effect in Pr
_{2}CuSi_{3}and Gd_{2}CuSi_{3}compounds. J. Alloys Compd.**2014**, 592, 63–66. [Google Scholar] [CrossRef] - Du, Q.; Chen, G.; Yang, W.; Wei, J.; Hua, M.; Du, H.; Wang, C.; Liu, S.; Han, J.; Zhang, Y.; et al. Magnetic frustration and magnetocaloric effect in AlFe
_{2−x}Mn_{x}B_{2}(x = 0–0.5) ribbons. J. Phys. D-Appl. Phys.**2015**, 48, 335001. [Google Scholar] [CrossRef] - Balli, M.; Fruchart, D.; Zach, R. Negative and conventional magnetocaloric effects of a MnRhAs single crystal. J. Appl. Phys.
**2014**, 115, 203909. [Google Scholar] [CrossRef] - Kolat, V.S.; Izgi, T.; Kaya, A.O.; Bayri, N.; Gencer, H.; Atalay, S. Metamagnetic transition and magnetocaloric effect in charge-ordered Pr
_{0.68}Ca_{0.32−x}Sr_{x}MnO_{3}(x = 0, 0.1, 0.18, 0.26 and 0.32) compounds. J. Magn. Magn. Mater.**2010**, 322, 427433. [Google Scholar] [CrossRef] - Phan, M.H.; Morales, M.B.; Bingham, N.S.; Srikanth, H.; Zhang, C.L.; Cheong, S.-W. Phase coexistence and magnetocaloric effect in La
_{5/8−y}Pr_{y}Ca_{3/8}MnO_{3}(y = 0.275). Phys. Rev. B**2010**, 81, 094413. [Google Scholar] [CrossRef] - Patra, M.; Majumdar, S.; Giri, S.; Iles, G.N.; Chatterji, T. Anomalous magnetic field dependence of magnetocaloric effect at low temperature in Pr
_{0.52}Sr_{0.48}MnO_{3}single crystal. J. Appl. Phys.**2010**, 107, 076101. [Google Scholar] [CrossRef] - Szalowski, K.; Balcerzak, T. Normal and inverse magnetocaloric effect in magnetic multilayers with antiferromagnetic interlayer coupling. J. Phys. Condens. Matter
**2014**, 26, 386003. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Midya, A.; Khan, N.; Bhoi, D.; Mandal, P. Giant magnetocaloric effect in magnetically frustrated EuHo
_{2}O_{4}and EuDy_{2}O_{4}compounds. Appl. Phys. Lett.**2012**, 101, 132415. [Google Scholar] [CrossRef] - Moya, X.; Kar-Narayan, S.; Mathur, N.D. Caloric materials near ferroic phase transitions. Nat. Mater.
**2014**, 13, 439–450. [Google Scholar] [CrossRef] [PubMed] - Guillou, F.; Porcari, G.; Yibole, H.; van Dijk, N.; Bruck, E. Taming the First-Order Transition in Giant Magnetocaloric Materials. Adv. Mater.
**2014**, 26, 2671–2675. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Gong, Y.-Y.; Wang, D.-H.; Cao, Q.-Q.; Liu, E.-K.; Liu, J.; Du, Y.-W. Electric Field Control of the Magnetocaloric Effect. Adv. Mater.
**2014**, 27, 801–805. [Google Scholar] [CrossRef] [PubMed] - Nalbandyan, V.B.; Zvereva, E.A.; Nikulin, A.Y.; Shukaev, I.L.; Whangbo, M.-H.; Koo, H.-J.; Abdel-Hafiez, M.; Chen, X.J.; Koo, C.; Vasiliev, A.N.; et al. New Phase of MnSb
_{2}O_{6}Prepared by Ion Exchange: Structural, Magnetic, and Thermodynamic Properties. Inorg. Chem.**2015**, 54, 1705–1711. [Google Scholar] [CrossRef] [PubMed] - Tkac, V.; Orendacova, A.; Cizmar, E.; Orendac, M.; Feher, A.; Anders, A.G. Giant reversible rotating cryomagnetocaloric effect in KEr(MoO
_{4})_{2}induced by a crystal-field anisotropy. Phys. Rev. B**2015**, 92, 024406. [Google Scholar] [CrossRef] - Tamura, R.; Ohno, T.; Kitazawa, H. A generalized magnetic refrigeration scheme. Appl. Phys. Lett.
**2014**, 104, 052415. [Google Scholar] [CrossRef] [Green Version] - Tamura, R.; Tanaka, S.; Ohno, T.; Kitazawa, H. Magnetic ordered structure dependence of magnetic refrigeration efficiency. J. Appl. Phys.
**2014**, 116, 053908. [Google Scholar] [CrossRef] [Green Version] - Li, G.; Wang, J.; Cheng, Z.; Ren, Q.; Fang, C.; Dou, S. Large entropy change accompanying two successive magnetic phase transitions in TbMn
_{2}Si_{2}for magnetic refrigeration. Appl. Phys. Lett.**2015**, 106, 182405. [Google Scholar] [CrossRef] - Von Ranke, J.P.; Alho, B.P.; Nóbrega, B.P.; de Oliveira, N.A. Understanding the inverse magnetocaloric effect through a simple theoretical model. Phys. B
**2009**, 404, 056004. [Google Scholar] [CrossRef] - Von Ranke, J.P.; de Oliveira, N.A.; Alho, B.P.; Plaza, E.J.R.; de Sousa, V.S.R.; Caron, L.; Reis, M.S. Understanding the inverse magnetocaloric effect in antiferro- and ferrimagnetic arrangements. J. Phys. Condens. Matter
**2009**, 21, 3045–3047. [Google Scholar] [CrossRef] [PubMed] - Reis, M.S. Oscillating adiabatic temperature change of diamagnetic materials. Solid State Commun.
**2012**, 152, 921–923. [Google Scholar] [CrossRef] - Reis, M.S. Oscillating magnetocaloric effect on graphenes. Appl. Phys. Lett.
**2012**, 101, 222405. [Google Scholar] [CrossRef] - Reis, M.S. Step-like features on caloric effects of graphenes. Phys. Lett. A
**2014**, 378, 918–921. [Google Scholar] [CrossRef] [Green Version] - Reis, M.S. Magnetocaloric cycle with six stages: Possible application of graphene at low temperature. Appl. Phys. Lett.
**2015**, 107, 102401. [Google Scholar] [CrossRef] - Alisultanov, Z.Z.; Reis, M.S. Oscillating magneto- and electrocaloric effects on bilayer graphenes. Solid State Commun.
**2015**, 206, 17–21. [Google Scholar] [CrossRef] - Ma, N.; Reis, M.S. Barocaloric effect on graphene. Sci. Rep.
**2017**, 7, 13257. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Peña, F.J.; González, A.; Nunez, A.S.; Orellana, P.A.; Rojas, R.G.; Vargas, P. Magnetic Engine for the Single-Particle Landau Problem. Entropy
**2017**, 19, 639. [Google Scholar] [CrossRef] - Mehta, V.; Johal, R.S. Quantum Otto engine with exchange coupling in the presence of level degeneracy. Phys. Rev. E
**2017**, 96, 032110. [Google Scholar] [CrossRef] [PubMed] - Azimi, M.; Chorotorlisvili, L.; Mishra, S.K.; Vekua, T.; Hübner, W.; Berakdar, J. Quantum Otto heat engine based on a multiferroic chain working substance. New J. Phys.
**2014**, 16, 063018. [Google Scholar] [CrossRef] [Green Version] - Chotorlishvili, L.; Azimi, M.; Stagraczyński, S.; Toklikishvili, Z.; Schüler, M.; Berakdar, J. Superadiabatic quantum heat engine with a multiferroic working medium. Phys. Rev. E
**2016**, 94, 032116. [Google Scholar] [CrossRef] [PubMed] - Dong, C.D.; Lefkidis, G.; Hübner, W. Quantum Isobaric Process in Ni
_{2}. J. Supercond. Nov. Magn.**2013**, 26, 1589–1594. [Google Scholar] [CrossRef] - Dong, C.D.; Lefkidis, G.; Hübner, W. Quantum Magnetic quantum diesel in Ni
_{2}. Phys. Rev. B**2013**, 88, 214421. [Google Scholar] [CrossRef] - Hübner, W.; Lefkidis, G.; Dong, C.D.; Chaudhuri, D. Spin-dependent Otto quantum heat engine based on a molecular substance. Phys. Rev. B
**2014**, 90, 024401. [Google Scholar] [CrossRef] - Abah, O.; Roßnagel, J.; Deffner, S.; Schmidth-Kaler, F.; Singer, K.; Lutz, E. Single-Ion Heat Engine at Maximum Power. Phys. Rev. Lett.
**2012**, 109, 2033006. [Google Scholar] [CrossRef] [PubMed] - Mani, R.G.; Smet, J.H.; von Klitzing, K.; Narayanamurti, V.; Johnson, W.B.; Umansky, V. Zero-resistance states induced by electromagnetic-wave excitation in GaAs/AlGaAs heterostructures. Nature
**2002**, 420, 646–650. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Prance, J.R.; Smith, C.G.; Griffiths, J.P.; Chorley, S.J.; Anderson, D.; Jones, G.A.C.; Farrer, I.; Ritchie, D.A. Electronic Refrigeration of a Two-Dimensional Electron Gas. Phys. Rev. Lett.
**2009**, 102, 146602. [Google Scholar] [CrossRef] [PubMed] - Hübel, A.; Held, K.; Weis, J.; Klitzing, K.V. Correlated electron tunneling through two separate quantum dot systems with strong capacitive interdot coupling. Phys. Rev. Lett.
**2008**, 101, 186804. [Google Scholar] [CrossRef] [PubMed] - Hübel, A.; Weis, J.; Dietsche, W.; Klitzing, K.V. Two laterally arranged quantum dot systems with strong capacitive interdot coupling. Appl. Phys. Lett.
**2007**, 91, 102101. [Google Scholar] [CrossRef] - Donsa, S.; Andergassen, S.; Held, K. Double quantum dot as a minimal thermoelectric generator. Phys. Rev. B
**2014**, 89, 125103. [Google Scholar] [CrossRef] - Muñoz, E.; Peña, F.J.; González, A. Magnetically-Driven Quantum Heat Engines: The Quasi-Static Limit of Their Efficiency. Entropy
**2016**, 18, 173. [Google Scholar] [CrossRef] - Muñoz, E.; Peña, F.J. Magnetically driven quantum heat engine. Phys. Rev. E
**2014**, 89, 052107. [Google Scholar] [CrossRef] [PubMed] - Peña, F.J.; Muñoz, E. Magnetostrain-driven quantum heat engine on a graphene flake. Phys. Rev. E
**2015**, 91, 052152. [Google Scholar] [CrossRef] [PubMed] - Kumar, J.; Sreeram, P.A.; Dattagupta, S. Low-temperature thermodynamics in the context of dissipative diamagnetism. Phys. Rev. E
**2009**, 79, 021130. [Google Scholar] [CrossRef] [PubMed] - Jacak, L.; Hawrylak, P.; W’ojs, A. Quantum Dots; Springer: Berlin/Heidelberg, Germany, 1998. [Google Scholar]
- Muñoz, E.; Barticevic, Z.; Pacheco, M. Electronic spectrum of a two-dimensional quantum dot array in the presence of electric and magnetic fields in the Hall configuration. Phys. Rev. B
**2005**, 71, 165301. [Google Scholar] [CrossRef] - Reis, M.S. Oscillating magnetocaloric effect. Appl. Phys. Lett.
**2011**, 99, 052511. [Google Scholar] [CrossRef] - Grujić, M.; Zarenia, M.; Chaves, A.; Tadić, M.; Farias, G.A.; Peeters, F.M. Electronic and optical properties of a circular graphene quantum dot in a magnetic field: Influence of the boundary conditions. Phys. Rev. B
**2011**, 84, 205441. [Google Scholar] [CrossRef]

**Figure 1.**Pictorial representation of the systems. The left panel depicts the Landau problem. We recall that in our formulation, we do not consider the edge effects. Red arrows represent the external magnetic field perpendicular to the sample. The right panel depicts an electron with spin (blue arrow) trapped in a parabolic potential that represents an electron in a quantum dot.

**Figure 2.**General description of the idea of the magnetocaloric effect (MCE). (Left panel) Standard MCE: We start with our system at $T={T}_{1}$ and $B={B}_{1}$. By performing an adiabatic stroke to $B={B}_{2}$, the system heats up reaching $T={T}_{2}$, the system is in contact with a thermal reservoir reaching a temperature of $T={T}_{3}$. Now, we proceed with an adiabatic demagnetization stroke back to $B={B}_{1}$; therefore the system cools down to $T={T}_{4}$. Then the system is in contact with a sample to cool down; therefore our system reaches again a temperature of $T={T}_{1}$. (Right panel): Inverse MCE: We start with our system at $T={T}_{1}$ and $B={B}_{1}$. By performing an adiabatic stroke to $B={B}_{2}$, the system cools down to $T={T}_{2}$ (this is due to the decrease of the entropy of the system’s phonons). Here, the system is in contact with a sample to cool down; therefore our system reaches $T={T}_{3}$. Now, we proceed with an adiabatic demagnetization stroke back to $B={B}_{1}$; therefore the system now heats up to $T={T}_{4}$. The system is in contact with a thermal reservoir, therefore reaching again a temperature of $T={T}_{1}$.

**Figure 3.**(Left panel) Entropy (in ${k}_{B}$ units) for the non-degenerate case of the Landau problem, as a function of temperature (in Kelvin) and for different values of external magnetic field intensity (measured in Tesla) from 0.1 T to 5 T. The straight horizontal line represents the adiabatic line $S(T,B)/{k}_{B}=2$, cutting different magnetic fields’ entropies at different temperatures. The inset shows the entropy in the temperature range from $T=0$ K to $T=10$ K. (Right panel) Entropy change, $-\Delta S$, according to Equation (16), as a function of the temperature for the non-degenerate case, where $\Delta B={B}_{f}-{B}_{i}$. We display this figure with an initial value of the external magnetic field of ${B}_{i}=0.01\phantom{\rule{4pt}{0ex}}T$ to ${B}_{f}$ (from 0.1 T to 5 T) and for temperatures from $T=0.01$ K to $T=10$ K. The inset depicts the values for $\Delta S$ from $T=40$ K to $T=300$ K.

**Figure 4.**MCE for the non-degenerate case of the Landau problem as a function of temperature. We display $\Delta T$ with an initial value of the external magnetic field ${B}_{i}=$ 0.5 T to ${B}_{f}$ from 1 T (blue) to 2 T (red). The inset shows the values for $M(T,B)$ for external magnetic fields ranging from $B=$ 1 T to $B=$ 2 T.

**Figure 5.**(Left panel) Behavior of the entropy ${S}_{L}(T,B)$ for the Landau problem with degeneracy. We show ${S}_{L}(T,B)$ in the range of B between 0.1 T to 5 T and for a temperature up to 10 K (the area used was $\mathcal{A}\equiv 1$ mm

^{2}). It is clearly observed that the entropy grows with the magnetic field and approximately collapses to the same value at high temperatures, as we see in the inset graphic. (Right panel) $\Delta {S}_{L}$ for the degenerate scenario of the Landau problem. We show $-\Delta {S}_{L}(T,{B}_{i},{B}_{f})$ in a range of the external magnetic fields, for ${B}_{i}=$ 0.01 T and ${B}_{f}=$ 0.1 T to ${B}_{f}=$ 5 T and up to 10 K in temperature. The inset figure shows the variation of entropy for a range of temperatures between 10 K and 100 K, where we can clearly observe that the variation of $-\Delta S$ decreases approximately by a factor of 100 as compared to the low temperature behavior, as expected.

**Figure 6.**MCE for the Landau problem with degeneracy as a function of temperature. The main figure shows $\Delta T$ in a range of the external magnetic fields, from ${B}_{f}=$ 0.1 T to ${B}_{f}=$ 20 T at fixed ${B}_{i}=$ 0.01 T, and up to 10 K in the temperature scale. The inset shows the values for $M(T,B)$ for external magnetic fields ranging from $B=$ 0.01 T to $B=$ 5 T.

**Figure 7.**Spinless electrons in a quantum dot. Entropy ${S}_{d}(T,B)$ (left panels), $M(T,B)$ (middle panels) and ${C}_{B}(T,B)$ (right panels) as a function of temperature $\left(T\right)$ from 0 K to 30 K for different values of magnetic fields in the range of 0.1 T to 5 T. In the middle panels of the graphs, we selected the representative value for the characteristic frequency of the harmonic trap in ${\omega}_{d}=1.6407\times {10}^{12}$ s

^{−1}, which in terms of energy represents an approximate value of 1.07 meV. For the top panels, we use the value $\frac{{\omega}_{d}}{4}$ and in the bottom panels the case of $4{\omega}_{d}$ for the characteristic frequency of the dot structure (${\omega}_{B}=\frac{eB}{0.067{m}_{e}}\equiv 2.63\times {10}^{12}\phantom{\rule{4pt}{0ex}}B$ s

^{−1}, where B is in Tesla units for comparison).

**Figure 8.**MCE effect for spinless electrons in a quantum dot. $\Delta T$ for: (i) ${\omega}_{B}<{\omega}_{d}$ (left panel), (ii) ${\omega}_{B}\sim {\omega}_{d}$ (middle panel) and (iii) ${\omega}_{B}>{\omega}_{d}$ (right panel). The insets of all graphics in this figure: $-\Delta S$ in units of ${k}_{B}$ as a function of temperature for: (i) ${\omega}_{B}<{\omega}_{d}$ (left panel), (ii) ${\omega}_{B}\sim {\omega}_{d}$ (middle panel) and (iii) ${\omega}_{B}>{\omega}_{d}$ (right panel). For all graphics presented in this figure, we have selected the value of $\hslash {\omega}_{d}\sim 1.07$ meV (i.e., ${\omega}_{d}=1.6407\times {10}^{12}$ s

^{−1}).

**Figure 9.**Electrons with spin trapped in a quantum dot. Entropy ${S}_{dS}(T,B)$ (left panel), $M(T,B)$ (middle panel) and ${C}_{B}(T,B)$ (right panel) as a function of temperature $\left(T\right)$ from different regions of temperature between 0 K to 80 K for different values of the magnetic field in the range of 0.1 T to 5 T. In the middle panels of the graphs, we selected the representative value for the characteristic frequency of the harmonic trap in ${\omega}_{d}=1.6407\times {10}^{12}$ s

^{−1}, which in terms of energy represents an approximate value of 1.07 meV. For the top panels, we use the value $\frac{{\omega}_{d}}{4}$, and in the bottom panels, the case of $4{\omega}_{d}$ for the characteristic frequency of the dot structure is used. The insets show the thermodynamic quantities (in the left and right panel) in an extended temperature range between $T=0$ K and $T=100$ K.

**Figure 10.**$-\frac{\Delta {S}_{dS}}{{k}_{B}}$ as a function of temperature for different values of characteristic dot frequencies. The middle panel corresponds to a value of ${\omega}_{d}=1.6407\times {10}^{12}$ s

^{−1}. For the left panel, we selected the value of $\frac{{\omega}_{d}}{4}$ and for the right panel the value of $4{\omega}_{d}$. The range of the external magnetic field values, ${B}_{f}$, is between 0.02 T and 5 T, and ${B}_{i}=$ 0.01 T. The temperature range is from $T=0.01$ K to $T=30$ K.

**Figure 11.**Entropy function ${S}_{dS}(T,B)$ of an electron with spin trapped in a quantum dot. We plot the entropy for two different values of the external magnetic field as a function of temperature. We use the characteristic value of ${\omega}_{d}=1.6407\times {10}^{12}$ s

^{−1}. The dot-dashed line corresponds to the external magnetic field $B$ = 0.01 T, and the dashed line corresponds to the value of $B$ = 5 T.

**Figure 12.**MCE effect for electrons with spin trapped in a quantum dot. $\Delta T$ as a function of temperature for different values of characteristic frequencies. The middle panel corresponds to a value of ${\omega}_{d}=1.6407\times {10}^{12}$ s

^{−1}. The left panel corresponds to the value of $\frac{{\omega}_{d}}{4}$, and the right panel depicts the results using the value of $4{\omega}_{d}$. The range of the external magnetic field values, ${B}_{f}$, is between 0.02 T and 5 T, and ${B}_{i}=$ 0.01 T. The temperature range is from $T=0.01$ K to $T=120$ K. The insets show $\Delta T$ values zoomed in to a smaller range of temperatures and for near room temperature (left and middle panel).

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Negrete, O.A.; Peña, F.J.; Florez, J.M.; Vargas, P.
Magnetocaloric Effect in Non-Interactive Electron Systems: “The Landau Problem” and Its Extension to Quantum Dots. *Entropy* **2018**, *20*, 557.
https://doi.org/10.3390/e20080557

**AMA Style**

Negrete OA, Peña FJ, Florez JM, Vargas P.
Magnetocaloric Effect in Non-Interactive Electron Systems: “The Landau Problem” and Its Extension to Quantum Dots. *Entropy*. 2018; 20(8):557.
https://doi.org/10.3390/e20080557

**Chicago/Turabian Style**

Negrete, Oscar A., Francisco J. Peña, Juan M. Florez, and Patricio Vargas.
2018. "Magnetocaloric Effect in Non-Interactive Electron Systems: “The Landau Problem” and Its Extension to Quantum Dots" *Entropy* 20, no. 8: 557.
https://doi.org/10.3390/e20080557