Control of Magnetocaloric Effect in Quantum Dots Using Electrical Field at Low Temperatures

Abstract

In this work, we consider a quantum dot in the presence of electric and magnetic fields simultaneously. Given the importance of studying the magnetocaloric effect (MCE), we have tried to investigate MCE in this system and obtain the influence of system parameters on the MCE. For this purpose, we obtained the energy levels and eigenstates analytically. Our purpose is to study the effects of temperature and electric field on the entropy change ($\Delta S$). To this end, we have employed the non-extensive thermodynamic to study the MCE. We have found that the non-extensive parameter, quantum dot radius and the electric field are important factors to control the MCE. The obtained results also show that our system presents the direct MCE. Also, by applying the non-extensive thermodynamics, we cannot see the oscillating MCE at low temperatures.

Appendix

In this part, we present the details of calculation of energy levels. The Schrödinger equation is expressed by

$$\frac{1}{2{m}^{*}}{\left({\varvec{P}}-\frac{e}{c}{\varvec{A}}\right)}^{2}\Psi -e\epsilon z\Psi +{V}_{conf}(\mathbf{r})\Psi ={E}_{t}\Psi$$

Here,

$$V\left(r\right)=\frac{1}{2}{m}^{*}{\omega }_{0}^{2}{r}^{2}=\frac{1}{2}{m}^{*}{\omega }_{0}^{2}\left({\rho }^{2}+{z}^{2}\right)$$

In cylindrical coordinates, we have

$$-\frac{{\mathrm{\hslash }}^{2}}{2{m}^{*}}\left[\frac{1}{\uprho }\frac{\partial }{\partial\uprho }\left(\uprho \frac{\partial }{\partial\uprho } \right)+\frac{{\partial }^{2}}{\partial {z}^{2}}+\frac{1}{{\uprho }^{2}}\frac{{\partial }^{2}}{\partial {\varphi }^{2}}\right]\Psi -\frac{i\hslash {\omega }_{c}}{2}\frac{\partial\Psi }{\partial \varphi }+\frac{{m}^{*}{\Omega }^{2}{\uprho }^{2}}{8}\Psi -e\epsilon z\Psi +\frac{{m}^{*}{\omega }_{0}^{2}{\mathrm{z}}^{2}}{2}\Psi ={E}_{t}\Psi$$

here ${E}_{t}={E}_{\rho }+{E}_{z}$.

Consider the wave function as

$$\Psi \left(\rho ,\varphi ,z\right)=f(\rho ,\varphi )g(z)$$

By using the above relation, we have

$$-\frac{{\mathrm{\hslash }}^{2}}{2{m}^{*}}\left[\frac{1}{\uprho }\frac{\partial }{\partial\uprho }\left(\uprho \frac{\partial }{\partial\uprho } \right)+\frac{1}{{\uprho }^{2}}\frac{{\partial }^{2}}{\partial {\varphi }^{2}}\right]f-\frac{i\hslash {\omega }_{c}}{2}\frac{\partial f}{\partial \varphi }+\frac{{m}^{*}{\Omega }^{2}{\uprho }^{2}}{8}f={E}_{\rho }f$$

$$-\frac{{\mathrm{\hslash }}^{2}}{2{m}^{*}}\frac{{d}^{2}g}{d{z}^{2}}+\frac{{m}^{*}{\omega }_{0}^{2}{\mathrm{z}}^{2}}{2}g-e\epsilon zg={E}_{z}g$$

The solutions of the above equations are as

$$f\left(\rho ,\varphi \right)=A{e}^{im\varphi }{e}^{-\frac{{\rho }^{2}}{4{a}^{2}}}{\rho }^{\left|m\right|}F(-{n}_{\rho },\left|m\right|+1;\frac{{\rho }^{2}}{2{a}^{2}})$$

$$g\left(z\right)=B{e}^{-\frac{{m}^{*}{\omega }_{0}}{\hslash }{\left(z-{z}_{0}\right)}^{2}}{H}_{{n}_{z}}\left[\sqrt{\frac{{m}^{*}{\omega }_{0}}{\hslash }}\left(z-{z}_{0}\right)\right]$$

here $A$ and $B$ are the normalization constants, $a=\sqrt{\frac{\hslash }{{m}^{*}\Omega }}$, ${z}_{0}=\frac{e\epsilon }{{m}^{*}{\omega }_{0}^{2}}$, $F(\mu ,\lambda ;x)$ is the confluent hypergeometric function and ${H}_{n}(x)$ is the Hermite function. Also, ${n}_{\rho }$, $m$ and ${n}_{z}$ are the quantum numbers, respectively.

Using the above wave function, the total energy is obtained as

$${E}_{{n}_{\rho },m,{n}_{z}}=\left({n}_{\rho }+\frac{\left|m\right|+1}{2} \right)\mathrm{\hslash \Omega }+\frac{m\mathrm{\hslash }{\omega }_{c}}{2}+\left({n}_{z}+\frac{1}{2} \right)\mathrm{\hslash }{\omega }_{0}-\frac{{e}^{2}{\epsilon }^{2}}{2{m}^{*}{\omega }_{0}^{2}}$$