Electrocaloric effect in quantum dots using the non-extensive formalism

Abstract

In this work, we consider a quantum dot in the presence of electric and magnetic fields simultaneously. The Schrodinger equation is analytically solved, and energy levels and eigenstates are determined. Then, we have calculated the changes in entropy using the non-extensive formalism. To this end, we have used the Tsallis entropy formalism to study the electrocaloric effect (ECE). Here, we have investigated the ECE of the system and determined the influence of system parameters on the ECE. Our attention of the work is to study the effect of temperature, electric field, magnetic field and, non-extensive parameter in the ECE. It is found that the aforementioned parameters have important roles on the ECE. We could obtain both normal and inverse the ECE by changing the parameters. This behavior can be employed for tuning the properties for particular applications.

Appendix

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

$$ \frac{1}{{2m^{*} }}\left( {{\varvec{P}} - e{\varvec{A}}} \right)^{2} {\Psi } - eFz{\Psi } + V_{conf} \left( {\mathbf{r}} \right){\Psi } = E_{t} {\Psi } $$

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 the cylindrical coordinates, we have

$$ - \frac{{\hbar^{2} }}{{2m^{*} }}\left[ {\frac{1}{{\uprho }}\frac{\partial }{{\partial {\uprho }}}\left( {{\uprho }\frac{\partial }{{\partial {\uprho }}}{ }} \right) + \frac{{\partial^{2} }}{{\partial z^{2} }} + \frac{1}{{{\uprho }^{2} }}\frac{{\partial^{2} }}{{\partial \varphi^{2} }}} \right]{\Psi } - \frac{{i\hbar \omega_{c} }}{2}\frac{{\partial {\Psi }}}{\partial \varphi } + \frac{{m^{*} {\Omega }^{2} {\uprho }^{2} }}{8}{\Psi } - e \in z{\Psi } + \frac{{m^{*} \omega_{0}^{2} {\text{z}}^{2} }}{2} {\Psi } = E_{t} {\Psi } $$

where $E_{t} = E_{\rho } + E_{z}$.

Consider the wave function as

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

By using above relation, we have

$$ - \frac{{\hbar^{2} }}{{2m^{*} }}\left[ {\frac{1}{{\uprho }}\frac{\partial }{{\partial {\uprho }}}\left( {{\uprho }\frac{\partial }{{\partial {\uprho }}}{ }} \right) + \frac{1}{{{\uprho }^{2} }}\frac{{\partial^{2} }}{{\partial \varphi^{2} }}} \right]f - \frac{{i\hbar \omega_{c} }}{2}\frac{\partial f}{{\partial \varphi }} + \frac{{m^{*} {\Omega }^{2} {\uprho }^{2} }}{8}f = E_{\rho } f $$

$$ - \frac{{\hbar^{2} }}{{2m^{*} }}\frac{{d^{2} g}}{{dz^{2} }} + \frac{{m^{*} \omega_{0}^{2} {\text{z}}^{2} }}{2}g - e \in zg = E_{z} g $$

The solutions of the above equations are (Atoyan et al. 2006)

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

$$ g\left( z \right) = Be^{{ - \frac{{m^{*} \omega_{0} }}{\hbar }\left( {z - z_{0} } \right)^{2} }} H_{{n_{z} }} \left[ {\sqrt {\frac{{m^{*} \omega_{0} }}{\hbar }} \left( {z - z_{0} } \right)} \right] $$

where $A$ and $B$ are the normalization constants, $a = \sqrt {\frac{\hbar }{{m^{*} {\Omega }}}}$, $z_{0} = \frac{e \in }{{m^{*} \omega_{0}^{2} }}$, $F\left( {\mu ,\lambda ;x} \right)$ is the confluent hypergeometric function, and $H_{n} \left( x \right)$ 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

$$ E_{{n_{\rho } ,m,n_{z} }} = \left( {n_{\rho } + \frac{\left| m \right| + 1}{2}{ }} \right)\hbar {\Omega } + \frac{{m\hbar \omega_{c} }}{2} + \left( {n_{z} + \frac{1}{2}{ }} \right)\hbar \omega_{0} - \frac{{e^{2} \in^{2} }}{{2m^{*} \omega_{0}^{2} }} $$