Magnetocaloric effect, magnetic susceptibility, heat capacity and optical properties of wedge-shaped quantum dots

Abstract

In this work, a GaAs wedge-shaped quantum dot (WSQD) is considered. In the first step, we theoretically investigated the magnetocaloric effect, magnetic susceptibility and heat capacity of WSQD without considering impurity. In the next step, we calculated the binding energy and optical properties of the WSQD under various factors, such as impurity position, temperature and pressure. For our results, we first solved the Schrödinger equation to determine the energy levels and eigenstates of the system. Then, the influence of impurity position, temperature and pressure on optical absorption coefficients (ACs) and refractive index changes (RICs) were investigated. The temperature dependence of the magnetic entropy changes (magnetocaloric effect) shows a pronounced minimum at low temperatures. The magnetic susceptibility shows both positive (paramagnetism) and negative values (diamagnetism) for different parameters. The impurity position has a key role in the electronic properties, ACs and RIs of a WSQD. The ACs and RICs enhance and shift towards lower energies under pressure. Also, the optical properties reduce in the presence of impurity.

Appendix A

The single-particle states are given by

$$\begin{aligned} H_0\mathrm {\Psi }=E_0\mathrm {\Psi }, \end{aligned}$$

(A.1)

where $\mathrm {\Psi }$ is the single-particle wave function and E is the energy eigenvalue. The Hamiltonian of eq. (A.1) in the cylindrical coordinates is given by

$$\begin{aligned} H_0{\Psi }\ {}= & {} -\frac{{\hslash }^2}{2m^*}\left[ \frac{1}{\rho }\frac{\partial }{\partial \rho }\left( \rho \frac{\partial }{\partial \rho }\ \right) +\frac{{\partial }^2}{\partial z^2}+\frac{1}{{\rho }^2}\frac{{\partial }^2}{\partial {\varphi }^2}\right] {\Psi }\ \nonumber \\{} & {} -\frac{i\hslash {\omega }_c}{2}\frac{\partial {\Psi }\ }{\partial \varphi }+\frac{m^*{{\omega }^2_c}{\rho }^2}{8}{\Psi }=(E_{\rho }+E_z){\Psi }.\nonumber \\ \end{aligned}$$

(A.2)

Let us present the wave function of the electron as

$$\begin{aligned} {\Psi } \left( \rho ,\varphi ,z\right) =f(\rho ,\varphi )h\left( z\right) . \end{aligned}$$

(A.3)

Inserting eq. (A.3 ) into eq. (A.2), separating variables in eq. (A.2), we shall obtain the following equations:

$$\begin{aligned}{} & {} -\frac{{\hslash }^2}{2m^*}\frac{\textrm{d}^2h\left( z\right) }{\textrm{d}z^2}=E_zh(z) \end{aligned}$$

(A.4)

$$\begin{aligned}{} & {} -\frac{{\hslash }^2}{2m^*}\left[ \frac{1}{\rho }\frac{\partial }{\partial \rho }\left( \rho \frac{\partial f}{\partial \rho }\ \right) +\frac{1}{{\rho }^2}\frac{{\partial }^2f}{\partial {\varphi }^2}\right] \nonumber \\{} & {} -\frac{i\hslash {\omega }_c}{2}\frac{\partial f\ }{\partial \varphi }+\frac{m^*{{\omega }_c}^2{\rho }^2}{8} f=E_{\rho }f. \end{aligned}$$

(A.5)

The solutions of eqs (A.4) and (A.5) appear as

$$\begin{aligned} f\left( \rho ,\varphi \right) =S\textrm{e}^{-\frac{{\rho }^2}{4a^2}}{\rho }^{\left| l\right| }F\left( -n,\left| l\right| +1;\frac{{\rho }^2}{2a^2}\right) \textrm{e}^{im\varphi } \end{aligned}$$

(A.6)

and

$$\begin{aligned} h\left( z\right) =C\,{\textrm{sin}\, \textrm{ K}}z+D\,{\textrm{cos} \,\textrm{ K}z,\ } \end{aligned}$$

(A.7)

where

$$\begin{aligned} {K}=\sqrt{\frac{2m^*E_z}{{\hslash }^2}}. \end{aligned}$$

Also, the parameters C,  D and S are constants. Now, by imposing the boundary conditions on eqs (A.6) and (A.7), we can obtain eqs (8)–(10) in the main text.