1 Introduction

It is fully known that research on semiconductor quantum dots (QDs) has attracted considerable attentions in recent years because QDs have great potential applications in micro-electronic devices such as quantum dot lasers, solar cells, single electron transistors and quantum computers [1, 2].

It is clear that the realistic profile of the confinement potentials has an important role to formulate physical properties of QDs. In the past two decades, the various confinement potentials have been usually employed in theoretical determination of physical properties of QDs. Examples of these models are parabolic potential [3], Rosen–Morse potential [4], Tietz potential [5] and spherical potential [6]. Adamowski et al. [7, 8] and Xie [9, 10] have considered a new confinement potential in QDs, the spherical Gaussian potential. Ciurla et al. [11] have proposed a new class of the confinement potentials, called the power-exponential potentials. We have proposed a new confinement potential in QDs which is called the modified Gaussian potential [12]. Jia et al. [13,14,15] have studied thermodynamic properties of real molecule systems using Manning–Rosen potential, Rosen–Morse potential and Tietz potential. In addition, these potentials have been employed in the literature for the practical applications [16, 17].

It is obvious that the study of thermodynamic properties of low-dimensional semiconductor structures such as entropy, specific heat and the average energy is one of the most essential forefront fields in Physics. In the past two decades, a lot of attentions have been focused on the study of thermodynamic properties of semiconductor nanostructures. In the studies, authors have used either a square well model, a parabolic potential model or double ring-shaped model for the confining potential [18,19,20].

Although different physical properties of QDs have been investigated widely, to our knowledge, there are few theoretical studies about thermodynamic properties of QDs using various confinement potentials. For instance, Boyacioglu and Chatterjee [21] have investigated the heat capacity and entropy in a GaAs quantum dot with Gaussian confinement in the presence of a magnetic field. Muller and Koonin [22] have calculated the heat capacity associated with the rotational degree of freedom of deformed many-body states. In another work, Ibragimov [23] has studied magnetic susceptibility, specific heat and entropy of parabolic quantum wires in tilted magnetic fields. Oh et al. [24] have studied the heat capacity of quantum wires and dots, and they have found that the calculated heat capacities at low temperatures show an oscillating behavior. On the other hand, Nguyen and Peeters [25] have studied the thermodynamic quantities in the presence of a single magnetic ion and a perpendicular magnetic field. Maksym and Chakraborty [26] have calculated the electronic heat capacity taking into account the electron–electron interaction. The goal of the present paper is to investigate the heat capacity, entropy and the average energy in a GaAs quantum dot with an effective-parabolic potential in the presence of the spin-Zeeman interaction. We compare our calculations with the results of Boyacioglu and Chatterjee results [21].

2 Model and the Theory

We consider an electron moving in a two-dimensional confining potential under a magnetic field along the z-direction. Including the spin-Zeeman term, the Hamiltonian of the system can be written as

$$ H = H_{e} + \frac{1}{2}\hbar \omega_{c} g^{*} \hat{S}_{z} $$
(1)

where \( g^{*} \) is the Landé g-factor of the electron and \( \omega_{c} = eB/m^{*} \) is the cyclotron frequency. The first term in Eq. (1) is

$$ H_{e} = \frac{1}{{2m^{*} }}\left( {\varvec{P} + e\varvec{A}} \right)^{2} + V_{\text{conf}} \left( \varvec{r} \right). $$
(2)

Here \( m^{*} \) is effective mass of the electron and \( \varvec{A} \) is the vector potential corresponding to the magnetic field. We choose the symmetric gauge vector potential as \( \varvec{A} = \left( { - B\frac{y}{2},B\frac{x}{2},0} \right) \). The second term in Eq. (2) is the confining potential which we take as the modified Gaussian potential (MGP) as [12]:

$$ V_{\text{conf}} \left( r \right) = - V_{0} {\text{sech}}^{2} \left( {\frac{r}{R}} \right), $$
(3)

where \( V_{0} > 0 \) and \( R \) are the depth and range of the confinement potential, respectively. One can consider \( R \) as the radius of the quantum dot. It is worth mentioning that we have proposed the modified spherical Gaussian potential which is flexible enough to calculate spectral energy and wave function, analytically. Also, this potential and its derivative are continues functions. In addition, our aim is to model quantum dots theoretically.

Inserting the vector potential into Eq. (2), we can obtain the Hamiltonian as

$$ H = - \frac{{\hbar^{2} }}{{2m^{*} }}\nabla_{r}^{2} + \frac{1}{8}m^{*} \omega_{c}^{2} r^{2} - V_{0} {\text{sech}}^{2} \left( {\frac{r}{R}} \right) + \frac{1}{2}\hbar \omega_{c} \left( {\hat{L}_{z} + g\hat{S}_{z} } \right), $$
(4)

where \( L_{Z} \) is the z-component of the angular momentum of the electron. Here, we have applied the time-independent perturbation theory. In this regard, we consider the modified Gaussian confining potential as a parabolic potential plus a perturbation. We have assumed that the deviation of the modified Gaussian confining potential from the parabolic potential is small enough \( (r/R \le 1) \) so that it can be treated as an effective-parabolic potential. Since the variable r is small in a quantum dot, our consideration is a fairly good approximation. So we rewrite the Hamiltonian (1) as

$$ H = H_{0} + H_{1} , $$
(5)

where

$$ H_{0} = - \frac{{h^{2} }}{{2m^{*} }}\nabla_{r}^{2} + \frac{1}{2}m^{*} \varOmega^{2} r^{2} + \frac{1}{2}\hbar \omega_{c} \left( {\hat{L}_{z} + g\hat{S}_{z} } \right) - V_{0} , $$
(6)

and

$$ H_{1} = \alpha \left[ {V_{0} - \frac{1}{2}m^{*} \omega^{2} r^{2} - V_{0} \text{sech}^{2} \left( {\frac{r}{R}} \right)} \right], $$
(7)

where \( \omega^{2} = \frac{{2V_{0} }}{{m^{*} R^{2} }} \), \( \varOmega^{2} = \omega^{2} + \frac{{\omega_{c}^{2} }}{4} \), and \( \alpha \) is constant value. For \( \alpha = 0 \) and \( \alpha = 1 \), we have a parabolic model and a modified Gaussian model, respectively. Using the first-order perturbation theory at the mean field level, we can write

$$ H_{1} = \alpha \left[ {\frac{{V_{0} }}{{r^{2} }} - \frac{1}{2}m^{*} \omega^{2} - \frac{{V_{0} }}{{r^{2} }}\left\langle {\text{sech}^{2} \left( {\frac{r}{R}} \right)} \right\rangle } \right]r^{2} , $$
(8)

where \( r^{2} \) is calculated with respect to ground state wave function of the harmonic oscillator of frequency \( \varOmega \). Now, the modified Gaussian potential problem can be considered by an effective-parabolic potential problem [27, 28]

$$ H = - \frac{1}{{2m^{*} }}\nabla^{2} + \frac{1}{2}m^{*} \varpi^{2} r^{2} + \frac{1}{2}\hbar \omega_{c} \left( {\hat{L}_{z} + g\hat{S}_{z} } \right) - V_{0} , $$
(9)

where

$$ \varpi^{2} = \left[ {\left( {1 - \alpha } \right)\omega^{2} + \frac{{\omega_{c}^{2} }}{4} + 2\alpha V_{0} \frac{\varOmega }{{\hbar + m^{*} \varOmega R^{2} }}} \right]. $$
(10)

A more detail between Eqs. (8) and (9) has been presented in the “Appendix.”

With respect to Eq. (9), one can readily solve the Schrödinger equation and obtain the following wave function [29]

$$ \begin{aligned} \phi_{\text{nls}} \left( {r,\theta ,\sigma } \right) & = \sqrt {\frac{{m^{*} \varpi }}{\pi \hbar }} \sqrt {\frac{n!}{{\left( {n + \left| l \right|} \right)!}}} \left( {\frac{{m^{*} \varpi }}{\hbar }} \right)^{{\frac{\left| l \right|}{2}}} r^{\left| l \right|} L_{n}^{\left| l \right|} \left( {\frac{{m^{*} \varpi }}{\hbar }r^{2} } \right) \\ & \quad \times \exp \left[ { - \frac{1}{2}\frac{{m^{*} \varpi }}{\hbar }r^{2} + il\theta } \right]\chi_{s} \left( \sigma \right), \\ \end{aligned} $$
(11)

where \( n \) and \( l = 0, \pm 1, \pm 2 \ldots \) are the radial and azimuthal angular momentum quantum numbers, respectively. Also, \( L_{n}^{\left| l \right|} \left( x \right) \) is the associated Laguerre polynomials and \( \chi_{s} \left( \sigma \right) \) is the eigenstate of the spin operator \( S_{z} \) with eigenvalues \( s = \pm 1/2 \). The energy levels of Hamiltonian (9) is given by

$$ E_{nls} = \left( {2n + \left| l \right| + 1} \right)\hbar \varpi + \frac{1}{2}\hbar \omega_{c} \left( {l + g^{*} s} \right) - V_{0} . $$
(12)

3 Entropy and Heat Capacity

The thermodynamic properties of a system can be obtained by the derivatives of the free energy of system [30, 31]. In this work, we have neglected electron–electron and spin–orbit interactions. Therefore, the partition function for the present system is exactly given by [30, 32]

$$ Z\left( {T,B} \right) = \frac{1}{2}\frac{{\cosh \left( {\frac{{\hbar g^{*} \omega_{c} }}{{4k_{B} T}}} \right)}}{{\sinh \left[ {\frac{\hbar }{{2k_{B} T}}\left( {\varpi + \frac{{\omega_{c} }}{2}} \right)} \right]\sinh \left[ {\frac{\hbar }{{2k_{B} T}}\left( {\varpi - \frac{{\omega_{c} }}{2}} \right)} \right]}}, $$
(13)

where \( k_{B} \) and \( T \) are the Boltzmann constant and temperature, respectively. It is seen from above equation that the partition function depends on temperature and magnetic field. First, we obtain free energy by using the relation \( F = - k_{B} TlnZ \). Then, we can calculate the entropy

$$ \begin{aligned} S & = - \left( {\frac{\partial F}{\partial T}} \right) = \frac{1}{2}k_{B} \ln \left[ {\frac{{\cosh \left( {\frac{{\hbar g^{*} \omega_{c} }}{{4k_{B} T}}} \right)}}{{\sinh \left[ {\frac{\hbar }{{2k_{B} T}}\left( {\varpi + \frac{{\omega_{c} }}{2}} \right)} \right]\sinh \left[ {\frac{\hbar }{{2k_{B} T}}\left( {\varpi - \frac{{\omega_{c} }}{2}} \right)} \right]}}} \right] \\ & \quad + \frac{\hbar }{2T}\left[ {\left( {\varpi + \frac{{\omega_{c} }}{2}} \right)\coth \frac{\hbar }{{2k_{B} T}}\left( {\varpi + \frac{{\omega_{c} }}{2}} \right) + \left( {\varpi - \frac{{\omega_{c} }}{2}} \right)\coth \frac{\hbar }{{2k_{B} T}}\left( {\varpi - \frac{{\omega_{c} }}{2}} \right)} \right] \\ & \quad - \frac{{\hbar g^{*} \omega_{c} }}{4T}\tanh \left( {\frac{{\hbar g^{*} \omega_{c} }}{{4k_{B} T}}} \right), \\ \end{aligned} $$
(14)

and heat capacity

$$ \begin{aligned} C & = - T\left( {\frac{{\partial^{2} F}}{{\partial T^{2} }}} \right) = \frac{{g^{*2} T^{2} k_{B}^{3} }}{{16\cosh^{2} \left( {\frac{{\hbar g^{*} \omega_{c} }}{{4k_{B} T}}} \right)}}\left( {\hbar \omega_{c} } \right)^{2} \\ & \quad + \frac{{\hbar^{2} T^{2} k_{B}^{3} }}{4}\left[ {\frac{{\left( {\varpi + \frac{{\omega_{c} }}{2}} \right)^{2} }}{{\sinh^{2} \left[ {\frac{\hbar }{{2k_{B} T}}\left( {\varpi + \frac{{\omega_{c} }}{2}} \right)} \right]}} + \frac{{\left( {\varpi - \frac{{\omega_{c} }}{2}} \right)^{2} }}{{\sinh^{2} \left[ {\frac{\hbar }{{2k_{B} T}}\left( {\varpi - \frac{{\omega_{c} }}{2}} \right)} \right]}}} \right]. \\ \end{aligned} $$
(15)

The free energy can be applied to obtain the average energy of the system as

$$ \begin{aligned} U & = E = - T^{2} \left[ {\frac{\partial }{\partial T}\left( {\frac{F}{T}} \right)} \right] = \frac{\hbar }{2}\left[ {\left( {\varpi + \frac{{\omega_{c} }}{2}} \right)\coth \frac{\hbar }{{2k_{B} T}}\left( {\varpi + \frac{{\omega_{c} }}{2}} \right)} \right. \\ & \quad \left. { + \left( {\varpi - \frac{{\omega_{c} }}{2}} \right)\coth \frac{\hbar }{{2k_{B} T}}\left( {\varpi - \frac{{\omega_{c} }}{2}} \right)} \right] - \frac{1}{4}g^{*} \hbar \omega_{c} \tanh \left( {\frac{{\hbar g^{*} \omega_{c} }}{{4k_{B} T}}} \right) \\ \end{aligned} $$
(16)

As mentioned, thermodynamic properties of the QD have been calculated using energy levels [Eq. (12)]. To obtain a better understanding of the obtained results, molecular dynamics (MD) simulation has been employed in this work. The details of MD simulation have been presented in the following section.

4 Computation Methods

In this work, we could not find experimental data for comparing with our results. Therefore, we have tried to obtain new results using density functional theory (DFT). Thus, DFT has been employed to determine the thermodynamic properties of GaAs semiconductor QDs. To this end, molecular dynamics simulations were carried out using Materials Studio v4.3, which was developed by Accelrys Software Inc. The first-principles and the plane-wave pseudo potential total energy calculations have been performed using the Cambridge Serial Total Energy Package (CASTEP) code [33]. We have represented the electrostatic interaction between valence electron and ionic core by ultra-soft pseudo potentials [34,35,36]. Also, we have treated the electronic exchange–correlation energy under the local density approximation (LDA) [37]. The plane-wave basis set cutoff is 330 eV for GaAs semiconductor QD. The Monkhorst–Pack method [38] with different special k-point mesh has been used for the special k-points sampling integration over the Brillouin zone. These parameters were sufficient in leading to well-convergence of total energy and geometrical configurations. By increasing the plane-wave cutoff energy and the k-point mesh, the total energy changes by less than 0.03 eV/atom and the lattice constants by less than 0.01%. In our calculations, we have following thresholds: energy change per atom less than 0.5 × 10−6 eV and displacement of atoms during the geometry optimization less than 0.002 Å.

In the CASTEP code, the results of a linear response calculation of the phonon spectra can be used to compute energy, entropy, free energy, and lattice heat capacity as functions of temperature. When we perform a vibrational analysis with CASTEP, the results of the thermodynamic calculations can be visualized using the thermodynamic analysis tools. The temperature dependence of the enthalpy is given by

$$ E\left( T \right) = E_{\text{tot}} + E_{\text{zp}} + \int {\frac{\hbar \omega }{{\exp \left( {\frac{\hbar \omega }{{k_{B} T}}} \right) - 1}}} F\left( \omega \right){\text{d}}\omega , $$
(17)

where \( E_{zp} \) is the zero point vibration energy of the lattice, \( k_{B} \) is Boltzmann’s constant, \( \hbar \) is Plank constant and \( F\left( \omega \right) \) is the phonon density of states. \( E_{zp} \) can be evaluated as

$$ E_{\text{zp}} = \frac{1}{2}\int {F\left( \omega \right)\hbar \omega {\text{d}}\omega } . $$
(18)

The contribution to the free energy is given by

$$ F\left( T \right) = E_{\text{tot}} + E_{\text{zp}} + k_{B} T\int {F\left( \omega \right)} \ln \left[ {1 - \exp \left( { - \frac{\hbar \omega }{{k_{B} T}}} \right)} \right]{\text{d}}\omega . $$
(19)

The contribution to the entropy can be obtained by

$$ S\left( T \right) = k_{B} \left\{ {\int {\frac{{\frac{\hbar \omega }{{k_{B} T}}}}{{\exp \left( {\frac{\hbar \omega }{{k_{B} T}}} \right) - 1}}} F\left( \omega \right){\text{d}}\omega - \int {F\left( \omega \right)} \left[ {1 - \exp \left( { - \frac{\hbar \omega }{{k_{B} T}}} \right)} \right]{\text{d}}\omega } \right\}. $$
(20)

The lattice contribution to the heat capacity is

$$ C\left( T \right) = k_{B} \int {\frac{{\left( {\frac{\hbar \omega }{{k_{B} T}}} \right)^{2} \exp \left( {\frac{\hbar \omega }{{k_{B} T}}} \right)}}{{\left[ {\exp \left( {\frac{\hbar \omega }{{k_{B} T}}} \right) - 1} \right]^{2} }}} F\left( \omega \right){\text{d}}\omega . $$
(21)

The total specific heat of a crystal is the sum of all phonon modes over the Brillouin zone. The temperature dependence of heat capacity at constant volume has been calculated under the quasi-harmonic approximation. The quasi-harmonic approximation is a phonon-based model of solid-state physics. It is based on the assumption that the harmonic approximation holds for every value of the lattice constant, which is to be viewed as an adjustable parameter. The quasi-harmonic approximation expands upon the harmonic phonon model of lattice dynamics. In the approximation, from a phonon point of view, the phonon frequency becomes volume-dependent. Recently, we have applied the CASTEP code to predict physical properties of XO (X = Am, Cd, Mg, Zr) compounds using density functional theory [39,40,41].

5 Results and Discussion

In this section, we have studied the effects of temperature and the magnetic field on the entropy, average energy and heat capacity of our system. Here, we report our results for the case of a GaAs quantum dot with \( g^{*} = - 0.44 \), R = 10 nm, \( m_{e}^{*} = 0.067m_{0} \), and \( V_{0} \) = 50 meV.

Figure 1 shows the entropy (\( S/k_{B} \)) as a function of temperature for three different magnetic fields. As we see, the entropy is increased with increasing the temperature at a fixed magnetic field. We should note that the entropy behaviors at small and high temperatures are qualitatively different. At small temperatures, \( k_{B} T \le 0.4 \) the entropy is enhanced monotonically with increasing temperature for all the values of the magnetic fields and it becomes essentially independent of the magnetic field. But, at high temperatures, the entropy behavior is different and it crucially depends on the magnetic field. It is seen from the figure that the entropy is increased with decreasing magnetic field at a fixed temperature.

Fig. 1
figure 1

The entropy \( \left( {S/k_{B} } \right) \) versus temperature \( \left( T \right) \) of a GaAs quantum dot for different magnetic fields (Color figure online)

Full size image

In Fig. 2, we have plotted the entropy (\( S/k_{B} \)) as a function of magnetic field for three different temperatures as 100 K, 200 K and 300 K. It is observed from the figure that the entropy is decreased with increasing magnetic field at a fixed value of temperature. It is worth mentioning that there are two kinds of energies, kinetic energy due to confinement and heat energy due to the thermodynamic disorder. At low and high temperatures, these energies are competed.

Fig. 2
figure 2

The entropy as a function of magnetic field \( B \) for various temperatures (Color figure online)

Full size image

Figure 3 displays the average energy E as a function of temperature for three different applied magnetic fields. It is clear from Eq. (16) that the average energy depends on both the temperature and magnetic field. It is seen from the figure that the average energy is enhanced with increasing temperature at a fixed magnetic field. One can see that at low temperatures, the average energy appears weakly dependent on magnetic field. In Fig. 4, we have plotted the average energy as a function of magnetic field for three different temperatures. One can observe that the average energy is increased with enhancing magnetic field at a fixed temperature. With increasing the magnetic field and temperature, the system disorder increases and thereby the average energy enhances.

Fig. 3
figure 3

Average energy as a function of temperatures for different magnetic fields (Color figure online)

Full size image
Fig. 4
figure 4

Average energy versus magnetic field for different temperatures (Color figure online)

Full size image

Figure 5 shows the heat capacity \( C/k_{B} \) versus temperature for three different magnetic fields. We observe that the heat capacity and thereby system disorder are increased with enhancing temperature at a fixed magnetic field. It is to be noted that the heat capacity saturates at high temperatures. In Fig. 6, we have presented the heat capacity (\( C/k_{B} \)) as a function of the applied magnetic field for three values of the temperatures. The figure shows that the heat capacity is a decreasing function of magnetic field. It is obvious from the figure that the heat capacity does not change much with magnetic field at T = 200 K and 300 K. However, at relatively lower temperatures such as T = 100 K, the heat capacity is decreased with increasing magnetic field.

Fig. 5
figure 5

Heat capacity \( \left( {C/k_{B} } \right) \) versus temperature \( \left( T \right) \) of a GaAs quantum dot for different magnetic fields (Color figure online)

Full size image
Fig. 6
figure 6

Heat capacity \( (C/k_{B} \)) as a function of magnetic field for different temperatures (Color figure online)

Full size image

Since we could not find experimental data to compare with our results, we have DFT to calculate heat capacity and average energy. Figure 7 shows heat capacity as a function of temperature at zero magnetic fields. It is seen from the figure that the values of heat capacity obtained using DFT are lower than the obtained data from Eq. (15). At high temperatures, the results become the same. In Fig. 8, the average energy has been plotted as a function of temperature at zero magnetic fields. The difference between DFT and obtained data from Eq. (16) becomes large at higher temperatures.

Fig. 7
figure 7

Heat capacity as a function of temperature at zero magnetic fields. Our results have been compared with DFT (Color figure online)

Full size image
Fig. 8
figure 8

Average energy versus temperatures at zero magnetic fields. Our results have been compared with DFT (Color figure online)

Full size image

It is noteworthy that we can obtain the magnetization using the average energy and the relation \( M = - \frac{\partial U}{\partial B} \). In Figs. 9 and 10, we have plotted the reduced magnetization (\( M/\mu_{B}^{*} \)) as a function of magnetic field and confinement range (R), respectively. \( \mu_{B}^{*} = e\hbar /2m^{*} \) is the effective Bohr magneton. We have compared our results with the parabolic potential [41]. We can see from Fig. 9 that the magnitude of the magnetization \( \left( {\left| M \right|} \right) \) increases with the enhancement in the magnetic field for our work and the parabolic potential. The used parameter in this figure is for \( V_{0} \) = 36.7 meV, T = 5 K and R = 10 nm. Both curves show almost a linear increase in magnetization with increasing magnetic field. However, the parabolic potential model underestimates the magnetization, particularly so, for large magnetic fields. Therefore, at sufficiently large magnetic fields, the difference in the magnetization between our work and parabolic potential model may be quite significant.

Fig. 9
figure 9

The reduced magnetization (\( M/\mu_{B}^{*} \)) as a function of the magnetic field for \( V_{0} \) = 36.7 meV, T = 5 K and R = 10 nm (Color figure online)

Full size image
Fig. 10
figure 10

The reduced magnetization (\( M/\mu_{B}^{*} \)) as a function of the confinement range (R) for \( V_{0} \) = 36.7 meV, T = 5 K and \( B \) = 2 T. Our results have been compared with the parabolic potential [42] (Color figure online)

Full size image

From Fig. 10, it is observed that reduced magnetization decreases with decreasing size of quantum dot for our work and parabolic potential. The variation is more or less linear. However, we observe that the parabolic potential model underestimates the reduced magnetization. The used parameters are \( V_{0} \) = 36.7 meV, \( T \) = 5 K and \( B \) = 2 T.

Figure 11 shows the magnetic susceptibility as a function of magnetic field for \( V_{0} \) = 36.7 meV, \( T \) = 5 K and \( R \) = 10 nm. It is seen from the figure that the magnetic susceptibility increases with increasing magnetic field.

Fig. 11
figure 11

The magnetic susceptibility as a function of the magnetic field for \( V_{0} \) = 36.7 meV, T = 5 K and R = 10 nm (Color figure online)

Full size image

In general, we can say that in the case of quantum dots, the energy levels are not quasi-continuous but they are discrete and therefore the entropy, heat capacity and the average energy can depend on the distribution of energy levels. Consequently, the properties should depend both on the energy level distribution and the temperature dependence of the occupation probability of the states. Applying the magnetic field, we expect to change the energy level distribution. Also, using the temperature, the occupation probability of the states is changed. With respect to these circumstances, the behavior of thermodynamic properties at low and high temperatures and magnetic fields can be different. We have attempted to provide explanations for the behavior of the heat capacity, entropy and average energy in this work.

6 Conclusion

We have used the canonical ensemble approach and study the entropy, heat capacity and average energy of a GaAs quantum dot by considering under the full Zeeman term. The entropy at low and high temperatures has different behaviors. At low temperatures, the entropy is monotonically increased. The average energy is enhanced with increasing temperature and magnetic field. But, at low temperatures, the average energy weakly depends on magnetic field. The heat capacity is reduced with increasing magnetic fields at low temperatures. The results show that the variation of heat capacity with magnetic field is not much at high temperatures. We have also used DFT to compare our results. The results show that our analytical results and DFT give approximately similar results. We have also obtained the magnetization and compared with the parabolic potential.