Effects of Spin Orbit Interaction (SOI) on the Thermodynamic Properties of a Quantum Pseudodot

Abstract

We investigate the effect of spin orbit interaction (SOI) on thermodynamics quantities of a quantum pseudodot. The energy levels have been derived and thermal properties are evaluated using the Tsallis formalism. Compared to BG, Tsallis formalism consists in evaluating the thermodynamic quantities only on the accessible states. The results show that spin orbit interaction (SOI) has a great effect on entropy since it increases the number of accessible states causing the splitting on internal energy. Our results also show that chemical potential influences the spin-states in a quantum pseudodot system and creates splitting of some thermodynamic quantities. SOI and chemical potential are prominent parameters to modulate the spin alignment and perform the thermodynamics of electronic devices.

1 Introduction

The realization and improvement of new electronic devices required a deeper knowledge of their intrinsic and thermodynamic properties, this in order to powered their efficiency and reduce the loss of informations. The study of nanostructures (quantum dot, quantum wire, quantum well, quantum pseudodot, etc.) has emerged as a very interesting field in condensed matter physics in the recent years [1,2,3,4,5,6,7,8,9,10,11,12,13,14]. The reason is that, nanostructures have a large number of application in nanoelectronic devices such as quantum laser, quantum pseudodot laser and quantum computers. To perform those nanostructures and semiconductors, their thermodynamic properties need to be infinitely studied. This is one more reason to investigate their intrinsic and extrinsic physical properties. Since intrinsic properties of those nanostructure still have dark zones, it becomes a fascinating area of the century. In the recent years, the physical properties of a quantum pseudodot [15, 16] have been deeply studied due to their potential application in photonic devices [17, 18]. Till now, many researchers investigated the electronic and optical properties of quantum pseudodot (QPs) [19,20,21,22,23,24,25]. Although their different properties have been studied widely, however, the effect of spin orbit interaction (SOI) on their thermodynamic properties (heat capacity, entropy, internal energy and free energy) need to be studied. Since statistical mechanics or statistical thermodynamics is considered as a field of physics which studied the molecular-level interpretation of microscopic and macroscopic thermodynamic quantities. Thermodynamics described by Boltzmann Gibbs was based on the repartition of particles in the whole system, evaluating the energy possessed by those particles at each level of the system. But this statistical thermodynamic done by BG have been enhanced by Tsallis after 1902, he proposed a generalized version of BG statistical mechanics. Tsallis proposed in 1988 to study the thermodynamic quantities just at the accessible states after deriving the probability of microstates [26,27,28,29,30]. Based on the interaction of spins with itself motion, the Rashba SOI has known the remarkable progress after the researchers discovered that spin and SOI are some related phenomena in electronics, spintronic and nanotechnology, such as single electron coherent oscillating [31] a spin effect transistor made by Datta and Das [32] and the spin current are relating transport phenomena [33]. Thus spintronic opened a new field of application of spin and SOI both in science and technology. In this way, the new perspective for spin orbit interaction have been proposed by Manchon et al. [34]. Based on the fact that an electron’s spin can be chosen as the best way for creating a qubit because it is formed by two different states up $\left( \uparrow \right)$ and down $\left( \downarrow \right)$ (two levels-system). Some semiconductors such as InAs is characterized by a strong spin orbit interaction, it has played an important role in spintronic devices because it can control the electron’s spin degree-of-freedom [35,36,37]. Entropy has been defined differently by many researchers in the past few years, many works have been done about thermodynamic quantities of nanostructures and this is due to their large applications in technology [38,39,40,41,42,43,44,45]. Thermodynamic properties of GaAs quantum dot have been studied by Boyacioglu and Chartterjee, they found that at low temperature, the heat capacity presents a schottky like anomaly while it reaches at a saturation value of $2K$[46]. The heat capacity of quantum wire (QW) and quantum Dot (QD) have been investigated by Oh et al. and their calculation shows an oscillating behavior. B Boyacioglu and A. Chartterjee studied the thermpodynamic quantities (heat capacity and entropy) in GaAs quantum wire (QW) taking Gaussian potential as confinement in the presence of magnetic field using the canonical ensemble approach. They found that at high temperature, entropy is at a monotonic increasing function of temperature but decreases with an increase in magnetic field at a sufficient low temperature. Gumber Surkirty studied the thermal behavior of Rashba quantum dot under magnetic field and he showed that, the SOI has a great effect on the partition function and heat capacity. Alubado et al. [47] studied the effect of Rashba SOI on electronic, thermodynamic and transport properties of a single electron quantum dot, their investigation shows that all those quantities are strongly influenced by the spin orbit interaction. In the last decade, several works have been done on Rashba effect, both experimental and theoretical. An analytical method has been used to study the three-electron quantum dot in the presence of spin orbit interaction [48]. B.S and D.Akay devoted their work on the pseudo-Zemann splitting in grapheme cones by magnetic field and they found that, magnetic field is useful for controlling the level of splitting in topological cones [49, 50]. In the presence of spin orbit interaction, Xin et al. investigated the ground state energy of polaron using the variation method of Pekar [51]. B.Donfack et al. studied the cumulative effect of temperature, magnetic field and SOI on the properties of magnetopolaron in RbCl quantum well [52], they found that SOI has a considerable effect on the properties of magnetopolaron and that, there is an appearance of an inflexion point in the variation of ground state binding energy as function of SOI and temperature. The effect of external field on thermodynamic quantities has been studied in some previous works [53,54,55,56,57,58,59,60,61,62,63,64,65,66], presently now the effect SOI on thermodynamic properties in quantum system has not been deeply studied, although the important role plays the SOI as well in quantum properties as in thermodynamic quantities of those quantum system. In this end, we decided to direct the present investigation on the effect of SOI on thermal properties of quantum pseudodot, in order to carry out the impact of SOI on thermal properties of quantum pseudodot using Tsallis formalism. In the current work, the energy levels, wave function and thermal properties are clearly derived, analytical expressions for the Tsallis formalism and thermal properties are obtained in Sect. 2, the results and discussion have been done in Sect. 3 and finally the conclusion in the last section.

2 Model and formulation

The formulation of the system which describes a charge carrier in two-dimensional quantum pseudodot can be written as [67, 68].

$$H = H_{0} + H_{{{\text{SOI}}}}$$

(1)

$${\text{Where}}\;\left\{ \begin{gathered} H_{0} = \frac{1}{{2\mu }}\left( {P + \frac{{eA}}{c}} \right)^{2} + V\left( \rho \right) \hfill \\ H_{{{\text{SOI}}}} = \frac{{\alpha _{{\text{r}}} }}{\hbar }\left[ {\sigma \times \left( {P + eA} \right)} \right] \hfill \\ \end{gathered} \right.$$

(2)

with $\mu$ the effective mass of charge carrier, $A$ the vector potential of magnetic field, $P$ represents the momentum, $e$ the elementary charge of electron and $V\left( \rho \right)$ represents the two-dimensional potential. It is given as:

$$V\left( \rho \right) = V_{0} \left( {\frac{\rho }{{\rho _{0} }} - \frac{{\rho _{0} }}{\rho }} \right)^{2}$$

(3)

where $V_{0}$ represents the chemical potential.

Lets follow the geometrical shape of the present structure by taking the cylindrical coordinate system in order to deeply study the effect of SOI on thermodynamics of pseudodot, the Hamiltonian of this coordinates system is given as:

$$\begin{aligned} H_{0} & = \frac{1}{{2\mu }}\left[ { - \hbar ^{2} \frac{1}{\rho }\left( {\frac{\partial }{{\partial \rho }}\left( {\rho \frac{\partial }{{\partial \rho }}} \right)} \right) - \hbar ^{2} \frac{1}{\rho }\left( {\frac{\partial }{{\partial \theta }}\left( {\rho \frac{\partial }{{\partial \theta }}} \right)} \right) + \frac{{eB\rho }}{2}\left( { - i\hbar \frac{\partial }{{\partial \rho }} - i\frac{\hbar }{\rho }\frac{\partial }{{\partial \theta }}} \right) + e^{2} A^{2} } \right] \\ & \quad + V_{0} \left( {\frac{{\rho ^{2} }}{{\rho _{0} ^{2} }} + \frac{{\rho _{0} ^{2} }}{{\rho ^{2} }} - 2} \right) \\ \end{aligned}$$

(4)

where $\hbar$ represents the Planck constant, $\rho _{0}$ the pseudo harmonic potential and $B$ the magnetic field.

The final form of $H_{0}$ in the cylindrical coordinates is

$$H_{0} = \frac{{\partial ^{2} }}{{\partial \rho ^{2} }} + \frac{1}{\rho }\frac{\partial }{{\partial \rho }} - \frac{{e^{2} B^{2} }}{{4c^{2} }}\rho ^{2} + \frac{{\partial ^{2} }}{{\partial \theta ^{2} }} - i\frac{{eB}}{{2\hbar }}\frac{\partial }{{\partial \theta }} - \frac{{2\mu }}{{\hbar ^{2} }}V_{0} \left( {\frac{{\rho ^{2} }}{{\rho _{0} ^{2} }} + \frac{{\rho _{0} ^{2} }}{{\rho ^{2} }} - 2} \right)$$

(5)

Taking in account the following partial wave function,we have

$$\psi \left( {\rho ,\theta } \right) = f\left( \rho \right)\frac{{e^{{im\theta }} }}{{\sqrt \pi }},$$

$$H_{0} \psi \left( {\rho ,\theta } \right) = E_{0} \psi \left( {\rho ,\theta } \right)$$

(6)

Using Eq. (6), we obtain the following equation

$$\frac{{\partial ^{2} }}{{\partial \rho ^{2} }}f\left( \rho \right) + \frac{1}{\rho }\frac{\partial }{{\partial \rho }}f\left( \rho \right) - \frac{{e^{2} B^{2} }}{{4c^{2} }}\rho ^{2} + \left( {\alpha ^{2} - \frac{{\beta ^{2} }}{{\rho ^{2} }} - \gamma ^{2} \rho ^{2} } \right) = 0$$

(7)

With

$$\left\{ \begin{gathered} \alpha = \frac{{2\mu }}{{\hbar ^{2} }}\left( {E_{0} + 2V_{0} } \right) \hfill \\ \beta = m^{2} + \frac{{2\mu V_{0} \rho _{0} ^{2} }}{{\hbar ^{2} }} \hfill \\ \gamma = \frac{{2\mu V_{0} }}{{\hbar ^{2} \rho _{0} ^{2} }} + \frac{{e^{2} B^{2} }}{{4\hbar ^{2} c^{2} }} \hfill \\ \end{gathered} \right.$$

(8)

The solutions of Eq. (7) are given in the following form:

$$f\left( \rho \right) = \rho ^{{\left| \beta \right|}} e^{{\frac{{ - \gamma \rho ^{2} }}{2}}} L\left( \rho \right)$$

(9)

Doing this change in variable $\zeta \to \gamma \rho ^{2}$, we have the following equation:

$$\zeta \frac{{d^{2} }}{{d\zeta ^{2} }}L\left( \zeta \right) + \left[ {\left| \beta \right| - n + 1} \right]\frac{{d^{{}} }}{{d\zeta }}L\left( \zeta \right) + nL\left( \zeta \right) = 0$$

(10)

Solving Eq. $\left( {10} \right)$, its solution is given as:

$$\psi \left( {\rho ,\theta } \right) = A\rho ^{{\left| \beta \right|}} e^{{ - \frac{{\gamma \rho ^{2} }}{2}}} F\left( { - n,\left| \beta \right| + 1;\,\lambda \rho ^{2} } \right)e^{{im\theta }}$$

(11)

Knowing $\psi \left( {\rho ,\theta } \right)$, we can now solve analytically the Schrödinger equation

$$H_{0} \psi \left( {\rho ,\theta } \right) = E_{0} \psi \left( {\rho ,\theta } \right)$$

The first part of energy is given as:

$$E_{0} = \hbar \left( {n + \frac{{\beta + 1}}{2}} \right)\sqrt {\omega _{{\text{c}}} ^{2} + \frac{{8V_{0} }}{{\mu \rho _{0} ^{2} }}} + \frac{{m\hbar \omega _{c} }}{2} - 2V_{0}$$

(12)

Let’s evaluate the energy corresponding to the spin orbit interaction part by solving the following equation:

$$H_{{{\text{SOI}}}} \psi \left( {\rho ,\theta } \right) = E_{{{\text{SOI}}}} \psi \left( {\rho ,\theta } \right)$$

(13)

Since in cylindrical coordinates, the SOI is expressed as:

$$\left\{ \begin{gathered} \sigma _{\rho } = \sigma _{{\text{x}}} \cos \phi + \sigma _{{\text{y}}} \sin \phi \hfill \\ \sigma _{\phi } = - \sigma _{{\text{x}}} \sin \phi + \sigma _{{\text{y}}} \cos \phi \hfill \\ \end{gathered} \right.\,\,\,\,\,\,\,\,\,\,\,{\text{and}}\,\,\,\,\,\,\,\,\,\,\,\left\{ \begin{gathered} \sigma _{x} = \frac{1}{2}\left( {\sigma _{ + } + \sigma _{ - } } \right) \hfill \\ \sigma _{y} = \frac{i}{2}\left( {\sigma _{ + } - \sigma _{ - } } \right) \hfill \\ \end{gathered} \right.$$

(14)

Using Eq. (14), the final form of $H_{{SOI}}$ in cylindrical coordinates is written as:

$$\begin{gathered} H_{{SOI}} = \frac{{\alpha _{r} }}{{2\hbar }}\left\{ { - \frac{\partial }{{\partial \rho }}\left( { - i\sigma _{ + } e^{{ - i\phi }} - \sigma _{ - } e^{{i\phi }} } \right)} \right\} + \frac{{eB\alpha _{r} \rho }}{{4\hbar }}\left( {\sigma _{ + } e^{{i\phi }} + \sigma _{ - } e^{{ - i\phi }} } \right) \hfill \\ \quad \quad = \frac{{\alpha _{r} }}{2}\left[ {\frac{\partial }{{\partial \rho }}\left( {i\sigma _{ + } e^{{ - i\phi }} - \sigma _{ - } e^{{i\phi }} } \right)} \right] + \frac{{eB\alpha _{r} \rho }}{{4\hbar }}\left( {\sigma _{ + } e^{{i\phi }} + \sigma _{ - } e^{{ - i\phi }} } \right) \hfill \\ \end{gathered}$$

(15)

From Eq. (15), we have

$$\left\{ \begin{gathered} H_{{{\text{SOI}}}} ^{ + } = \frac{{\alpha _{r} }}{2}\left( {i\frac{\partial }{{\partial \rho }}e^{{ - i\phi }} + \frac{{eB\rho }}{{2\hbar }}e^{{i\phi }} } \right) \hfill \\ H_{{{\text{SOI}}}} ^{ - } = \frac{{\alpha _{r} }}{2}\left( {\frac{\partial }{{\partial \rho }}e^{{ - i\phi }} + \frac{{eB\rho }}{{2\hbar }}e^{{ - i\phi }} } \right) \hfill \\ \end{gathered} \right.$$

(16)

Solving Eq. (13), we have the second part of energy as:

$$E_{{{\text{SOI}}}} ^{ \pm } = \pm \frac{{A\alpha _{{\text{r}}} }}{2}\left( {1 - \frac{{\left| \beta \right|}}{\gamma }} \right)$$

(17)

The final energy is obtained adding Eqs. (12) and (17)

$$E^{ \pm } = \hbar \left( {n + \frac{{\beta + 1}}{2}} \right)\sqrt {\omega _{{\text{c}}} ^{2} + \frac{{8V_{0} }}{{\mu \rho _{0} ^{2} }}} + \frac{{m\hbar \omega _{{\text{c}}} }}{2} - 2V_{0} \, \pm \frac{{A\alpha _{{\text{r}}} }}{2}\left( {1 - \frac{{\left| \beta \right|}}{\gamma }} \right)$$

(18)

3 Thermodynamic Properties

After obtaining our energy in Eq. (18), thermodynamic quantities will be derived according to Tsallis formalism and BG description. Obviously statistical mechanics is also qualified as the theory of probability. In this part, we derived the thermodynamic quantities such as; generalized entropy, internal energy, free energy and heat capacity.

Entropy can be written as $S = K_{{\text{B}}} \ln \Omega$ where $K_{{\text{B}}}$ is the Boltzmann constant and $\Omega$ the total number of accessible states. Boltzmann–Gibbs (BG) described a thermodynamic physics and proposed a common form of an entropy as:

$$S = - K_{{\text{B}}} \sum\limits_{{i = 1}}^{w} {p_{{\text{i}}} \ln p_{{\text{i}}} }$$

(19)

With $\sum\limits_{{i = 1}}^{w} {p_{{\text{i}}} } = 1$ and $p_{{\text{i}}}$ the probability of the 1st state.

After BG, Tsallis proposed in 1988 a new formalism for thermodynamic and propose a new way to define an entropy as:

$$S_{{\text{q}}} = k\frac{{1 - \sum\limits_{{i = 1}}^{w} {p_{{\text{i}}} ^{{\text{q}}} } }}{{1 - q}}\,\,\,\,\,q \in R$$

(20)

Using an expression of energy defined in Eq. (18), the final expression of entropy is given by:

$$\begin{aligned} S_{{\text{q}}} & = \beta \frac{{E_{{}}^{ + } \left[ {1 - \beta \left( {1 - q} \right)E_{0}^{ + } } \right]^{{\frac{1}{{1 - q}}}} + E_{{}}^{ - } \left[ {1 - \beta \left( {1 - q} \right)E_{{}}^{ - } } \right]^{{\frac{1}{{1 - q}}}} }}{{\left[ {1 - \beta \left( {1 - q} \right)E_{{}}^{ + } } \right]^{{\frac{1}{{1 - q}}}} + \left[ {1 - \beta \left( {1 - q} \right)E_{{}}^{ - } } \right]^{{\frac{1}{{1 - q}}}} }} \\ & \quad + \ln \left[ {\left[ {1 - \beta \left( {1 - q} \right)E_{{}}^{ + } } \right]^{{\frac{1}{{1 - q}}}} + \left[ {1 - \beta \left( {1 - q} \right)E_{{}}^{ - } } \right]^{{\frac{1}{{1 - q}}}} } \right] \\ \end{aligned}$$

(21)

where k is a conventional positive constant and the real number $q$ is an entropy index that characterize and degree of non-extensivity, $E_{{}}^{ + }$ and $E_{{}}^{ - }$ are, respectively, an energy for different spin states.

Since an internal energy can be defined as the total energy of accessible states in a system, it is evaluated as:

$$U_{{\text{q}}} = \sum\limits_{j}^{w} {p_{{\text{j}}} \varepsilon _{{\text{j}}} }$$

(22)

using Eq. (22), the internal energy is written as:

$$U_{{\text{q}}} = \left( {1 - T\beta } \right)\frac{{E_{{}}^{ + } \left[ {1 - \beta \left( {1 - q} \right)E_{{}}^{ + } } \right]^{{\frac{1}{{1 - q}}}} + E_{{}}^{ - } \left[ {1 - \beta \left( {1 - q} \right)E_{{}}^{ - } } \right]^{{\frac{1}{{1 - q}}}} }}{{\left[ {1 - \beta \left( {1 - q} \right)E_{{}}^{ + } } \right]^{{\frac{1}{{1 - q}}}} + \left[ {1 - \beta \left( {1 - q} \right)E_{{}}^{ - } } \right]^{{\frac{1}{{1 - q}}}} }}$$

(23)

where $\varepsilon _{{\text{j}}}$ is eigenvalue of the Hamiltonian, $w$ is the total number of microscopic state and $U_{{\text{q}}}$ represents the generalized internal energy of system.

Since the Tsallis formalism is a probabilistic method used to really describe thermodynamic of quantum system, the following formalism allows to describe the probability of all accessible state.

$$p\left( {\varepsilon _{{\text{j}}} } \right) = \left\{ \begin{gathered} g_{{\text{n}}} \left[ {1 - \beta \left( {1 - q} \right)\varepsilon _{{\text{j}}} } \right]^{{\frac{1}{{1 - q}}}} \quad \quad {\text{if}}\quad g_{{\text{n}}} \left[ {1 - \beta \left( {1 - q} \right)\varepsilon _{{\text{j}}} } \right]^{{\frac{1}{{1 - q}}}} \succ 0 \hfill \\ 0\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad {\text{otherwise}} \hfill \\ \end{gathered} \right.$$

(24)

where $g_{{\text{n}}}$ represents the degeneracies of energy levels $\varepsilon _{{\text{j}}}$, and the total number of microscopic states is given by:

$$Z_{{\text{q}}} = \sum\limits_{j} {g_{{\text{n}}} \left[ {1 - \beta \left( {1 - q} \right)\varepsilon _{{\text{j}}} } \right]^{{\frac{1}{{1 - q}}}} }$$

(25)

$$Z_{{\text{q}}} = g_{{\text{n}}} \left\{ {\left[ {1 - \beta \left( {1 - q)} \right)E^{ - } } \right] + \left[ {1 - \beta \left( {1 - q} \right)E^{ + } } \right]} \right\}^{{\frac{1}{{1 - q\,}}}}$$

(26)

From the relation relating Tsallis entropy, heat capacity and temperature, we can obtain the generalized specific heat by the variation of generalized internal energy:

$$C_{{\text{q}}} = T\frac{{\partial S_{{\text{q}}} }}{{\partial T}} = \frac{{\partial U_{{\text{q}}} }}{{\partial T}}$$

(27)

$$\begin{aligned} C_{{\text{v}}} & = \frac{q}{{1 - q}}\frac{{E_{{}}^{{ + 2}} \left[ {1 - \beta \left( {1 - q} \right)E_{{}}^{ + } } \right]^{{\frac{1}{{1 - q}}}} + E_{{}}^{{ - 2}} \left[ {1 - \beta \left( {1 - q} \right)E_{{}}^{ - } } \right]^{{\frac{1}{{1 - q}}}} }}{{\left[ {1 - \beta \left( {1 - q} \right)E_{{}}^{ + } } \right]^{{\frac{1}{{1 - q}}}} + \left[ {1 - \beta \left( {1 - q} \right)E_{{}}^{ - } } \right]^{{\frac{1}{{1 - q}}}} }} \\ & \quad - \frac{\beta }{{1 - q}}\left[ {\frac{{E_{{}}^{ + } \left[ {1 - \beta \left( {1 - q} \right)E_{{}}^{ + } } \right]^{{\frac{{2q - 1}}{{1 - q}}}} + E_{{}}^{ - } \left[ {1 - \beta \left( {1 - q} \right)E_{{}}^{ - } } \right]^{{\frac{{2q - 1}}{{1 - q}}}} }}{{\left[ {1 - \beta \left( {1 - q} \right)E_{{}}^{ + } } \right]^{{\frac{1}{{1 - q}}}} + \left[ {1 - \beta \left( {1 - q} \right)E_{{}}^{ - } } \right]^{{\frac{1}{{1 - q}}}} }}} \right]^{2} \\ \end{aligned}$$

(28)

Being the parameter used to characterize the exchange of energy between the system and its environment, whether in the thermodynamics’ description given by BG or Tsallis, free energy is measured using the following equation.

$$F_{{\text{q}}} = - KT\ln Z_{{\text{q}}}$$

(29)

$$\begin{aligned} F_{{\text{q}}} & = \left( {1 - T\beta } \right)\frac{{E_{{}}^{ + } \left[ {1 - \beta \left( {1 - q} \right)E_{{}}^{ + } } \right]^{{\frac{1}{{1 - q}}}} + E_{{}}^{ - } \left[ {1 - \beta \left( {1 - q} \right)E_{{}}^{ - } } \right]^{{\frac{1}{{1 - q}}}} }}{{\left[ {1 - \beta \left( {1 - q} \right)E_{{}}^{ + } } \right]^{{\frac{1}{{1 - q}}}} + \left[ {1 - \beta \left( {1 - q} \right)E_{{}}^{ - } } \right]^{{\frac{1}{{1 - q}}}} }} \\ & \quad - \ln \left[ {\left[ {1 - \beta \left( {1 - q} \right)E_{{}}^{ + } } \right]^{{\frac{1}{{1 - q}}}} + \left[ {1 - \beta \left( {1 - q} \right)E_{{}}^{ - } } \right]^{{\frac{1}{{1 - q}}}} } \right] \\ \end{aligned}$$

4 Results and Discussion

In this section, we bring out the numerical results for quantum pseudodot by investigating the effect of SOI on the thermal properties of quantum pseudodot. Since Tsallis formalism allows to evaluate thermal properties at accessible states in the system, that is why is started by calculating the probability so using Tsallis formalism, we are ensured where we are evaluating thermal properties, is accessible for particles, while the GB formalism is used to evaluate thermal properties in the whole system even at non-accessible states of system. Then in order to have accurate values of our study, we found that Tsallis formalism is the most appropriate for our investigation.

Figures 1, 2 and 3 present the variation of internal energy, respectively, with SOI constant $\alpha _{{\text{r}}}$, cyclotron frequency $\omega _{{\text{c}}}$ and chemical potential with. Varying those parameters, we observe in Fig. 1 an increase in internal energy with a cyclotron frequency in one hand and on the other hand, the decreasing of internal energy with chemical potential. Basically, an internal energy represents the summation of the accessible states. On those figure, we also observe the dependence of an internal energy on the spin state. For example, in both cases, we have an increase in internal energy when spin is in down-state and a decreasing in the up-state, which means the most accessible state is occupied by the down-state spin. A similar behavior is presented in Fig. 3, but adding on it the effect of spin orbit interaction, it affect considerably an internal energy for $V_{0} \ge 7\,\,$. This effect is also observed by an appearance of splitting on Figs. 1, 2 and 3. Since the internal energy is defined as the summation of all accessible states, this appearance of splitting directly leads to an increase in the number of accessible states.

Fig. 1

Variation of internal energy $U$ as function of spin orbit interaction constant $\alpha _{{\text{r}}}$ for different values of chemical potential $V_{0}$

Fig. 2

Internal energy $U$ as function of cyclotron frequency $\omega _{{\text{c}}}$ for different values of chemical potential $V_{0}$

Fig. 3

Variation of internal energy $U$ as function of chemical potential $V_{0}$ for different values of spin orbit interaction constant $\alpha _{{\text{r}}}$

Figure 4 plots internal energy as function of $\alpha _{r}$ for different values of the real number $q$. In this figure, we observe an increase in internal energy $U_{{\text{q}}}$ with the SOI constant $\alpha _{{\text{r}}}$. $U_{{\text{q}}}$ Increases with the decreasing of the real number $q$, the reason of this is that: an increase in $q$ reduces the number of accessible states created by the effect of SOI. This analysis is in accordance with the results found by [55].

Fig. 4

Internal energy $U_{{\text{q}}}$ as function of spin orbit interaction constant $\alpha _{{\text{r}}}$ for different $q$

Figures 5 and 6 show the dependence of entropy, respectively, with the magnetic field for different temperatures and with chemical potential $V_{0}$ for different values of SOI constant $\alpha _{r}$. In Fig. 5, we observe that entropy decreases with an increase in magnetic field but remains an increasing function of temperature. At very weak regime magnetic field, we have the maximum entropy. The effect of temperature on entropy is more sensitive at weak magnetic field because its confinement effect is not more pronounced, but an increase in temperature increases the kinetic energy too where an increase in entropy because of it. The more we sheaf from weak to strong regime of magnetic field, the more entropy decreases. The monotonic behavior of entropy with magnetic field for different temperatures is due to the great effect of magnetic field on the properties of a quantum pseudodot, moreover, the decreasing in curve spacement from weak to strong regime marks the effect of magnetic field.

Fig. 5

Entropy $S$ as function of cyclotronic frequency $\omega _{{\text{c}}}$ for different temperature $T$

Fig. 6

Entropy as function of chemical potential $V_{0}$ for different values of spin orbit interaction constant $\alpha _{{\text{r}}}$

The dependence of entropy on chemical potential for different values of SOI is displayed on Fig. 6. We observe an increase in entropy with those parameters, from this, we can conclude that an increase in chemical potential reduce the confinement of particles, this is the justification of an increase in entropy it is also because an increase in chemical potential releases spin and add their number of degree-of-freedom. Since an increase in SOI constant increase the number of accessible states, entropy grows at a definite value of temperature. Those results are similar to those found by [69].

We characterized the exchange of energy of the system with its environment of Fig. 7 as function of spin orbit interaction constant different temperature taking $q = 0.3,\,\,V_{0} = 10\,,\,\,\,\,\omega _{{\text{c}}} = 10$. We observe an increase in free energy $F_{q}$ with these parameters $\alpha _{r} \,,\,\,T$. Since an increase in SOI constant increases the number accessible state, then the more particles are moving from one state to another, the more they are exchanging energy with environment. This behavior for $q = 0.3,\,\,V_{0} = 10\,,\,\,\,\,\omega _{c} = 10\,\,;\rho _{{}} = 2\,nm\,;\,\,\rho _{0} = 1\,nm$, an increase in temperature accelerates the motion of particles then increases free energy too.

Fig. 7

Free energy $F_{q}$ as function of spin orbit interaction constant different temperature taking $q = 0.3,\,\,V_{0} = 10\,Mev,\,\,\,\,\omega _{{\text{c}}} = 10\,\,;\,\,\rho _{{}} = 2\,nm\,;\,\,\rho _{0} = 1\,nm$

Figure 8 displays the dependence of heat capacity $C_{q}$ on temperature for different values of chemical potential $V_{0}$ with $q = 0.23,\,\,\,\omega _{{\text{0}}} = 10,\,\,\,\alpha _{{\text{r}}} = 0.06$. At low temperature $T \le 4K$ we observe a weak effect of chemical potential and temperature, since a heat capacity is the response of the system with temperature. From this analysis, the energy to vary the temperature in quantum pseudodot system is considerable affected by the variation of chemical potential and at certain values of chemical potential like $V_{0} \le 10$ a quantum pseudodot cannot store enough of energy this why there is appearance of negative values of heat capacity. In this case, the energy produced should be consumed in changing of temperature. This figure shows an appearance of splitting caused by different values of chemical potential, this appearance of splitting is because an increase in chemical potential modifies the spin state in a quantum pseudodot.

Fig. 8

Dependence of heat capacity on temperature for different values of chemical potential taking $q = 0.23,\,\,\,\omega _{{\text{c}}} = 10,\,\,\,\alpha _{{\text{r}}} = 0.06$

In Fig. 9, we observe the same behavior of heat capacity with chemical potential, but for $\omega _{{\text{c}}} = 10,\,\,\,T = 10\,K,\,\,\alpha _{{\text{r}}} = 0.06$, we observe on one hand side an increase in heat capacity with some values of chemical potential. Here for $q$ out of this range $\left[ {0.95 - 0.96} \right]$, the heat capacity is zero for all values of chemical potential and temperature. Contrary to commonly meet results in many works on thermodynamic properties, a temperature of system with negative heat capacity will further increase at it lost heat. Moreover, this result can appear paradoxal at first, but let’s recall that for inhomogeneous system we can found a negative heat capacity, which is not always in accordance with thermodynamic equilibrium. An example of such system is a quantum pseudodotsystem which presents for a certain values of chemical potential a negative heat capacity. Then for $q$ taken in that range, the kinetic energy of particles and levels spacing increase Fig. 10.

Fig. 9

Variation of heat capacity as function of q for different values of chemical potential taking $\omega _{{\text{c}}} = 10,\,\,\,T = 10\,K,\,\,\alpha _{{\text{r}}} = 0.06$

Fig. 10

Variation of energy as function of SOI constant $\alpha _{{\text{r}}}$ for different values of magnetic field and principal quantum number $n$

In this figure, we observe that far from the fact that an energy is an increasing function of magnetic field and principal quantum number, one can note here a great effect of SOI constant on an energy of a quantum pseudodot. Combined with the effect magnetic field, SOI interaction can create more accessible in quantum system, this can be justified throughout an appearance of splitting in the previous figure.

5 Conclusion

In the present work, we have studied the effect of SOI and chemical potential on the thermodynamics of a quantum pseudodot. From our analysis, we deduce: (i) there is a splitting caused by SOI constant and some values of chemical potential since an increase in these parameters modifies the spin orientation of a quantum pseudodot. We conclude that internal energy and entropy are continuous functions of SOI and chemical potential at finite temperature. (ii) An increase in accessible states created by a SOI causes an increase in entropy and free energy which marks a great effect of SOI on the thermodynamics of a quantum pseudodot. (iii) For some value of the real number $q$, the heat capacity is zero at finite temperature and spin interaction coupling constant.