Effects of Rashba spin–orbit interaction and topological defect on the magnetic properties of an electron confined in a 2D quantum dot

The energy spectra of Hamiltonian of a single electron confined in a parabolic quantum dot (QD) have been investigated, taking into considerations different external effects such as the Rashba spin–orbit interaction term, applied uniform magnetic field and topological defect. The results show that the topological defect enhances the energy levels, while the increment in the strength of Rashba parameter removes the spin degeneracy and reduces the energy levels. The obtained QD energy spectra are used to study the statistical mean energy from which we consider the behaviour of the magnetization and magnetic susceptibility on the QD. It is found that the topological defects and Rashba terms play important roles in flipping the sign of the magnetic susceptibility from negative to positive sign . The present results are in very good qualitative agreement compared with the reported ones.


Introduction
Quantum dot (QD) is a nanostructured material that confined the carriers in all three directions, reducing the degrees of freedom to zero and exhibiting a discrete energy spectrum, as in a natural atom [1][2][3]. The addition of a magnetic field and Rashba term will make the Hamiltonian problem very interesting research problem in the field of nanoscience and technology. Semiconductor nanostructure QDs have become very hot research subject due to their potential device applications such as QD lasers, quantum computation, QD solar cells, biological and medical application such as the image cancer treatment [4][5][6]. The spin-orbit coupling phenomena in semiconductor QDs system is very important interaction energy term as it plays an essential role in controlling the properties of the QD which makes the QD an excellent basis in the new emerging field of technology, called spintronics. The QD-spin transistor is considered to be an important electronic device in the area of spintronics. Different authors had, very recently, studied the thermodynamic and magnetic properties of the QD systems in the presence of a magnetic field [7,8]. Baghdasaryan et al. [7] had considered the thermal and magnetic properties of an electron confined in a toroidal QD in the presence of an applied magnetic field. Sedehi and Khordad had presented a theoretical study on the magnetic and thermal properties of QD/ring with different potentials: parabolic-inverse confinement potential and modified Gaussian potential under the effect of an applied uniform magnetic field [9]. Edet et al. had investigated the position-dependent mass Schrodinger equation for the screened potential with Aharanov-Bohm (AB) and external magnetic field. The thermodynamic properties and magnetic susceptibility of the system at various temperature range had been studied [10]. The combined effects of pressure, and the spin-orbit interaction on the QD energy spectra had also been considered [11,12].
In this work, we studied, in detail, the combined effects, both topological defects and the spin-orbit interaction term on the properties of the QD. We consider a single electron QD confined in two dimensions in the presence of a magnetic field, taken to be along zaxis, in addition to the spin-orbit interaction term. The total Hamiltonian of QD system is solved and the corresponding energy spectra are obtained in analytical form. The obtained energies are used to calculate the partition functions and investigate the variation of the energy levels, magnetic and thermal properties of the zero-dimensional semiconductor QD systems with the presence of Rashba and topological effect terms. We display the significance of the topological defect and Rashba strength in the change of the magnetic phase of the QD-material from diamagnetic to paramagnetic type and the flip in the sign of magnetic susceptibility (χ) from negative (−) to positive (+). The rest of this work is organized as follows. The Hamiltonian of a single electron confined in a QD, taking into consideration the combined effects of Rashba spin-orbit interaction and topological defect, are presented in the theory-part. We have displayed the computed numerical results of the magnetic quantities in results and discussion-part. Our conclusions are given in the final part.

Theory
The Hamiltonian of an electron confined in QD by a potential, V c (r), and under the effect of an applied uniform magnetic field and spin-orbit interaction term can be given asĤ where m * is the electron effective mass of the material of GaAs QDs, P refers to the electron momentum operator, e is the elementary charge, c is the speed of light, A is the vector potential is chosen to be in the symmetric gauge as V c (r) is the confining potential, modelled as a parabolictype-like, where ω 0 is the strength of the confinement potential frequency, r is the position vector of an electron in the QD and its equal (x 2 + y 2 ) 1 2 , where H R is the Rashba term, α R is the Rashba parameter strength of SOI, σ are the Pauli matrices, {σ x , σ y },h is Plank's constant [13,14]. Topological effect is the surface of the QD which has a topological defect described in polar coordinates (ρ, ϕ) by the metric [15]: where ρ = α −1 r, ϕ = αθ, then the metric becomes while α is a kink parameter and it controls the cut off, whereas 0 < α < 1, if α = 1 means that there is no effect of topological defect, with ϕ belongs to 0 < ϕ < 2πα, and 0 < θ < 2π. The total QD Hamiltonian, H, can be reduced to a solvable harmonic oscillator one with analytical energy spectra expression. The eigen energy spectra are defined, in terms of the quantum numbers (n, l) and other physical functions [15], as where is the effective frequency and defined as ω c denotes the cyclotron frequency, s is the spin of the electron, γ is the Rashba spin orbit parameter, p is the topological parameter and it equal the inverse of kink parameter (α −1 ), n = 0, 1, 2, . . . . is the radial quantum number, l = 0, ±1, ±2, . . ., is the angular quantum number and g * is the effective Lande g-factor.
The obtained eigen energies of the QDs will be used as input essential data to calculate the statistical average energy, where β = 1 k B T and k B is the Boltzmann constant. Having computed the average energy, all the thermal and magnetic quantities of the QD can be obtained using the well-known relations: Magnetization [13,14,16], and Magnetic Susceptibility, The dependence of the thermal and magnetic functions will be investigated as functions of confinement strength, magnetic field cyclotron frequency, Rashba term, topological factor and temperature. In addition, the phase diagram for the magnetic susceptibility will be plotted as function of topological defect, Rashba term, magnetic field strength and temperature to show the magnetic transition of the GaAs from diamagnetic to paramagnetic.

Results and discussion
In this section, we discuss the obtained results for the energy spectra of an electron confined in a parabolic QD in presence of RSOI term, topological effect and an applied uniform magnetic field. We study the physical properties of the GaAs QDs material by computing the statistical energy, magnetization and susceptibility. For GaAs QD, we used the following physical parameters; effective electron mass: m * = 0.067m e , Figure 1. The E versus # of bases with different temperature (T) range, from 10 K to 100 K, T at ω c = 3 R * , ω 0 = 2.5 R * , γ = 0.5a * .R * and p = 1.
effective Lande factor: g * = − 0.44, effective Rydberg energy:R * = 5.694 meV, effective Bohr radius: a * = 9.8 nm, Rashba parameter γ (1a * .R * = 55.8 meV.nm) and magnetic field ω c (ω c (R * ) = 0.296 × (B in Tesla (T)). In Figure 1, as a significant convergency test to our computational scheme, we have displayed the calculated average energies against the number of basis (#) for different temperature range, from 10 K to 100 K, and physical QD functions. As the energy of the QD-system increases, the electron average energy E enhances due to the available thermal energy, E th ≈ k b T, which is gained by the electron as an excitation energy. The average energy plot clearly shows a numerical stability for various values of Hamiltonian functions. Based on the following two important issues, first, the energy spectra (E n,l,s ) expression is an exact one, given by Equation (7), second, the present convergency test for the average energy < E > will produce accurate computed numerical results for magnetic (M) and susceptibility (χ) quantities.
In Figure 2(a), we have used equation (7) to plot the QD Fock-Darwin states as a function of a magnetic field strength for p = 1, ω 0 = 2.5 R * and zero Rashba parameter γ = 0.The energy spectra show a similar behaviour with the corresponding ones given by Ref. [14], The labeled of the Fock-Darwin energy states |n, l, s were plotted in Figure 2(a) from the bottom at ω c = 1R * are: |0, 0, + , |0, 0, − , |0, −1, + , |0, −1, − , |0, 1, + and |0, 1, − . The plot clearly shows that a converging average energy value can be obtained for small number of bases ≈ 20, at low temperature, while at high temperature, we need around 40 bases. Figure 2(c) displays the effect of topological defect (p) on the QD energy spectra, we observe that the energy state increases at constant ω c for spin up and down. For l = 0 and 1, the energy level increases as ω c increases, while the energy decreases for the case of l = −1 at ω c = 3R * and then increases, it is attributed to the electron confinement in a small region which leads to an enhancement in the momentum, and in turn the kinetic energy, by uncertainty principle [13] and eventually the energy of the electron increases. In Figure 2(d), the Rashba effect removes the spin degeneracy for l = +1 and l = −1 energy states by showing wide energy splitting, E, for spin up and spin down states.
The behaviour of the energy spectra due to the Rashba and topological effects are shown in Figure 2(d).
To investigate the effects of external parameters such as Rashba and topological defect on the energy levels of the QD, we have plotted in Figure 3 < E > against ω c for ω 0 = 2.5 R * and various selected values of temperature.
If we apply, jointly, RSOI and p as in Figure 3(d), the effect of γ is higher than p until ω c ≈ 1.8 R * that is the energy level decreases. However, for ω c > ≥ 1.8 R * the p effect is higher than γ and the energy increases.
The variation of magnetization as a function of ω c for different cases of T, p and γ is plotted in Figure 4. The magnetization M has a peak structure and it sign is negative for the derivative of E with respect to ω c , This magnetization behaviour can be understood from the average energy plot shown in Figure 3 At p = 1 and γ = 0a * .R * , we show in in Figure 4(a), the effect of changing γ on the M − ω c − plot. We see that M increases until ω c ≈ 0.8R * for T = 5 K and then decreases, but it is decreasing when the temperature is increasing. However, in Figure 4(b) for γ = 0.7a * .R * , the magnetization, M, decreases with increasing ω c while M decreases as increases T at constant ω c . In Figures 4(c,d), we show the effect of Rashba effect, for p = 1.5, on the M-magnetic field curve. For example, Figure 4(c) shows that magnetization, M decreases, as shown in Figure 4(c).
The variation of the susceptibility as a function of ω c for various values of T, p and γ is shown in Figure 5, where χ is the derivative of M with respect to B. We can see from the Figure 5(a) that at p = 1 and γ = 0a * .R * , the magnetic susceptibility for T = 5 and 10 K is positive (χ > 0) until ω c ≈ 0.8 R * and then it changes to negative (χ < 0). This sign flipping in χ means that the material can change its type from diamagnetic to paramagnetic. In Figure 5(b) with Rashba coupling (γ = 0.7a * .R * ) the material is diamagnetic for different temperature. When p = 1.5 in Figure 5(c), χ is negative for all ω c range at T = 10, 20 and 30 K, while its positive  until ω c = 0.4 R * and then negative at T = 5 K, with both Rashba and topological effects as in Figure 5(d), the effect of Rashba is higher and χ is negative for various ω c and T. Figure 8. Computed magnetic phase diagram of GaAs QD (p = 1, γ = 0, ω 0 = 6.3R * ) as a function of T and ω C which corresponding one given by Ref. [17].
We search for the magnetic phase diagrams of the GaAs nanomaterial for a wide range of QD physical parameters. To achieve our objective, we present a contour-plot (Figures 6(a-d) for the magnetic susceptibility of the QD as function of the QD-physical parameters. Figure 6(a) shows the phase transition obtained by equation (12), without any external affect. The obtained contour plot, for various QD-parameters, is in a very good agreement with the corresponding one given by Refs. [11,17]. To investigate further the effects of topological defect, we have presented additional contour plot, Figure 7, for a wide range of defect parameter (p = 1.0-2.0), small confinement (ω 0 = 0.5R * ) and quite high temperature (T ≈ 250K). The figure shows obviously the roles of topological functions: p, T and ω 0 in controlling the magnetic type transition of the QD-material from diamagnetic (χ < 0) to paramagnetic (χ > 0) phase diagrams. Furthermore, the present magnetic susceptibility contour plot shown in Figure 8, and for p = 1 case, is compared, against the corresponding one shown in Ref. [17], where the authors had obtained the contour plot of the magnetic susceptibility by approximating the gaussian QD model (GQD) as a parabolic type confining potential, in terms of gaussian confining potential strength, V 0 , and QD radius, R, as ω 2 h = V 0 m * R 2 .

Conclusion
In this work, the Hamiltonian of a single electron confined in a parabolic QD including various external factors like: Rashba spin orbit interaction term effect, applied uniform magnetic field and topological defect had been used in a closed form. We have studied the dependence of the energy spectra for our QD as a function of: confinement frequency (ω 0 ), magnetic field (ω c ), Rashba (γ ) and topological defect (p). Our energy results obtained from a close energy expression are in very good agreement with reported works [15]. The computed results show that if we increase the topological factor (p), the E increases also and this result affects significantly M and χ of the QD-material. The behaviour of the magnetic susceptibility χ is shown against the QD functions. We found that the QD material changes its magnetic phase from diamagnetic (χ < 0) to paramagnetic (χ > 0) type [13]. These magnetic transition phase diagrams of the QD are shown explicitly in the contour plots as function of various physical functions of the QD-Hamiltonian.