ENERGY LEVELS OF AN ELECTRON IN A CIRCULAR QUANTUM DOT IN THE PRESENCE OF SPIN-ORBIT INTERACTIONS

The two-dimensional circular quantum dot in a double semiconductor heterostructure is simulated by a new axially symmetric smooth potential of finite depth and width. The presence of additional potential parameters in this model allows us to describe the individual properties of different kinds of quantum dots. The influence of the Rashba and Dresselhaus spin-orbit interactions on electron states in quantum dot is investigated. The total Hamiltonian of the problem is written as a sum of unperturbed part and perturbation. First, the exact solution of the unperturbed Schrödinger equation was constructed. Each energy level of the unperturbed Hamiltonian was doubly degenerated. Further, the analytical approximate expression for energy splitting was obtained within the framework of perturbation theory, when the strengths of two spin-orbit interactions are close. The numerical results show the dependence of energy levels on potential parameters.


Introduction
The motion of an electron in an inner layer of a double semiconductor heterostructure is usually treated as two-dimensional in the ( , ) x y plane. In addition, the planar motion is also restricted if an electron is placed in a quantum dot localized in the middle layer of heterostructure. The Rashba R V [1] and Dresselhaus D V [2] interactions are presented by the formulas = ( ) / , = ( ) / , where x  and y  are the standard Pauli spin-matrices. The strengths of these interactions depend on the materials used. The contributions of two spin-orbit interactions can be measured within various experimental methods [3,4]. In the general case the whole spin-orbit interaction has the form R D V V  . At the same time, considerable attention is paid to the special case [3,5,6], when the spin-orbit interactions of Rashba and Dresselhaus have equal strength = R D   . It can be experimentally achieved due to the fact that the Rashba interaction strength can be controlled by an external electric field, and the Dresselhaus interaction strength can be varied by changing the width of quantum well along the z axis [3,7].
As a rule, circular quantum dots are simulated with the help of axially symmetric confinement In [8,9], a simple but sufficiently adequate rectangular potential of finite depth was proposed. This model with a discontinuous potential describes the main properties of circular quantum dots but without taking into account the individual characteristics. In [10], the smooth confinement potential of a new type which has finite depth and width was applied in the case of equal strengths = R D   . The presence of additional potential parameters allows us to simulate different kinds of circular quantum dots. In the actual paper, we use this potential in order to calculate the energy levels of electron for unequal but close strengths R D    .

Methods and results
The circular quantum dot of radius 0  is described by means of the confinement potential , where 0 V is the depth of the potential well. The function ( ) v r depends on ratio 0 = / r   in the following way The functions 1 ( ) v r and 2 ( ) v r have the following forms: The parameters g and s change within ranges 0 < < 1 g and < < 1 g s . The function ( ) v r and its first derivative are continuous in the inflection points = r g , = r s and = 1 r . The total Hamiltonian of the problem can be written as a sum M is the effective electron mass which characterizes the motion in a semiconductor.
We shall solve the full Schrödinger equation = H E   in two stages. First, we obtain an exact solution of the unperturbed Schrödinger equation  and then we shall take into account the perturbation 1 H within the framework of the perturbation theory. By analogy with [10] it is easy to show that the required solutions of the unperturbed Schrödinger equation admit a factorization where = 0, 1, 2, m    is the angular momentum quantum number. Here we use the polar coordinates , we get the radial equation It is seen that the wave function depends only on the combination 0 2 e a  . In the region 0 < < r g , the finite at 0 r  solution of radial equation is expressed via the Bessel function [11] by means of the formula In the region < < g r s , it is simple to obtain two solutions in terms of the confluent hypergeometric functions [11]: In the region < < 1 s r , it is easy to show that two solutions are Note that the functions 4 ( ) w r and 5 ( ) w r are real if In the region > 1 r , the decreasing solution is expressed via the modified Bessel function [11] with the help of the formula   c w r r g c w r c w r g r s w r c w r c w r s r c w r r The coefficients i c are found from the continuity condition for function ( ) w r and its first derivative ( ) ' w r at three inflection points = r g , = r s , and = 1 r . The fulfilment of this condition and the continuity of the potential and its first derivative guarantee the continuity of the second and the third derivative of the wave function.
w g w g w g w g w g w g w s w s w s w s T g s v m a e w s w s w s w s w w w w w w Then the dependence of dimensionless energy 0 0 ( , , , , ) e g s v m a on three dimensionless potential parameters g , s , and 0 v is determined by the transcendental equation

Conclusion
The confinement model potential for a quantum dot considered in the present paper is smooth, has finite depth and width and permits the exact solutions of the separated unperturbed Schrödinger equation for electron states in the presence of the spin-orbit interaction of Rashba and Dresselhaus. The contribution of perturbation is really small in comparison with the unperturbed energy 0 e if the strength R  is sufficiently close to the strength D  ( 1   ). Further, we intend to construct higher-oder corrections to the energy levels.