Electronic States in a Doubly Eccentric Cylindrical Quantum Wire

Electronic states of a single electron in doubly eccentric cylindrical quantum wire are theoretically investigated in this paper. The motion of electron in quantum wire is free along axial direction in a cylindrical quantum wire and restricted in annular regions by three different parallel finite cylindrical barriers as soft wall confinement. The effective mass Schrödinger equation with effective mass boundary conditions is used to find energy eigenvalues and   corresponding wavefunctions. Addition theorem for cylindrical Bessel functions is used to shift the origin for applying boundary conditions at different circular boundaries. Fourier expansion is applied after addition theorem to get wavefunctions in analytical form. A determinant equation is obtained as a result of applications of effective mass boundary conditions which roots gives energy of various electronic states. The lowest root gives ground state energy. The variation in ground state energy with eccentricity is obtained numerically and presented graphically. Electronic states in massive wall confinement and hard wall confinement is further obtained as limiting behavior of the states obtained in soft wall confinement. The knowledge of electronic states in such cylindrical hetrostructures semiconductor material can lead to improve the efficiency of many quantum devices.


Introduction
The semiconductor hetrostructures are very useful in construction of many quantum devices like quantum laser where active medium is a quantum wire or dot. A lot of properties of quantum wire has been studied. Optical properties of a cylindrical quantum wire has been investigated theoretically to study variation of absorption coefficient and refractive index [1,2]. Binding energies of impurity in quantum dot structures have been studied as function of geometry [3]. Effect of shape and size on electron energy spectrum in various shaped quantum structures have been investigated [4][5][6][7][8]. The properties of quantum wire and dots with cylindrical geometry are of special interests among various researchers due to its wide range of applications [9][10][11][12][13][14][15]. Theoretical study of electrical properties of single eccentric cylindrical structures with C 1v and C ∞v symmetries have been studied [16]. We have investigated electrical properties of a new structure-doubly eccentric cylindrical quantum wire. There is no report on electronic states of a doubly eccentric cylinder in nano regime to our knowledge. A doubly eccentric cylindrical quantum wire consists of two eccentric cylindrical quantum wire nested inside a cylindrical quantum wire of larger radius. A cross-section of such quantum wire is shown in Fig.1. Such doubly eccentric cylindrical wire in nano regime can help to develop semiconductor material to enhance the efficiency of many quantum devices. The electronic states in a doubly cylindrical hetrostructures with C 2v symmetry in soft wall confinement (SWC) are studied theoretically in this paper. As its limiting behavior, the corresponding electronic states in hard wall confinement (HWC) and massive wall confinement (MWC) are also obtained. An exact solution of effective mass Schrödinger equation is obtained by using addition theorem for shifting origin of cylindrical Bessel functions in hetrostructures to satisfy corresponding boundary conditions for wavefunctions. The validity of addition theorem in doubly eccentric cylinder imposes a restriction on the range of its eccentricity but that doesn't lead to an issue as allowed range of eccentricity to cover most of practical problems. Section 2 of this paper presents general geometrical structure of the problem and its exact solution in soft wall confinement (SWC). At first, the solution of effective mass Schrödinger equation is obtained in terms of cylindrical Bessel functions for various regions. Effective mass boundary conditions are then applied across each pair of boundaries. Fourier expansion is applied on the expression obtained after applying appropriate boundary condition to get the solution in analytical form. As a result, these give a set of infinite simultaneous equations which is discussed in section 3. A determinant equation is obtained for non-trivial solution of the infinite simultaneous equations. The roots of this determinant equation give various energy eigenvalues. The lowest root of determinant equation is obtained numerically to get ground state energy. The separation between axes of two eccentric cylinders is termed as eccentricity of structure. The ground state energy is calculated by varying eccentricity of structure. The result in variation of ground state energy with eccentricity is presented graphically. The corresponding limiting solution for massive wall confinement (MWC) and hard wall confinement (HWC) is discussed in section 4. Section 5 presents concluding remarks. Fig. 1 shows annular cross-section of a doubly eccentric cylindrical structure. The length is along Z-axis which is perpendicular to plane of figure. The axes of region I, II and III are parallel. Radii of regions I and III are 'a' and that of region II is 'b'. The separation between centres of region II and III is d 1 and that between I and II is d 2 . In the structure, it is kept d 1 = d 2 = d so that structure has C 2v symmetry which is more useful in physical applications. However, this methodology can be used to study the electronic states in doubly eccentric cylinder with d 1 ≠ d 2 as well. The parameter 'd' represents separation between axes of inner cylinder and is called eccentricity. We have considered a system in which an electron is free to move along axis of cylinder in region II and its annular motion is confined by different cylindrical barriers shown as regions I, III and IV in Fig. 1. Using the theory of effective mass approximation, each region can be considered as region of constant potential with corresponding effective mass of the electron. Let effective mass of the electron in th l region is * , l ml  1, 2, 3 and 4. We have set the value of potential energy of region II equal to zero as the reference of potential energy. Let the barrier potential in th l region is   0, l Ul  1, 3 and 4.Taking

Solution
as cylindrical polar co-ordinates with respect to origin O 1 , the effective mass Schrödinger equation [17][18] for hetrostructures is given as For uniform distribution of effective mass * l m in the th l region, * In region I, the solution of Eq. (2) with origin at O 1 comes as where m A are unknown constants,   In region II, the solution of Eq. (2) with origin at O 1 comes as In region III, the solution of Eq. (2) with origin at O 1 comes as Eq. (3) to Eq. (9) give the wavefunctions with some unknown constants in region I to IV respectively. These unknown constants and hence energy eigenvalues can be calculated using the fact that  and * 1 m   (effective mass boundary condition) has to be continuous at each of the boundaries [19]. To apply boundary conditions across boundary between region I and II, wavefunctions I  and II  in terms of  

Boundary conditions between region I and II
The two boundary conditions required to be satisfied at 1 a   for all 1  and 1 z are where and  Fig.3) comes as

Results and Discussion
The Eq. (24) For non-trivial solution of C m 's, the determinant of co-efficient in above equations should be equal to zero. Therefore, Where, P and Q are transpose of corresponding matrices given by the Eq. (25) and Eq. (36) respectively. For a given SWC, the solution of Eq. (38) will be a polynomial equation in energy eigenvalues E which various roots will give energy eigenvalues of the system. By increasing the order of determinates, the roots converges to specific values. The lowest root gives ground state energy. The lowest root is obtained numerically which converges to third decimal places for 10×10 determinant entries in Eq. (40). Considering region II is made of GaAs and all other regions I, III and IV are made of Ga 0.7 Al 0. 3

Hard wall confinement (HWC)
Under the limit that barrier potential energies (potential energies of region I, III and IV) are infinite, wavefunction II   

Conclusion
A general technique to study electronic states in doubly eccentric cylindrical quantum wire for soft wall confinement has been presented. Massive wall confinement and hard wall confinement is presented as its limiting case. The ground state energy of the structure is calculated. The variation in ground state energy with eccentricity is obtained numerically. The ground state energy is found to be decreases with increase in eccentricity of structure. One can similarly find numerically other higher roots of determinant equation to study excited states as well. The discussed method can be also further used to study having more than two similar or different cylindrical barriers nested in it in nano regime.