Electronic properties of multi-ablooms flower-like quantum dots: a 2-D finite element study

In this paper, we investigate the electronic properties of two-dimensional multi-abloom nanoflower. Also, we approximate a typical form of the Schrödinger equation through an appropriate 2-D finite element approach, and then, we investigate the effects of different parameters such as number of ablooms, number of rose, variety geometric domain shape and systems size on the energy levels, eigenfunctions, and dipole matrix elements.


Introduction
Because of vast applications in science and technologies, during last few years, quantum dots (QDs) have been an appealing field study. During last few years, a wide variety of shapes, such as quantum dots arrays (García et al. 2013), conical (Melnik and Willatzen 2004), rectangular (Zhou and Li 2005), cylindrical (Bester and Zunger 2005), disc-shaped (Wei and Xie 2010), elliptical (Sadeghi et al. 2013), lens-shaped (Rodrıguez and Ramırez 2008), pyramidal (Hwang et al. 2004), spherical (Bose 1998), T-shaped (Wu and Cao 2006), Y-shaped (Fang and Wang 2007), etc., for the quantum dot systems have been studied. But as we know up to now, the electronic inter-sub-band structures of nanoflower systems (Zhao et al. 2012) has not so far been addressed. It is clear that due to the complicated form of the quantum dot shapes, the calculation of energy levels and wave functions, analytically, are a nontrivial task (Fang and Wang 2015).
The finite element method (FEM) are a numerical technique for solving PDEs and were originally applied to problems in engineering, structural mechanics applied sciences and are effective for problems with nonregular and complicated domain geometry. In this method, there is a natural variational approach to the numerical solution of elliptic boundary value problems (BVP) such as Helmholtz form of the Schrodinger, which minimize residual over some suitable finite-dimensional subspace defined on the elements.
The FEM involves four basic steps: first, we must partition the solution region of the PDE into a finite number of nonoverlap elements which the rectangular or triangular shapes and h or p refining are most common for planar and 2-D problems. The collection of all regular or congruent elements should cover and resemble the original region as closely as possible. In the second step, we derive interpolation and governing equations for a typical element by defining a complete polynomial on an element. The smooth degree of assembling elements, determine continuity of global approximation. This step, determine blocks and entries of the problem coefficient stiffness matrix. In the third step, we need assemble all partition elements of the solution region to obtain the global coefficient matrix. Finally, the resulting sparse system of equations will be solved.
In the present work, we solve a typical 2-D Schrodinger equation by means of the FEM and we evaluate the energy eigenvalues and corresponding eigenfunctions of a multiabloom nano-flower. Then we compute the corresponding dipole matrix elements M 12 that is an important parameter in evaluation of optical properties of the semiconducting system through density matrix approach.

Finite Element Solution of the 2-D Schrödinger Equation
In the envelope function approach, we assume a 2-D Schrödinger equation as, In fact, this 2-D time-independent Schrödinger equation is a linear eigenvalue problem with Dirichlet boundary conditions (BCs). The problem domain is rectangular where in the normalized case, this domain can be scaled and transformed to the form Ω = [0, 1] × [0, 1] . In Eq. (1), ℏ , m * and V(x, y) are the Plank constant, effective mass and geometrical confining potential respectively and also set = ℏ 2 2m * . (1) As explained, in the FEM, we discretize domain space into some rectangular element subdomains and build bilinear base functions. If we assume n x and n y number of discretization points along x and y direction axes, the steps along the x and y directions will be h x = 1 n x , h y = 1 n y . Then, we have 0 = x 0 < x 1 < ... < x n x = 1 , 0 = y 0 < y 1 < ... < y n y = 1 , x i = a x + ih x , y i = a y + jh y . The bilinear polynomial basis functions defined on the rectangular element can be written as, where i = 0, 1, 2, ..., n x and j = 0, 1, 2, ..., n y . The total number of points are equal (1 + n x )(1 + n y ) . Since the wave functions degenerate on the boundaries, the boundary nodes delete from computing and are not necessary. Therefore, the number of unknown variables (and hence the number of algebraic equations) are equal to the number of internal discretization points (n x − 1)(n y − 1). Now, we construct the finite element approximation of the wave-function through linear combination of basis functions Eq. (2) on all internal nodes, where u i,j are unknowns. For simplicity, we use normalized and iso-parametric elements by scaling = x−x i h x and = y−y j h y . Therefore, we can rewrite the Eq. (2) as an alternative, Since, in the vicinity of the point (x i , y j ) there are only 9 neighborhood points and naturally 4 proper related elements, thus in matrix form of the Hamiltonian, there are only 9 non-zero diagonal layers of elements and we will have a resulting tri-diagonal block Hamiltonian matrix. Each of these blocks is 3 by 3 and they are symmetric. Now, we define residual through substituting the linear expansion (3) into the Eq. (1), The approximated wave-function ̃(x, y) is in the generalized and distributional form (Marti 1986), thus we can use the Galerkin weighted residual method as, where, in a more precise manner we have, And in the normalized space we have, In order to evaluate the Eq. (6), we need to use green theorem of second kind as (Marti 1986;Brenner and Scott 2007), Due to homogeneous Dirichlet boundary condition, this equation can be simplified into, The Fig. 1 represents a simple description about the relating basis functions in the neighborhood of the point (x i , y j ). Now, through integral (6), we have three types of blocked tri-diagonal matrices. For the matrix M 1 we have the matrix elements as Fig. 2. and M 2 matrix elements are as Fig. 3. And M 3 matrix elements are as Fig. 4. When we evaluate the above mentioned sub-matrices, we will have the following eigenvalue equation system, where E are the energy eigenvalues.

Numerical results and discussions
By using a finite element method and compact density matrix formalism, we have investigated the electronic properties of multi-abloom nano-flowers. Figure 5 shows the few multi abloom potential profile.
In the Fig. 6 we have presented the variation of the first 20 lowest eigen-energies of a 5-abloom nano-flower as a function of the domain size. In this figure, we see that, as the domain size increases the energies decreases.  In the next four Fig. 8a-d, we have presented the 9 eigen-functions of the system with 2, 3, 5, and 8 number of ablooms. These figures indicate the distribution of the probability densities inside the quantum structure. This probabilities show the probability of finding an electron in an assumed structures and therefore can provide us information about the free careers distribution.  (2 ) ). b First 9 eigen-functions of the system with 3 ablooms. (the 3 ablooms rose can be found by r = Cos(3 ) or r = Sin (3 ) ). c First 9 eigen-functions of the system with 5 ablooms. d First 9 eigenfunctions of the system with 8 ablooms ∫ a x |x| 1 (x, y) 2 (x, y)dxdy , respectively. As Fig. 9 shows, when the system length increases, the dipole matrix elements M 21 also increases. However, the Fig. 10 indicates that the variation of the dipole matrix elements M 21 as a function of the number of ablooms is an oscillatory function.

Conclusion
In this work, we investigated the electronic properties of multi-abloom nanoflower. By using detailed finite element method, we showed that as the domain size increased, the energies decreased. A nonlinear behavior in the energy values observed when the number of ablooms increased. When the system length increased, the dipole matrix elements M 21 also increased. The variation of the dipole matrix elements M 21 as a function of the number of ablooms was an oscillatory function.