Eigen-equation of Electronic Energy in Quantum Dot

Abstract In this paper, we present one simple model of quantum dot to describe the potential. Based on the boundary continuity of wave function and its derivative, using the Chebyshev polynomial of the second kind and matrix theory, we deduced one eigen-equation of electronic energy which can clearly describe the relationship between the energy level and the surface potential in quantum dot. The further study shows that the eigen-equation of electronic energy is different when the material of quantum dot is different.


Introduction
Semiconductor quantum dot is composed of a small number of atoms. The number of atoms is usually about a few to hundreds of atoms, and the size of the three dimensions is less than 100 nm. In the three dimensions of the quantum dot, because the motion of the carrier is limited by the size effect and the quantum effect is very significant. Because of the special energy, the quantum dot exhibits the unique physical properties, such as quantum size effect and quantum tunneling effect, et, al. it has very important significance in the research of basic physics and new electronic and optoelectronic devices. Now a large number of studies have been reported on the energy level of the quantum dot [1][2][3][4][5][6][7][8][9][10][11][12][13]. For example, on the base of the adiabatic approximation, the adiabatic approximation with averaging and full numerical solution, Yiming Li, Jinn-Liang Liu et al solved the three dimensional Schrödinger equation, and gave qualitative as well as quantitative trends in electronic properties with various parameters [14].By the method of integrating directly the Schrödinger equation, Xiao-Yan Gu [15] gave the calculated energy spectra for two electrons in quantum dot given. K G Dvoyan [16] used perturbation theory and limiting potential to study the energy states of electron in ellipsoidal quantum dot. Analytical expressions for particle energy spectrum have been obtained taking into account that electron effective masses are different in medium and in quantum dot. However, few reports completely describe the relationship of energy band or band gap with the surface potential, the interior periodic potential and the structure parameters.
In this paper, we try to study the dependence of the electronic energy on the quantum surface potential and other structure parameters. Our aim is to deduce an eigen-equation theoretical for describing the relationship of electronic energy with the surface potential, the internal periodic potential and structure parameters, which will be used to calculate the electronic energy in quantum dot.

Theory
In this paper, one simple model is presented (shown Figure.
x is the potential, ) (x ψ is the wave function, E is the energy, m is the electronic mass,  is a Plank constant. By solving one-dimensional Schrödinger equation, we can get, respectively For the above model with the atom layer of 2 1 N + in Fig.1 Based on the property of the transfer matrix, (3) can be changed into Here 11 12 1 1 cos cos sin sin , cos sin sin cos sin cos cos sin , sin sin cos cos (defined as chebyshev polynomials ) and 1 cos cos sin sin sin sin cos cos By substituting both 0 x = and 2 x l Nb b = = + into(2)and it's derivative, we have, respectively, Thus, the substitution of (7) into (4) produces Now by multiplying (8)

Eigen-equation of Electronic Energy in Quantum Dot
The expansion of (9) can be changed into After the expansion, we have   (13) Here N is even number. The substitution of (11) and (12) into (9) produces 2 0 0 0 cos sin sin cos

The Eigen Equation of Electronic Energy in Different Quantum
The solution of (13) is In order to demonstrates directly the relation of the energy level with the surface potential, by putting Putting ( )   here N is the total periodic number, n is the quantum number.

The Eigen Equation of the Quantum Dot with
Eq. (16), (20)and (23) are defined as the eigen equation of electronic energy in quantum dot, which can clearly describe the relationship of electronic energy with the surface potential, the internal periodic potential and structure parameters. They can more clearly describe the electronic state under different quantum number compared with that