正切平方势单量子阱的本征值和本征函数

鉴于“方形”势阱描述量子阱中的电子运动行为过于简单、过于理想,引入了正切平方势来代替,使结果得到了改善。在量子力学框架内,利用正切平方势把电子的Schrdinger方程化为超几何方程,利用系统参数和超几何函数严格地求解了电子的本征值和本征函数,并以Ga_1-xAl_xAs-GaAs-Ga_1-xAl_xAs量子阱为例计算了电子的能级和能级之间的跃迁。结果表明,电子在量子阱中的能量是量子化的,而相邻能级之间的跃迁给出与实验进一步符合的结果。

tan2x Potential and Eigenvalue and Eigenfunction for Single Quantum Well

The special structure of superlattice leads to strong interest from 1970's,because it has the parti-cular photo-electric characteristics.Along the development of the film technique,the component and the thinckness for the superlattice material can be controlled artificially,then gap engineering and doping engineering are formed.The well width can be regulated by means of controlling the thinckness of the film for the superlattice material;the well deep can be regulated also by means of controlling the component for the superlattice material.If the barrier layer of the multi-quantum well or superlattice was slight enough,the small size effect must consider.Of course,if the barrier layer of the multi-quantum well or superlattice was think enough,to such an extent as to the interaction between quantum wells can be neglected,the superlattice can be regarded as the single repeat of the single quantum well,then problem of multi-quantum well reduced to the problem of the single quantum well.Usual interaction potential is the squared potential well to describe the motion behaviour of an electron in the single quantum well.The new interaction potential with tan2x form is leaded.In the frame of quantum mechanics,equation to describe the particle motion is reduced to the hypergeometric equation by this potential.The energy eigenvalue and the eigenfunction problem of the equation are transformed to the eigenvalue and the eigenfunction problem of the hypergeometric equation.The eigenvalue and the eigenfunction of the system are calculated,and the distribution of level in quantum well are discussed.As an example,the level and the transition of an electron in the single quantum well for material Ga_ 1-xAl_xAs-GaAs-Ga_ 1-xAl_xAs are calculated.It shows that coincidence between the theory and the experiment is improved futher.