Validity of the connection-matrix approach to GaAs-${\mathrm{Al}}_{\mathit{x}}$${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$As quantum wells

The abruptness of the semiconductor heterojunction has been dealt with by many past researchers by incorporating the single-band effective-mass equation with a connection matrix at the interface. This connection-matrix method has been used to predict the eigenenergies of the GaAs-${\mathrm{Al}}_{\mathit{x}}$${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$As quantum wells. In this paper, the validity of this approach to quantum wells is examined by varying the location of the interface within a lattice constant to reveal the eigenenergy dependence on the location. The eigenenergy should be independent of the interface location if the single-band effective-mass equation combined with the connection matrix is a consistent theoretical entity. However, as the interface is displaced from -a to a (where a is the lattice constant) relative to the location where the numerical values of the connection matrices are usually given in the literature, the first and the second eigenenergies of the GaAs-${\mathrm{Al}}_{0.3}$${\mathrm{Ga}}_{0.7}$As quantum wells can deviate by about 6 and 20 meV, respectively. Therefore, it is concluded that the connection-matrix approach to quantum wells is not sufficiently consistent. However, the eigenenergy deviations become less signficant for the quantum well with lower barrier energy and wider well width.