Calculation of the spectrum of a gallium arsenide semiconductor with a triangular potential function by the power series method

The paper investigates the quantum characteristics of the widely used semiconductor gallium arsenide in the modern promising field of microelectronics. For the energy levels in a triangular potential well, analytical expressions are obtained using the zeros of the Airy function. In addition, the corresponding Schrödinger equation with this potential function is solved by the power series method and both the energy spectrum of the lower levels and the corresponding wave functions are calculated. Satisfactory agreement between the values of the energy levels obtained in both approaches is found, but the prospects of calculating the quantum characteristics directly using the Schrödinger equation are noted. The solution of the Schrödinger equation is sought as a linear combination of two linearly independent solutions in the form of power series. The coefficients of this linear combination depend on the total energy as a parameter. Taking into account the boundary conditions on the boundary of the integration segment leads to a linear algebraic system of equations. Nontrivial solutions of this system determine both the energy spectrum and the corresponding wave functions. Due to the sharp dependence of the energy levels on the type of wave functions, a careful choice of boundary points is necessary, as well as the number of terms in the series of wave functions. Optimal values of the specified fitting parameters allow us to obtain the values of the energy levels with the desired high accuracy.

heterostructures, computer modeling, schrödinger equation, gallium arsenide, energy spectrum, wave functions, power series method, Airy function