Computation of Energy Dispersion Relation for an Electron in a one Dimensional Periodic Potential Using Intel Visual Fortran 17.0 Update 3 for Windows

Aims: To calculate the time independent Schrӧdinger’s equation for an electron in a one dimensional periodic potential so as to obtain the eigenvalues of the energy.

Place and Duration of Study: Department of Physics, Bayero University Kano, Nigeria. Jan 2016 and July 2017.

Methodology: In this work, Intel Visual Fortran 17.0 update 3 for windows contained in Intel Parallel Studio XE 2017 Cluster Edition for windows was used to solve the required Schrödinger’s equation with periodic potentials together with Visual Studio Community 2015 using the nearly free approximation.

Results: Here we present the Electron Energy Bands in a one dimensional periodic potential presented in a reduced zone scheme based on nearly free approximation for rectangular, sawtooth, cosine, harmonic and interpolated periodic potential. The result shows that an increase in the potential height causes an increase in the band gap and vice versa. The result shows good agreement when compared with similar results in the same model.

Conclusion: Our model can be used to solve the Schrödinger’s equation. The energy dispersion relation for all the potentials shows that the energy gaps increases with increase in the potential where the potential was varied from 10 to 15Ry whilst keeping the spacing fixed to a = 1.5ao. In Solid State Physics, a high energy gap between a filled band and an empty band corresponds to an insulator. Hence decreasing the potential height indicates that it is changing from an insulator to semiconductor.