Energy gap renormalization and diamagnetic susceptibility in quantum wires with different cross-sectional shape

In this study, we investigate the effect of the cross-sectional shape on the energy gap renormalization and diamagnetic susceptibility in various quantum wires. To this end, we consider quantum wires with different cross-sectional shapes such as circular, square, hexagonal, and triangular. First, we employ the finite-element method and Arnoldi algorithm to solve the Schrödinger equation. Then, we calculate the energy levels, wavefunctions, binding energy, energy gap renormalization, and diamagnetic susceptibility. Our numerical results show that the binding energy decreases when the cross-sectional area is increased for all the quantum wires. Moreover, it is inferred that the cross-sectional shape is not important for large cross-sectional area when calculating the binding energy. Indeed, the main parameter is the cross-sectional area rather than the length of a side. The energy gap renormalization decreases with increasing cross-sectional area, regardless of the impurity concentration. We observe that the highest and lowest energy gap renormalization correspond to triangular and circular quantum wires, respectively. The absolute value of the diamagnetic susceptibility increases with increasing cross-sectional area for all the quantum wires investigated.