Conditions for a direct band gap in Si quantum wires

Abstract

The condition for a direct band gap in Si quantum wires is investigated within the effective mass theory for wires coherent in every direction perpendicular to the wire direction. The condition is calculated using the equivalent condition that the minimum energy of the one-dimensional sub-bands of electrons is at the Γ-point, because holes always have a minimum energy at this point. It is shown that only Si quantum wires on the {100} plane can have a direct band gap. In particular, 〈100〉-oriented Si quantum wires have a direct band gap regardless of their cross-sectional shape. For wires other than 〈100〉-oriented ones, the condition of a direct band gap for rectangular, elliptic, triangular and trapezoidal cross-sectional shapes is investigated, assuming that the confinement potential is infinitely high in order to have the condition determined only by their cross-sectional shape and direction independently of their cross-sectional size. In all cases, wires have a direct band gap when the ratio of wire height (size perpendicular to the {100} plane) to width (size parallel to the {100} plane) decreases, and 〈110〉-oriented wires have the largest ratio of height to width for a direct band gap.