Quantum Freeze‐Out of Carriers in Semimetallic and Semiconducting Thin Wires

A semimetal—semiconductor transition is predicted to take place due to the quantum freeze‐out of carriers in semimetallic thin wires. These wires are assumed to be cylinders of radius R 0 although this quantum freeze‐out effect is also predicted to take place in thin wires of rectangular cross section as well. In cylindrical wires, the onset of semiconducting behavior is expected to occur at a critical radius R * 0 = α 01 ħ /(2 M Δ) 1/2 , where α 01 = 2.405 is the first zero of the Bessel function of the zeroth order, M = m e m h /( m e + m h ) is the reduced mass of the electron—hole system, and Δ is the overlap of the conduction and valence bands. For wires of radii smaller than R * 0 , the electrical conductivity of the thin wire will exponentially decrease with temperature because of the presence of an effective band gap E g = (α ħ 2 /2 M R ) ‐ Δ > 0.