Optical properties of a hollow cylindrical quantum wire in a parallel applied magnetic field

Abstract In this theoretical study, we investigate optical properties of a hollow cylindrical quantum wire in the presence of a homogeneous magnetic field. The magnetic field is applied parallel to the axis of the quantum wire. The investigations were achieved by solving the Schrödinger equation within the effective mass approximation. The optical properties studied were absorption coefficient (AC) of electromagnetic radiation and changes in refractive index (CRI) of the hollow cylinder. The parallel applied magnetic field lifts the degeneracy of states of opposite angular momentum (the $$\pm |m |$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>±</mml:mo> <mml:mo>|</mml:mo> <mml:mi>m</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> </mml:math> states, m being the angular momentum quantum number) known as the Zeeman splitting. As such, in the presence of magnetic field, AC splits into two branches, one corresponding to a transition involving the positive m states and the other corresponding to a transition involving the negative m states. Increase in intensity of the electromagnetic radiation reduces the magnitude of the AC. Presence of inner radius of the hollow cylindrical wire lowers transition energies. Apart from absorption due to the $$(m=0 \rightarrow \pm 1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>→</mml:mo> <mml:mo>±</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> transitions, the presence of the inner radius also facilitates absorption due the $$(m=-1\rightarrow 0)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>→</mml:mo> <mml:mn>0</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> transition as the magnetic field strength increases, a phenomenon that does not happen in solid cylindrical quantum wires. The presence of the inner radius also enhances AC in strong magnetic field. Increase in intensity of the electromagnetic radiation affects the CRI (here considered up to third order), introducing a normal dispersion region in an otherwise anomalous region. The parallel magnetic field also splits the CRI depending on the sign of the m states involved in the transitions.