The quantum wire is modeled via a lateral electrostatic confinement potential in the

direction

, leaving free electron propagation in the

direction (which is the direction of the wire). The Rashba spin-orbit interaction that couples electron spin with its momentum arises due to lack of symmetry within the nanostructure and is induced by a macroscopic electric field directed along the

direction. Zeeman interaction, which couples electron spin to applied magnetic field, is induced by a varying external magnetic field

in three independent spatial directions

,

, and

, where

and

are in-plane and

is perpendicular to the nanostructure, as shown in the schematic. When the Rashba and Zeeman interactions are zero, the subbands are degenerate (shown for 10 subbands,

being the ground state). The subbands undergo spin-splitting in the presence of either Rashba or Zeeman interaction.

The parameter

models the Rashba spin-orbit interaction strength, while

,

, and

model the magnetic field strength in their respective directions.