spin_potential_shape.rst

More interesting systems: spin, potential, shape

Each of the following three examples highlights different ways to go beyond the very simple examples of the previous section.

Matrix structure of on-site and hopping elements

We begin by extending the simple 2DEG-Hamiltonian by a Rashba spin-orbit coupling and a Zeeman splitting due to an external magnetic field:

H = \frac{-\hbar^2}{2 m} (\partial_x^2+\partial_y^2) -
  i \alpha (\partial_x \sigma_y - \partial_y \sigma_x) +
  E_\text{Z} \sigma_z +  V(y)

Here \sigma_{x,y,z} denote the Pauli matrices.

It turns out that this well studied Rashba-Hamiltonian has some peculiar properties in (ballistic) nanowires: It was first predicted theoretically in Phys. Rev. Lett. 90, 256601 (2003) that such a system should exhibit non-monotonic conductance steps due to a spin-orbit gap. Only very recently, this non-monotonic behavior has been supposedly observed in experiment: Nature Physics 6, 336 (2010). Here we will show that a very simple extension of our previous examples will exactly show this behavior (Note though that no care was taken to choose realistic parameters).

The tight-binding model corresponding to the Rashba-Hamiltonian naturally exhibits a 2x2-matrix structure of onsite energies and hoppings. In order to use matrices in our program, we import the Tinyarray package. (NumPy would work as well, but Tinyarray is much faster for small arrays.)

For convenience, we define the Pauli-matrices first (with \sigma_0 the unit matrix):