External potentials

The Device Builder supports the application of the following external fields to the simulation:

1. Constant electric field: Defined by its magnitude (in V / Angstrom) and direction.
2. Constant Magnetic field: Defined by its magnitude (in Tesla) and direction. Moreover, we implemented three gauges for the vector potential entering the Hamiltonian: "minimal", "planar", and "symmetric".
3. Confinement potential: Defined by its (symmetric) 3 x 3 stiffness matrix (in eV / Angstom^2).

Gauges

To understand the three different gauges for the vector potential, we start bye the "symmetric" gauge, which is defined by

$${\vec{A}}_{\mathrm{symmetric}}=\frac{1}{2}\vec{B}\times \vec{r}=\frac{1}{2}\left(\begin{array}{ccc}0& -B_z& B_y\\ B_z& 0& -B_x\\ -B_y& B_x& 0\end{array}\right)\cdot \vec{r}={\underline{\mathcal{B}}}_{\mathrm{symmetric}}\cdot \vec{r}=\frac{1}{2}\left(\begin{array}{c}{B}_{y}z-B_{z}y\\ B_{z}x-B_{x}z\\ B_{x}y-B_{y}x\end{array}\right).$$

In the equation above we have re-written the vector product as application of a anti-symmetric matrix, ${\underline{\mathcal{B}}}_{\mathrm{symmetric}}$, which allows us to motivate the alternative gauges by changing this matrix.

The "planar" gauge is defined by

$${\vec{A}}_{\mathrm{planar}}={\underline{\mathcal{B}}}_{\mathrm{planar}}\cdot \vec{r}=\left(\begin{array}{ccc}0& -\frac{1}{2}{B}_z& 0\\ \frac{1}{2}{B}_z& 0& 0\\ -B_y& B_x& 0\end{array}\right)\cdot \vec{r}=\left(\begin{array}{c}-\frac{1}{2}{B}_{z}y\\ \frac{1}{2}{B}_{z}x\\ B_{x}y-B_{y}x\end{array}\right),$$

which implies that the vector potential only depends on the coordinates $x$ and $y$.

Finally, the "minimal" gauge is given by

$${\vec{A}}_{\mathrm{minimal}}={\underline{\mathcal{B}}}_{\mathrm{minimal}}\cdot \vec{r}=\left(\begin{array}{ccc}0& 0& 0\\ B_z& 0& 0\\ -B_y& B_x& 0\end{array}\right)\cdot \vec{r}=\left(\begin{array}{c}0\\ B_{z}x\\ B_{x}y-B_{y}x\end{array}\right),$$

where the vector potential still depends on the coordinates $x$ and $y$ in the general case, but only on $x$ if the magnetic field is in the $y-z$ plane.

While the different gauges do not affect the result for physical quantities, the simulation size can be reduced if the vector potential, entering the Hamiltonian via the Peierl's substitution, only depends on the coordinate $x$, for example when considering a magnetic field along the $z$ axis in the "minimal" gauge.