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7. Magnetic Fields

Static \(\mathbf{B}\)-fields are an important case of lattice modifications that can be performed automatically by KITE. This feature is of particular interest to the study of Landau levels, magneto-transport and magneto-optical effects, among others.

Uniform B-fields in KITE: overview

The automated \(\mathbf{B}\)-field functionality works by the addition of Peierls phases in the Hamiltonian and can be used in conjunction with other lattice modifications, including disorder. This is a new feature under development that currently allows for:

  • Uniform \(\mathbf{B}\) fields in 2D lattices (with the B-field perpendicular to the plane)
  • Uniform \(\mathbf{B}\) fields in 3D lattices (with the B-field collinear to the third primitive lattice vector)

The \(\mathbf{B}\)-field is added by using the following KITE modification:

modification = kite.Modification(magnetic_field = mag)

where (mag) is the magnetic field strength (given in Tesla). When used with periodic boundary conditions, \(|\mathbf{B}|\) is restricted to be a multiple of a minimum magnetic field, which is determined internally when generating the configuration file (see details below).

For example, to compute the DOS of a disordered system subject to the \(\mathbf{B}\)-field modification outlined above, one may use

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calculation.dos(
    num_points=100,
    num_moments=5000,
    num_random=10,
    num_disorder=10
)
mod_mag_field = kite.Modification(magnetic_field = mag)
kite.config_system(
    lattice,
    configuration,
    calculation,
    modification=mod_mag_field,
    disorder=disorder,
    filename="B_field.h5"
)
Here, the configuration file requests a DOS calculation with 5000 Chebyshev moments, 10 random vectors and 10 disorder realizations. Note the addition of the lattice modification (modification=mod_mag_field) with respect to the settings as discussed in the section about disorder.

Implementation details

Units

Lattice parameters must be given in nanometers (see example dos_dccond_square_lattice.py in the bash kite/examples/-folder).

The magnetic fields considered in KITE are uniform, so the corresponding vector potential is linear. It is naturally expressed in terms of the primitive reciprocal lattice vectors (\(\mathbf{b}_{i=1,2,3}\)) in the Landau gauge

\[ \mathbf{A}\left(\mathbf{r}\right)=\frac{h}{\left(2\pi\right)^{2}e}\frac{n}{N_{2}}\left(\mathbf{r}\cdot\mathbf{b}_{2}\right)\mathbf{b}_{1} \]

where \(h\) is Planck's constant, \(e>0\) is the elementary charge, \(N_{2}\) is the number of unit cells along the \(\mathbf{a}_{2}\) direction (primitive vector of the direct lattice) and \(n\) is an integer. The corresponding magnetic field points along the \(\mathbf{a}_{3}\) direction for 3D systems and perpendicularly to the basal plane (\(\mathbf{e}_{\perp}\equiv\hat{\mathbf{z}}\)) for 2D systems:

\[ \mathbf{B}=\begin{cases} \frac{h}{e\Omega_{c}}\frac{n}{N_{2}}\mathbf{a}_{3} & (\textrm{3D})\\ \\ \frac{h}{e\Omega_{c}}\frac{n}{N_{2}}\hat{\mathbf{z}} & (\textrm{2D}) \end{cases} \]

and is restricted to be a multiple of a minimum field

\[ B_{\textrm{min}}=\frac{h}{e\Omega_{c}}\frac{1}{N_{2}}\times\begin{cases} |\mathbf{a}_{3}| & (\textrm{3D})\\ \\ 1 & (\textrm{2D}) \end{cases} \]

where \(\Omega_{c}\) is the 3D/2D volume of the unit cell. When the user requests a magnetic field strength \(|\mathbf{B}|\) (given in Tesla), KITE calculates \(B_{\textrm{min}}\) first and then uses that to determine the required \(n\) to achieve the closest possible value of \(|\mathbf{B}|\) by rounding \(|\mathbf{B}|/\mathbf{B}_{\textrm{min}}=n\) to the nearest integer. If \(n\) rounds down to zero, it means that the system is too small to support the requested magnetic field. When determining \(B_{\textrm{min}}\), KITE assumes that the primitive vectors in the Python configuration script are given in nanometers.