blg_strain - calculations for magnetoelectric effect in strained bilayer graphene

This is a Python package used to perform the calculations described in the article "Electrically tunable and reversible magnetoelectric coupling in strained bilayer graphene " (arXiv:2103.04124).

Installation

Download this repository, navigate to the directory in a command prompt, and run pip install -e . to install the module and enable editing.

Quickstart

Below is a description of the basic usage of this package, lacking however explicit details about the strained bilayer graphene (sBLG) model. For a detailed explanation of the model, please see the accompanying article. Please also see the example.ipynb notebook, which has the below commands and accompanying plots of the calculated quantities.

Define the lattice

The StrainedLattice class stores information about the sBLG lattice, where the strain is parameterized by the variables eps (strain magnitude) and theta (strain angle relative to the $x$ axis). Individual hopping parameters can be turned off using the turn_off keyword argument.

from blg_strain.lattice import StrainedLattice

sl = StrainedLattice(eps=0.01, theta=0)  # 1% uniaxial tensile strain along x
sl.calculate(turn_off=['gamma3', 'gamma4'])  # calculate with gamma3 and gamma4 hopping turned off

The most relevant calculated attributes of the sl object are:

• sl.K and sl.Kp: locations of the K and K' points in momentum space
• sl.gamma0s, sl.deltas, etc.: modified hopping parameters and bond vectors for the strained lattice

Calculate the band structure

The BandStructure class stores the energy, wavefunctions, Berry curvature, and orbital magnetic moment for sBLG calculated over a window of momentum space surrounding the K point. The quantities at the K' point can be calculated considering the relevant symmetry operations (see article for discussion). The BandStructure class takes as arguments:

• sl, an instance of the StrainedLattice class
• window, width of the window of sampled momentum space
• Delta, the interlayer asymmetry

from blg_strain.bands import BandStructure

bs = BandStructure(sl=sl, window=0.1, Delta=0.01)
bs.calculate(Nkx=200, Nky=200)  # 200x200 grid

The most relevant calculated attributes of the bs object are:

• bs.kxa (length Nkx), bs.kya (length Nky): arrays defining the momentum-space grid surrounding the K valley.
• bs.Kxa, bs.Kya (Nkx x Nky): 'ij' indexed meshgrid made from bs.kxa and bs.kya
• bs.E, bs.Omega, bs.Mu (4 x Nkx x Nky): energy eigenvalues, Berry curvature, and orbital magnetic moment for each of 4 bands at each coordinate in momentum space. Bands indexed 1 and 2 are the valence and conduction bands, respectively.
• bs.Psi (4 x 4 x Nkx x Nky): eigenstates for each of 4 bands (dimension 0) with 4 components (dimension 1) at each coordinate in momentum space
• bs.splE, etc.: length-4 array of bivariate splines for each quantity, allowing for interpolation onto a finer grid. bs.splE[2](kxa, kya) returns a two-dimensional array of conduction band energies calculated over a grid defined by the one-dimensional arrays kxa and kya.

Calculate properties of bands with filled states

The FilledBands class stores the carrier density, displacement field, and magnetoelectric coefficient for a band structure filled up to a given Fermi level EF (EF=0 defined as the middle of the band gap). These quantities are computed by integrating over the window defined in the BandStructure instance bs, which can describe the entire Brillouin zone as long as the Fermi surface lies completely within the window.

from blg_strain.bands import FilledBands

fb = FilledBands(bs=bs, EF=0.01)
fb.calculate(Nkx=500, Nky=500)  # resampled to 500x500 grid

The most relevant calculated attributes of the fb object are:

• fb.alpha: $x$ and $y$ components of linear magnetoelectric susceptibility
• fb.D: electric displacement field
• fb.n: carrier density

Saving and loading results

Each of the above classes inherits from blg_strain.utils.saver.Saver, which enables straightforward saving and loading to the HDF5 file format. Because of the hierarchical nature of the three classes (FilledBands depends on BandStructure depends on StrainedLattice), BandStructure objects are saved in a subdirectory of the directory where the corresponding StrainedLattice object is saved, and FilledBands objects are saved in a subdirectory of the directory where the corresponding BandStructure object is saved. The following code can be used to save the objects created in the examples above:

sl.save('example')
bs.save()
fb.save()

This creates the following file structure:

/
└───example
    │   StrainedLattice_eps0.010_theta0.000_Run0.h5   
    └───StrainedLattice_eps0.010_theta0.000_Run0
        │   BandStructure_Nkx200_Nky200_Delta10.000.h5
        └───BandStructure_Nkx200_Nky200_Delta10
            │   FilledBands_Nkx500_Nky500_EF15.000.h5

The objects can be reloaded at any time using (for example) sl = StrainedLattice.load(filename).

It is often useful to create files consisting of only the final derived quantities resulting from a series of BandStructure and FilledBands calculations. Loading such a "summary" file can take considerably less time than loading the individual FilledBands files. The function blg_strain.utils.saver.load creates a summary file in the StrainedLattice directory. Subsequently calling the load function will load from summary.h5 instead of re-making the summary file:

from blg_strain.utils.saver import load
Deltas, EFs, ns, Ds, alphas = load(sl_path)

Project structure

The blg_strain package contains classes and methods used in calculating the electronic properties of strained bilayer graphene (sBLG). The plots directory contains a jupyter notebook used to generate figures for the paper from the simulation results in plots/data. This directory also contains final PDF files for each of the figures (some of which have been edited in Adobe Illustrator).

Briefly, the files under blg_strain are responsible for the following:

• bands.py: contains classes to contain results of band structure calculations for combinations of parameters
• berry.py: calculates Berry curvature and orbital magnetic moment from the eigenstates of the Hamiltonian
• hamiltonian.py: defines the $4\times4$ tight-binding Hamiltonian
• lattice.py: contains all calculations related to the geometry of the sBLG lattice
• macroscopic.py: calculations for "macroscopic" quantities: carrier density, electric displacement field, magnetoelectric coefficient (an integral of the Berry curvature and orbital magnetic moment over momentum space)
• microscopic.py: calculations for "microscopic" quantities, namely the Fermi-Dirac distribution
• strayfield.py: calculations for the stray field from finite-width/finite-length current-carrying wires and a uniformly magnetized rectangle. These calculations are used to imagine how to best study the magnetoelectric effect in sBLG experimentally.
• Utilities (/utils)
  • const.py: physical constants and values for hopping parameters, etc. for BLG
  • plotting.py: helper functions to plot band structure
  • saver.py: defines a class to aid with saving and loading calculation results
  • utils.py: miscellaneous utilities, mostly used for defining the calculation grid

Contributing

Pull requests are welcome. Please contact the author to discuss potential applications of this code towards other computational studies.

License

MIT