Reconciling Different Viewpoints of Landau Level Fan Chart

This blog serves as a memorial for the project titled “A Landau level simulator” to keep notes of why i started and what i learned from it.

Background and motivation

Landau level

In solid-state physics, the famous band theory describes how electrons are distributed favorably over energy and momentum in electronic materials. The states (energy, momentum, spin and etc.) of electrons collectively form an electronic band structure. When subjecting to a magnetic field, these electronic states would shift in energy according to its interactive terms with the field. As this field gets very strong, the interactive term (orbital contribution) becomes dominant and start to form dense bands with narrow energy dispersion (A lot of states have almost the same energy). These bands are also energetically discrete and referred to as “Landau level”.

Experimental Landau fans and different perspectives of theorists

If scanning the Landau levels in energy (shift the observation window energetically in y-direction) for a series of increasing magnetic fields (x-direction), we would be able to revealing the energy information of various Landau levels and how they evolve with the field. However, in experimental measurements, this scan is not so straightforward to implement, because that we don’t control the energy of observation window (chemical potential) directly. The variation of energy is actually realized via controlling the population of electrons (electron density) as tuning the voltage applied to one side of a capacitor placed on top of a device (gate). Importantly, the highest energy of populated electrons (chemical potential) is not linearly dependent on the electron density. This difference (nonlinear transition) between theoretic picture and physical realization always causes disputes between theorists and experimentalists (Don’t ask why I know this ;P).

An experimental attempt: A Landau-level simulator

To win our theorist friends and keep ourselves happy, we need to connect our experimental data (Landau levels evolves in electron density) with more favorable physical picture in theorist’s mind (Landau levels evolve in energy). A more challenging task is to interpret our experimental data and translate it into an electronic band structure. A simulator of Landau levels is thus aimed to provide theoretic prediction that is directly comparable to experiments for a predefined electronic band structure at zero magnetic field. If this prediction aligns with experiments well, we would be able to conclude the electronic band structure in an error-and-try manner. At worst, this Landau level simulator will hopefully provide some insights into making choice between multiple reasonable guesses.

Overview

The design of this project can be broken down into four steps: Firstly, construct an electronic band structure comprised of multiple single bands. These bands need to be defined by setting up parameters, e.g., choice of band (linear or quadratic), effective mass for quadratic bands, Fermi velocity for linear bands and zero-field energy point for each band. Secondly, we calculate the Landau level splitting as function of magnetic field. Thirdly, we translate this energy versus field into a density versus field plot. This process involves calculating the total density at each energy level (a loose definition of density of state), in another word, constructing an one-to-one correspondence between energy and density. The last step is to collect this transformed results of Landau levels at the chemical potential position in a density-versus-field plot for a series of discrete density values corresponding to experimentally attainable range. The collection of results will be then ready for a comparison with experiments (Landau fan chart).

Introduction to key aspects of a CLI software

The source code of this open-source project is hosted in Github

Electronic band structure

The electronic band structure (energy versus momentum) of materials can be defined before entering actual calculations. For simplicity, it is assumed to be isotropic. Though this assumption seems to be oversimplified, we still can reach amazingly good match with experiments. This indicates that decorating the electron band by adding more terms and symmetries to its Hamiltonian is not as crucial as expected at least in the material system we studied.

Configuration

Default settings such IO, plotting styles and some modelling parameters (Landau level broadening) are stored in config.py.

Functional descriptions

Add a band to the system

python addband.py [-density] {-is_dirac} {-is_cond} [-gfactor GFACTOR] [{-dp VF D} {-cp MEFF SPIN}]

Remove band(s) from the system

python delband.py [-i INDEX or all]

Snapshot the system

python peeksys.py

Landau levels manifested in energy versus field

python run.py [-enplot] {-dir DIR} {-fnm FNM} [--enrange Estart Eend Enum] [--bfrange Bstart Bend Bnum] {-nmax NMAX} {-angle ANGLE}

Landau levels manifested in density versus field

python run.py [-denplot] {-dir DIR} {-fnm FNM} [--enrange Estart Eend Enum] [--bfrange Bstart Bend Bnum] {-nmax NMAX} {-angle ANGLE}

Simulation of Landau fan chart for given densities

1. Plot the density versus field relationship for a set of given densities (specified by --allden and -nos) as a simulation of Landau fan chart.

   python run.py [-simu] [--allden "NS1 NE1 NS2 NE2 ..."] [-nos NOS] {-dir DIR} {-fnm FNM} [--enrange Estart Eend Enum] [--bfrange Bstart Bend Bnum] {-nmax NMAX} {-angle ANGLE}
   

2. Alternatively, one can load densities stored in a csv file (no header) under a directory DIR and with a filename FNM :

   python run.py [-simu] [-loadden path-to-csvfile] {-dir DIR} {-fnm FNM} [--enrange Estart Eend Enum] [--bfrange Bstart Bend Bnum] {-nmax NMAX} {-angle ANGLE}
   

Density of state (DOS)

1. Plot the DOS versus field relationship at a fixed chemical potential

   python run.py [-dos] {-dir DIR} {-fnm FNM} [--enrange Estart Eend Enum] [--bfrange Bstart Bend Bnum] {-nmax NMAX} {-angle ANGLE}
   

2. Plot the DOS as function of density and field

   python run.py [-dosm] [-loadden path-to-csvfile] {-dir DIR} {-fnm FNM} [--enrange Estart Eend Enum] [--bfrange Bstart Bend Bnum] {-nmax NMAX} {-angle ANGLE}
   

3. Plot the data from csv file

   python plotfile.py [-f path-to-csvfile]
   

Introduction to modules

module addband

python addband.py [-option][parameters] Add a new band into the system. A system will be automatically initiated upon the creation of the first band.

module peeksys

python peaksys.py Peek into the system you created. Once you’d like to see what is in your system, how many bands and their parameters. The above command will print out a summary of bands in table-like format.

module delband

python delband.py Remove bands from a system

It will prompt a dialogue which band to delete? Input the index number: Or press 'e' to exit. Type the number in front of all the parameters within a row for that band you want to delete. Or quit by typing e. In some cases, you need to delete bands in multiple-line scripts, then just using python toybands/delband.py -i [INDEX or all] ‘all’ will remove all.

module run

python run.py [-option][parameters] Carry out various tasks in the existing material system.