Computer Lab

Python and package installation

• Install Anaconda
• Install KWANT package:

conda install -c conda-forge kwant

• Install PythTB package:

pip install pythtb --upgrade

• Install and test ASE package:

pip install --upgrade --user ase
python -m ase test

Python references

• J. M. Stewart, Python for Scientists (Cambridge University Press, Cambridge, 2014). (PDF from UD library)
• Scipy lecture notes
• NumPy for Matlab users
• Timing Python script performance

KWANT references

• KWANT documentation
• Selecting matrix solvers in KWANT
• KWANT tools
• VIDEO: Introduction to KWANT
• Learning KWANT through examples
• Interactive tutorials using Jupyter Notebooks
• Discretization of continuous Hamiltonians
• Conductance of graphene nanoribbons
• Conductance of topological insulators built from graphene nanoribbons
• Spin Hall effect in four terminal devices
• Topological Hall effect in four terminal devices

Charge and spin densities in quantum dots via equilibrium density matrix

• charge_density.m
• spin_density.m
• poisson_schrodinger.m and Fhalf.m

DOS of 1D disordered nanowire using eigenvalues + visualization of Anderson localization of eigenfunctions

• disordered_nanowire.m

Density of states using equilibrium Green functions

• dos_negf_closed.m computes DOS for finite 1D wire
• dos_negf_open.m computes DOS for finite 1D wire attached to one or two macroscopic reservoirs
• graphene_dos.m computes DOS for a supercell of graphene with periodic boundary conditions

Subband structure of graphene nanoribbons using tight-binding models

• 8zgnr.m (poor man's script follows literally the lecture slide, so it works only for 8-ZGNR)

Quantum transport in 1D nanowires using NEGF

• How to use NEGF matrix formulas: Step-by-step in pictures
• qt_1d.m (computes conductance, as well as total and local density of states for 1D nanowire modeled as tight-binding chain, with possible potential barriers or impurities, attached to two semi-infinite leads)

Tunneling magnetoresistance in tight-binding models of magnetic tunnel junctions using NEGF

• mtj_1d.m (computes TMR of F/I/F MTJs modeled using 1D tight-binding chain)
• mtj_3d.m (computes TMR of F/I/F MTJs modeled using mixed real space and k-space tight-binding model of 3D junctions

Quantum transport in graphene nanostructures via NEGF

• gnr_cond_recursive.m,bstruct.m, blocktosparse.m, sparsetoblock.m, h_zigzag.m, invnn.m, Invb.m

Self.m, (code to compute the conductance of a finite graphene nanoribbon attached to two semi-infinite graphene electrodes)

MATLAB functions

• matrix_exp.m (Exponential, or any other function with small changed in the code, of a Hermitian matrix)
• visual_graphene_H.m (For a given tight-binding Hamiltonian on the honeycomb lattice, function plots position of carbon atoms and draws blue lines to represent hoppings between them; red circles to represent on-site potential between them; and cyan lines to represent the periodic boundary conditions; it can be used to test if the tight-binding Hamiltonian of graphene is set correctly); This function calls another three function which should be placed in the same directory (or in the path): atomCoord.m, atomPosition.m, and constrainView.m
• self_energy.m (Self-energy of the semi-infinite ideal metallic lead modeled on the square tight-binding lattice - the code shows how to convert analytical formulas of the lead surface Green function into a working program)
• transmission.m (Transmission function for 1D tight-binding chain with spin-dependent terms)

Density functional theory with GPAW package

How to run GPAW on ulam

GPAW has been installed on ulam with the OS installed python 2.6.

• in order to use the serial version of GPAW type:

 python your_gpaw_program.py

• in order to use the parallel version of gpaw use the following syntax (replace 8 with the number of cores you want to use):

 mpirun -np 8 gpaw-python_openmpi your_gpaw_program.py

Getting started with GPAW

• VIDEO Tutorial: Electronic structure calculations with GPAW
• VIDEO Tutorial: Overview of GPAW
• VIDEO Tutorial: CO molecule
  • Plotting CO wavefunction
  • relax.py used in VIDEO Tutorial:

from ase import Atoms
from ase.io import write
from ase.optimize import QuasiNewton
from gpaw import GPAW

d = 1.10  # Starting guess for the bond length
atoms = Atoms('CO', positions=((0, 0, 0),
                               (0, 0, d)), pbc=False)
atoms.center(vacuum=4.0)
write('CO.cif', atoms)
calc = GPAW(h=0.20, xc='PBE', txt='CO_relax.txt')
atoms.set_calculator(calc)

relax = QuasiNewton(atoms, trajectory='CO.traj', logfile='qn.log')
relax.run(fmax=0.05)

• ASE tutorials
• Basics of GPAW calculations
• Band structure of Ni plotted in three steps
• Lattice constant, DOS, and band structure of Si
• STM simulations

GPAW Exercises Related to Midterm Project

• Band structure of bulk graphene
• Subband structure of graphene nanoribbons
• Subband structure of carbon nanotubes

References

• Electronic structure calculations with GPAW: a real-space implementation of the projector augmented-wave method, J. Phys.: Condens. Matter 22, 253202 (2010). [PDF]
• Parameters selection in GPAW scripts
• LCAO basis set in GPAW
• Parameters selection in ASE to define atomic coordinates
• Density of States in GPAW
• About k-point sampling

First-principles quantum transport calculations using NEGF+DFT within GPAW package

Theory Background

• Crash course on NEGF+DFT codes
• NEGF+DFT within GPAW - see also J. Chen, K. S. Thygesen, and K. W. Jacobsen, Ab initio nonequilibrium quantum transport and forces with the real-space projector augmented wave method, Phys. Rev. B 85, 155140 (2012). [PDF]
• M. Strange, I. S. Kristensen, K. S. Thygesen, and K. W. Jacobsen, Benchmark density functional theory calculations for nanoscale conductance, J. Chem. Phys. 128, 114714 (2008). [PDF]
• D. A. Areshkin and B. K. Nikolić, Electron density and transport in top-gated graphene nanoribbon devices: First-principles Green function algorithms for systems containing a large number of atoms, Phys. Rev. B 81, 155450 (2010). [PDF]

GPAW Exercises

• Pt-$\mathrm {H} _{2}$-Pt nanojunction.
• Annulene molecule between two graphene nanoribbon electrodes nanojunction