Carbon Nanotube#

This example demonstrates how to obtain electronic structure information as inputs for quatrex and how to perform different kinds of transport simulations for a roughly 10 nm long (8, 0) single-wall carbon nanotube (CNT).

Geometry and electronic structure#

We can create the geometry of the CNT using the ase Python package, which has a built-in function to generate nanotubes (ase.build.nanotube). After exporting the geometry to a POSCAR file, one can set up a DFT calculation using VASP to obtain the electronic structure in plane-wave basis. VASP is then run with the LWANNIER90 = .TRUE. flag to generate the necessary input files for Wannier90. Projecting onto initial guesses of one \(p_z\) orbital per carbon atom, we then run Wannier90 and get the Hamiltonian in the Wannier basis, which is used as input for quatrex.

A procedure for generating quatrex input data from Wannier90 outputs is described in the electronic structure data section. The resulting device structure (structure.xyz) and Hamiltonian (hamiltonian.h5) are stored in the ./examples/w90/carbon-nanotube/inputs directory.

You can find more detailed information on the input data provenance in the README.md file in the example directory ./examples/w90/carbon-nanotube/

Here you can see the resulting Wannier centers of the CNT structure. Note that the orbital centers are not necessarily coincident with the atomic positions.

The actual orbital coordinates are stored in an .xyz file that looks like

768
Lattice="102.84851504839999 0.0 0.0 0.0 50.0 0.0 0.0 0.0 50.0" Properties=species:S:1:pos:R:3 pbc="F F F"
X        0.72770969      21.80861750      25.06696100
X        2.01508769      22.02025450      25.89201500
X        0.15518369      22.76798050      27.19112700
X        4.05541969      25.00283250      27.35868900
X        4.34182130      25.04734650      28.15860700
X        2.81802769      26.25013450      27.98060700
X        4.24380469      27.06711950      27.31367300
X        2.22266969      27.93073350      26.16808300
X        1.18530469      28.19138250      25.05384000
X        2.06710169      27.73589650      23.58711800
X        0.45901869      27.26246150      22.75888700
X        2.09800069      26.07841450      22.18612000
X        4.21870469      24.88363450      21.84139300
X        1.95921469      24.37907150      22.02629800
...

The band structure of the device around its equilibrium Fermi level is captured very well by this Wannierization.

Computing the transmission function#

Say we are interested in seeing the transmission spectrum through this CNT. This kind of coherent transport is most efficiently treated in the wavefunction formalism, so we set formalism = "wf". We will not be employing a self-consistent solution of the Hartree potential for this purpose, so we do not include a [scsp] section in the config here.

Next we need to define the contact regions of the device. The whole carbon nanotube is made up of 24 repeated unit cells along the transport direction "x". From the DFT simulation, we know that each of these cells has a length of 4.27615261 Å and from the structure file above, we can see that the orbital center with the smallest x-coordinate sits at (0.15518369, 22.76798050, 27.19112700). Together, this allows us to get a contact definition as follows:

[[device.contacts]]
name = "left"
origin = [0.1551, 0.0, 0.0]
lattice_vectors = [[4.27615261, 0, 0], [0, 50, 0], [0, 0, 50]]
direction = "a"
fermi_level = -3.6

The Fermi level is the one from DFT. Similarly for the right contact, we set

[[device.contacts]]
name = "right"
origin = [102.694, 0.0, 0.0]
lattice_vectors = [[-4.27615261, 0, 0], [0, 50, 0], [0, 0, 50]]
direction = "a"
fermi_level = -3.601

Here we added a small chemical potential difference of 1 meV to get a small current flowing across the device. We will compute the transmission function at 1000 energy points between -6.5 eV and -1.0 eV, which will yield an energy grid of 5.5 meV. Note that this may still be too coarse to resolve sharp resonances in the transmission function, but it is sufficient for this example. We set

energy_window_min = -6.5
energy_window_max = -1.0
energy_window_num = 1000

Finally, for QTBM we need to employ the spectral OBC algorithm. For robustness, and since this is a very small system, we choose the obc.nevp_solver= "full".

formalism = "wf"

[device]
transport_direction = "x"

    [[device.contacts]]
    name = "left"
    origin = [0.1551, 0.0, 0.0]
    lattice_vectors = [[4.27615261, 0, 0], [0, 50, 0], [0, 0, 50]]
    direction = "a"
    fermi_level = -3.6

    [[device.contacts]]
    name = "right"
    origin = [102.694, 0.0, 0.0]
    lattice_vectors = [[-4.27615261, 0, 0], [0, 50, 0], [0, 0, 50]]
    direction = "a"
    fermi_level = -3.601

[electron]

energy_window_min = -6.5
energy_window_max = -1.0
energy_window_num = 1000


obc.algorithm = "spectral"
obc.nevp_solver = "full"

You can run this simulation by invoking

quatrex run <path/to/config.toml>

After quatrex completes, you will find the file transmission_lr.npy in the output folder.

Including phonon pseudo-scattering#

Next we may want to introduce a small amount of scattering with a phonon model. To accomplish this, we need to move beyond the wavefunction picture and set formalism = "negf".

The self-consistent Born approximation (SCBA) is used to ensure consistency between the scattering self-energy and the Green's function. The SCBA loop, including only phonon pseudo-scattering, is typically very stable, so we do not need any under-relaxation here (i.e. mixing_factor = 1.0). The [scba] section of the config will look like this:

[scba]
max_iterations = 15
mixing_factor = 1.0
phonon = true

As stated above, the full structure, encompassing 768 Wannier orbitals, is made up of 24 transport cells that contain 32 orbitals each. The [device] section of the config therefore now looks like this:

[device]
transport_direction = 'x'
block_size = 32

Lastly, besides the phonon = true flag in the SCBA section, to configure the actual phonon interaction, we can set the following parameters:

[phonon]
interaction_cutoff = 5.0 # Angstrom

model = "pseudo-scattering"
phonon_energy = 40e-3         # eV
deformation_potential = 15e-3 # eV

formalism = "negf"

[scba]
max_iterations = 15
mixing_factor = 1.0
phonon = true

[device]
transport_direction = 'x'
block_size = 32

[electron]
left_contact.name = "left"
left_contact.fermi_level = -3.6

right_contact.name = "right"
right_contact.fermi_level = -3.601

energy_window_min = -6.5
energy_window_max = -1.0
energy_window_num = 1000

obc.algorithm = "spectral"
obc.nevp_solver = "full"


[phonon]
interaction_cutoff = 5.0 # Angstrom

model = "pseudo-scattering"
phonon_energy = 40e-3         # eV
deformation_potential = 15e-3 # eV

After running the simulation (quatrex run <path/to/config.toml>), among other quantities, you will find the spectral device current in the outputs, taking into account thermal scattering.