A non-interacting equilibrium 2D quantum transport simulation framework

## Project description

# Transport In a Non-Interacting Equilibrium simulation framework (TINIE)

## Overview

We present to you a code that calculates conductance and electric current
running through 2D cavities, quantum dots or potential wells with arbitrarily
placed reservoirs in a perpendicular and constant magnetic field. The code can
be used in a wide range of calculations involving 2D electron transport. The
main difference between this code and its competitors is the fact that TINIE
does not need to be provided with free parameters. The code is parallelized
with `mpi4py`

, allowing for computational tasks distribution across multiple
processors.

## Installation

The code is a Python package (written using Python 3.6). Installation of the
package is as easy as typing `python3 setup.py install`

. Thorough unit testing
for the package has been implemented and can be launched via `python3 setup.py test`

.
The package has scripts with a parser user interface, which need to be given
appropriate permissions prior to their execution. To that end, the following commands
must be run from the git root repository:

```
$ source dev_setup.sh
$ scripts/tinie_prepare -dE 1e-7 ...
$ scripts/tinie -i "transport_equilibrium/test_files/preptinie_test.h5" -dw
$ 1e-5 ...
```

## Package Functionality

This package contains tools that could be used to calculate coupling of a specific system that contains a central region (2DEG) and some leads. After the coupling is calculated one could proceed to calculate transmission coefficients and partial currents in the leads. All the calculations are performed in Hartee atomic units. After installing the package, a simple test run can be launched from the git root directory as follows:

```
$ scripts/tinie_prepare
$ scripts/tinie
```

## Package Structure

The code is written using object-oriented programming, and its functionality
can be shortly described in the following way: first, `Lead`

, and `Center`

objects are created and passed as inputs for the `Coupling`

object and then all
of them are passed into a `SystemDump`

object, which calculates all the
couplings and Hamiltonians and dumps the data into an hdf5 file. `SystemFetch`

is then used to read the data from that hdf5 file. That data is passed into the
`Calculator`

interface, where `SelfEnergy`

interface calculates the
self-energies $`\Sigma_{L}(\omega)`

$ and rate operators $`\Gamma_{L}(\omega)`

$
using the `Coupling`

and the eigenenergies of the `Lead`

.

After that, `SelfEnergy`

for all the leads and the eigenenergies of the
`Center`

are passed into the `GreenFunction`

interface that evaluates the
advanced ($`G^{A}(\omega)`

$) and retarded ($`G^{R}(\omega)`

$) Green's
functions, finalizing the initialization of the `Calculator`

. From there the
code is able to compute the transmission matrix
$`\mathcal{T}_{\alpha\beta}(\omega)=\mathrm{Tr}[G^{R}(\omega)\Gamma_{\beta}(\omega)G^{A}(\omega)\Gamma_{\alpha}(\omega)]`

$
and the partial currents

```
i_{\alpha\beta}=2\int_{\omega}\mathrm{d}\omega\frac{1}{2\pi}[f(\omega-V_{\alpha}-\mu)-f(\omega-V_{\beta}-\mu)]\mathcal{T}_{\alpha\beta}(\omega)
```

in the lead, where $`f(E)`

$ is the Fermi-Dirac energy distribution. Furthermore,
it is then possible to compute other transport properties, such as conductance
at a specific temperature. Additionally, we include the possibility of computing
density of states and local density of states.

Modular structure of the code allows for the implementation of your own custom
type of conducting channel ("lead"), quantum dot ("central region") and
coupling via implementation of a class that inherits from `Lead`

, `Center`

or
`Coupling`

. The details of how exactly the classes should be implemented will
be explained in the sections to follow.

## Currently Implemented System Classes

As of now, the following system classes are implemented:

`Center`

objects, located in`transport_calculator/systems/central_region`

`Itp2dCenter`

: itp2d-compatible interface.`CustomCenter`

: container for a custom predefined central region Hamiltonian $`\mathbf{H}^{C}`

$.

`Lead`

objects, located in`transport_calculator/systems/leads`

. Note that wavefunction normalization has been omitted for the sake of compactness of the expression. Wavefunctions in the code are all normalized.`FiniteHarmonicLead`

: lead described by a wavefunction $`\psi^{L}_{k,l}(x,q)=\cos(k(x-x^{L}_{max})+\frac{\pi}{2})e^{-\frac{1}{2}q^2}H_{l}(q)`

$, where $`H_{l}(q)`

$ is the the Hermite polynomial of order $`l`

$, $`q=\sqrt{\omega_{c0}y-\frac{l}{B}\frac{\omega^{2}_{c}}{\omega^{2}_{c0}}}`

$, $`\omega_{c}=B,\omega^2_{c0}=\omega^2_0+\omega^2_c`

$, with $`\omega_0`

$ being the frequency of quantum harmonic oscillator and $`B`

$ being magnetic field strength. The formula is provided in Hartee atomic units. $`x`

$ and $`y`

$ coordinate wavefunctions are interchangeable depending on the lead alignment.`BoxLead`

: particle in a box lead describe by wavefunction $`\psi^{L}_{k,l}(x,y)=\sin(\frac{k\pi}{L_{x}}(x-x^L_{max}))\sin(\frac{l\pi}{L_{y}}(y-y^L_{max}))`

$, where $`L_{x}`

$ and $`L_{y}`

$ are the length and width of the box correspondingly and $`k,l \in \mathbb{Z}_{+}\setminus \{0\}`

$.`CustomLead`

: container for a custom predefined lead region Hamiltonian $`\mathbf{H}^{L}`

$.

`Coupling`

objects, located in`transport_calculator/systems/couplings`

`OverlapCoupling`

: strong coupling of the type $`\mathbf{V}_{ij}=-\frac{1}{2}\int_{\Omega}\mathrm{d}\mathbf{r}\psi^{*}_{L,i}(\mathbf{r})\Delta\psi_{C,j}(\mathbf{r})`

$, where $`\psi_{L,i}`

$ is the $`i`

$th eigenfunction of the lead and $`\psi_{C,j}`

$ i s the $`j`

$th eigenfunction of the central region and $`\Omega`

$ is the overlap region of the lead and the quantum-dot.`TightBindingCoupling`

: weak coupling between non-overlapping lead and central regions of the type $`\mathbf{V}_{ij}=-\frac{1}{2}\int_{\Omega_{L}}\mathrm{d}\mathbf{r'}\psi^{*}_{L,i}(\mathbf{r'})\int_{\Omega_{C}}\mathrm{d}\mathbf{r}\frac{\psi_{C,j}(\mathbf{r})}{||\mathbf{r'}-\mathbf{r}||^2}e^{-i\theta}`

$, where $`\theta=-\frac{B}{2}(x'-x)(y'-y)`

$, $`\Omega_{L}`

$ is the lead region to be coupled and $`\Omega_{C}`

$ is the central region to be coupled.`OneLayerCoupling`

: weak coupling between the boundaries of a non-overlapping lead and central regions of the type $`\mathbf{V}_{ij}=-\frac{1}{2}\int_{\partial \Omega_{L}}\mathrm{d}\mathbf{r'}\psi^{*}_{L,i}(\mathbf{r'})\int_{\partial \Omega_{C}}\mathrm{d}\mathbf{r}\frac{\psi_{C,j}(\mathbf{r})}{||\mathbf{r'}-\mathbf{r}||^2}e^{-i\theta}`

$.`CustomCoupling`

: container for custom predefined coupling matrix $`\mathbf{V}`

$. Compatible only with`CustomCenter`

and`CustomLead`

objects.

The implementational details of these elements can be checked in the source code, which is rich with insightful and helpful comments.

## Adding Your Own Custom System Classes

As it has been mentioned before, the code has been designed in such a way as to
allow as much freedom in expansion as possible. In particular, you can
introduce additional types of central regions, lead regions and coupling
methods. All you have to do is to create your own class file in the
corresponding folder in `tinie/systems`

and make sure that the
class you are creating inherits from one of the basic abstract classes
(`Center`

, `Lead`

or `Coupling`

). Below you can find a list of functions you
would have to implement (correctly) in order for your custom class to be fully
integrated into the transport scheme:

`Center`

region:`__init__(*attrs)`

: initializer`get_type_specific_parameters()`

: retrieves child-specific extra parameters`get_energies()`

: retrieves central region Hamiltonian $`\mathbf{H}^{C}`

$`get_potential()`

: retrieves potential energy values in the central region`get_state(n)`

: retrieves $`n`

$th wavefunction`get_states()`

: retrieves all wavefunctions on the grid`get_number_of_states()`

: retrieves the number of states in the central region`get_sliced_state(n, width, side)`

: retrieves $`n`

$th wavefunction on a grid slice`get_sliced_states(width, side)`

: retrieves all wavefunctions on a grid slice`get_boundary_state(n, side)`

: retrieves nth wavefunction evaluated on some boundary`get_coordinate_ranges()`

: retrieves x and y coordinate ranges`get_coordinates()`

: retrieves the coordinate meshes`get_slice_coordinates(width, side)`

: retrieves the sliced coordinate meshes`get_boundary_coordinates(side)`

: retrieves the boundary coordinate range

`Lead`

region:`__init__(*attrs)`

: initializer`set_magnetic_field_strength(B)`

: sets magnetic field strength`set_energy_spacing(delta_E)`

: sets lead energy spacing`get_type_specific_parameters()`

: retrieves child-specific extra parameters`get_energies()`

: retrieves lead region Hamiltonian $`\mathbf{H}^{L}`

$`get_state_point(x, y, n)`

: evaluates the $`n`

$th state wavefunction at a single point $`(x, y)`

$`get_state(x_points, y_points, n, mode)`

: retrieves the $`n`

$th state wavefunction on a custom/discretized grid`get_number_of_states()`

: retrieves the number of states in the lead region`get_boundary_state(n, num_boundary_points)`

: retrieves $`n`

$th wavefunction evaluated on the lead boundary`get_boundary(num_boundary_points)`

: retrieves the boundary grid with specified discretization

`Coupling`

region:`__init__(Center_object, Lead_object, *attrs)`

: initializer, sets the center and lead objects ready for the coupling matrix calculations`get_coupling_matrix_element(i, j)`

: retrieves coupling matrix element $`\mathbf{V}_{ij}`

$, that is, the coupling between $`i`

$th lead state and $`j`

$th central state`get_coupling_matrix()`

: retrieves the coupling matrix $`\mathbf{V}`

$

Details about the input/output parameter types can be found in the source code. Upon implementing all of these functions correctly for the corresponding custom object and extending the parser interface accordingly, the code extension will be fully consistent with the original code!

## Scripts Included in the Package

tinie includes a few scripts that should ease the usage of the software:

## scripts/tinie_prepare

This script prepares the coupling system and saves it in a tinie_prepare hdf5 file, which contains the following attributes and datasets:

Attribute | Description |
---|---|

"type" | File type, must be "PREPTINIEFile" |

"center/type" | Type of the central region |

"center/num_states" | Number of states in the central region |

"center/parameters" | Type-dependent parameters of the central region |

"leads/num_leads" | Number of leads |

"leads/lead_n/type" | Type of the lead $`n` $ |

"leads/lead_n/num_states" | Number of states in lead $`n` $ |

"leads/lead_n/energy_spacing" | Energy spacing in lead $`n` $ |

"leads/lead_n/parameters" | Type-dependent parameters of lead $`n` $ |

"couplings/num_couplings" | Number of couplings |

"couplings/coupling_n/type" | Type of coupling between lead $`n` $ and the central region |

Dataset | Description |
---|---|

"center/hamiltonian" | Hamiltonian of the central region |

"center/potential" | Potential energy values in the central region |

"leads/lead_n/hamiltonian" | Hamiltonian of the lead region $`n` $ |

"leads/lead_n/x_axis_limits" | x-axis limits of lead $`n` $ |

"leads/lead_n/y_axis_limits" | y-axis limits of lead $`n` $ |

"leads/lead_n/energy_limits" | Energy limits of lead $`n` $ |

"couplings/coupling_n/coupling_matrix" | Coupling matrix between lead $`n` $ and the central region |

Some of these datasets are stored in chunked/compressed format for more data-intensive
simulations. All the simulation parameters are adjusted via a parser user
interface, which takes the following arguments (type `scripts/preptinie --help`

if you ever feel lost!):

Argument | Description |
---|---|

`-dE` ,`--delta-E` |
Lead energy spacing |

`-B` |
Magnetic field strength |

`-xlim` ,`--x-axis-limits` |
x-axis limits of each lead, typed in form `[x_min_0, x_max_0] [x_min_1, x_max_1] ...` |

`-ylim` ,`--y-axis-limits` |
y-axis limits of each lead, typed in form `[y_min_0, y_max_0] [y_min_1, y_max_1] ...` |

`-Elim` ,`--energy-limits` |
Energy limits of each lead, typed in form `[E_min_0, E_max_0] [E_min_1, E_max_1] ...` |

`-ctr` ,`--center-type` |
Central region type, typed in form `"ctr_type(*ctr_params)"` |

`-l` ,`--lead-number` |
Number of leads |

`-ld` ,`--lead-types` |
Lead region types, typed in form `"ld0_type(*ld0_params)" "ld1_type(*ld1_params)" ...` |

`-cpl` ,`--coupling-types` |
Coupling region types, typed in form `"cpl0_type(*cpl0_params)" "cpl1_type(*cpl1_params)" ...` |

`-o` , `--output-file` |
Path, where preptinie file is saved |

## scripts/tinie

This script reads the preptinie hdf5 file, performs the transport calculation and saves the results in a tinie hdf5 with the following attributes and datasets:

Attribute | Description |
---|---|

"type" | File type, must be "TINIEFile" |

"evaluated_chemical_potential" | Chemical potential $`\mu` $ of the system |

"evaluated_bias_voltage" | Bias voltage in the leads of the system |

"evaluated_temperature" | Temperature of the system |

"omega_spacing" | Probe energy spacing |

"lead_energy_spacing" | Lead energy spacing |

"eta" | Small number eta used in the Green's function |

"number_of_couplings" | Number of couplings in the system |

Dataset | Description |
---|---|

"partial_currents" | Matrix of partial currents between each lead |

"total_currents" | Total currents in each lead |

"omega_dependent_partial_currents" | Energy profile of the partial current matrix |

"omega_dependent_total_currents" | Energy profile of the total currents |

"transmission" | Transmission matrix as a function of probe energy |

"transmission_error" | Imaginary component of transmission |

"conductance" | System conductance matrix values |

Some of these datasets are stored in chunked/compressed format for more data-intensive
simulations. All the transport calculation parameters are adjusted via a parser user
interface, which takes the following arguments (type `scripts/tinie --help`

if
you ever feel lost!):

Argument | Description |
---|---|

`-dw` ,`--delta-omega` |
Probe energy spacing |

`-eta` |
Small imaginary constant used in calculating Green's function |

`-mu` ,`--chem-pot` |
Chemical potential, at which system is evaluated |

`-V` ,`--lead-bias` |
Lead biases, at which system is evaluated, typed in form `V_0 V_1 ...` |

`-T` ,`--temperature` |
Temperature, at which system is evaluated |

`-i` ,`--input-file` |
Path, from which preptinie file is read |

`-o` , `--output-file` |
Path, where tinie file is saved |

`--wide-band` , `--no-wide-band` |
Boolean flags user can specify if he wishes to use wide band approximation methods (or not) |

`-S` ,`--self-energy` |
Path, from which the array of custom self energies is read |

`-G` ,`--rate-operator` |
Path, from which the array of rate operators is read |

Note that if you wish to use the wide band approximation approach you must specify either self energies or rate operators or both!

## scripts/tinie_dos

This scripts reads the preptinie hdf5 file and the file containing the eigenfunctions of the central region, computes DOS/LDOS and saves the results in dostinie hdf5 file with the following attributes and datasets:

Attribute | Description |
---|---|

"type" | File type, must be "TINIEDOSFile" |

Dataset | Description |
---|---|

"dos" | Density of states values |

"ldos" | Local density of states values |

"x" | x-axis values of the system central region |

"y" | y-axis values of the system central region |

"omega_dos" | Energies at which DOS was evaluated |

"omega_ldos" | Energies at which LDOS was evaluated |

Some of these datasets are stored in chunked/compressed format for more data-intensive
simulations. All the DOS/LDOS calculation parameters are adjusted via a parser user
interface, which takes the following arguments (type `scripts/dostinie --help`

if
you ever feel lost!):

Argument | Description |
---|---|

`-w` , `--omega-ldos` |
Probe energies, at which LDOS should be evaluated |

`-dw` ,`--delta-omega` |
Probe energy spacing |

`-eta` |
Small imaginary constant used in calculating Green's function |

`-mu` ,`--chem-pot` |
Chemical potential, at which system is evaluated |

`-V` ,`--lead-bias` |
Lead biases, at which system is evaluated, typed in form `V_0 V_1 ...` |

`-T` ,`--temperature` |
Temperature, at which system is evaluated |

`-i` , `--input-file` |
Path from which preptinie file is read |

`-psi` , `--wf-file` |
Path from which central region wavefunctions are read |

`--wf-range` |
Range of the wavefunctions to read from the wavefunction file |

`-o` ,`--output-file` |
Path where the dostinie file is saved |

`--dos` , `--no-dos` |
Boolean, decides if DOS is computed |

`--ldos` , `--no-ldos` |
Boolean, decides if LDOS is computed |

## scripts/tinie_draw

This script reads data from the tinie hdf file, makes pretty transmission/backsacttering/current/density of states plots and saves them. This script has a parser user interface, where you can specify the following plot arguments:

Argument | Description |
---|---|

`-i` ,`--input-file` |
Path from which tinie file is read |

`-idos` , `--input-dos-file` |
Path from which dostinie file is read |

`-o` ,`--output-file` |
Path where the plots are saved |

`-E` , `--energy-rangs` |
Range of energies over which to draw the plot |

`--transmission` , `--no-transmission` |
Boolean, decides if transmission is plotted |

`--backscattering` , `--no-backscattering` |
Boolean, decides if backscattering is plotted |

`--partial-currents` , `--no-partial-currents` |
Boolean, decides if partial currents are plotted |

`--total-currents` , `--no-total-currents` |
Boolean, decides if total currents are plotted |

`--dos` , `--no-dos` |
Boolean, decides if density of states is plotted |

`--ldos` , `--no-ldos` |
Boolean, decides if local density of states is plotted |

`--norm-ldos` , `--no-norm-ldos` |
Boolean, decides if LDOS will be normalized by 1 or not |

`--ldos-E` |
Evaluate LDOS at probe energy closest to the one specified |

`--ldos-idx` |
Evaluate LDOS at an index corresponding to a probe energy |

`--stability` , `--no-stability` |
Boolean, decides if the numerical stability tests are plotted |

## scripts/make_system_files

This script generates custom Hamiltonians or coupling matrices and saves them
in a .npy file to be passed on as arguments for
`CustomCenter`

/`CustomLead`

/`CustomCoupling`

objects. They can also be used to
generate custom self-energy/rate operators for the wide-band approximation.
Run the script, follow the instructions and the rest is history.

Naturally, these scripts provide only some of the basic functionality extensions. Additional scripts/code modifications may be added based on the user's end goals.

## Project details

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