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.

Example: Usage of TINIE with ITP2D

To better demonstrate how TINIE is used, we will show it by means of an example problem. We will compute transport properties of a quantum-dot system with two leads in a magnetic field. Specifically, we shall procure the information about the central region from ITP2D, a Schrödinger equation eigensolver that interfaces with TINIE. The following workflow is typical for most transport problems solved with TINIE:

Step 0: computation of the Hamiltonian and wavefunctions of the central region. We may obtain this information from any eigensolver of our choosing, provided that it is TINIE-compatible. Quantum dot may be modeled by a radial harmonic potential of form $V(r)=\frac{1}{2}\omega_{0}\Vert r \Vert^{2}$. Solving the first 25 states of this model with magnetic field strength $B=1.0$ is done in ITP2D as follows:

$ ./itp2d -v -n 20 -l 12 -s 100 -p "harmonic(1)" -B 1.0 -o ITP2D_FILE_PATH

Here the central region occupies a $[-6,6]$ region in both x- and y-directions and is centered at the origin. More information about usage of ITP2D can be found on its bitbucket page.

Step 1: the transport system preparation step. In this step, the coupling matrices for the leads are computed. Suppose we wish to compute overlap coupling between the central region and the leads and we want to vary the probe energy within each lead in range $[0.0,2.0]$ with energy spacing of $\Delta E = 0.001$. In this case, the first 5 states of the central region are sufficient for the calculation. Our leads are such that:

• Lead 0 is confined to region $[-10.0,-4.0]$ in $x$-direction, $[-5.0,5.0]$ in $y$-direction and connects to the lead from the left;
• Lead 1 is confined to region $[4.0,10.0]$ in $x$-direction, $[-5.0,5.0]$ in $y$-direction and connects to the lead from the right.

Both leads in this case have harmonic potential of strength $\omega=1.0$ in $y$-direction, and particle-in-a-box potential in $x$-direction. This information is sufficient for us to commence the system preparation. For that, tinie_prepare script is used as follows:

$ tinie_prepare -dE 1e-3 -B 1.0 -ctr "itp2d(ITP2D_FILE_PATH,(0,4))" -l 2 -ld "finharm(left,1.0,dir)" "finharm(right,1.0,dir)" -xlim "[-10.0,-4.0]" "[4.0,10.0]" -ylim "[-5.0,5.0]" "[-5.0,5.0]" -Elim "[0.0,2.0]" "[0.0,2.0]" -cpl "overlap()" "overlap()" -o TINIE_PREPARE_FILE_PATH

This produces the PREPTINIEFile that contains the information about the coupling of the transport system which can be reused for different transport calculations of the next step.

Step 2: the transport calculation step. This is where the real fun begins, the steps before are in a sense just a preparation. To compute various transport properties of the system, such as transmission, conductance and current, we fix temperature of the system $T=1.0$, chemical potential $\mu=1.0$ and fix bias voltages in the leads to be $V_{0}=0.5$ in Lead 0 and $V_{1}=1.5$. Moreover, we adjust the energy spacing of probe electrons to $\Delta\omega=0.01$ and set the Green's function boundary parameter to $\eta=0.1$. With this information we can use tinie script as follows:

$ tinie -i TINIE_PREPARE_FILE_PATH -dw 1e-2 -eta 1e-1 -T 1.0 -mu 1.0 -V 0.5 1.5 -o TINIE_FILE_PATH

This produces the TINIEFile that contains all the above mentioned transport quantities and more, with detailed description of its contents outlined in the sections below.

In addition to the transport properties, we can compute local and standard density of states (LDOS/DOS) of the system via the tinie_dos script. To that end, in addition to the parameters specified above, user would want to specify the energies at which LDOS should be evaluated, as well as the location of the file with the central region wavefunctions. We then use the script as follows:

$ tinie_dos -i TINIE_PREPARE_FILE_PATH -psi ITP2D_FILE_PATH --wf-range 0 4 -w 1.0 2.0 3.0 -dw 1e-2 -eta 1e-1 -T 1.0 -mu 1.0 -V 0.5 1.5 -o TINIE_DOS_FILE_PATH

Here, we evaluated LDOS at probe energies $\omega\in\{1.0,2.0,3.0\}$. Results of this calculation are stored in TINIEDOSFile, with details about its contents available in sections below.

Step 3: visualizing the results. To that end, one can use the tinie_draw script. Suppose we want to plot transmission, conductance, total current and DOS of the system in the energy range $[0.0,5.0]$, as well as LDOS at probe energy $\omega=1.0$. We then use the following command:

$ tinie_draw -i TINIE_FILE_PATH -idos TINIE_DOS_FILE_PATH -E 0.0 5.0 --ldos-E 1.0 --transmission --conductance --total-currents --dos --ldos -o FIGURE_PATH

This will produce beautiful LaTeX-rendered plots of the aforementioned quantities. Below we show the example plots of conductance, DOS and LDOS produced by the script:

Conductance	DOS	LDOS

Quantum transport calculations in two-dimensional systems have never been as easy!

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:

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!):

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:

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!):

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:

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!):

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:

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.