3D - Conductance of a quantum point contact (gated two-dimensional electron gas)

Input Files:

• 3D_conductance_in_top_gated_2DEG_nnp.in - simulation of the potential in 2DEG

• 3D_conductance_in_top_gated_2DEG.py - generates all plots

• 3D_conductance_in_top_gated_2DEG_verification.py - does not generate conductance

• 3D_conductance_in_top_gated_2DEG_without_plot.py - generates only conductance

• 3D_conductance_in_top_gated_2DEG.ipynb - Jupyter Notebook for practicing the tutorial

The Python scripts and the Jupyter Notebook file are available on our GitHub

Scope of the tutorial:

• computing electrostatic potential using nextnano++

• interfacing nextnano++ with Kwant, for computing the conductance between two leads

Main adjustable parameters in the input file:

• the widths of the gates - $gate_width

• the gap beetween the gates - $gap_length

• lowest bias on the top gate - $top_gate_bias_min

• highest bias on the top gate - $top_gate_bias_max

• number os bias sweeps of the top gate - $top_gate_steps

• bias of the bottom gate - $bottom_gate_bias

Relevant output files:

• bias_xxxxx\bandedges_2d_2deg_slice.fld (potential energy profile)

Simulated Structure

Figure 2.5.2.167 presents the simulated structure, where a two-dimensional electron gas (2DEG) is formed at the interface of the AlGaAs and GaAs (the substrate) materials. The electron density in the 2DEG is enhanced by doping the region of the AlGaAs with n-type impurities only in the part close to the surface.

A GaAs layer over the n-AlGaAs region acts simply as a cap of the device. On the top of the surface metallic gates are deposited and can present different geometries. We will choose the gates in the Figure 2.5.2.168 as QPCs, to which negative bias will be applied in order to deplete electrons at the center of the 2DEG region. Although these gates pursue one of the simplest geometries, the method here described can also be used for gates with more complex shapes.

The dopant and surface charges concentrations used in this simulation are realistic, and were obtained by the calibration method described in [Chatzikyriakou_PhysRevResearch_2022]

Figure 2.5.2.167 Schematics of a side view of the simulated device

Figure 2.5.2.168 Top view of the gates deposited on the top of the simulations

The Simulation

The main objective of this tutorial is to simulate the conductance between two leads in the 2DEG region as a function of the applied bias in the gates deposited at the top of the structure.

Initially we will use nextnano++ to obtain the conduction band in the device changing the applied bias to the top gate in the range of -1.5 V and 0.0 V. The applied bias to the bottom gate will be kept constant (-1.1V), through the whole set of simulations. For this first phase of this tutorial, we will use the input file: 3D_conductance_in_top_gated_2DEG_nnp.in.

In order to obtain the trasmission coefficients between two leads in the 2DEG, we will import a slice of the conduction band in this region into the software Kwant, using the Python script: 3D_conductance_in_top_gated_2DEG.py

Kwant is an open-source tool that performs numerical calculations on tight-binding models. For the installation of Kwant in your computer, please, follow the instructions on the Kwant webpage.

Phase 1: Obtaining the conduction band in the 2DEG region using nextnano++

The conduction band in the whole device can be obtained as a solution of the 3D-Poisson equation.

For realistic devices, a large number of nodes in the grid is required to evaluate with high accuracy the voltage that depletes electrons at the center of the 2DEG region. The nextnano++ input file sweeps automatically the value of the top gate (\(V_{gate}\)) and generates 2D-slices of the band edges in the 2DEG plane that will be used in the next phase of the simulation.

Phase 2: Setting up Kwant

In order to setup Kwant in a consistent way with the configuration of nextnano++ we need to define the next variables:

• the effective mass of electrons in the 2DEG region ms = 0.067 * 9.109e-31

• lattice constant of the tight-binding system (nm) a = 1

• conversion constant from eV (output of nextnano++) to Kwant energy unit T = hbar*hbar/2/nm/nm/ms/e

where:

• e = 1.602e-19 is the electron charge (in C),

• hbar = 6.626e-34/2/np.pi is the Dirac constant (in Js),

• h = 6.626e-34 is the Planck constant (in Js),

• nm = 1e-9 is the conversion factor from 1 nanometer to 1 meter (in m),

Additionally, it is convenient to define a smaller portion of the slice of the potential obtained in the previous phase as the scattering region that will be used by Kwant. Here we will use a square scattering region with size of 400 nm x 400 nm, with the same center as before, the coordinates (0,0).

Phase 3: Computing the conductance coefficients with Kwant

Describing briefly the Kwant script, the program reads the file containing the potential of the 2DEG ( a 2D-slice ), whose path is especified in the script through the variable path_extracted_potential. Through interpolation, Kwant maps the values of the potential into each node of the corresponding 2D-square lattice defined in the previous phase.

This is the basic elements for building the system of equations to be solved under the tight-binding approach, whose the matrix elements and hoppings are set by discretization of the Hamiltonian:

\[H = -\frac{\hbar^2}{2ms} (\delta^2_x + \delta^2_y) + V(x,y),\]

where \(V(x,y)\) is the potential extracted from nextnano++.

The leads will be considered as ohmic contacts, and are attached to the left (lead 0) and to the right (lead 1) of the scattering region, as shown in Figure 2.5.2.169.

At this point it is convenient to verify the band edges of both leads, one of them plotted in the Figure 2.5.2.169. Finally the program solves the system of equations and the conductance from lead 0 to lead 1 is computed, for the especific potential imported. As example, when applying a voltage of -1.11 V to the upper gate of the structure, and -1.1 the the lower gate, the conductance between the two leads in the 2DEG is equal to 2.0074 \(2e^2/h\)

Figure 2.5.2.169 Imported conduction band when a bias of -1.11V is applied to the top gate.

Figure 2.5.2.170 Band structure of the lead 0 for top gate voltage equal to -1.11 V.

As we mentioned before, QPCs can be a very useful structure to control the conductance of electrons in a 2DEG region. In this example, we can verify how changes on the bias of one of the gates modifies the transport of electrons in the 2DEG region.

The Kwant script iteratively will import each potential simulated in nextnano and compute the correspondent conductance. This script requires that you have our package nextnanopy installed in your machine, that can be downloaded for free in our nextnanopy repository. As this process will process 101 files, it could take some minutes to perform the calculations. At the end of the process, a plot will be generated in your screen.

Figure 2.5.2.171 Conductance between lead 0 to lead 1 as function of the bias applied to the top gate.

The Figure 2.5.2.171 presents the channel conductance computed for each value of \(V_{gate}\). The steps in the curve show the expected quantization for this device.

Acknowledgment

This tutorial is a result on the nextnano GmbH collaboration in the scope of the UltraFastNano Project aiming at development of the first Flying Electron Qubit at the picosecond scale, and it is funded by the European Union’s Horizon 2020 research and innovation program under grant agreement No 862683. The tutorial contains part of results from a collaboration of Institut Néel (CNRS), CEA-IRIG and nextnano Lab in France, and nextnano GmbH in Germany.