This presentation shows briefly one of the exercises realized within the classes: Computational methods in nanosystems (syllabus, USOS) which takes place at the Academic Centre for Materials and Nanotechnology AGH in Krakow. The aim of this exemplary exercise is to numerically model the QPC nanodevice and show the effect of conductance quantization.
Conductance quantization in QPC is one of the prominent examples showing how modern technology can exploit quantum phenomena to obtain novel features when it comes to electron transport.
The QPC device is a narrow constriction created in the two dimensional electron gas (2DEG) formed at the interface between two semiconducting materials, for example GaAs/AlGaAs (see Fig. 1a). Electrons traveling from one ohmic contact to the other are repelled by the negatively charged electrodes placed on top of the system (point contact gates shown in Fig. 1a and b). That way the electrons must squeeze through the narrow channel created in between the two electrodes. The width of the channel is so small ($W\approx200\;$nm) that the energy quantization takes place in between the charged electrodes (similarly as in atoms or quantum dots), which manifests itself by the measured conductance quantization between the ohmic contacts. The number of conductance quanta that are going to be measured is equal to the number of quantum states created inside the narrow channel through which electrons travel from left to right. By changing the voltage at the point contact gates we can tune the width of the narrow channel and also change the number of occupied states in the channel. That is why the conductance as a function of gate voltage is quantized as shown in Fig. 1c. The value of the singe conductance quantum is determined by the fundamental constants: electron charge and Planck constant ($2e^2/h$).
Fig. 1. (a) Electrons in the 2DEG (marked by the green plane) experience a narrow constriction created by the point contact gates when traveling between the ohmic contacts; (b) Micrograph of the point contact gate structure at the top of the system (Image: Minoru Kawamura, RIKEN Center for Emergent Matter Science, link); (c) The measured conductance as a function of gate voltage (M. A. Topinka et al, Science 289, 2323 (2000)).
# We import the modules that will be useful. import numpy as np import holoviews as hv hv.extension('bokeh') import kwant import numpy as np from matplotlib import pyplot import scipy.linalg as la from scipy.constants import physical_constants import kwant.continuum import warnings warnings.filterwarnings("ignore", category=DeprecationWarning)