# Quantized Conductance in Gold Wires

This is going to be my first Physics related blog post. I just finished proofreading my last ever 6510 lab report. Physics 6510 is a laboratory course that every single graduate student at Cornell Physics department needs to take in order to graduate. If one happens to enter PSB or the Clark hall on a normal day, it is not uncommon to hear some graduate students complain about this course being mandatory or how old and painful the instruments in the lab are. In a typical semester, a student is required to perform three experiments, but (un)fortunately, due to COVID-19, we were able to do just two this time. My second lab was on conductance quantization in the context of Gold wires.

Quantization of conductance is a ubiquitous phenomenon in condensed matter physics. In the context of quantum Hall effect, it was first observed by von Klitzing in 1980. This observation gave rise to the application of ideas from topology in condensed matter systems; as a result, a new sub-discipline i.e. topological condensed matter was born. Conductance quantization in such Hall systems is observed only at low temperatures and high magnetic field which is mostly inaccessible to undergraduate/graduate laboratories.

In this post, I will discuss the quantization of conductance in the context of point contacts between two Au wires, which was first discovered by Wees et al. in 1988 . In contrast to the quantum Hall systems, this experiment does not require low temperature and high magnetic field making it ideal for studying in a simple laboratory setting. The main concept involved in this experiment is ballistic transport which occurs when the conduction distance is less than the mean free path of the electron so that it can move unimpeded without scattering.

If we have a 2-dimensional conductor with length L and width W, the conductance is given by \(G = \sigma \frac{W}{L}\). Based on this formula, decreasing the value of L should cause conductance to grow indefinitely. However, it has been experimentally observed that the measured conductance approaches a limiting value in the ballistic limit. This will be an integral multiple of \(G_0\). Such limiting value will be depend on the width W because it determines the number of transverse modes available for conduction. Assuming periodic boundary conditions, we can estimate the number of modes to be \(2W/\lambda_F\). Thus, the number of modes will increase in discrete steps with the width of the contact.

The point of this post is not to subject the readers to my entire lab report, but I’d like to describe the experiment very briefly and share my results. The experimental setup is pretty simple. The heart of the experiment is an apparatus with two Gold wires whose distance can be controlled by turning a screw. After bringing them to an appropriate distance, tapping the apparatus will cause point contacts to be formed or broken between the wires and this will be manifested in the voltage trace measured by an oscilloscope. In some of the traces, we should be able to observe discrete jumps which will ideally correspond to integral multiples of the conductance quantum. I was able to get the following histogram by taking 50 traces with discrete steps out of 1000 traces. My own data was very poor, but I couldn’t take more data due to closure of lab in response to COVID-19. Thanks to professor Anders Ryd for providing me with the data which I think is from a 6510 alumnus.

It might be confusing to the reader, how my voltage trace transforms into a conductance value but the voltage can be easily converted to resistance using simple circuit relations, and conductance is just the reciprocal of the resistance. The first useful peak is observed at \(G=G_0\) as expected. Although we expect the second peak at $G=2G_0$, it occurs at \(G=1.86G_0\). The third peak is seen at \(G=3G_0\), and the fourth peak is flatter and occurs before \(G=4G_0\). After the fourth peak, it is hard to discern the peaks so only the steps up to \(G=5G_0\) have been considered. Averaging over the first three distinct peaks, the value of quantized conductance obtained was \((0.98 \pm 0.16)G_0\), where \(G_0\) is the exact value \(7.748091729 \times 10^{-5}~S\).

To address the observation of conductance value at non-integer multiples of \(G_0\), it is useful to discuss the validity of the ballistic transport assumption. The provided Au wires are not perfectly pure, and they have accumulated impurities on the surface, so scattering due to impurities is inevitable resulting in non-ballistic contributions. As discussed earlier in the post, ballistic transport occurs when the transport length is smaller than the mean free path. For Au, the mean free path is 37.3 nm \cite{mfp}; thus the nanowire contacts of longer than this length will also contribute to the non-ballistic transport. These issues can be addressed by using a pure sample and cleaning the surface of the wires so that the nanoscale protrusion are removed.

Despite not having perfect data, it is remarkable that such magnificent quantum phenomenon can be demonstrated in the lab with a simple experimental setup. Immediately after making this post, I am sending off my report to my instructor, and that shall be the end of 6510 for me unless I fail the course for some inexplicable reasons.