Quantum Conductance in Gold Wire

Sylvia Griffitt and Gabriel Spahn

Abstract

Quantum conductance can be measured by breaking and reattaching a gold wire. The conductance of the cross section decreases in discrete steps of

as the number of conductance channels (n) decreases. Multiple quantum conductance levels appear during a single break, so combining the data from multiple breaks shows the repeated conductance channel numbers. Something remarkable is that fractional n values appear to emerge across multiple breaks. This is unphysical, as a fractional n value implies a fractional number of electrons moving through a break. Some of the fractional levels observed were n = 1.26, 1.59, and 2.62. A possible explanation of this is that multiple regions of quantum conductance with integer n values form in series, resulting in an apparent fractional n value.

Motivation and Theory

In very thin wires where the conduction channel cross section dimensions are roughly equal to the Fermi wavelength of an electron, the conductance of a wire becomes quantized, only proceeding in steps of

[1, 2, 3]. This quantized conductance was first observed experimentally in 1987, where small constrictions in two-dimensional electron gases were used to demonstrate the effect [4]. Conductance is the inverse of the resistance. As the cross sectional area reaches widths that are on the order of the electron De Broglie wavelength (

), the conductance no longer decreases linearly, instead it starts to follow decrease by steps proportional to the base unit of conductance,

.

Figure 1: Comparison of the quantum and classical conductance

Experimental Setup

The set up of this experiment is very simple. A DC circuit is made using a breadboard, a D cell battery, a external resistor, and a piece of gold wire mounted on a piece of spring steel that functions as a variable resistor. An oscilloscope measures the voltage over both the external resistor and the gold wire sample. The conductance can be found using a voltage divider for the two resistors in series,

Figure 2: the circuit which is used to measure the resistance over the gold wire. The break in the gold wire is modeled as a variable resistor. The external resistor is present because the resistance over the gold wire is very low until it begins to break.

The gold wire samples are stretched using the system displayed below. First, a small section of wire is wrapped around the ends of two wires that will connect it to the circuit. The center of the gold wire section is notched to encourage it to break at that location. Then, the ends of the section are epoxied down to a strip of spring steel, which is wrapped in electrical tape to prevent shorting. Finally, the spring steel is placed against two supports at its ends and bent in the middle by an optics translational stage. This bending is meant to stretch the gold wire until it eventually breaks; however, the gold wire often had to be broken manually.

Figure 3: a top down view of the set up that is used to break the wire. The gold wire is mounted to a flexible piece of spring steel, which can be used to break the gold wire in a controlled fashion.

Each sample of gold wire can be used to create many voltage traces, because (unlike many other conductors) gold does not react with the air to form oxides or rust; this allows the two halves of a broken wire sample to reconnect into a single conductor and then be re-broken an arbitrary number of times. Quantum conduction behavior can also be observed during reconnection of the two halves. Following the collection of a data trace on the oscilloscope, the data is saved to a computer using LabView as an intermediary. Additionally, significant metadata is recorded about each trace, including the value of the external resistor in use, the n values of the observed plateaus, the length of each plateau, as well as whether the trace was generated by the gold wire sample breaking or reconnecting.

Analysis and Discussion

Many individual traces displayed quantum conductance behavior, showing clear voltage and conductance plateaus. However, the location of these plateaus was often surprising. Many appeared at fractional integer values such as n=3/2 or n=5/2. Others appeared to have constant voltage offsets from the theoretically predicted values.

Figure 5: Relative frequency of different channel numbers n over a sample size of 30 traces, each of which displayed clear quantization plateaus.The n values cluster around a number of peaks, denoted by vertical dashed lines.

The histogram dataset was then split at the troughs between each marked peak, and each peak was analyzed using Gaussian statistics to determine its mean and standard deviation. The results are displayed in Figure 6a. Then, the same analysis was repeated using the segments of data centered around each integer multiple of n. As displayed in Figure 6b, the mean value in this analysis was offset significantly from the integer center for each segment, and the standard deviations observed were consistently worse for each section compared to their counterpart in the previous analysis.

Figure 6a (top): Gaussian analysis of the histogram data separating it at the observed troughs. 6b (bottom): similar analysis, this time separating the data so that each segment is centered on an integer multiple of n.

This dual analysis demonstrates clearly that enforcing the idea that the total conduction observed be an integer multiple of the base conductance unit results in worse fits to the experimental data. This fact, combined with the clear fractional conductance plateaus in Figure 4, is in direct tension with the single-constriction theory of quantum conductance. However, a few other experiments have observed fractional conductance in situations ranging from memristors to monatomic wire. One particular experiment proposed a compelling explanation for the observed behavior: the presence of multiple wire constrictions in series [5]. For two constrictions, each with their own integer number of conducting channels n and m, respectively, the resulting conductance of the combined system is given by

This conceptualization explains levels like the n=3/2 visible in Figure 4 as two wire constrictions, each with n=3 conduction channels, chained together in series. It would also explain the offset in Figure 4 as one wire constriction varying between the low values of n, with the other varying independently at some high values of n that effectively contribute a series resistance to the gold wire sample. This same effect would also explain the leftward shift of the integer peaks in Figure 5.

Conclusion

Quantum conductance is clearly a more complicated phenomenon than the simple model of one constriction on the scale of the De Broglie wavelength implies, as there are clearly non-integer n values that show up repeatedly across data sets. The integer nature of the conductance channels is a property of the quantum nature of electrons, so accepting the idea of the electrons taking on fractional values would require the restructuring of most of modern physics. Because it is exceedingly unlikely that this is the case, there is clearly something else at play that leads to these non-integer values. The proposed solution of there being multiple constrictions exhibiting quantum conductance that are separated by regions of bulk conductance nicely explains these fractional values.

References

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