S_18 Investigating Quantum Conductance in Gold Wire

INVESTIGATING QUANTUM CONDUCTANCE IN GOLD WIRE

Kaj Erickson, Hamlet Tanyavong

Methods of Experimental Physics, Spring 2018

Abstract

The quantum nature of conductance was observed. Voltage measurements across a gold wire with an atomically scaled constriction were used to observe this phenomenon. The fundamental conductance level was measured to be 7.9±0.8×10^-5 S, which is within 0.2 standard deviations of the theoretical value of 7.75×10^-5 S. The second and third conductance levels were measured to be 14.7±0.5×10^-5 S and 20.7±0.6×10^-5 S respectively, and were within 1.6 and 4.2 standard deviations of the theoretical values.

Introduction

Quantum conductance is a phenomena that occurs when an electrical current flows through an atomically scaled restriction^[1]. Because of the wavelike nature of electrons in the current, only electrons with specific momentums are allowed to pass, while the rest are not. Though this effect is negligible on scales much larger than that of the atom, it cannot be ignored on scales otherwise. Our investigation was motivated by the need for a complete understanding of this phenomena in mesoscopic systems; such an understanding is essential in the design and construction of nanoelectronics^[2].

In this particular experiment, we took voltage measurements across a single gold wire while it was being stretched across a strip of spring steel, which was in turn bent by a micrometer. While the constriction in the wire was larger than the scale of an atom, we did not detect any quantum effects and the conductance behaved classically, decreasing proportionally with the cross-sectional area of the wire.^[1] However, we were able to detect a transition into the regime of quantum conductance once the cross-sectional area was only several atoms thick. Voltage measurements taken while the wire was in this state allowed us to calculate the conductance and confirm that its quantum nature had been observed.

Theory

The quantum nature of conductance can theoretically be observed in almost all conductors^[3], but in this experiment we worked exclusively with gold. The ductile properties of gold allowed us to stretch the gold wire and create a constriction at the breaking point before the wire separated into two pieces. As we stretched the wire, we reduced the cross sectional area to the point where the width of the wire was on the same scale as the wavelength of electrons in the gold. At this point, the restricted cross-section forced a standing wave analogous to the waves present in a potential well^[1]. This restriction and the effect on the nature of the waves is shown in the diagram below (fig. 1).

Figure 1: The illustration depicts the standing waves created in the cross-sectional restriction of the gold wire. This restriction mimics an infinite potential well in which only integer numbers of nodes are allowed.

There exists a minimum electron wavelength in gold of 0.54nm, which corresponds to the fermi energy of gold - 5.2eV^[4]. The cross section of a single gold atom is approximately 0.3nm which restricts the wave function to only two nodes at the edges and corresponds to the n=1 state. This n=1 state corresponds to a single conductance channel, the n=2 state corresponds to two conductance channels, and so on. The conductance through n conductance channels can be described by^[1]

where h is the Planck constant, e is the charge of an electron, and T is the transmission coefficient. The transmission coefficient arises from the probability that the electron will pass through the conductance channel without scattering, and can be calculated with^[3]

where L₀ is the mean free path of the electron in gold, and L is the length of the conductance channel. If we assume that the larger n states maintain a longer constriction, we then expect this transmission coefficient is expected to decrease for larger n states.

Experimental Setup

The mechanical components of our setup were assembled on an optical breadboard. Specifically, we used an NRC Model 430 micrometer translational stage to bend a strip of 12” Starrett .009 gauge feeler stock (spring steel); our gold wire was attached attached across the convex side of the spring steel with hot glue. Electrical tape was used to insulate the gold wire from the conductive surface of the spring steel. The electrical components were all connected in series and included a battery with voltage measured to be 1.608±0.004V, a 98.4±0.2kΩ resistor, and a 99.995% pure 50µm gauge gold wire. The gold wire was notched with a razor blade at the desired break point. The voltage across the gold wire was measured with a National Instruments PCI-6034E ADC board and simultaneously monitored with a Tektronix 2024B Oscilloscope. A diagram of the combined electrical and mechanical components is shown below (fig. 2).

The images to above show the wire setup that produced successful measurements. The gold wire was wrapped around the braided red wire several times underneath the electrical tape. The two glue dabs were applied as close together as possible, in order to ensure that the wire reconnected after the initial break.

Measurement Process

Our measurement process required us to initially break the gold wire by bending it across the spring steel with the micrometer. We found that the initial break removed the tension from the wire and allowed for greater mechanical control over the duration of all following breaks. The wire was then recombined by dialing back the micrometer until the broken edges of the wire made contact. At this point, it was possible to break and recombine the gold wire at will with small movements of the micrometer. We used the attached oscilloscope to determine whether the circuit was open or closed, which indicated the state of the wire. During these small movements, we measured the voltage across the wire at a frequency of 200kHz with the ADC for an interval of ten seconds. These measurements were processed with a LabVIEW program and then exported to a CSV file for analysis. It was typical to observe two or three breaks per ten second interval that displayed quantum behavior.

Results

The raw data consisted of 64 million voltage measurements taken over 32 separate ten second runs. The conductance corresponding to these voltage measurements was calculated using^[3]

where V_b corresponds to the battery voltage, V_m to the measured voltage across the wire, and R_ext to the external resistance. Before advanced analysis was performed on this data, we needed to visually and qualitatively confirm the desired behavior of our conductance measurements. Such behavior is shown below in figure 3.

Figure 3: The graph above depicts conductance measurements made during three separate breaks of the gold wire. If the conductance behaved classically, the graph would depict three smooth downward slopes. However, we clearly observe discrete steps where the conductance dwells, and then an instantaneous jump down to the next conductance step. It is this behavior that indicates quantum conductance.

The quality of the breaks depicted in figure 3 could be described as good in comparison to an average measured break. Some breaks did not maintain a constant conductance, and sloped directly from a closed to an open circuit. Other breaks were not as ideal as those shown in figure 3, and the conductance sloped at times instead of stepping between constant values.

The data analysis was done exclusively using Matlab because of our large data sets. The first data reduction method involved removing every conductance measurement that was not taken during a break. Specifically, the conductance measurements obtained during the recombination of the wire were excluded from our analysis. We sought only to investigate the behavior of a breaking wire, not a recombining wire. The removal of these recombinations was done by visually differentiating between breaks and recombinations based on whether the conductance was increasing or decreasing over a few milliseconds and then removing the sections in which the conductance was increasing. After the removal of the recombinations, the data was binned by conductance with Matlab’s histogram function. The resulting histogram is included below (fig. 4)

Conclusions and Future Work

We were able to observe three conductance states in our experiment. The n=1 state was associated with a conductance of 7.9±0.8×10^-5 S, while the n=2 and n=3 states were associated with

14.7±0.5×10^-5 S and 20.7±0.6×10^-5 S, respectively. The first state was the most accurate, being only 0.2 standard deviations off from the predicted value; the others were 1.6 and 4.2 sigmas off. We mainly attribute the inaccuracies in the latter two values to unclean breaks and an altering transmission coefficient.These problems, however, are very difficult to work around due specifically to our experimental design.

One way to reduce the amount of unclean breaks is to have a larger mechanical reduction ratio. Even with the translational slide and spring steel we used, the states were still too sensitive and difficult to maintain. Having cleaner breaks would greatly reduce the noise we see in our data, figure 5, and make the conductance levels more prominent. Another way to get a good quality break is to notch the gold and ensure the slice is perpendicular to the axis of the wire. Examining it under a microscope gives a great insight on whether or not good data can be taken from the sample; and a post-break examination could be used to diagnose problems seen in the data.

Sources

^[1] R. Tooley, A. Silvidi, C. Little, and K. Eid, Conductance Quantization, A Laboratory Experiment in a Senior-Level Nanoscale Science and Technology Course (Oxford, 2012), Ohio.

^[2] D. K. Ferry, H. L. Grubin, C. Jacoboni, and A.-P. Jauho, Quantum Transport in Ultrasmall Devices, Vol. 342, NATO ASI B (Plenum Press, New York, 1995).

^[3] S. Datta, Electromic Transport in Mesoscopic Systems (John Wiley & Sons, Cambridge, 1997), 377 pp., New York.

^[4] W. M. Haynes, ed., CRC Handbook of Chemistry and Physics, edited by T. J. Bruno, 96th ed. (Taylor & Francis, Boca Raton, 2015) Chap. 10, 2677 pp., Florida.

^[5] M. P. Das, F. Green, The Landauer Formula: A Magic Mantra Revisited, Department of Theoretical Physics, Research School of Physical Sciences and Engineering, The Australian National University, (April 2003), arXiv: 0304573