F20QuantizedConductance
Quantized Conductance in a Gold Wire
Terrance Gray and Cameron Kahler
Abstract
Nanotechnology is an important and growing field, but there are limitations to what can be designed due to quantum mechanics. In particular, the conductance of a nanoscaled wire is a quantized function of its width rather than being a continuously variable quantity. The goal of this experiment was to reduce the width of a gold wire to just a few atoms and measure the resultant quantization of the conductance of the wire. A thin wire was gradually and repeatedly split and recombined while an oscilloscope was used to measure the jumps in voltage due to its changing width. The voltage measurements were finally converted into the corresponding conductances of the wire and compared with a theoretical model. This information can aid in the design of nanoelectronics by verifying the quantum properties of nanoscopic conductors.
Introduction
The phenomenon of quantum conductance occurs in all conductors as they approach widths on the order of several atoms. Nanotechnology needs to account for this as computers shrink and become more efficient. The behavior of conductors that have sizes on the order of nanometers has a variety of applications in microchips and other devices.
Observing and measuring quantized behavior requires nanowires. Initial experiments stretched a wire until it was very thin, only a few nanometers wide, and then created voltage differences across the wire to observe quantum behavior. Maintaining a nanowire required precise equipment and occasionally very low temperatures. Eventually, it was found that stretching a wire and leaving it at that width for extended periods of time was unnecessary. Separating and recombining a wire was enough to create a nanowire for an extremely brief period. In the moments before a wire breaks, since all atoms of the conductor are not brought into or out of contact simultaneously, a nanowire is created when and where only a few atoms are in contact. This experiment will be using this property to observe quantum conductance by separating and rejoining a gold wire several times and observing the resultant discrete voltage jumps on an oscilloscope.
Theory
Classically, the conductance of a wire is a function of the wire’s conductivity, cross sectional area, and length. However, when the cross sectional area approaches nanoscopic dimensions, consideration of quantum mechanical effects cause the conductance to become quantized and only dependent on the width of the wire and some fundamental constants. In particular, the conductance is independent of length and macroscopic resistivity in the quantum limit.
The key observation is that, in a sufficiently thin wire, electrons can only pass through a constriction if their de Broglie wavelengths correspond to standing waves that fit in the space. The wavelengths are therefore quantized as
λn=2w/n, n = 1, 2, 3, ...
where w is the wire's width at the constriction. Like all standing waves, the electrons in such a wire have a maximum wavelength given by twice the width of the wire. Crucially, the electrons also turn out to have a minimum wavelength. In a conductor in the 'low temperature limit' (which includes room temperature) we have that, to a good approximation, all electron energy states below the Fermi level are filled while all states above that level are empty. This results in the electrons having a minimum wavelength, called the Fermi wavelength, corresponding to their maximum energy. This quantity is essentially constant for a fixed conductor at fixed temperature.
Altogether, we have that electrons in a thin wire can only take on a finite set of discrete, distinct wavelengths. We can think of each valid wavelength as corresponding to a conductance channel in the wire, and compute the contribution to the wire's conductance from each. One can compute the conductance explicitly by computing the current per channel given some fixed voltage4; the result is that the conductance is quantized in integer multiples of 2e2/h, where e is the charge of an electron and h is Planck’s constant. Essentially, each conductance channel contributes this amount to the conductance, with more channels allowed in the wire as its width increases. Overall, for a wire with an integer number of conductance channels N, the conductance is given by
(1)
We calculated these levels in our particular experiment by measuring the voltage across a gold wire in a voltage divider circuit and computing them explicitly from the voltages in the setup and resistance of the secondary resistor. The general form of the voltage divider as was applicable is
(2)
where V0 is the voltage supplied by a battery, R is the resistance of the resistor being measured over, and ΔVR is the potential difference over that resistor. This voltage was necessarily quantized along with the conductance of the gold wire, allowing for a direct measurement of the quantum phenomena. Inverting the equation and writing the result in terms of the voltages across the wire and the resistor yields the equation we used to deduce the quantized conductance levels from voltage measurements as
(3)
where ΔVw is the potential difference across the wire. The latter form was primarily used for measurements since it was expected to be better for cancelling noise and oscillations that were present throughout the circuit; contributions picked up by both the wire and the resistor would cancel somewhat in the division. Additionally, the second form was convenient in that we didn't need information about the voltage across the battery during each particular trial—as this changed multiple times during the experiment when we swapped out a dead battery for a new one.
Setup
Figure 1. On the left is a circuit diagram of the setup used to supply voltage to the gold wire and measure its conductance as a function of time. The (quantized) voltage drops across the resistor and wire were measured by an oscilloscope and divided according to equation (3) in order to discern the conductance levels. On the right is a picture of the translational stage and spring steel setup actually containing the wire and used to stretch/unstretch it to very thin widths. A close up of the wire glued to the spring steel and insulated by electrical tape is also shown.
A gold wire through which quantum phenomena were to be observed was attached to a piece of spring steel by superglue droplets placed close to one another, but not touching. Then, a small notch was made in the wire between the glue drops using a razor to encourage the wire stretching and breaking at a single point. The wire was isolated electrically from the spring steel by electrical tape, and was attached on each end to a circuit forming a voltage divider. A 100k resistor and 1.5-3.5V battery (varied throughout the experiment) were used in the voltage divider circuit shown in Figure 1 on the left. An oscilloscope was used to measure the voltages across the resistor and wire as a function of time, allowing visualization of the jumps in conductance from one level to another as a function of time.
In order to instigate a change in conductance levels in the wire, a micrometer screw attached to a translational stage set to push on the spring steel was slowly turned to extend the stage. This setup allowed the curvature of the steel to be controlled very finely; as the micrometer screw was turned, the steel slowly bent and the wire attached to it was forced to stretch by minute amounts. The gold wire mainly stretched to a breaking point in the region where a notch was made, reaching nanoscopic widths once the steel was sufficiently bent. The end result of this process was then that all of the quantized conductance levels were reached through a full cycle of the wire being stretched and broken, and could be deduced through the recorded voltages. Note that we used an oscilloscope because the transitions between levels often occurred very rapidly, with a full series of transitions from continuous conductance to zero conductance often occurring within just a couple milliseconds. The oscilloscope allowed us to track these exact stretches of time by triggering on the quantized levels using the oscilloscope's normal mode, after which there was some time to save the voltage traces to a flash drive for analysis.
Initially, we observed a lot of noise in the voltages recorded in the experiment. There were oscillations picked up by the components in addition to random noise possibly from thermal excitations or otherwise. In order to combat the noise, particularly the oscillations, we tried several techniques:
We insulated the experimental setup with a “Faraday cage” composed of a cardboard box lined with aluminum foil to reduce the effects of incoming EM radiation.
We braided the oscilloscope probes about each other to reduce the area between them and partially cancel any induced EMFs from background EM radiation.
We connected the ground point of our circuit to a neutral conductor to keep ground at a constant 0V.
In the end, these techniques decreased the oscillation noise drastically, though we still ended up with a reasonable amount of noise in the traces that obfuscated some of the higher conductance levels. Dividing the voltages across the resistor and wire to get the conductance as discussed in the theory also helped, and in the final analysis we averaged over the noise in seemingly constant conductance regions to still recover many of the desired levels.
Results
After taking a ratio of the two voltages that we measured, we used equations 1 and 3 from the theory section to transform our voltage data into a plot of N vs. time. In these plots, clear levels were observable, with a few caveats.
Since the resistance (and therefore voltage) of the wire vary with the inverse of N (as conductance is linear in N), the voltage levels get closer and closer together as N increases. This makes distinguishing between them much more difficult for larger N.
Even after taking the ratio of our traces to cancel oscillations in combination with the noise cancellation methods outlined in the setup section we still observed small amounts of random noise. The small amount of noise is exacerbated at higher N, since the levels are so close together.
The ratio of VR/VW also leads to some difficulty. As N gets larger, VR increases and VW decreases. This leads to a division of a large number over a much smaller number, and any noise fluctuations in either voltage can be greatly increased. This is observed in our ratio plots by the noise getting much larger as N increases, leading to the higher levels (Generally N > 3) being much more difficult to identify.
The oscilloscope used was found to have a low bitrate, with precision in the recorded voltages only down to increments of 0.02 V. This caused the ratio VR/VW to become discretized by the bitrate, not just by the inherent quantization of the conductance. In particular, where the divided by VW was small in magnitude compared to 0.02 V, the discrete steps became large and imprecise, swallowing up the actual conductance levels.
Over the course of the experiment, we took over 200 traces while still refining our setup and methods. After separating traces with the clearest quantization, we had about 53 traces to analyze. Even with the above complications, we could generally identify regions of conductance corresponding to N = 1-3 for most of these traces. Some also had N = 4 and 5 regions visible, although much less frequently. By taking the average over each region, the positive and negative portions of any random leftover noise should be effectively cancelled out, and we are then able to extract an N-value for each identified region in a trace. An example of our process is shown in the figure below.
Figure 2. Some example plots of the analysis. The lefthand plot shows the voltages across the resistor and wire as a function of time as measured on the oscilloscope. The righthand plot shows the conductance in units of N obtained by dividing the voltages according to equation (3). Each of the bracketed regions of seemingly constant conductance were averaged over to obtain the shown values for the conductance level N.
After computing the N-values shown in each trace, we were able to get data points for N = 1-5, although the amount of points for each level decreased as the higher levels were less likely to be observed. A histogram that shows all the N-values found was created and is shown below. The bins were color coded according to the integer N value they were closest to. For example, any value that fell between 1.5 and 2.49 were assigned to the N = 2 group. This histogram shows a clear peak at N = 1 and N = 3, with more of a plateau at N = 2. N = 4 and N = 5 are not distinguishable from the surrounding data. Averages for each group were taken to get a final computed value for each N, which are displayed in Table 1 with their corresponding Standard Deviations and Errors.
Figure 3. A histogram of the conductance values obtained by averaging over approximately constant steps on the conductance plots analyzed, and the frequencies at which those values were observed. Note the well-defined peaks at N = 1 and N = 3. One can still observe something resembling a peak at N = 2, but it is more of a plateau, and the N = 4 and N = 5 levels are essentially indistinguishable from background.
From this histogram we are able to observe clear grouping of the data around N = 1 and N = 3. These peaks in the graph correspond to the N-values of the quantized conductance. The N = 1 peak is the most defined, and in general as the N-value increases, the peaks decrease in height and become less defined, following what we know about higher levels being less frequently observed. The N = 4 and N = 5 levels currently are unable to be discerned except from the dashed lines corresponding to the expected values.
An interesting observation is that the N = 2 peak is much more flat than the surrounding ones. It appears to stretch from the N = 1.75 bin to the N = 2.5 bin. This anomaly is also visible in the table, where we see that the error for N = 2 is actually higher than for N = 3, even though N = 2 was observed more often. This behavior has been reported by previous groups performing this experiment, although we do not currently have a physical explanation for why it occurred.
We also observed in our calculations of N that the N = 1 value is much lower than expected, especially when compared to how close the other N-values were. Since we only observed this effect for N = 1, it only occurred during the breaking of the wire. Our current theory for why this occurs is based on the inherent capacitance of the wire. In an ideal circuit, the battery goes from some positive voltage to 0 volts almost instantaneously, which is the discrete jump we expect. However, if the wire is not ideal and has some capacitance value, when the battery drops to 0 volts the wire will begin discharging its stored voltage. Rather than immediately reaching 0 volts, the voltage measured across the wire will decay exponentially, extending the time for the voltage to vanish and possibly leading to false positives in our data for N << 1.
Overall, even with the plateau effect around N = 2, we were still able to see the data grouped around N = 1, N = 2, and N = 3 with less points appearing between them. We expect that with more data traces taken and analyzed that these peaks would become more pronounced compared to their surrounding regions, and that similar peaks at N = 4 and N = 5 would become more pronounced as we add more data points to that region. It is likely that in order for any higher N-value to be observed, the apparatus and/or noise cancelling methods would need to be refined in order to combat the inherent difficulty with discerning those higher levels.
Conclusions
By slowly stretching and breaking a gold wire while measuring the time-varying voltage across it, we were able to discern many of the levels of quantized conductance exhibited by a thin conductor. We obtained clear peaks in our aggregate data around the expected N = 1 and N = 3 values, and obtained a plateau at N = 2 that still approximately averaged to the expected result. However, we couldn’t easily observe N = 4 and N = 5 peaks; they occurred quite infrequently and were often obscured by noise and therefore hard to recognize. Additionally, issues with the precision of our oscilloscope limited the accuracy attainable in our analysis by discretizing the conductances obtained via division, particularly in the regions of low voltage across the wire (i.e., in the regions of higher N). We still obtained much data that paints a clear picture regardless. Overall, we can confirm from our results that the phenomenon of quantized conductance indeed occurs in a thin gold wire. However, we can only be reasonably confident of the first three conductance levels, and leave the task of discerning the other levels more precisely to future groups interested in this project.
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