s19quantumconductance

Things that we learned that hopefully will cut down future project time. The biggest time drain involved notching the wire correctly. Try your hardest to acquire a microscope that is able to see how accurate your wire notch is. If you cut through the wire do not fret. Attempt to get the wires to touch using the head of a razor or some other very thin object. A single contact point is all that is necessary to see quantum conductance. Another thing to note is that our transmission coefficient is VERY large. Normally the transmission coefficient is very low, so the the quantum conductance corresponds to a voltage that is very small. Using a small battery voltage could help curtail this issue (potentially... aka try your hardest to find a 1V battery instead of the 9V battery that we used). Lastly, learn triggering as quickly as possible. Ask Kevin... he is very good at explaining things.

Investigating the Quantum Conductance of a Gold Wire

Samuel Eckstrom and Tanner Leighton directed by Professor Dan Dahlberg

PHYS 4052W

University of Minnesota, Twin Cities

The purpose of the experiment conducted was to observe the discrete nature of conductance that occurs when the width of a gold wire approaches the Fermi wavelength. An apparatus was designed to break and recombine the wire while an oscilloscope measured the voltage over an external resistance. One conductance channel, which is equivalent to one available wavelength able to pass through the wire, had a conductance of (1.339 ± 0.173) * 10-6 Siemens, which was 0.150 standard deviations away from the expected value. Two conductance channels had a conductance of (2.672 ± 0.264) * 10-6 Siemens, which was 0.174 standard deviations away from the expected value. This behavior continued to increasing conductance channels.

Introduction

One issue based on quantum mechanics is the behavior of electrical conductance at the quantum scale. Electrical conductance is the inverse of electrical resistance. Conductance is the ease with which electrons flow through a material. When a wire is pulled, the resistance goes up, as resistance is dependent on the surface area of the wire. Therefore, the conductance decreases simultaneously. The wire eventually gets to a point where there are very few atoms linked together before breaking. When this occurs, quantum mechanics predicts that the conductance will not behave in the classical way, which states that conductance is continuous, but instead states that the conductance changes in discrete levels. These discrete levels are associated with the number of conductance channels that are allowed through the wire. The aim of our experiment is to observe this phenomena. In addition to observing quantum conductance, Kaj Erickson and Hamlet Tanyavong, a previous MXP group that also performed research on quantum conductance, found that not only did quantum conductance occur, but also found that the conductance channels with an odd parity occurred more often than the even valued channels [1].

Theory

Figure 1: shows the three different limits to conductance. In the figure, the following variables are the following: L is the length of the constriction, w is the width of the constriction, l is the length of the mean free path of the electron, and lambda is the fermi wavelength. As shown, the classical regime occurs when the mean free path of the electron is much less than the width and length of the constriction. The ballistic regime occurs when the length of the constriction is smaller than the mean free path of the electron. Lastly, the quantum regime occurs when the width of the constriction approaches the fermi wavelength. This is when conductance should appear quantized.

The quantum regime is the limit this project focuses on. This occurs as the width of the constriction decreases to a size comparable to the de Broglie wavelength of an electron at the fermi surface

w = λF (1).

Assuming the width of the constriction is much smaller than the length of the constriction, a full quantum mechanical treatment must be applied. This will cause a discrete number of wavelengths, or conductance channels, to pass through the wire. The behavior is similar to that of a one dimensional infinite square well, where the motion in the x-direction is still continuous, yet the allowed conductance channels follow a quantized level n. Assuming all the states below the Fermi energy are occupied, and all the states above the Fermi energy are empty, then the shortest wavelength of an electron able to pass through the wire is equal to the fermi wavelength. Furthermore, the number of conductance channels depends directly on the width of the constriction

n = 2w / λF (2)

where w is the width of the constriction, and λF is the Fermi wavelength. Equation 2 illustrates how the number of allowed conductance channels decreases as the width of the constriction decreases. In addition, the fermi wavelength is a constant. Therefore, the only variable the number of channels depends upon is the width. The quantized conductance can be derived through relating the conductance to the current [2]. The final equation becomes

Gn = e2Tn / h = e2nl / h(l + L) (3)

where T is the transmission coefficient. This transmission coefficient can be broken down further into T = l / (l + L), where l is the mean free path of an electron in gold, and L is the length of the constriction.

Apparatus

The apparatus was built on an optical breadboard. Two bricks and an NRC Model 430 micrometer translational stage were mounted on top of an optical breadboard. The two bricks held in place a stainless steel feeler gauge that was being pushed on by the micrometer. A screw was attached perpendicular to the translational stage in order to apply pressure to a specific spot of the stainless steel. Electrical tape was wound around the stainless steel. This was necessary to ensure our 0.001 inch diameter gold wire could be epoxied without the risk of the circuit being shorted. Two drops of epoxy were applied, each being a sufficient distance apart to allow a notch to be created in the gold wire. The notch was created using a household razor. A notch was necessary to allow for easy breaking and reassembling of the wire when the micrometer applied pressure to the stainless steel. The gold wire was in series with a 9 volt battery and a 100 kOhm carbon resistor. An external breadboard was used to create the circuit. A DPO 2024B Digital Phosphor oscilloscope’s trigger function was used to measure the voltage change across the 100 kOhm resistor. The trigger level was set from anywhere between 3 and 6 volts, and the time scale was in units of 25 microseconds. Triggering was necessary, as the breaking or reformation of the gold wire was very quick: too quick for humans to accurately measure by hand.

Figure 2: shows the apparatus discussed. The labels represent the following: 1 - micrometer translational stage used to slowly apply pressure to the gold wire. 2 - blocks used to hold the stainless steel in place. The circles represent clips to ensure the stainless steel does not move around. 3 - the gold wire being pushed by the translational stage. This is the point where the gold wire is stretched apart or recombined. 4 - the 100 kOhm carbon resistor. Note that during experimentation the oscilloscope reads over the external resistor, and not the gold wire. This was shown in the figure, however, to represent that conversion between the two voltages is simple using Kirchhoff's voltage law. 5 - the 9V battery. 6 - the oscilloscope used to trigger and collect our data.

The images above show the experimental setup.

Procedure

The most basic step in the experiment was breaking the wire. As stated before, the thin gold wire was notched, making it easy to break and recombine when small pressures were applied. This pressure came from the translational stage. Turning the micrometer caused the translational stage to move either forward or backward very slightly. If the attached screw was in contact with the stainless steel, the pressure against the gold wire would either increase or decrease very slightly. This change in pressure was enough to cause the wire to break or recombine. A breaking or recombining wire caused a large change in voltage to occur. An oscilloscope was triggered when this sudden change in the voltage occurred. The trigger level was set anywhere between 3 and 6 volts. The trigger level was arbitrary as long as the entire event, ie. the wire went from completely broken to completely reformed or vice versa, was recorded. When triggered the oscilloscope captured the behavior on a time scale with units of 25 microseconds.

Results

Only recombination trials were recorded. This was intentional. It was impractical to take data as the wire was being pulled apart. Quantum conductance was rarely seen when wire was pulled apart. It is believed this occurred due to the tension in the wire as it was being pulled apart. When the wire was recombined there was no tension in the wire, so the wire’s width was able gradually increase in comparison to the quick split of a wire that had tension in it. The voltage across the external resistor was transformed into conductance using Ohm’s law and Kirchoffs law. The derivation is simple, and is as follows

GAu = 1 / RAu = VR / (Rext * (VB - VR)) (4)

where RAu is the resistance of the gold (this decreases as the wire’s width increases), VR is the voltage across the external resistor, VB is the voltage across the battery, and Rext is the voltage across the external resistor. Both VB and Rext are known constants, having values of 9 volts and 100 kOhms, respectively. Furthermore, equation 4 becomes the following when VB and Rext are plugged in

GAu = VR / (900000 - 100000VR) (5).

As the voltage across the resistor increases, the conductance of the gold wire increases. This makes logical sense, as more conductance channels should become available as the wire recombines and its width increases. Knowing equation 5 allows the voltage to be measured and transformed into a conductance.

Figures 3 & 4: show a recombination trial. Figure 3 represents the data we saw on the oscilloscope. Figure 4 is the plot that represent the dwell times of the same trial. Each count is equal to 2.5 microseconds.

Analysis

Equation 3 was used to find the voltages for 1 to 5 conductance channels. Equation 5 was used to seamlessly convert between voltage and conductance. Equation 3 could then be combined with equation 5 to form

Gn = e2Tn / h = e2nl / h(l + L) = VR / (900000 - 100000VR) (10).

The transmission coefficient was determined through experimentation. This was determined by setting the transmission coefficient to the value that caused the lowest magnitude voltage plateau to be equal to the conductance quantum, G0. The transmission coefficient for the data was determined to be approximately 1 / 59. The physical meaning of this is that about 1.7% of electrons are able to pass through the wire, while 98.3% are reflected. Once the correct transmission coefficient was found, the different theoretical voltages were calculated for various levels of n. As stated before, large values of n were both less common and difficult to determine. Therefore, only the integer values from 1 to 5 were determined. Theoretically, the following values were discovered for various integers: n1 = 1.045 volts, n2 = 1.872 volts, n3 = 2.544 volts, n4 = 3.099 volts, and n5 = 3.567 volts. Once the expected values were found, the experimental data was compared.

Table 1: shows the results of each wire, and its corresponding standard deviation for each conductance channel from 1 to 5. An N/A represents that no trials of a certain conductance channels were seen. A measured value with an N/A error represents that there was only 1 trial for a certain wire.

The following results summarize the findings: the ground level conductance channel n = 1 had a conductance of (1.339 ± 0.173) * 10-6 Siemens, which was 0.150 standard deviations away from the expected value of 1.313 * 10-6 Siemens; the second conductance channel n = 2 had a conductance of (2.672 ± 0.264) * 10-6 Siemens, which was 0.174 standard deviations away from the expected value of 2.626 * 10-6 Siemens; the third conductance channel n = 3 had a conductance of (3.943 ± 0.149) * 10-6 Siemens, which was 0.013 standard deviations away from the expected value of 3.941 * 10-6 Siemens; the fourth conductance channel n = 4 had a conductance of (5.106 ± 0.131) * 10-6 Siemens, which was 1.115 standard deviations away from the expected value of 5.252 * 10-6 Siemens; lastly, the fifth conductance channel n = 5 had a conductance of (6.370 ± 0.144) * 10-6 Siemens, which was 1.354 standard deviations away from the expected value of 6.565 * 10-6 Siemens.

Figure 7: shows the amount of times different n-values were observed. The different wires are represented with different colors. There was a slight decrease as the amount of conductance channels increased.

Conclusion

Four different gold wires were stretched to the point of breaking and recombined multiple times. During the reformation of the gold wire, an oscilloscope was used to measure the voltage across a resistor. The voltage measured was then converted into the equivalent conductance through the gold wire. Quantum conductance was observed, and the different conductance channels were calculated. One conductance channel had a conductance of (1.339 ± 0.173) * 10-6 Siemens, which was 0.150 standard deviations away from the expected value. Two conductance channels had a conductance of (2.672 ± 0.264) * 10-6 Siemens, which was 0.174 standard deviations away from the expected value. Three conductance channels had a conductance of (3.943 ± 0.149) * 10-6 Siemens, which was 0.013 standard deviations away from the expected value. Four conductance channels had a conductance of (5.106 ± 0.131) * 10-6 Siemens, which was 1.115 standard deviations away from the expected value. Five conductance channels had a conductance of (6.370 ± 0.144) * 10-6 Siemens, which was 1.354 standard deviations away from the expected value. These values illustrate that our model described quantum conductance effectively. In addition to the observed quantum conductance, it appeared that odd parity conductance channels were more common than even parity conductance channels. This observation, however, is rather weak, and should be the subject of further experimentation. There was a variety of problems within this experiment that could be improved upon. The biggest cause of concern is the transmission coefficient, which was determined to be 1 / 59. The transmission coefficient is equal to

T = l / (l + L) (11)

where l is the mean free path of an electron in gold and L is the length of the gold wire constriction. The mean free path of an electron in gold is 37.7 * 10-12 m [3]. This can then be inserted into equation 18, where the constriction length can then be calculated. The constriction length is calculated to be 2.187 * 10-9 m. The Van der Waals radius of gold is 166 * 10-12 m. Dividing the constriction length by the radius of gold returns the length of the constriction. Our transmission coefficient corresponds to a constriction length of approximately 13 atoms. This seems like a rather large number of atoms, as previous research in quantum conductance found that the transmission coefficient could be ignored due to the constriction length being only 1 or 2 gold atoms long. A very likely cause for the large constriction length can be attributed to the way the wires were notched. Although the notch was meant to be small, the four wires were all notched severely enough that they ended up breaking completely when stretched.

References

[1] Erickson, Kaj; Tanyavong, Hamlet Spring 2018, ‘Investigating Quantum Conductance in Gold Wire’, University of Minnesota Twin Cities, https://sites.google.com/a/umn.edu/mxp/student-projects/spring-2018/s18_investigating-quantum-conductance-in-gold-wire

[2] Tolley, R; Silvidi, A; Little, C & Eid K.F. 20 December 2012, ‘Conductance quantization: A laboratory experiment in a senior level nanoscale science and technology course’, American Journal of Physics, 81, 14, 2013

[3] Gall, D 23 December 2015, ‘Electron mean free path in elemental metals’, Journal of Applied Physics, 119, 085101