1

Quantum Conductance

A Major Qualifying Project Report

submitted to the Faculty of

WORCESTER POLYTECHNIC INSTITUTE

in partial fulfillment of the requirements for the

Degree of Bachelor of Science

By

_______________________________

Christopher Bruner

Date: April 24th, 2008

Project number: NAB MQP8

_______________________ ________________________

Professor Nancy Burnham Professor Rafael Garcia

2

Abstract

Quantum conductance is the observed change, in discrete steps, of the electrical conductance

in an electrical element. The purposes of the project are to build a circuit similar to Foley et al.

to show quantum conductance, to build a display for the circuit to show future students quantum

conductance, and to design a laboratory experiment for PH 2651 so that students can repeat the

experiment for themselves. The histogram of the conductance steps showed that majority

occurred in integral multiples of the quantum conductance, which is 52

10*75.72 h

e siemens,

but half and quarter multiple steps were also seen. Due to time constraints, the designing of the

PH 2651 laboratory experiment was not done.

3

Introduction

Electrical conductance is a measure of how easily electricity flows along a certain path

through an electrical element. It is the inverse of electrical resistance, which is a measure of the

degree to which an object opposes an electrical current.

Quantum conductance is an observed change, in discrete steps, of how well a material

can pass electrons. The conduction steps are thought to be caused by one of two ways. The first

idea is that the discrete nature of the contact of a few atoms in a nano-sized contact and the

rearrangement of the atoms caused by that contact makes a change in the force in the contact,

and thus the conductance steps are related to the force change. The second idea is that the contact

acts as a waveguide for electrons to travel through and as the size of contact changes, the number

of allowed modes jumps in integral steps. It has not been confirmed which idea is the correct

one and there is much controversy among physicists on what the true cause is. But which ever is

the correct idea, this phenomenon is fundamental in that it is a property of nanostructures that

can be observed at room temperature. An ideal graph of quantum conductance is shown in

figure 1.

4

Figure 1: This is the idealized pattern of the conductance steps

There were three goals in my project. First was to build a circuit to show Quantum

Conductance using a setup done by a group at the University of Massachusetts at Amherst (Foley

1999). Second was to put my circuit into a display in order for people to see quantum

conductance in action. The third goal was to design a laboratory experiment for PH 2651 so that

future students could build circuits to show quantum conductance.

Classical conductance was formalized from the idea of an electric current made from a

voltage potential, making the conductance equal to the current divided by the applied voltage.

This formula has no bearing on the dimensions of the electrical element. However, when the

size of the electrical element approaches nano-sized, the conductance becomes dependent upon

the dimensions of the element and the classical formula is replaced by the Landauer Formula.

The Landauer Formula is

2

1,

0

cN

mn

mntGG (1)

(Tao, 2005) with G being the conductance, Nc the total number of channels, and tmn the

transmission coefficient of channel m with respect to n, and G0 is 52

10*75.72 h

e siemens.

5

The value of G0 comes from the fact that the electron wave can only pass through the

constriction if it interferes constructively, which for a given size of constriction only happens for

a certain number of modes N. The current carried by such a quantum state is,

)(****2 21 nnFn EgveI (2)

where e is the electric charge, vFn is the group velocity at the Fermi level n, gn(Ef) is the density

of the state n, which equal Fnvh *

1, and (μ1 – μ2) is the chemical potential energy across the

contact (Ott, 1998). Then by dividing by the voltage across the circuit which equals (μ1 – μ2) / e,

we get the value of G0 from above.

6

Literature Review: Quantum Conductance

Quantum conductance was first observed in an experiment measuring the electron

properties of a two dimensional electron gas (van Wees, 1988). After the initial discovery,

research has continued on quantum conductance in various materials and using different methods

and continues to be an important topic in the areas of quantum mechanics, nanochemistry, and

electrochemistry. It could even lead to a method for finding contaminants in materials (Li, 2000),

and for molecular electronics (Yoon, 2003).

Quantum conductance has also been observed in carbon nanotubes. Many papers, both

theoretical (Lu, 200) (Lu, 2005) (Sanvito, 2000) (Yoon, 2003) and experimental (Frank, 2007)

(Urbina, 2003), have been submitted. The theoretical papers show that the conductance through

a nanotube is dependant only on the boundary of the nanotube with the voltage source or the

boundary with another nanotube with a different configuration. The experimental papers show

the conductance steps at integer values of G0 or at half G0.

In fact, there are still arguments on whether or not quantum conductance is an actual

physical phenomenon or a consequence of the atomic arrangement and rearrangement of the

contact where the conductance is being measured. This chapter examines literature on research

in quantum conductance.

There are various methods for measuring quantum conductance in many different

materials, each with advantages, disadvantages, costs, and conclusions. This review will

describe in detail each method, and the materials that can be used for each method.

Two Dimensional Electron Gas

7

As mentioned above, the first system for showing quantum conductance is the two

dimensional electron gas. Electron leads were put onto a material, namely GaAs-AlGaAs, at

0.6K, the voltage between the leads was varied, and the conductance was measured (van Wees,

1988). It was found that the conductance varied as nearly discrete jumps of G0 as the voltage

changed. At the time, it was hypothesized that that discrete jumps were seen as a special case of

the Landauer formula, see equation 1, with no mixing of electron transmission channels. Being

the first system to show quantum conductance, it has no real advantages in either showing the

cause of quantum conductance, or in the cost of setting up the experiment, because the

experiment is done at 0.6K: all it shows is the conductance steps can be seen.

Wire to Wire Contact

The next method to measure quantum conductance is the wire contact method. It consists

of setting a voltage potential between two wires and bringing them in and out of contact with

each other (Foley, 1999) (Costa-Kramer, 1995) (Ott, 1998, 1) (Ott, 1998, 2). See Figure 1 below.

Figure 2: Experimental setup of wire to wire contact system (Foley, 1999)

As the macroscopic wires come into contact, a nanowire is formed between them, and as

they move apart, the nanowire stretches as the atoms that make up the nanowire rearrange

themselves until a single chain of electrons forms and then the nanowire breaks. While the

rearrangement occurs, the conductance was shown to vary in steps of G0. As with the 2-D

8

electron gas system, there is little that can be drawn from this method in terms of cause of

quantum conductance; the movement of the wires is not on the microscopic level which would

be needed to see if the conductance jumps are due to atomic rearrangements or if it is a unique

phenomenon entirely. However, this method is highly cost effective, especially if you use a

manual method for bringing the wires in and out of contact and use batteries as the voltage

source (Foley, 1999). The setup is also relatively easy to construct (Ott, 1998, 2). The most

important and noteworthy feature of this method, however, is that it can be done at room

temperature. This is one of the few quantum mechanical properties that can be observed at room

temperature.

Compression and Elongation

The next method is the compression/elongation method. It consists of setting a voltage

potential through a tip and pressing the tip into a substrate and then pulling it away (Costa-

Kramer, 1997) (Cuevas, 1998) (Landman, 1996) (Krans, 1995) (Rubio, 1996) (Urbina, 2003).

See Figure 2 below.

9

Figure 2: Experimental setup of compression and elongation system (Costa-Kramer, 1997)

As the tip separates from the substrate, a nanowire is formed the same way as in the wire

to wire contact method. This is usually done with a scanning tunneling microscope (STM) so

that the motion of the tip can be controlled to within an error of a few hundred femtometers. The

advantage of this method is that you can begin to tackle the question of whether or not atomic

rearrangements are the cause of quantum conductance because the motion of the tip into the

substrate can be controlled; the question of atomic rearrangements is especially questioned in

(Costa-Kramer, 1997) as a Bi tip is displaced 80 nm, much larger than the average electron

spacing, and the conductance remained constant at G0. However, this method uses some

expensive equipment, such as the tunneling microscope, and these experiments are usually done

at near 0 K temperatures.

Mechanically Controlled Break Junction

The next method is the mechanically controlled break junction. It consists of

mechanically breaking a nanowire and measuring the conductance as it stretches (Waraich,

2006) (Krans, 1994) (Rodrigues, 2002) (Scheer, 1997) (Yanson, 1997) ((Zhou, 1995)). See

Figure 3 below.

Figure 3: Experimental setup mechanically controlled break junction system (Zhou, 1995)

10

A voltage potential is set across a wire that is placed onto a substrate and is then

mechanically bent at the center. As the wire stretches, the conductance is measured. It has the

advantage of being more accurate than any previously mentioned method because the stretch of

the nanowire can be controlled better than the compression and elongation method. This

improvement showed that there is mixing of conductance channels in the breaking, but the sum

was at integer values of G0 (Scheer, 1997). It was also the method used in (Rodrigues, 2002)

that showed for silver, the conductance was in agreement with theoretical predictions of the

relationship of conductance to atomic structure. The materials needed are much less expensive

than an STM, but more than the wire to wire method.

Electrochemical

The next method is the electrochemical method. This was the first method discovered by

chemists, not physicists. There are several ways of setting up this system (Tao, 2005). The first

way was to place an STM tip into a CuSO4 ion solution and vary the voltage potential to build

the copper nanowire (Tao, 1998); see Figure 5 below.

Figure 5: Experimental setup of electrochemical system number 1 (Tao, 1998)

The immediate advantage of this method was that there was considerably less motion of

the tip involved, making precise measurement taking easier. Also a specific nanowire formation

11

could be held for a long time. But soon after a second method was found that is even better.

This approach consisted of placing a wire in an ion solution, only exposed at the center of the

wire, and then electrochemically etching the wire (Bogozi, 1999) (Li, 2000) (Li, 2002) (Tao,

2005), as seen in Figure 6.

Figure 6: The second electrochemical setup Tao, 2005)

This approach eliminates the need for an STM tip and there is no motion of the contact.

One discovery found by this approach was that the placement of different impurities onto the

wire caused a shift in the conductance (Li, 2000). Soon after a third approach was discovered.

This one had the advantage of a self-terminating mechanism (Boussaad, 2002) (Tao, 2005) as in

Figure 7.

Figure 7: The third electrochemical setup (Boussaad, 2002)

12

This approach puts an electrode connected to an external resistor into an electrolyte and

placing a voltage potential across the electrode. This builds a specifically sized nanowire

dependant on the external resistor. This makes the wire making much easier.

Material Dependence

Each method can use various materials for the wires. The 2-D electron gas method must use a

material with one valence electron in order to make the electron gas. The wire to wire contact

method has been shown to work for metals, metallic glass, and metallic oxides. This method can

be made to work even on a frugal budget. The compression and elongation method works best

with metals because they compress and elongate better than most other materials. The

mechanically controlled break junction method also works best with metals for the same reason,

unless you want to see the change in conductance due to the actual break of the material (Ott,

1998, 1). The electrochemical method must use metals because it requires an ion solution.

Conclusion

In summary, recent research in quantum conductance is that it is a quantum mechanical

property that can be seen in various materials, at room temperature, and with macroscopic

substances. The debate of whether it is a true phenomenon or a consequence of atomic

configurations still goes on even today.

13

Experimental Chapter

One of the two main parts of this project was to build a display of a circuit that shows

quantum conductance. This display is similar to an experiment done by a group at the

University of Massachusetts at Amherst (Foley 1999). Figure 2 shows a diagram of the setup

and figures 3a through 3f are photos and diagrams of the display. The display that I made does

not have the voltmeter or the computer because the display is not to take measurements from, but

just to show the conductance jumps.

The circuit was built from a standard breadboard (see figure 3a).

Figure 3a: The display setup of the wire to wire contact system. The oscilloscope is not present in this

picture.

14

A 9V battery was used with a 10kΩ variable resistor, a 3.9 kΩ resistor, and a 50Ω resistor

to generate a voltage up to 100mV( see figures 3b and 3c)

Figure 3b: The voltage source. The black square is the variable resistor, the brown, striped cylinder is the

3.9kΩ resistor, the blue cylinder is the 47Ω resistor, and the red and black wires are the positive and

negative sides of the battery respectively.

Figure 3c: The variable resistor consists of a metal track with a lead on each end and a third lead with a

metal connector that can slide along the metal track. The track has a total resistance of a certain value and

the metal connector varies how much of the track’s resistance is used.

The current through the resistors and the gold wire setup (see figure 3d) was sent through

the LM358N operational amplifier where the current was converted to a voltage signal (see

figures 3e and 3f) that was then seen on the Tektronic TDS 210 digital oscilloscope.

15

Figure 3d: The gold wire is soldered to the copper wire split so there is no noise at the copper-gold

interface.

Figure3e: The current to voltage converter. The black square is the operational amplifier, the brown,

striped cylinder on top of the op-amp is the 100kΩ resistor, the yellow wire in pin 7 connects to the

oscilloscope probe, the blue wire on the left grounds the op-amp, and the red and black wires on the right

are the positive and negative sides of the battery respectively. The yellow wire in pin 6 leads to the gold

qire. The negaive lead of the battery must be grounded for the op-amp to work. The pin numbers are in

figure 3f; the top of the opamp in figure 3f is the left side of the op-amp in this figure

16

Figure 3f: The LM358N operational amplifier receives current from the inverting input where a voltage

potential is made on the non-inverting input and this voltage is then amplified and sent out the output. It is

powered by a 9V battery connected at the Vcc+ and Vcc

-.

(http://www.datasheetcatalog.org/datasheet/stmicroelectronics/2163.pdf)

There were several problems that were encountered in the making of this display that

anyone wanting to reproduce this circuit would run into. The first problem is connecting the fine

gold wire to the much thicker copper wire used in the circuit. The connection must be clean in

order to minimize voltage noise at the gold-copper boundary. I split the copper wire in half long

ways about 1cm, placed the gold wire in the slit, and soldered the split closed, as seen in Figure

4.

This was successful in making a clean interface.

The second problem was a faulty operational amplifier. The results can be seen in figure

5.

17

Figure 5: Results from faulty amplifier.

There is a shift from positive current to negative current that should not be there because

the current can vary only between 0 amps and a positive value, and there is also a linear increase

back to the positive side that could not be accounted for.

The third problem was shielding the circuit. By placing a cardboard box wrapped in

aluminum foil around my circuit and a sheet of aluminum foil grounded to the oscilloscope

under my circuit, the amount of external electrical noise in the circuit was greatly reduced. See

figures 6a and 6b for the comparison of before and after the box was put into place.

18

Figure 6a: Circuit noise before shielding Figure 6b: Circuit noise after shielding

The last problem was making the wires come in and out of contact. By placing the gold

wires very close to each other, tapping the table the circuit was on made the wire go in and out of

contact. This may need to be adjusted if the circuit is moved to a different surface.

This display can show quantum conductance, one of the few quantum mechanical effects

that can be seen at room temperature. If made like mine or the one in (Foley, 1999), it is an

inexpensive and effective way for undergraduates to show and begin to understand quantum

conductance.

19

Results

Figures 7a and 7b show representative samples of the conductance for copper and gold

respectively, all of the traces can be seen in the appendix. To get the conductance change

relative to G0 from the voltage change seen on the oscilloscope, divide the voltage change by the

resistance connected to the op-amp to get the current, and then divide that value by the driving

voltage of the voltage source to get the actual conductance change, finally divide that value by

G0 to get the relative conductance change. For example, for a measured voltage of 0.8V, a

driving voltage for 0.1V and resistance of 100kΩ, you would get a relative conductance change

of 1.04. Figure 8 shows a histogram of the conductance jumps. The histogram shows that the

majority of the jumps occur near the quantum conductance, but there are also half and quarter

integer jumps as well. Due to an error in Microsoft Excel, all of my histogram points are shifted

up by 0.3 G0.

Figure 7a: Representative sample of gold Figure 7a: Representative sample of copper

20

0

20

40

60

80

100

120

140

160

180

00.

20.

40.

60.

8 11.

21.

41.

61.

8 22.

22.

42.

62.

8 33.

23.

43.

63.

8 44.

2

Conductance Value (G/G0)

Fre

qu

en

cy

of

Va

lue

Se

en

Figure 8: Conductance Histogram

21

Conclusion

With the addition of a working op-amp, my circuit showed that most of the conductance

occurred in steps of the quantum conductance. Also with the addition of a more permanent

shielding box, the display will be complete. Due to time constraints, the designing of the PH

2651 laboratory experiment was not done.

Acknowledgements

I would like to thank Professor Mark Tuominen from the University of Massachusetts at

Amherst for showing me his circuit design, Professors Nancy Burnham and Rafael Garcia for

advising me during the project, and the DeGraf family for funding my project.

22

References

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a self-terminated electrochemical method, Applied Physics Letters, 80, 2398-2400.

Costa-Kramer, J. L., Garcia, N., Garcia-Mochales, P., Serena, P. A. (1995). Nanowire formation

in macroscopic metallic contacts: quantum mechanical conductance tapping a table top,

Surface Science, 342, 1144-1149

Costa-Kramer, J. L., Garcia, N., Olin, H. (1997). Conductance quantization in bismuth nanowires

at 4 K, Physical Review Letters, 78, 4990-4993.

Cuevas, J. C., Yeyati, A. L., Martin-Rodero, A., Bollinger, G. R., Untiedt, C., Agrait, N. (1998).

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Foley, E. L., Candela, D., Martini, K. M., Tuominen, M. T. (1999). An undergraduate laboratory

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389-393.

Frank, S., Poncharal, P., Wang, Z. L., de Heer, W. A. (1998). Carbon nanotube quantum

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23

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24

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25

Appendix

Copper Traces

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

1.6 16 160 2.07

0.8 8 80 1.04

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

0.8 8 80 1.04

1.6 16 160 2.07

26

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

0.8 8 80 1.04

1.6 8 160 2.07

0.4 4 40 .52

0.8 8 80 1.04

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

0.6 6 60 0.77

1.0 10 100 1.29

27

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

1.4 14 140 1.81

0.8 8 80 1.04

1.0 10 100 1.29

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

0.8 8 80 1.04

0.8 8 80 1.04

28

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

0.8 8 80 1.04

0.6 6 60 .77

0.8 8 80 1.04

0.8 8 80 1.04

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

0.8 8 80 1.04

1.2 12 120 1.55

0.8 8 80 1.04

29

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

0.8 8 80 1.04

0.4 4 40 0.52

0.6 6 60 0.77

0.8 8 80 1.04

0.8 8 80 1.04

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

1.0 8 80 1.29

1.2 12 120 1.55

0.8 8 80 1.04

0.8 8 80 1.04

0.8 8 80 1.04

30

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

0.8 8 80 1.04

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

1.2 12 120 1.55

1.2 12 120 1.55

0.4 4 40 0.52

1.0 10 100 1.29

0.8 8 80 1.04

31

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

0.8 8 80 1.04

1.2 12 120 1.55

0.6 6 60 0.77

0.6 6 60 0.77

0.8 8 80 1.04

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

2.0 20 200 2.58

0.8 8 80 1.04

32

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

0.8 8 80 1.04

1.6 8 160 2.07

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

0.8 8 80 1.04

0.8 8 80 1.04

0.8 8 80 1.04

33

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

0.8 8 80 1.04

0.6 6 60 0.77

0.6 6 60 0.77

0.8 8 80 1.04

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

0.4 4 40 0.52

1.2 12 120 1.55

0.4 4 40 0.52

34

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

1.6 16 160 2.07

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

2.0 20 200 2.58

2.0 20 200 2.58

0.6 6 60 0.77

0.6 6 60 0.77

0.8 8 80 1.04

35

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

0.8 8 80 1.04

0.8 8 80 1.04

1.2 12 120 1.55

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

0.4 4 40 0.52

0.4 4 40 0.52

0.8 8 80 1.04

0.8 8 80 1.04

1.2 12 120 1.55

0.8 8 80 1.04

36

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

1.2 12 120 1.55

1.2 12 120 1.55

0.4 4 40 0.52

0.4 4 40 0.52

0.4 4 40 0.52

0.4 4 40 0.52

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

1.2 12 120 1.55

0.8 8 80 1.04

37

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

0.8 8 80 1.04

0.8 8 80 1.04

0.8 8 80 1.04

0.6 6 60 0.77

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

2.0 20 200 2.58

0.4 4 40 0.52

0.4 4 40 0.52

0.4 4 40 0.52

0.4 4 40 0.52

0.4 4 40 0.52

38

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

1.2 12 120 1.55

0.4 4 40 0.52

0.8 8 80 1.04

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

0.8 8 80 1.04

0.4 4 40 0.52

0.4 4 40 0.52

0.4 4 40 0.52

39

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

1.6 16 160 2.07

0.6 6 60 0.77

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

0.8 8 80 1.04

1.2 12 120 1.55

0.8 8 80 1.04

40

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

1.2 12 120 1.55

0.8 8 80 1.04

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

1.6 16 160 2.07

0.8 8 80 1.04

0.8 8 80 1.04

41

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

0.4 4 40 0.52

0.8 8 80 1.04

0.4 4 40 0.52

0.8 8 80 1.04

0.4 4 40 0.52

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

0.8 8 80 1.04

0.8 8 80 1.04

1.2 12 120 1.55

42

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

0.8 8 80 1.04

0.8 8 80 1.04

0.4 4 40 0.52

0.8 8 80 1.04

0.4 4 40 0.52

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

0.4 4 40 0.52

0.8 8 80 1.04

0.8 8 80 1.04

43

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

2.0 20 200 2.58

1.2 12 120 1.55

0.8 8 80 1.04

0.8 8 80 1.04

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

1.2 12 120 1.55

1.2 12 120 1.55

0.8 8 80 1.04

44

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

2.4 24 240 3.10

1.2 12 120 1.55

0.8 8 80 1.04

0.8 8 80 1.04

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

0.8 8 80 1.04

0.8 8 80 1.04

45

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

1.2 12 120 1.55

1.2 12 120 1.55

0.8 8 80 1.04

0.8 8 80 1.04

0.4 4 40 0.52

0.4 4 40 0.52

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

1.2 12 120 1.55

1.2 12 120 1.55

0.8 8 80 1.04

0.8 8 80 1.04

0.8 8 80 1.04

0.8 8 80 1.04

46

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

1.2 12 120 1.55

0.8 8 80 1.04

0.8 8 80 1.04

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

1.2 12 120 1.55

0.8 8 80 1.04

0.8 8 80 1.04

0.8 8 80 1.04

47

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

0.8 8 80 1.04

0.8 8 80 1.04

0.6 6 60 0.77

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

0.8 8 80 1.04

1.2 12 120 1.55

0.8 8 80 1.04

0.2 2 20 .26

0.2 2 20 .26

0.4 4 40 .52

48

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

1.6 16 160 2.07

0.8 8 80 1.04

0.4 4 40 .52

0.8 8 80 1.04

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

1.6 16 160 2.07

0.8 8 80 1.04

0.6 6 60 0.77

0.8 8 80 1.04

49

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

1.2 12 120 1.55

0.8 8 80 1.04

0.8 8 80 1.04

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

2.0 20 200 2.58

0.8 8 80 1.04

50

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

0.8 8 80 1.04

0.8 8 80 1.04

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

2.0 20 200 2.58

1.6 16 160 2.07

0.8 8 80 1.04

0.8 8 80 1.04

0.8 8 80 1.04

1.6 16 160 2.07

51

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

1.6 16 160 2.07

0.6 6 60 0.77

0.8 8 80 1.04

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

0.8 8 80 1.04

0.8 8 80 1.04

52

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

1.6 16 160 2.07

0.8 8 80 1.04

0.4 4 40 .52

0.8 8 80 1.04

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

0.8 8 80 1.04

0.8 8 80 1.04

0.8 8 80 1.04

0.8 8 80 1.04

0.8 8 80 1.04

53

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

0.8 8 80 1.04

1.2 12 120 1.55

54

Gold Traces

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

2.0 20 200 2.58

0.8 8 80 1.04

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

0.8 8 80 1.04

1.2 12 120 1.55

55

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

1.2 12 120 1.55

1.2 12 120 1.55

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

0.8 8 80 1.04

0.8 8 80 1.04

56

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

0.8 8 80 1.04

1.4 14 140 1.81

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

0.8 8 80 1.04

0.8 8 80 1.04

1.6 16 160 2.07

57

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

0.8 8 80 1.04

0.8 8 80 1.04

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

0.6 6 60 0.77

0.8 8 80 1.04

0.4 4 40 0.53

0.6 6 60 0.77

58

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

1.2 12 120 1.55

1.6 16 160 2.07

0.8 8 80 1.04

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

0.8 8 80 1.04

0.8 8 80 1.04

1.6 16 160 2.07

59

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

0.8 8 80 1.04

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

0.8 8 80 1.04

60

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

0.8 8 80 1.04

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

0.8 8 80 1.04

0.8 8 80 1.04

0.8 8 80 1.04

0.4 4 40 .53

0.8 8 80 1.04

61

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

0.4 4 40 .53

0.8 8 80 1.04

0.4 4 40 .53

0.4 4 40 .53

0.8 8 80 1.04

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

1.2 12 120 1.55

0.8 8 80 1.04

62

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

0.8 8 80 1.04

1.6 16 160 2.07

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

1.2 12 120 1.55

2.0 20 200 2.58

63

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

0.8 8 80 1.04

0.6 6 60 0.77

0.6 6 60 0.77

0.8 8 80 1.04

0.8 8 80 1.04

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

1.2 12 120 1.55

0.8 8 80 1.04

64

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

2.4 24 240 3.10

0.8 8 80 1.04

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

1.6 16 160 2.07

0.8 8 80 1.04

0.8 8 80 1.04

65

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

1.6 16 160 2.07

0.8 8 80 1.04

0.8 8 80 1.04

0.4 4 40 0.52

1.2 12 120 1.55

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

2.0 20 200 2.58

0.8 8 80 1.04

2.4 24 240 3.10

66

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

3.2 32 320 4.13

0.4 4 40 0.52

0.8 8 80 1.04

0.8 8 80 1.04

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

0.8 8 80 1.04

0.4 4 40 0.52

0.8 8 80 1.04

0.4 4 40 0.52

0.8 8 80 1.04

67

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

1.4 14 140 1.81

0.8 8 80 1.04

0.8 8 80 1.04

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

0.8 8 80 1.04

0.8 8 80 1.04

1.2 12 120 1.55

0.8 8 80 1.04

0.4 4 40 0.52

68

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

0.8 8 80 1.04

0.8 8 80 1.04

0.4 4 40 0.52

0.4 4 40 0.52

0.8 8 80 1.04

0.8 8 80 1.04

0.4 4 40 0.52

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

0.8 8 80 1.04

0.8 8 80 1.04

1.2 12 120 1.55

0.8 8 80 1.04

0.8 8 80 1.04

69

Voltage change

(V)

Current Change

(μA)

Conductance

Change (μS)

Relative

Change

1.2 12 120 1.55

0.8 8 80 1.04

0.8 8 80 1.04

0.8 8 80 1.04