S14QuantizedNatureofConductance

Testing the Quantized Nature of Conductance

Abstract

We worked to measure and characterize the quantum nature of electrical conductance in gold nanowires stretched to atomic diameters. We expected to see a relationship such that the conductance was in discrete multiples of G=2*n*e^2/h where n=2*w*/lambda. Our results show that for expected quantum numbers of n=1,2and 3 our experimental data was .99 +/- 1.52x10^-4, 1.84 +/- 2.53x10^-4 and 3.03 +/- 4.23x10^-4 which were off by 53.5, 625.1 and 74.3 sigma respectively

Introduction

The concept of a current as flowing electrons is something and understood in the realm of physics, especially on the large scale. Creating a difference in electrical charge causes electrons to flow across the difference, which results in a current of electricity. However, most often we consider many electrons moving in through a wire or material that is far larger than they are, which makes changes in current effectively continuous. Since we understand that this current is a flow of individual electrons it must be though that there is discretization of the charges if observed on appropriate scales. As was mentioned above, most experiments are on a large scale relative to the electrons, and the result is that this phenomenon hasn't been widely explored.

The goal of this experiment was to observe changes in the current through a gold wire as it decreased in size from the nanometer scale to the Angstrom scale, at which point the size of the wire should have become comparable to the wave function of the electrons. This changed the electrons' ability to flow from a group scale, with many flowing at once, to an individual scale, where very few electrons can flow through a given area. By observing the behavior of the current through this region and the voltage across it as the wire became increasingly narrow we attempted to characterize the quantized nature in which the electrons flowed.

This experiment is a good one to perform at the undergraduate level, especially for anyone interested in quantum or solid-state physics, as the physics of it are fairly straightforward but still quite interesting. It is also less complicated in setup and working than many other potential experiments, although it was found to be difficult to troubleshoot, as will be discussed below.

Theory

When the electric potential in a region is non-zero the electrons present begin to accelerate opposite the direction of the electric field. The exhibited average velocity, or the drift velocity, is:

vd=e*E*m (1)

where e is the electron charge, E is the electric field, is the average time between electron collisions and m is mass.[1]

This leads directly to the classical notion of the electrical current density which can be described by Ohm's law. Consider the equation for current:

I=(delta)Q/(delta)t=e*N*A*vd (2)

Where I is the current defined as the change in charge, Q, over the change in time, t. This can be broken down as the product of the electric charge e, electron density N=nV, area A, and the drift velocity. Substituting drift velocity from equation (1) into equation (2) and dividing by area gives us the current density

J=I*A=(e^2)*N*E*m=(sigma)E (3)

where (sigma) is now the conductivity. These characteristics are also known as the Drude Model.[1] This is of interest to us since we can then find the conductance of a wire to be:

G=(sigma)*A/L (4)

which is dependant on the cross sectional area, A, and the length, L, of the wire. This method starts to break down as soon as the geometry of the wire approaches nanometer-scales, where the conductance of the wire is no longer dependent on the length.

The above idea can be broken general observations into two specific categories. The classical limit (also known as the diffusive limit) and the quantum limit, each defined by the size of the conductor. If we define the mean-free-path of an electron to be l, The width as w and the length as L, each of these categories is then split by the following, respectively: L,w >> l and w~(lambda)F,L << l, where (lambda)F is the fermi wavelength of the electron.

F.1: a) Classical depiction of a wire where the length and width of the wire is far larger than the mean free path of the electron. b) Quantum case where the width of the wire is now comparable to the debroglie wavelength, conductance here can only be characterised using quantum relationships. [1]

In the first category, the classical limit, the length and width of the wire is far greater than the mean-free-path of the electron. This means that the electron can scatter as it usually does inside the conductor and classical methods can be used to characterise the conductance.

The final configuration sets the width of the wire comparable to the de Broglie wavelength of the electron. This is when we must consider the wave nature of the electron and is purely quantum. This is similar to the one dimensional infinite square well of width w. The relationship to the width and the wavelength of the particle is:

(lambda)n=2*w/n (5)

Where is the debroglie wavelength and it is quantized. Using this same argument, the smaller the width of the wire becomes, the more restricted the electrons become. Similar to the allowable wavelengths in a well, the wire begins to only allow a certain number of electrons and thus we can define these as conduction channels which are analogous to different energy levels in a well and we see quantization in conductance. If all the states below the Fermi energy are occupied, the shortest De Broglie wavelength is fixed at the fermi wavelength:

(lambda)F=h/sqrt(2*m*(eF)) (6)

Where h is Planck's constant and (eF) is the fermi energy. Using equations 5 and 6, we can then find the number of conduction channels n [1]:

n=2*w/(lambda)F (7)

Which is quantized since the allowable wavelengths are also quantized.The Fermi energy is defined by the highest energy level of the electrons in a metal at 0 Kelvin, so as we are at room temperature for this experiment all states at and below the threshold are occupied. This allows us to make the assumptions necessary to arrive at the criterion for conduction channels stated above [2].

We can see that as we decrease the width, we are essentially decreasing the amount of conduction channels that are available for the electrons to travel. In a gold nanowire, this can be reduced down to one atom and only one conduction channel is then available. The expression of current in one conduction channel is expressed as[3]:

integral(Ik)=2*e* [integral from 0 to eV](Vk(epsilon)[rho(kL(epsilon))-rho(kR(epsilon))]depsilon) (8)

Vk is the fermi velocity in channel k. epsilon is the one-dimensional density of states.kL(epsilon) is the density of states below the fermi energy level and kR(epsilon) is the density of states above the fermi energy level. The density of states can be simplified as[3]:

rho=swrt(m/2*h^2*epsilon)=1/h*v ; epsilon<epsilonF (9)

rho=0 ; epsilon>epsilonF

Using this with equation (8), we can simplify our current expression for one conduction channel as:

Ik=2*e^2/h (V) (10)

This expression can then be generalized for n number of conduction channels and dividing by voltage will give us our equation for conductance:

Gn=2*n*e^2/h (11)

We can see that the conductance is quantized and as the width of the wire becomes smaller n changes in integer steps from equation (7) and therefore changing the conductance in discrete values.[1]

We measured both the classical case and the purely quantum case.

Setup

A relatively simple setup is required for this experiment. Our first setup was as follows. We began with a moveable stage with a pin on the top, and one post on either side of it. A thin piece of flexible steel with a gold wire epoxied down on front of it was placed in front of the pin and against the posts. The stage moved forward so that the pin flexed the steel to stretch the gold wire slowly. A current was passed through the wire and the voltage across it was measured out through a DVM and recorded. However, there were many difficulties with this setup, and we were ultimately forced to make several changes.

The final setup we used was even simpler. A current was passed through the wire as before and the voltage was read into the DVM, but this time it fed into an oscilloscope that was set to trigger when the wire broke and save the graph of voltage values. To stretch the wire we actually just held the steel in hand and flexed it gradually so that it stretched and eventually broke. We had previously attempted many different ways to contact the gold to the DVM and read out values, but all of them proved unsuccessful. Kurt Wick then supplied us with indium to flatten out and epoxy onto the steel at the ends of the wires. With this change we were finally able to consistently read values to the DVM, and therefore to the oscilloscope. The graphs on the oscilloscope were saved as .csv files so the data could be taken from Excel files and analyzed in Matlab.

Results

Analyzing through Matlab we were able to obtain quantum numbers or channels of .99 +/- 1.52x10^-4, 1.84 +/- 2.53x10^-4 and 3.03 +/- 4.23x10^-4 for n = 1,2 and 3 respectively. As soon as we exported our data into Matlab, we converted the o-scope voltage reading to a conductance reading. This conductance was found using the equation that related with out circuit voltage as well as its resistance. Through a simple voltage divider and we were able to find the conductance as:

G=I/V=(Vb-Vw)/(Vw*Rext) (12)

Where Vb is the power supply voltage. Vw is the voltage reading from the o-scope and Rext is the external resistor. This conductance was then plotted on a histogram seen below. We further analyzed the peaks and determined where they were as a function of the quantum numbers or channels.

Our data ended up being 53.5, 625.1 and 74.3 sigma off respectively. The large source of error is partly due to lack of time and the acquisition of data only beginning late into the project. Another likely source of error would take into account the increasing resistance of the gold wire when the wire is being stretched to such small scales. One possible solution to the deviation would be to look at the resistance just before the wire breaks and see if it becomes comparable to the 100K resistor that is in series with it. If this holds true, our results would need a correction factor and this will perhaps give better results.

Suggestions for future groups

There were many difficulties in our attempt at this experiment, to the point that we only became able to get data as we approached the very end of the project. The usage of an oscilloscope greatly increased our ability to actually read values, as did the addition of the indium dots, and both of these should be considered for future projects. A suggestion our advisor had at one point was to get a wire puller and use that to stretch the wire, as that is the express purpose of the machine and would give consistent and easily controllable stretching that would greatly simplify things from the start. Using Matlab to analyze the data is a great idea, although we have run into a few difficulties with doing so (we believe it's due to the quantity of data in some runs), so that's something to be aware of.

Conclusion

Once it worked this was an interesting experiment, and one that we would definitely suggest future groups consider. If you do so, please keep in mind the difficulties that we had and use some of the fixes that we used or suggested. Far more of the semester was spent trying to figure out what to change to try and get data than actually getting any successfully. Kurt is a fantastic resource, as he was the source of the fix that ultimately allowed us to get results.

-- Main.rich0934 - 06 May 2014