Quantised Conductance

Transcription

1 Quantised Conductance Introduction Nanoscience is an important and growing field of study. It is concerned with the science and applications of structures on a nanometre scale. Not surprisingly the confinement of particles, usually electrons, on this scale leads to quantum affects. The electrons may be confined in 3D as in a "box", in 2D as in a plane or in 1D as in a wire. If the 3D "box" is made small enough it is usually called a "quantum dot" and is considered to be the zero D case. In this experiment you will examine the effects on the conduction of electrons through a narrow metal wire as the diameter of the wire is made very small. There are two main approaches to making nanometre scale structures -- start with a larger scale structure and remove material until the desired size is obtained or grow the structure from scratch. Both of these approaches require elaborate apparatus unsuited to an undergraduate laboratory. Instead, in this experiment you will use a much cruder but very simple method to produce a very fine wire and that is to just open a relay. The relay is a switch consisting of two metal pads that can be in contact or separated. As the switch is opened one or several "necks" may form. For a brief period, before it breaks, of the "neck" is in effect a very narrow wire and may be expected to display quantised conductance. Theory of quantised conductance in a 1D wire. Energy levels. Suppose the wire has a square cross-section of side a and has length L as shown in figure 1 with a<< L. Figure 1 The size and shape is such that the electrons are confined only in the xy plane. The potential in this plane may be considered as an infinite square well and hence the allowed electron energies,e, are given by E = (n x 2 + n y 2 ) ħ 2 π 2 /( 2ma 2 ) + ħ 2 k z 2 /2m where n x and n y are integers with a minimum of 1(why not zero?)

2 Figure 2 shows these energy levels but not to scale as the levels are not equally spaced. Figure 2. Several points are worth noting about this figure 1. Each curve corresponds to given value of n x 2 + n y 2 and is sometimes called a sub-band or channel. 2. For a given k z the spacing of the two lowest sub-bands is 3ħ 2 π 2 /( 2ma 2 ) (derive this). Show that for a 1nm the spacing is 0.12 ev and therefore much greater than kt at room temperature. Transitions between sub-bands are therefore very unlikely for very thin wires and so the electrons can be treated as existing in independent sub-bands. 3. The wire has a finite length and therefore the allowed values k z are not continuous but discrete as indicated by the dots in figure 2; however the energy difference between adjacent states in the same sub-band is much less than that between the sub-bands. 4. At T=0K all states below E F are filled and all above are empty. In figure 2 only the states in N sub-bands are occupied. Quantised conductance. Suppose that the wire is connected at each end to an electron reservoir (bulk metal) as shown in figure 3. Figure 3 Suppose further that a potential difference V is applied as shown with reservoir 2 earthed. If for simplicity we assume that all the potential drop occurs at end 1 then the energies of all electrons at that end will have their energies raised by ev. Electrons can enter the wire only into appropriate empty states with energies in the range E F to E F + ev. For the 1D wire the number of states per unit length per sub-band per unit energy range at E F for both spin directions, ie the density of states, D(E F ), at E F is 2/(π ħv F ) where v F is the Fermi velocity and ½ mv F 2 = E F.

3 If the length of the wire is much less than the electron mean free path then the electrons travel through it without scattering ie ballistically. In this case, if states in N sub-bands contribute to the current, I, then I = ½ e 2 V v F N D(E F ) (why?) so I = (2e 2 N / h ) V The conductance, G (=I/V), is therefore given by G= 2e 2 N / h = NG o The conductance is therefore independent of the length of the channel (in the above limit) and is quantised in units of 2e 2 / h ( work out this value in (ohm) -1 ). As mentioned earlier, in your experiment you will form the 1D wire by opening the separating te contacts of a relay. As the contact spacing is increased the wire will become narrower and hence the spacing of the sub-bands will increase; this in turn will reduce the value of N and so the conductance should decrease in steps of size G o. The aim of your experiment is to observe and record these steps and check whether their size is G o or an integer multiple thereof. Procedure Use the circuit as shown in figure 4. Figure 4 R1 is the resistance between the two contacts of the relay. OP37 is an operational amplifier. Show that, for the above circuit, G = 1/R1 = - V out / ( R2 V in ) Note the values of V, R2, R3 and R4.Open the relay contacts and use the digital scope to capture the output voltage transient; it should resemble that shown in figure 5.

4 Figure5 Figure 6 The procedure for transferring the scope data to the computer is described in the Appendix; read this before taking the data. Print out one or two of the transients and calculate, and mark on the trace, the size of the steps in units of G o. You will probably find that many of the steps do not correspond to integer values of G o ; gives possible reasons for this. You will also find that successive transients are not the same why? To have confidence that one or more of the steps do occur at integer G o values record many transients and use the PC to calculate and plot a histogram, as illustrated in figure 6,of the number of steps as the function of step size. Summarise your results and give your conclusions. APPENDIX How to transfer an Oscilloscope trace to the Computer: Switch computer on first, LEAVE OSCILLOSCOPE OFF, log on with your trinity username and password. When bootup is complete, switch on oscilloscope. To check settings are correct on the oscilloscope press the print/utility button on the main controls. Select the RS-232 option under the display. The following settings should appear: Connect to Factors Resolution Baud Handshake Computer Off Low 9600 DTR Once the trace has been triggered and captured on the oscilloscope screen (press stop button on the top left of oscilloscope controls if necessary) it can be imported to the computer as follows:

5 Click the Quantum folder icon. Make a new folder under your name (check no one else has a folder with the same name!) eg. joe Now go back to the desktop and run the QC acquisition icon. You will be asked for the name of dat file : PUT YOUR FOLDER NAME HERE(not the filename!!) You will then be asked for the number of the file: put in 0 for your first file etc. The data file will in fact be called joe1.txt (not joe0.txt!) and is located at: C:\Quantum\Joe\Joe1.txt [This file contains two columns with the x and y data corresponding to each data point measured on the oscilloscope.] [Keep a careful record in your lab manual of which files have captured useful data.] An image of the screen capture should appear. You cannot print from here. Exit the screen and run the easyplot program. 1. Select file. 2. Click open 3. Type the appropriate pathname: example C:\Quantum\Joe\Joe1.txt 4. Rescale / label axes as desired by double clicking on axis lines 5. Now check that the printer switch box beside the printer is set to your experiment ( Quantized conductance ) then select file followed by print to print. Only print one or two of the oscilloscope traces from which to calculate the step size in units of G O there is no need to print them all. Analyzing the data: To analyze the files: 1. Run QC Process. 2. Enter the folder name into the box for nom du fichier 3. Enter the file number into the box for numero coubre 4. Select Keep for Histogram 5. Open the next file you wish to include in histogram and again click Keep for Histogram 6. When all files to be included have been processed click Histogram 7. Click Save Histogram It will be saved into the folder C:\Quantum\Joe with the name Joe_hist.txt. Use easyplot to view and print the histogram.