Laboratory experiment on quantized conductance Autumn 010 IH654 Nanoelectronics Ilse de Moffarts April 010 1
1. Introduction The first experiment on ballistic transport goes back to 1965, when Yuri Sharvin (Moscow) used a pair of point contacts to inject and detect a beam of electrons in a single-crystalline metal. In his experiment he used a two-dimensional electron gas in a GaAs-AlGaAs heterojunction to create a quantum point contact. In 1988, the Delft-Philips and Cambridge groups reported the observation of a sequence of steps in the conductance of a constriction in a D electron gas, as its width W was varied by means of the voltage on the gate. The steps are near integer multiples of e²/h 1/13k. A far more simple experiment however, was proposed by Costa-Krämer et al [1] in 1995, placing in contact two metallic wires and separating them, while forming a ballistic nanowire in between. It is on the latter experiment that this lab is based. 1.1 The physics of quantized conductance Let us first consider the conductance of a one-dimensional wire of length L. The conductance is the reverse of the resistance and is equal to the current divided by the voltage I G 1 (1). R V The current is given by the following formula ven I () L for which v is the velocity of the electrons, e is the elementary charge (= absolute value of the charge of an electron) and N is the total number of electrons that contribute to the current. The drop in potential energy for 1 electron going from one end of the wire to the other is E ev (3). Substituting both equation () and (3) in equation (1), gives ve N G (4). EL The problem reduces now to computing the number of contributing electrons N. At each of the two terminals (call them A and B), which are connected to the two ends of the one-dimensional wire, the quantum states are filled up to the Fermi energy, E FA respectively E FB, with two electrons per state (Pauli exclusion principle). At terminal A, the Fermi energy level is higher by E due to the applied voltage and this causes the current flow. So from terminal A all the electrons that occupy the states in a range E below E FA, can flow to unoccupied states in terminal B. The number of contributing electrons N equals twice the number of quantum states S in [E FA - E, E FA ]. Recall from chapter 1, section 1.7, that for particles in a one-dimensional box of length L, the wavenumber is restricted to values n k n n 0, 1,... (5) L and thus the de Broglie wavelength of an electron can only take on discrete values L n n n 0, 1,... (6). The formula of the velocity of an electron is
p h v (7), m m with m the effective mass of an electron, so the electron velocity in a one-dimensional wire can only have the following discrete values nh v n n 0, 1,... (8). Lm The number of electrons N, which occupy the S states in [E FA - E, E FA ], travel at different velocities depending on the quantum state. This velocity range ( x S) h xh Sh (9) v vx S vx Lm Lm Lm converts to a range of kinetic energy m v v vsh E kin mv v (10). L The potential energy E, provided to the system by the applied voltage, equals the kinetic energy E kin of the electrons that propagate through the wire. Isolating the number of quantum states S from (10), results in EkinL EL EL S (11) N S (1). vh vh vh Substituting the number of contributing electrons N (1) into equation (4) immediately gives e G (13). h Now consider the conductance of a nanosized contact, like we will make in this experiment. As seen in Figure 1, a nanosized contact acts as a ballistic conductor, for which the scattering of electrons at the sample boundaries limits the current, rather than impurity scattering. Although one might expect the current I to be infinitely large in the ballistic transport regime, it is actually finite, because electrons are scattered back at the entrance of the constriction. The nanocontact acts like a waveguide for electrons, with a finite, integer number of occupied modes above cutoff, that are able to propagate *. As the size of the contact is increased (decreased), the number of allowed modes increases (decreases) discontinuously. For each allowed mode, the nanosized contact acts like a one-dimensional wire with conductance G = e /h. Figure 1 Electron trajectories characteristic for diffusive (l < W, L), quasi-ballistic (W < l < L) and ballistic (W, L < l) transport regimes with l the electron mean free path, W the width and L the length of the channel * This is a rather rough argument. For a rigorous explanation read C. W. J. Beenakker and H. van Houten, Quantum Transport in Semiconductor nanostructures, Solid State Phys. 44, 1-8 (1991). 3
1. The experimental set-up The goal of this lab is to observe the quantization of electrical conductance G in multiples of e /h with a simple, room-temperature experiment (Figure ). The set-up consists of two metal wires that are gently pulled out of each other. A constant voltage in the range of millivolts is applied to one of the wires. The other wire is connected to the virtual-ground input of a transimpedance amplifier (current-to-voltage converter). Finally the output voltage of the op amp is monitored with a digital oscilloscope. - 9 V R 1 Gold wires R F R R 3 + 9 V - R 4 + OP-07 Oscilloscope - 9 V Figure Experimental setup. Two 9 V batteries are used for the voltage source and the current to voltage converter. The op amp type OP-07 is selected for its low noise and high input impedance. The critical part of the experiment is when the metal wires are pulled out of each other. At this moment a nanocontact will form between the two macroscopic wires (Figure 3), with decreasing diameter as they are pulled further out of each other, causing quantized steps in the conductance. It appears that gold wires give the best results, probably due to the high malleability (~ softness) and the freedom from surface oxidation of gold. Figure 3 Illustration of the formation of a nanocontact between two gold wires, in the extreme case for which the contact only exists of a few bridging atoms 4
. Preparation tasks In order to understand the lab setup, calculate the answer of the following questions and write the results in the first column of Table 1. Use the scheme of Figure with following values: R 1 = 10 k, R = 3.6 k, R 3 = 690, R 4 = 49.9 and R F = 0 k. 1. Calculate V, the voltage over resistor, and V 4, the voltage over resistor 4, which is the input voltage of the nanocontact.. Calculate the value of one conductance step. Using the input voltage calculated in the first question, calculate one quantized current step. 3. Consider the transimpedance amplifier. The input current flows through the feedback resistor, resulting in an output voltage V out = - R F * (-I in ). Calculate one voltage step. Table 1 Comparison between expected calculated values and experimental measured values Expected values Measured values V = voltage over R V 4 = voltage over R 4 = input voltage (when the gold contact is broken) 1 conductance step 1 current step I in 1 voltage step at output of opamp V out 5
3. Laboratory Part I: Building and testing the circuit First, let us construct the first part of the circuit to obtain a suited input voltage for the gold wire nanocontact. For the voltage supply, two batteries of 9 V are used to reduce the noise. By connecting high resistance to the battery, the current flow is reduced. We want to obtain 0-40 mv as input voltage; much larger voltages would potentially lead to electron-heating effects whereas much smaller voltages would make the steps too shallow compared to the noise. The building blocks you have to make the circuit according to Figure are: 9 V batteries, connectors for the 9 V batteries, a solderless breadboard (Figure 4) with jump wires and 4 resistors (Figure 5). 1 Figure 4 Solderless breadboard (1) 6 interconnected round tie points per row in horizonal array () power bus: of 4 each of interconnected tie points with black line printing, 4 of 1 each of continual interconnected tie points with red line printing Figure 5 Color code for resistors http://www.standrews.ac.uk/~www_pa/scots_guide/info/comp/passive/res istor/colourcode/colourcode.html Make the circuit, measure the values for the first 7 rows of Table 1 and fill them in the second column. Now calculate with this measured voltage V 4 the expected voltage step. Secondly, make the amplifier circuit. Connect all pins of the op amp (except for offset null 1 & ) and implement the feedback path. Be sure not to touch any of the pins as this can generate voltage spikes which could damage the op amp! The correct functioning of the transimpedance amplifier can be tested with an impulse generator and the digital oscilloscope. Finally, connect the leads to a gold wires. Figure 6 Pin connections (top view) for OP07 from datasheet 6
4. Laboratory Part II: Digital oscilloscope and computerized data acquisition From the preparation task you should be able to select a suitable vertical scale setting for the oscilloscope to observe some conductance steps in the end of the transient. The optimal time base setting for the oscilloscope depends on how fast you pull the wires out of each other ( 0.1 ms/division). The oscilloscope must be set for normal- or single-mode dc triggering. Move the gold wires very slowly in and out of contact with each other and try to find the right settings for the oscilloscope to see the quantized steps. 1. Try to stop the oscilloscope at a nice steplike transient. It is preferred to have as less overshoot as possible. Save the data on the computer using the ScopeExplorer software. Repeat this 4 times.. Replace the gold wires by wires of a soldering alloy. This material is very soft and that makes it good for forming the nanocontact. What will be the effect on the value of the voltage step seen in the oscilloscope? 3. Repeat question 1 for the soldering wires. 5. Lab Report Instructions Each group member must write an individual report about this lab containing the following parts: 1. Introduction: containing a short introduction about the laboratory setup and the purpose of the lab. Experimental results: containing the 4 transients for gold wires and the 4 transients for soldering wires Additionally, you have to derive the actual voltage stepsize that we have observed in the oscilloscope. Do this for the first and the second conductance step by calculating the average of the zero, the first and the second level and subtracting them from each other. Compare this experimental result with the value calculated in the preparation. 3. Conclusion: write a conclusion about the lab results All results must be cited with the correct units. Always write the corresponding property and its unit next to the graphs axis. Figures and tables should always be accompanied by a caption with their number and name. References Beenakker, H. v. (1996). Quantized conductance. Retrieved May 1, 010, from Instituut-Lorentz for theoretical physics: http://www.lorentz.leidenuniv.nl/beenakkr/mesoscopics/topics/qpc/physics_today/node.html A. I. Yanson, G. R. (1998). Formation and Manipulation of a Metallic Wire of Single Gold Atoms. Nature. E. L. Foley, D. C. (1999). An undergraduate laboratory experiment on quantized conductance in nanocontacts. Am. J. Phys., Vol 67, No. 5. Houten, C. W. (1991). Quantum Transport in Semiconductor nanostructures. Solid State Phys., Vol. 44, &-8. 7