1. 1
Laboratory experiment on quantized conductance
Autumn 2010
IH2654 Nanoelectronics Ilse de Moffarts
April 2010

2. 2
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 2D electron gas, as its width W
was varied by means of the voltage on the gate. The steps are near integer multiples of 2e²/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
V
I
R
G 
1 (1)
.
The current is given by the following formula
L
veN
I  (2)
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
eVE  (3)
.
Substituting both equation (2) and (3) in equation (1), gives
EL
Nve
G


2
(4)
.
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, EFA respectively EFB, 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 EFA, can flow to unoccupied states in
terminal B. The number of contributing electrons N equals twice the number of quantum states S
in [EFA - E, EFA].
Recall from chapter 1, section 1.7, that for particles in a one-dimensional box of length L, the
wavenumber is restricted to values
L
n
kn
2
 ...2,1,0 n (5)
and thus the de Broglie wavelength of an electron can only take on discrete values
n
L
n  ...2,1,0 n (6)
.
The formula of the velocity of an electron is

3. 3
m
h
m
p
v

 (7)
,
with m the effective mass of an electron, so the electron velocity in a one-dimensional wire can
only have the following discrete values
Lm
nh
vn  ...2,1,0 n (8)
.
The number of electrons N, which occupy the S states in [EFA - E, EFA], travel at different
velocities depending on the quantum state. This velocity range
Lm
Sh
Lm
xh
Lm
hSx
vvv xSx 

 
)( (9)
converts to a range of kinetic energy
L
vSh
vmv
vvm
Ekin 


2
2 (10)
.
The potential energy E, provided to the system by the applied voltage, equals the kinetic energy
Ekin of the electrons that propagate through the wire. Isolating the number of quantum states S
from (10), results in
vh
EL
vh
LE
S kin 


 (11)

vh
EL
SN


2
2 (12)
.
Substituting the number of contributing electrons N (12) into equation (4) immediately gives
h
e
G
2
2
 (13)
.
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 =
2e2
/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, 1-228 (1991).

4. 4
1.2The experimental set-up
The goal of this lab is to observe the quantization of electrical conductance G in multiples of
2e2
/h with a simple, room-temperature experiment (Figure 2). 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.
-
OP-07
- 9 V
R1
R2 R3
R4
Gold wires
+
- 9 V
+ 9 V
RF
Oscilloscope
-
OP-07
- 9 V
R1
R2 R3
R4
Gold wires
+
- 9 V
+ 9 V
RF
Oscilloscope
Figure 2 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

5. 5
2. 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 2 with following values: R1 = 10
k, R2 = 3.6 k, R3 = 690 , R4 = 49.9  and RF = 20 k.
1. Calculate V2, the voltage over resistor 2, and V4, the voltage over resistor 4, which is the
input voltage of the nanocontact.
2. 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 Vout = - RF * (-Iin). Calculate one voltage step.
Table 1 Comparison between expected calculated values and experimental measured values
Expected values Measured values
V2 = voltage over R2
V4 = voltage over R4 =
input voltage
(when the gold contact
is broken)
1 conductance step
1 current step Iin
1 voltage step at
output of opamp Vout

6. 6
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 20-40
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 2 are: 2 9 V batteries, 2
connectors for the 9 V batteries, a solderless breadboard (Figure 4) with jump wires and 4
resistors (Figure 5).
1
2
2
1
2
2
Figure 4 Solderless breadboard
(1) 6 interconnected round tie points per row
in horizonal array
(2) power bus: 2 of 24 each of interconnected
tie points with black line printing, 4 of 12 each
of continual interconnected tie points with red
line printing
Figure 5 Color code for resistors
http://www.st-
andrews.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 V4 the expected voltage step.
Secondly, make the amplifier circuit. Connect all pins of the op amp (except for offset null 1 & 2)
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

7. 7
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.
2. 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
2. 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 21, 2010, from Instituut-Lorentz
for theoretical physics:
http://www.lorentz.leidenuniv.nl/beenakkr/mesoscopics/topics/qpc/physics_today/node2.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 , &-228.