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                             WHITE
                             PA P E R




                                                  An Improved Method for
                                           Differential Conductance Measurements
                                                         by Adam Daire

                             Introduction
                             As modern electronics continue to shrink, researchers are increasingly looking
                             to nanotechnology as the basis for the next breakthrough in device size and
                             power consumption. Indeed, as semiconductor structures are made smaller and
                             smaller, the distinction between small silicon geometries and large molecules
                             becomes blurred. Approached from either direction, the consequences are the
                             same. Quantum behavior such as tunneling begins to play an important role in
                             the electrical characteristics. In the macroscopic world, conductors may have
                             obeyed Ohm's Law (Figure 1a), but in the nanoscale, Ohm's definition of
                             resistance is no longer relevant (Figure 1b). Because the slope of the I-V curve
                             is no longer a fundamental constant of the material, a detailed measurement
                             of the slope of that I-V curve at every point is needed to study nanodevices.
                             This plot of differential conductance (dG = dI/dV) is the most important
                             measurement made on small scale devices, but presents a unique set of
                             challenges.




                                     Figure 1a. Macroscopic scale     Figure 1b. Nanoscale
                                     (Classical)                      (Quantum)



Keithley Instruments, Inc.
28775 Aurora Road
Cleveland, Ohio 44139
(440) 248-0400
Fax: (440) 248-6168
www.keithley.com
                                     A   G r e a t e r     M e a s u r e   o f   C o n f i d e n c e
Who Uses Differential Conductance?
Differential conductance measurements are performed in many areas of research, though
sometimes under different names. When used to measure the electron energy structure of
small devices such as quantum dots, nanoparticles, or artificial atoms, it is sometimes referred
to as electron energy spectroscopy. When used to perform non-contact measurements on
surfaces within a scanning tunneling microscope, it may be called tunneling spectroscopy.
When studying ultra-small semiconductor structures or nanotubes with semiconducting
properties, it might be called a density of states measurement. Still other researchers refer
to it more mathematically, as the derivative of the I-V curve, or simply dI/dV or G. It can
be used to understand conduction phenomenon in cryogenic environments or to observe
and predict the conditions when tunneling is more likely to occur. A few examples of the
device types that can be studied this way are Schottky diodes, tunnel diodes, and single-
electron transistors. The fundamental reason differential conductance is interesting is that the
conductance reaches a maximum at voltages (or more precisely, at electron energies in eV)
at which the electrons are most active. This explains the common use of the name energy
spectroscopy. This is also why dI/dV is directly proportional to the density of states and is
the most direct way to measure it.


Existing Methods of Performing Differential Conductance
While there is no standardized technique to obtain differential conductance, almost all
approaches have followed one of two methods:

       (1) Perform a current-voltage sweep (I-V curve) and take the mathematical derivative,
           or

       (2) Use an AC technique of applying a sinusoidal signal superimposed on a DC bias
           to the sample. Then use a lock-in amplifier to obtain the AC voltage across and the
           AC current through the DUT (device under test).


I-V Technique
The I-V sweep technique has the advantage that it is easier to set up and control. It only
requires one source and one measurement instrument, which makes it relatively easy to
coordinate and control. The fundamental problem is that even a small amount of noise
becomes a large noise when the measurements are differentiated.




                A   G r e a t e r   M e a s u r e     o f   C o n f i d e n c e
Figure 2a. I-V curve              Figure 2b. Differentiated I-V      Figure 2c. 100 curves, averaged
                                             curve                              together


         Figure 2a shows an I-V curve, a series of sourced and measured values (V1, I1), (V2,
I2), etc. Several techniques can be used to differentiate this data, but the simplest and most
common uses the slope between every pair of consecutive data points. For example, the first
point in the differential conductance curve would be (I2

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