Low-current Noise Measurement Techniques
F.R. Leon, A.F. Hebard
University of Florida, Gainesville, FL
(August 4, 1999)
ABSTRACT
The noise of a resistive sample is a measurable physical property. The noise signature can depend on
such variables as the materials used in preparing a sample and the method used to prepare it. A versatile
experimental technique is needed to measure these noise signatures, one which can be applied to any form
of resistive structure. In addition, many samples with unique physical properties, such as tunnel junctions,
are destroyed by large currents. We require a method which is sensitive enough to measure noise at low
currents. Such a technique is explained here.
WHAT IS NOISE?
Although the term can be used to describe anything that obscures a desired signal, noise can be divided
into two main types: interference and random noise. Interference occurs when an undesired signal obscures
a desired one, a good example being 60Hz interference occurring from proximity to power lines. Random
noise arises from physical sources, such as from thermal fluctuations. It is often associated with resistance
and is expressed in terms of voltage fluctuations across, or current fluctuations in parallel with, resistors.
There are two main characteristics of a noise source. The first is its frequency distribution. The second is
the physical phenomena which is producing it.
NOISE UNITS AND NOISE ARITHMETIC
Noise is commonly measured in units of energy per frequency, much like the measurement of electro2
/Hz. The variable
magnetic radiation. For a circuit element, this quantity is often expressed in units of Vrms
for noise voltage is “E,” which distinguishes it from “V,” the variable for signal and power voltage. In
addition, noise adds in quadrature. If there are two independent noise sources A and B, then the total noise
energy created by them is
2
2
2
= EA
+ EB
.
Etot
1
SOURCES OF NOISE
Random noise arises from many physical processes, some of which are understood and some which are
not. Johnson, or “thermal,” noise is the most common form of explained noise and is seen in all resistors,
regardless of their composition. It is given by
2
Erms
= 4kB T R
∆f
2
is the noise power, ∆f is the frequency bandwidth, kB is Boltzmann’s constant, T is the
where Erms
temperature and R is the resistance value.
Johnson noise sets a lower limit on the noise of a resistor, since it is independent of the current through
the resistor. This noise is due to the fluctuations inherent in any mechanism of energy loss. Viscous friction
in a liquid is another example, as it is associated with the random Brownian motion of the particles.
Shot noise is another well-understood form of noise, and arises from the discrete charge of the electron. It
is often explained as “rain on a tin roof” since its relative magnitude decreases with increasing current. It
is given by:
2
Erms
= 2qRVdc .
∆f
Where q is the charge of the electron, Vdc is the constant voltage across the resistor and R is the resistance
value.
Unexplained random noise sources not only lack a physical explanation; they also differ in two major ways
from Johnson noise and Shot noise. First, unexplained sources tend to be ohmic, meaning
2
,
E 2 ∝ Idc
although the proportionality constant is not equal to the resistance R of the device and might not be linear
with R. Second, they have an inverse dependence on frequency
E2 ∝
1
fα
where α is a number greater than 1. For off-the-shelf metal film resistors, α ≈ 1, while α ≈ 2 for carboncomposite resistors. This type of noise has equal power per decade of frequency and is often referred to as
“pink” noise.
WHY MEASURE NOISE?
The conversion of electrical energy to heat causes noise. For resistors of different compositions, this process
occurs in physically different ways. Each dissipative process is accompanied by characteristic fluctuations
2
which can be measured with a spectrum analyzer. This yields a “noise signature” much like the spectral
emission lines of an atom or molecule.
The tunnel junction is a resistive structure with interesting physical properties. A tunnel junction consists
of two conducting thin films with a thin layer of insulating dielectric in between. Placing an electrical
potential across the conducting films causes electrons to tunnel through the dielectric. If impurities are
introduced into the dielectric, the tunneling rate can be affected, either positively or negatively, thereby
affecting the resistance. In addition, impurities affect the noise signature by introducing peaks and valleys
into an otherwise smooth curve. By studying the noise spectra of samples, insight can be gained into the
different mechanisms by which electrons travel through the substance.
DC NOISE MEASUREMENT CIRCUITS
A
C
R
I
D
B
FIG. 1. Simple DC noise measurement
In the most basic configuration, the noise of a resistor or any conducting device can be measured using
the four-terminal circuit shown in Figure 1. A DC current I is fed through the resistor R from lead A to lead
B. A voltage is measured across leads C and D. The measured voltage will fluctuate due to the noise of the
resistor. The resistance itself will therefore appear to change. The power spectrum (energy per frequency)
of these fluctuations can be determined by putting the signal through a spectrum analyzer, which takes the
Fourier transform of the fluctuations. Although in theory this is the most direct method of measuring the
noise of a device, in practice this configuration has a number of drawbacks. Luckily, however, each of these
can be solved through modifications.
The first problem with the four-terminal circuit is the fact that the noise voltage E will be miniscule
compared to the constant Vdc = Idc R which is seen across the resistor. The voltage across the resistor is
given by
3
Vtot =
p
(Idc R)2 + E 2 .
If the frequency spectrum is taken of Vtot , the inputs of the spectrum analyzer will be overloaded by the
constant voltage and unable to detect the tiny variations in the noise voltage. What is needed is a way to
subtract the DC voltage from the signal before we amplify and Fourier transform it. The balanced bridge
circuit in Figure 2 accomplishes the task quite nicely. The RB ’s are ballast resistors that provide a stable
current source through the samples, denoted by RS .
Vin
t
RB
...
.........
.....
..
...........
.....
.
....
...
.........
.....
..
...........
.....
.
....
RB
RV
...
........
.....
..
...........
.....
..
.
...
...
........
.....
..
...........
.....
..
.
...
RV
..
.........
......
..
..........
.....
..
...
..
.........
......
..
..........
.....
..
...
VA
RS
t
t
t
VB
RS
FIG. 2. Balanced bridge circuit
By adhering to the relationship
RB = 10RS ,
we can ensure that the noise of the ballast resistors will be negligible compared to the noise of the samples.
A stable, DC voltage is applied to Vin . Once the variable resistors are adjusted so that the average voltage
across the bridge, VA − VB , is zero, we can measure the differential noise voltage VAB :
VAB = VA − VB =
4
√
2E.
NOISE MEASUREMENT USING AN AC LOCK-IN TECHNIQUE
Lock-In Amplifier
Bridge Circuit
Vin
Reference
t
Out
Demodulated
RB
..
.........
......
..
..........
.....
..
...
..
.........
......
..
..........
.....
..
...
RB
RV
...
.........
......
..
.
.
.........
.....
..
...
...
.........
......
..
.
.
.........
.....
..
...
RV
..
........
......
...
.
.
.........
.....
..
....
..
........
......
...
.
.
.........
.....
..
....
Noise Out
VA
Differential
VB
Input
RS
t
t
t
RS
FIG. 3. AC Bridge Circuit
The DC bridge circuit, though an improvement over a simple four-terminal measurement, still has a
serious drawback. Even the best differential amplifiers are significantly noisier at DC than they are at
AC frequencies, at which they are almost noiseless. To avoid the noise from DC amplification, we use an
alternating Vin to drive the bridge circuit and a lock-in to detect and demodulate the noise. The differential
amplifier we use to measure VAB will then be run at its optimum low-noise frequency. The circuit is pictured
in Figure 3, and it operates much like an AM radio. A steady constant-amplitude, constant-frequency sine
wave is provided by the lock-in as a reference and fed into the bridge at Vin . The samples represented as
resistors RS , create noise which modulates the amplitude of the incoming sine wave. The lock-in amplifier
demodulates this noise by multiplying VAB by the reference channel and outputting a slowly-varying voltage
which can then be analyzed. The lock-in is necessary because it singles out the in-phase component of the
signal, thereby eliminating any capacitive effects of the samples being analyzed.
The AC bridge circuit is sensitive enough to measure noise at µA current levels. The data presented below
were taken with a maximum current of 300µA. A tunnel junction breaks down at a maximum current of
50µA and therefore requires an order of magnitude more sensitivity. Through improved shielding and data
averaging, this sensitivity can be achieved.
5
POWER SPECTRUM ANALYSIS OF NOISE DATA
Once the demodulated noise is obtained from the output of the lock-in, it must be Fourier-transformed
to determine its frequency distribution. The spectrum analysis of noise is a fine art, particularly so for low
currents, in which case the 1/f flicker noise disappears into the thermal background after only a few Hertz.
A time-domain signal that does not start and stop at the same value over the sampling period will have
a discontinuity if it is copied and placed end-to-end. Fourier transforming such a discontinuous function
will erroneously leak amplitude into all of the frequency bins. Before data can be transformed, it must be
windowed, and an appropriate windowing function must be chosen. By first multiplying the time-domain
signal by a smooth function which goes to zero at its endpoints, this amplitude leakage can be avoided. There
are various windowing functions to choose from. The Hanning window is very common and is normally the
best to choose for noise measurements since it has the lowest noise floor, with moderate amplitude accuracy
and frequency selectivity. It is given by:
i
ωi = 1 − cos 2π
N
However, since 1/f noise has a sharp peak at zero Hertz, the Hanning window is not the best since it will
not accurately measure the amplitude of the peak. The Flat-top window function,
i
i
i
i
+ 1.29 cos 4π
− 0.388 cos 6π
+ 0.028 cos 8π
ωi = 1 − 1.93 cos 2π
N
N
N
N
is the best to use when amplitude accuracy is needed. The Flat-top is particularly suited to data whose
frequency distribution is rough and it is therefore ideal for 1/f noise analysis. Frequency resolution and
amplitude accuracy cannot both be obtained simultaneously. Other window functions, such as the BMH
window, offer a balance between the two and provide a large dynamic range.
Since the amplitude of 1/f noise is largest at low frequencies, the most interesting data occurs at just a few
Hertz. Unfortunately, the lower the frequency, the longer it takes to acquire data. A reasonable frequency
to probe to is 31.25 mHz, requiring a time record of 32 seconds. In addition, noise tends to be noisy. The
Fourier transform of a single time trace will not give an accurate noise curve. Many traces must be acquired
and averaged in order to yield a smooth curve. 10 averaged traces give a reasonably smooth noise curve.
6
DATA AND RESULTS
0.0001
’carb.dat’ using 1:2
’carb.dat’ using 1:(0.9*$3*$1**(-1.5))
1e-05
1e-06
1
FIG. 4. Carbon-Composite Resistor Noise
Figure 4 is a plot of the noise of a 1kΩ carbon-composite resistor up to a frequency of 1 Hz in units of
2
of power versus Hertz of frequency. The data is shown with logarithmic scales for both the x and y
4pVrms
axes so that the values for α and β in the following relationship can be easily determined:
y=β
1
xα
On logarithmic scales, α is the negative slope of the straight line fit, and β is the y-axis offset. The solid line
is the data for a current of 300µA. The dotted line is the thermal background scaled by the above equation.
For a carbon-composite resistor, α = 1.5 and β = 0.9 gives a decent fit. Figure 5 gives a comparable plot
for a 1kΩ metal-film resistor. For this type of resistor, α = 0.8 and β = 1.1 provide decent overlap. These
plots are actually averages of Fourier transforms for 10 time traces. By averaging traces, the data can be
smoothed considerably. Ideally, the solid and dotted plots would be overlapping straight lines.
7
0.0001
’metal.dat’ using 1:2
’metal.dat’ using 1:(1.1*$3*$1**(-0.8))
1e-05
1e-06
1
FIG. 5. Metal-Film Resistor Noise
8