E341 Spring 2013
Rev 7-2007.04.18
Modified by: AR,AB
Noise Measurement Lab
Overview:
The goal of this lab is to measure noise parameters of the fabricated piezoresistive cantilevers.
Noise is important since it places an ultimate limit on the measurement resolution of the device.
There are two types of noise mechanism in the cantilever, i.e. Mechanical and Electrical. For the
cantilevers the mechanical noise floor is far lower than the electrical, hence here we concentrate on the
electrical. The Mechanical noise becomes measurable at the resonance frequency since it gets amplified
by the quality factor. The Mechanical noise can actually be used to calibrate the sensitivity of the
cantilevers. The major portion of Mechanical noise is simply due to the Brownian motion of the air
molecules colliding with the cantilever.
The electrical noise is mainly due to the peizoresistor noise. Noise in piezoresistors has two main
components: Johnson (white or thermal) noise and 1/f noise. At lower frequencies, usually less than 1
kHz, 1/f noise tends to dominate while Johnson noise dominates at higher frequencies.
In our piezoresistors, we are interested in the low frequency region (e.g. static force
measurements) and so 1/f noise is the main concern in the noise studies of our devices.
Johnson noise is universal to all resistors. Thermal noise does not depend on type of material that
the resistor is made of. It just depends on the resistance value and the temperature. Thermal noise is
uniform across all frequencies, hence the adjective “white”. Thermal noise is also known as Johnson
noise, as it is caused by the Brownian movement of the carriers and is directly related to temperature as
shown below
VJ = 4kTR
where VJ is the Johnson noise, k is Boltzmann’s constant (Joule/Kelvin), T is the absolute temperature,
and R is the resistance of the piezoresistor. Note that it does not depend on frequency and so it is constant
across the whole spectrum.
1/f noise, as its name implies, is inversely proportional to frequency and is not yet completely
understood. In particular, 1/f noise in piezoresistors has been shown to be dependent on processing
parameters such as implant dose and anneal. 1/f noise is also a good measure of the quality of resistors.
Noise in excess of the normal value is indicative of poor process quality.
Practical examples of noise:
Figure 1 shows a scope trace of white noise. The amplitude distribution of this noise is Gaussian
as shown in the figure. When talking about noise we always need to talk about its bandwidth as well. For
example if we took the spectral density of the noise shown it would be flat from DC to day light, i.e. at
every frequency it has the same power. It is useful to memorize the noise of a 1 KΩ resistor. A 1 kΩ
resistor has 4 nV/√Hz at room temperature. This means in a 1 Hz bandwidth, we will measure 4 nV RMS.
It is much easier to manipulate this number than think about Boltzmann’s constant and Kelvin
temperature. We can always use the noise of 1 kΩ resistor to calibrate our system and setup.
Noise add in root means squared (RMS) fashion that is if we have a 2 kΩ resistor the noise will be
4^2+4^2 = 5.6 nV/√Hz . The reason for RMS addition is because noise sources are uncorrelated. Another
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E341 Spring 2013
Rev 7-2007.04.18
Modified by: AR,AB
important number to remember is 3 dB noise bandwidth equivalent or π/2 * 3dB bandwidth. That is the
effective bandwidth of a single pole filter for white noise is π/2 * 3dB bandwidth. For example if we did
not have a spectrum analyzer available and wanted to know the noise of our resistor in a 100 Hz
bandwidth, we would need to build a very sharp brick wall filter at 100 Hz. Instead a simple single pole
buffered resistor and capacitor filter at 100/π/2 would give us the same result. That is √(100Hz)*4
nV/√Hz = 40 nV RMS.
40 nV RMS is a very small voltage and there are not many digital voltmeter that have such resolution.
Hence to really see the noise of the resistor we need to amplify it. We need an amplifier with lower noise
than that of the resistor. For this lab we are using AD622 due to its availability and cost. AD622 the
instrumentation amplifier that you are using has an input referred noise floor of 10 nV/√Hz above 10 Hz.
A 10 kΩ resistor will have √10*4 nV = 12.6 nV/√Hz of noise. Now if we amplify the total noise by a
gain of 1000 we will have the output noise of (10^2+12.6^2)^0.5*1000 = 16.1 µv/√Hz. This number is
above the noise floor of the spectrum analyzer and in 1 kHz bandwidth will result in ~0.5 mV RMS
(√1000 * 16.1 µv/√Hz) of noise which is easily measurable.
Noise is part of a signal, and thus has power spectral density (PSD). The plot in the figure 2
below is a typical voltage noise spectrum of a piezoresistor where it shows the voltage noise versus
frequency on a log-log scale. The roll-off region at the high end of the frequency range is due to the
bandwidth limitations of the amplifier used to magnify the noise signal and is not part of the signal of the
actual device.
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E341 Spring 2013
Rev 7-2007.04.18
Modified by: AR,AB
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Johnson noise
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Roll off region
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Frequency (Hz)
Figure 2. Typical noise spectrum of a piezoresistor.
To find the RMS noise for a given bandwidth, the data has to be squared (to convert to power)
and then integrated over the bandwidth of interest. Note that on a log-log scale, the PSD has a slope of -1,
which translates into a slope of ½ on the voltage noise plot since it is the square root of the PSD. You
should check this every time you finish a measurement as a simple check that you are taking the correct
data.
The simplest model for 1/f noise is an empirical one called the Hooge model, which predicts that
1/f noise will increase with decreasing carriers according to the relation:
S1 / f =
Vb2α
Nf
V1 / f = Vb
α
Nf
S1/f is the Hooge noise power density, V1/f is the Hooge noise, f is the frequency, N is the number
of carriers, Vb is the bias voltage, and α is a non-dimensional fitting parameter often ascribed to the
“quality of the lattice.” α is typically in the range of 10-3 to 10-7. Note that because of the 1/f shape of the
PSD, the voltage noise per decade is constant.
There is a fundamental difference between the thermal noise and the 1/f noise. The thermal noise
is really a voltage noise and the 1/f noise is a conductance fluctuation. It does not matter whether or not
you power the resistor, you will still measure the thermal noise. In some hostile environment the thermal
noise is used as a measure of temperature. The 1/f noise is, however, a fluctuation in conductivity, i.e. the
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E341 Spring 2013
Rev 7-2007.04.18
Modified by: AR,AB
resistance is actually changing. In order to measure the resistance you need to apply a current to turn it
into a voltage. Typically we see 40 µΩ/√Hz at 1 Hz. The origin of this fluctuation is still under
investigation; however Hooge’s attributes the 1/f noise to mobility and lattice scattering.
Experiment Setup:
AD622 Pre-AMP
With 10uf high pass filter
HP 3562a Spectrum
Analyzer
Piezoresistor
circuit setup
Matlab
Dumpdata.m
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1/f noise region
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Johnson noise
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Frequency (Hz)
Piezoresistor circuit Setup:
The first step is setting up the piezoresistor is to make a full Wheatstone bridge configuration.
Piezoresistor are very good temperature sensors, i.e. their resistance change with temperature. A typical
PRT (Piezo Resistor Transducer) has about 3000 ppm/C depending on process parameters. For example a
10 kΩ resistor will change 30 Ω in 1 C. Now remember we are trying to measure 40 µΩ.
To isolate the result of temperature effect we can use two piezoresistor and subtract their temperature
contribution. However the 1/f noise will be √2 higher since two piezoresistors contribute to the noise in an
uncorrelated manner. As a practical matter we also need two fixed resistors to complete the bridge. See
figure 3 below. We assume that the fixed resistors do not have 1/f noise and that their thermal noise is
much lower in 1 Hz region (a reasonable assumption if you use metal film resistors). Due to temperature
coefficients mismatch or thermal proximity, there can be enough of a variation that can saturate the
spectrum analyzer and cause erroneous steps in frequency response.
Hence we use a 10 µF capacitor to make sure we do not see drifts or noise components below 0.016 Hz.
This high pass cutoff is due the 10 µF multiplied by the input resistance (1 MΩ) of the spectrum analyzer.
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E341 Spring 2013
Rev 7-2007.04.18
Modified by: AR,AB
Vbridge
Vbridge
Fixed R
Fixed R
Piezo
Piezo
-Vbout+
Bridge Nulling:
1) Measure the piezoresistor with a multimeter.
2) Select the same value metal film resistor to complete the bridge. The resistor do not have to
match exactly, roughly 10% is good enough. Please use 5 V bridge voltage.
3) Write down these values
4) Use a digital voltmeter and the potentiometer to set Vb out as close as possible to zero (< 1 mV).
We need to null the bridge. Record this offset voltage. You may need to move the potentiometer
to the other side. Also if the right value of potentiometer is not available you can use a fixed
resistor in series with the pot.
5) Measure –vbout and +vbout node to ground and record them they should be very close to each
other (<1 mV) and half of Vbridge ~2.5.
6) You need to record the Vbout, Vbridge, all the four resistor values, also –Vbout and Vbout+
referenced to ground.
Please note the PRT are very light sensitive. After you have nulled the bridge make sure you cover
them with an enclosure (cardboard box). Otherwise you will measure erroneous results.
Amplifier Connection:
1) Connect the bridge to the AD622 amplifier and you have designed it.
2) Measure the gain of your circuit
3) You can do that by changing the input voltage and measuring the change in the output
voltage. Make sure you use changes of voltage and not absolute voltage to calculate gain.
4) The easiest way to do this is to measure the current value of Vbout and Output of the AD622
(Vout). Then change the Vbout by 1 mV using the potentiometer and record the Vout. Now
the gain is ΔVout/ΔVbout, you should measure 1000.
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E341 Spring 2013
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Modified by: AR,AB
5) Make sure you record these values and you null the bridge back. (Vout should be <1 V,
closer to 0 the better)
Connection to the spectrum Analyzer:
1) Connect the AD622 through a 10 µf capacitor to channel 1 of the spectrum analyzer.
HP3562A Setup and measuring noise:
HP 3562A is a Fast Fourier Transform based analyzer, i.e. it samples the input time data and digitally
process it. It is a very useful instrument since it can also double as a scope or be used as a network
analyzer. If the analyzer has been previously setup then you can skip these steps; however in case of a
power outage or reset it is good to know how to get back the conditions.
1
2
S1
S2
4
3
HP3562A
1) At first we need to select our measurement mode, either linear or log. We can do this by pressing
the measurement mode button(1) and the soft menu button (S2)
2) Then we need to select the measurement type by pressing “select measurement” (2) and then
pressing the soft key button (S2) for power spectrum
3) Now we need to get our units in V/√Hz. Press “Units” (3) and in the soft menu button select “P
spec units” (S1) and again in soft menu select “V/√Hz PSD”.
4) Now Press “Coord” (4) and then soft menu button MAG(log)
5) The next step is to setup the frequency range Press the “Freq” button and set the start freq at 1 Hz
and span of 4 decades. That is from 1 Hz to 10 kΩ.
6) Press “start” to see the spectrum.
7) Or prior to start press “view input” and press the soft button ‘view input time’ (s1) to see the
noise in time. You can then press start to see the spectrum. It takes a few minutes for the
spectrum to appear.
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E341 Spring 2013
Rev 7-2007.04.18
Modified by: AR,AB
8) The laptop is connected to the HP by GPIB cable. You should use the program Dumpdata.m to
record your result. Dumpdata file generates three files a datafile, an .EPS file and a .Fig file.
Make sure when you type in the Dumpdata in Matlab you are working in the noise directory.
9) The file name is very important to figure out who has what. So one unified format should be
followed. Please follow the example below. Make sure you substitute your wafer number and die
number the measured the resistance, the voltage across the resistor, your initials, date, and ME342
at the end)
10) Dumpdata(‘ 2007W19die07x09R10kVr2p5byAJRApril222007ME342’)
11) Make sure the PRT does not see light. Also do not forget to press the ‘start’.
First noise measurement:
For the first noise measurement you should calibrate the system and just measure the noise of the AD622.
You can easily do that by shorting the output of the bridge.
You should get a similar plot as below. Make sure you give it the file name with the measured gain
AD622OnlyME342GainXXX
Now that you know the setup is calibrated and you have the right tools you should measure the noise of
the piezoresistors.
Make sure you measure the noise at three different bias voltages (2, 4, 6 Volts) and submit your plots.
Best of luck and hope you will be having fun.
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