Download 2015.3.18

IEE5668 Noise and Fluctuations
03/11/2015 Lecture 3:
Mathematics and Measurement
Prof. Ming-Jer Chen
Dept. Electronics Engineering
National Chiao-Tung University
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Content
1. Introduction
- Purposes of the course
- Projections from the course
2. Theoretical Framework and Experimental Setups
- Random Events and Random Walk
- Probability Distributions (steady versus unsteady)
- Mathematics of Stochastic Processes
- Autocorrelation Function and Power Spectral Density
- Wiener-Khintchine Theorem
- Equivalent Circuitry and Transformation
- Measurement Issues
3. Random Telegraph Signals (specific RTS) in a MOS System
- Origin of a Single Oxide or Interface Trap
- Single Electron Capture and Emission Kinetics
- Energy of the System
- Coulomb Energy
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4. 1/f Noise as in a MOS System
5. Thermal Noise in any Electronics Devices
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Microscopic Theory of Thermal Noise: Einstein’s Approach
Macroscopic Theory of Thermal Noise: Nyquist’s Approach
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7. Generation-Recombination (G-R) Noise (Trap related; General RTS)
8. Other Key Issues
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6. Shot Noise in any Electronics Devices
- Random Number of Carriers via Thermionic Injection
- Random Number of Carriers via Field Injection (Tunneling)
RTS, BTI, and 1/f Noise in NanoFETs (percolation, variability)
RTS and Shot noise in Quantum Dot Devices
Noise and Fluctuations in Nanowires
Noise in Bioelectronics
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Buckingham, Chapter 2
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Time Averaging versus Ensemble Averaging
X: open circuit voltage across
a 2-terminal conductor
T: Observation or measurement
time (Device related)
N: Number of observation or
measurement (Device related)
(Time Averaging)

T
(Ensemble Averaging)
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(Slide 9 of Lecture 2)
At upper level, random variable x = I/2;
At lower level, x = -I/2.
<x(s)x(s+t)> = (I/2)2Probability of an even number of transitions in t
- (I/2)2 probability of an odd number of transitions in t
Autocorrelation
function
= (I/2)2(p(0; t)+p(2; t)+p(4; t)+…..)
- (I/2)2(p(1; t)+p(3; t)+p(5; t)+….)
= (I/2)2 exp(-2t)
(Note: no dc term in this case)
The task to derive the following power spectral density is straightforward:
I 2
) /

I 2 
2
S ( f )  4  x( s) x( s  t )  cos tdt  4( )  exp( 2t ) cos tdt 
0
0
2
1   2 / 4 2
2(
Application Example:
Thermal Noise in a Conductor
No measurements conducted
Intrinsic thermal noise
Thought Experiment
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Important Historical Events
Events associated with Thermal Noise:
1. (1828) A Scottish botanist, Robert Brown, observed an irregular
motion of pollen grains in the water.
2. (1906) Einstein’s microscopic random walk model of Brownian
motion, also leading to the diffusion and its coefficient, the Einstein’s
relation, Statistically non-stationary, and the Avogadro’s number.
3. (1908) Jean Perrin (France) measured out Avogadro’s number NA
based on Einstein’s paper. (diffusion constant D  NA)
4. (1908) Langevin (France)’s solving of Brownian motion equation.
5. (1927, Nature; 1928, Physical Review) Johnson (Bell Labs.)
observed thermal noise in an electrical signal.
6. (1927 & 1928, Physical Review) Nyquist (Bell Labs.)’s
thermodynamics approach.
7. (1974) Buckingham’s EE approach.
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Buckingham, Appendix 3
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SVn(w) = 4KBTR/(1 + 2<>2)
Applying Wiener-Khintchine Theorem
We can derive the autocorrelation function for the
terminal voltage fluctuations
Vn() = Vn(t)xVn(t +) = (KBTR/<>) exp(-/<>)
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Discussions
1. Thermal noise is a statistical stationary process
(Let <> = 1)
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Discussions
2. Usually, <> << 1 (Why?). This leads to a common
expression:
SVn(w) = 4KBTR!
Some Useful Calculation in a silicon 2DEG layer:
s = 11.9 x 8.854x10-14 F/cm
 = 1/qn;
typical  = 300 cm2/V-s;
carrier density of a 2DEG ~ 7x1012 /cm2
thickness of a 2DEG ~ 2-3 nm;
thus, n ~ 2.3-3.5x1019 /cm3
So  ~10-3 ohm-cm;
 Dielectric Relaxation Time <> (= s ) ~ 10-15 sec.
(a few femto seconds)
GHz: 109 Hz; THz:1012 Hz; PHz: 1015 Hz.
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Discussions
3. A fast method for calculating power spectral density
of shorted-circuit current fluctuations SIn():
I = V/R. Therefore,
SIn() = SVn()/R2 = (4KBT/R)/(1 + 212)
(Let <> = 1)
4. Because of equal probability of finding  in the
forward direction and finding the same  in the reverse
direction, we can conclude that Vn(t) = 0.
5. Since Vn() = Vn(t)xVn(t +) = (KBTR/1) exp(-/1),
we have Vn(t)2 = KBTR/1.
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Discussions
6. From above 4. and 5., we can reasonably approximate the
thermal noise by a Gaussian or Normal distribution function:
p(x) = (1/square root of 22) exp(- (x – x(t))2/22))
where x is the mean of the variable x
2 (= x2(t) –x(t)2) is the variance of the x in a stochastic
process.
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1. Measurement Effects
(Extrinsic Thermal Noise)
2. Circuit Application
with Thermal Noise as
Noisy Source
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On Bandwidth, i.e.,
the effect of C, a very
practical problem
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Frequency Domain
Equivalent Circuit of Noise Spectra
Measurement Setup
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