1 IEE5668 Noise and Fluctuations Prof. Ming-Jer Chen Dept. Electronics Engineering National Chiao-Tung University 03/11/2015 Lecture 3: Mathematics and Measurement

2 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 4. 1/f Noise as in a MOS System 5. Thermal Noise in any Electronics Devices Microscopic Theory of Thermal Noise: Einstein’s Approach Macroscopic Theory of Thermal Noise: Nyquist’s Approach 6. Shot Noise in any Electronics Devices - Random Number of Carriers via Thermionic Injection - Random Number of Carriers via Field Injection (Tunneling) 7. Generation-Recombination (G-R) Noise (Trap related; General RTS) 8. Other Key Issues 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

3 Buckingham, Chapter 2

4 Time Averaging versus Ensemble Averaging T 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) (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. = (  I/2) 2  Probability of an even number of transitions in t - (  I/2) 2  probability of an odd number of transitions in t = (  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: Autocorrelation function

Application Example: Thermal Noise in a Conductor No measurements conducted Intrinsic thermal noise Thought Experiment 10

11 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 N A based on Einstein’s paper. (diffusion constant D  N A ) 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.

12 Buckingham, Appendix 3

13 S Vn (w) = 4K B TR/(1 +  2 2 ) Applying Wiener-Khintchine Theorem We can derive the autocorrelation function for the terminal voltage fluctuations  Vn (  ) = V n (t)xV n (t +  ) = (K B TR/ ) exp(-  / )

14 Discussions 1. Thermal noise is a statistical stationary process (Let =  1 )

15 Discussions 2. Usually,  << 1 (Why?). This leads to a common expression: S Vn (w) = 4K B TR! Some Useful Calculation in a silicon 2DEG layer:  s = 11.9 x 8.854x F/cm  = 1/q  n; typical  = 300 cm 2 /V-s; carrier density of a 2DEG ~ 7x10 12 /cm 2 thickness of a 2DEG ~ 2-3 nm; thus, n ~ x10 19 /cm 3 So  ~10 -3 ohm-cm;  Dielectric Relaxation Time (=  s ) ~ sec. (a few femto seconds) GHz: 10 9 Hz; THz:10 12 Hz; PHz: Hz.

16 Discussions Discussions 5. Since  Vn (  ) = V n (t)xV n (t +  ) = (K B TR/  1 ) exp(-  /  1 ), we have V n (t) 2 = K B TR/  A fast method for calculating power spectral density of shorted-circuit current fluctuations S In (  ): I = V/R. Therefore, S In (  ) = S Vn (  )/R 2 = (4K B T/R)/(1 +  2  1 2 ) (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 V n (t) = 0.

17 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 (= x 2 (t) –x(t) 2 ) is the variance of the x in a stochastic process.

18 1. Measurement Effects (Extrinsic Thermal Noise) 2. Circuit Application with Thermal Noise as Noisy Source

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20 On Bandwidth, i.e., the effect of C, a very practical problem

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22 Equivalent Circuit of Noise Spectra Measurement Setup Frequency Domain