Tobias Märkl. November 16, 2009

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1 ,, Tobias Märkl to 1/f November 16, / 33

2 Content 1 duction to of Statistical Comparison to Other Types of Noise of of 2 Random duction to Random General of, to 1/f / 33

3 , to 1/f 3 / 33

4 What are we going to talk about?, In 1925, J. B. Johnson examined fluctuations in electron emission from a heated filament For decreasing frequencies in low frequency regime, the fluctuation strength increased So far only white noise had been observed (W. Schottky, 1918) to 1/f This low frequency noise obeys a power law (noise intensity 1/f γ, γ 0) 4 / 33

5 as Part of Total Noise, Classification: Spectral density function of power law noise can be decomposed into different parts: S(f ) = c 0 f 0 + c 1 f 1 + c 2 f This is not an approximation; every term represents different type of noise! to 1/f Most important terms are f 0, f 1 and f 2 (there are systems, where also f 1 2 or f 3 2 appear) 5 / 33

6 Names and Terminology Other common names for 1/f noise in literature: 1/f type noise, S(f ) f γ, with γ 1 low frequency noise pink noise (analogy to optic spectra) excess noise flicker noise What do we mean by low frequencies?, to 1/f 1/f type noise is typically dominant below corner frequency f c Hz. 6 / 33

7 of Where does 1/f noise occur? What quantities are affected? Answer: Almost everything you can imagine! Examples: measured quantities of electric circuits and components (I, U, R) frequency of quartz crystal oscillators; affects time measurement precision rate of traffic flow on highways astronomy: number of sunspots apart from regular cycles, light intensity of stellar objects (e.g. quasars) loudness and pitch of music and speech economic and financial data biological systems and many, many more..., to 1/f 7 / 33

8 Statistical of, Paradox of infinite power: variance = total power contained in fluctuations of x σ 2 x(f l, f h ) = fh f l S x (f )df ln f h f l total power diverges at both frequency limits f l 0 and f h this paradox has not yet been resolved! upper limit not a problem, since f h never accessible to measurement due to dominant white noise for lower limit, no cutoff frequency was ever observed; analyses have shown no deviations down to Hz in operational amplifiers [Caloyannides, 1974] to 1/f 8 / 33

9 Correlation Function and Memory of the System, Consider measurement with finite frequency bandwidth between f 1 and f 2, f 2 f 1 : Correlation function is then ψ x (t) σ 2 x 1 1 ln f 2 /f 1 [C + ln(2πf 2 t)], C = constant to 1/f ψ x (t) is very slowly (logarithmically) decaying with time system has a very long memory present state strongly dependent on the past 9 / 33

10 Comparison to Other Types of Noise Comparison of white noise, pink noise and red noise: Let s look at and listen to some signals (from length: 1 second) White noise (γ = 0); constant variance, correlation function is δ(t) Pink noise (γ = 1); variance increases like 1 + ln(t/τ), correlation function decreases only slowly, to 1/f Red noise (γ = 2, Random Walk ); variance increases linearly with time, constant correlation function 10 / 33

11 Spectra of Different Noises Spectral density functions of above noise samples: notice log-scale on both axes slope gives exponent γ different power law behavior of noise samples is apparent numerical analysis confirms exponents that are suggested by creator, to 1/f 11 / 33

12 of, What causes 1/f noise? fundamentally different systems exhibit 1/f noise highly improbable that identical mechanism causes noise in all of them however, mechanism gives rise to similar or identical mathematical properties for most systems, origin of 1/f noise is completely unknown or at least subject to (controversal) debate to 1/f for some systems, there exist theories; none of them capable of explaining all details 12 / 33

13 Resistance in Electric Circuits Any resistance shows fluctuations, often with 1/f Is this due to temperature fluctuations? ( ) S R (f ) =? R 2 S T (f ) T A model by [Voss & Clarke, 1976] for temperature fluctuations as source of resistance 1/f noise was investigated in wide temperature range ( K) [Eberhard & Horn, 1977] but finally refuted, to 1/f Others, too, [Scofield, 1981] found no dependence of voltage 1/f noise on temperature fluctuations in thermally coupled thin metal films temperature fluctuations are unlike to cause 1/f noise! Consider fluctuations of resistance itself, then / 33

14 Example: Conductivity of Semiconductors, The Hooge parameter α In an effort to systematically collect data on 1/f noise, F. H. Hooge introduced empirical relation: S R R 2 = α fn, α: normalized measure for relative noise; N: number of conductance electrons. First estimates gave α to 1/f Since noise heavily depends on sample preparation (growing, doping, surface properties, contacting), α is only meaningful, if samples are somehow similar! 14 / 33

15 Example: Conductivity of Semiconductors, A few comments are necessary: noise is bulk effect, not surface effect specimen should be rather homogeneous relation between α and electron mobility µ was found to 1/f (remember mobility: µ E = v D ) Why are we interested in µ? The conductivity can be written as σ = q n µ [Hooge, 1994] 15 / 33

16 Results, Does resistance fluctuation come from electron number or mobility? Again, several experiments were performed. Analyses of the effect of n and µ on α showed that experimental results were much better described by mobility fluctuations. Noise obeying the Hooge relation is also called α noise. Further investigation showed that it is caused by lattice scattering. to 1/f Since electron mobility is linked to lattice vibrations through scattering, one could finally interpret 1/f-like conductivity noise as phonon number fluctuations. 16 / 33

17 of, Is there anything useful that we can do with 1/f noise? Yes, there is: remember: 1/f noise power ln f h fl for constant frequency ratio we get constant power every decade contains same amount of power to 1/f used for calibration of high fidelity audio equipment (too heavy load for high frequency speakers with white noise) 17 / 33

18 Random, to 1/f image from 18 / 33

19 Do you like Popcorn? What is random telegraph noise? commonly used to describe resistance fluctuations that show random switching between several, often only two, discrete values in literature you will often find: Random Telegraph Noise () or Random Telegraph Signal (RTS) also termed burst noise or popcorn noise signal resembles telegraph signals with two different states ON and OFF [Yuzhelevski, 2000], to 1/f 19 / 33

20 Noise in Small Samples, often occurs in very small specimen of: semiconductor components, e.g. MOSFETs, p-n junctions and resistors (electrical quantities: U, I, R) metal contacts, e.g. nanobridges, metal-insulator-metal tunnel junctions to 1/f quantum dots (light intensity) 20 / 33

21 How small is small?, Examples of some experiments: MOSFETs: length scales µm [Ralls et al., 1984] Cu nanobridges: volume V = nm 3, width nm [Ralls & Buhrmann, 1988] tunnel junctions: active cross-section A = µm 2, thickness d 1nm [Farmer, 1987] to 1/f 21 / 33

22 Some general properties, Let s look at the temporal behaviour! time between switching processes is random but signal values are time-independent to 1/f time scale of actual switching process much shorter than time interval during which system remains in one state future state of system only depends on present state, not on history system has no long-term memory 22 / 33

23 Statistical Consider a two level fluctuation of quantity x: w 1, w 2 : probability of finding system in state (1) or (2); τ 1 = total rate of transitions between (1) and (2) correlation function: [Yuzhelevski, 2000] ψ x (t) = Θ(t)w 1 w 2 (x 1 x 2 ) 2 e t/τ spectral density function (Lorentzian shape): S x (f ) = 4w 1 w 2 (x 1 x 2 ) 2 τ 1 + (2πf ) 2 τ 2 mathematical tool to describe : discrete Markov processes, to 1/f 23 / 33

24 What causes? In general, different mechanisms let s look at an example: MOSFETs What is a MOSFET? Metal-Oxide-Semiconductor Field Effect Transistor left: MOSFET schematically, 1) without and 2) with applied voltage G SS gate voltage leads to bending of band structure, crossing of Fermi level, to 1/f channel at interface with free charge carriers source drain current can be controlled by voltage at gate, not by current! 24 / 33

25 in MOSFETs, fluctuations between constant discrete values assume single electron processes electrons are captured and released again by traps (e.g. impurities, lattice defects) in nearby oxide layer, within few Å from interface trapping and releasing causes conduction electron number N to change by 1 to 1/f (electrostatic field of trapped electrons changes mobility of other electrons in addition to number fluctuations, magnitude difficult to estimate) 25 / 33

26 What causes?, Consequences of number fluctuations N number fluctuations N directly affect conductivity σ = nqµ result: current through structure changes amplitude of fluctuation: R/R = N/N MOSFETs [Ralls et al., 1984]: R/R 10 3 N to 1/f Ratio vs. background noise: R/R backgr = In larger systems, disappears and often 1/f noise arises is it caused by number fluctuations after all? 26 / 33

27 , to 1/f 27 / 33

28 , occurs almost everywhere, universal type of noise not well understood or explained, unresolved fundamental problems 1/f type noise in resistance of semiconductors described by empirical relation and can be attributed to mobility fluctuations random switching between discrete values observed in very small structures interesting statistics, system has no long-term memory single electron trapping processes likely to cause in MOSFETs to 1/f 28 / 33

29 , to 1/f 29 / 33

30 I, [Kogan] Kogan, Electronic Noise and Fluctuations in Solids, Cambridge University Press, 1996 [Web resource] [Hooge, 1994] Hooge,, IEEE Transactions on Electron Devices, Vol. 41, No. 11, November 1994 [Hooge, 1981] Hooge, Kleinpenning, Vandamme, Experimental Studies on Rep. Prog. Phys., Vol. 44, pp , 1981 to 1/f [Caloyannides, 1974] Caloyannides, Microcycle Spectral Estimates of in Semiconductors, J. Appl. Phys., Vol. 45, No. 1, pp , / 33

31 II, [Eberhard & Horn, 1977] Eberhard, Horn, Temperature Dependence of in Silver and Copper, Phys. Rev. Lett., Vol. 39, No. 10, p. 643, 1977 [Voss & Clarke, 1976] Voss, Clarke, Flicker (1/f) Noise: Equilibrium Temperature and Resistance Fluctuations,Phys. Rev. B, Vol. 13, No. 2, p. 556, 1976 [Keshner, 1982] Keshner,, Proceedings of the IEEE, Vol. 70, No. 3, March 1982 [Scofield, 1981] Scofield, Darling, Webb, Exclusion of Temperature Fluctuations as the Source of in Metal Films, Phys. Rev. B, Vol 28, No. 12, p. 7450, 1981 to 1/f 31 / 33

32 III, [Kleinpenning, 1990] Kleinpenning, On and Random in Very Small Electronic Devices, Physica B, Vol. 164, pp , 1990 [Uren, 1985] Uren, Day, Kirton, 1/f and Random Telegraph Noise in Silicon Metal-Oxide-Semiconductor Field-Effect Transistors, Appl. Phys. Lett., Vol. 47, pp , 1985 [Yuzhelevski, 2000] Yuzhelevski, Yuzhelevski, Jung, Random telegraph noise analysis in time domain, Review of Scientific Instruments, Vol. 71, No. 4, p. 1682, 2000 [Ralls & Buhrmann, 1988] Ralls, Buhrmann, Defect Interactions and Noise in Metallic Nanoconstrictions, Phys. Rev. Lett., Vol. 60, No. 23, p. 2434, 1988 to 1/f 32 / 33

33 IV, [Ralls et al., 1984] Ralls, Skocpol, Jackel et al., Discrete Resistance Switching in Submicrometer Silicon Inversion Layers: Individual Interface Traps and Low-Frequency (1/f) Noise Phys. Rev. Lett., Vol. 52, No. 3, p. 228, 1984 [Farmer, 1987] Farmer, Rogers, Buhrmann, Localized-State Interactions in Metal-Oxide-Semiconductor Tunnel Diodes, Phys. Rev. Lett., Vol. 58, No. 21, p. 2255, 1987 to 1/f 33 / 33

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