PHYS 352. Shot Noise, 1/f Noise. current (which is rate of charge)

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1 PHYS 352 Shot Noise, 1/f Noise Shot Noise current (which is rate of charge) current does not flow completely smoothly when charges arrive as quanta e.g. charge collection in a photodiode is a random, independent process in each slice of time, a variable amount of charge arrives or is collected the variation makes what is called shot noise when charge flow is a stochastic (random) process it leads to a Gaussian distribution of the amplitude of the fluctuations unlike Johnson noise, does not happen on its own i.e. current needs to flow to have shot noise whereas Johnson noise is present when there is no current 1

2 from Wikipedia Shot Noise A Visual Cousin of Shot Noise graininess of digital photos under low light conditions - each pixel collects a low level of photons - adjacent pixels: some collect a few more, some a little less - relative size of the fluctuation is larger when fewer photons Derivation: Shot Noise in time T, with average current I 0, N electrons are collected in a Poisson process, the standard deviation of the number of electrons collected: 1/ 2 T 0 σ = N N = I qe where q e is the electron charge the standard deviation of the current: qe σ I = σ N T qe = I0 T 1/ 2 standard deviation is the square root of the squared deviation from the mean: σ = ( 2 I I = ) I 0 I rms so describe the actual current from the device I, to be equal to I 0, the average, with a noise source I rms : I I + = 0 I rms 2

3 Shot Noise Current Formula time slice T is equivalent to bandwidth Δf, by: 1 more discussion in later lectures in the course Δf = 2T related to Nyquist sampling theorem higher bandwidth resolves smaller time structure in the current fluctuations lower bandwidth averages over large time shot noise rms amplitude of current fluctuations: I rms = e 0 2q I Δf shot noise also has a flat frequency power spectrum independent of f (but depends on Δf) hence is also white noise like Johnson noise Is Shot Noise Always Present? no Always Significant? no the relative amplitude goes as: I N rms = = 1 I0 N N larger current larger shot noise rms but smaller, relative amount (i.e. S/N improves) correlations between electrons alter purely stochastic process e.g. conduction in a macroscopic metal has no shot noise because long-range correlations exist for diffusive conduction at mesoscopic scales, metals exhibit sub-poissonian shot noise levels figure from de Jong 3

4 1/f Noise is a common noise present in numerous systems, not just in electrical signals noise power spectral density goes as 1/f noise voltage goes as 1/ f many systems in nature exhibit 1/f power spectrum e.g. sound of a waterfall, electrical potentials in neurons, traffic flow, voltage noise in transistors, etc. see also called flicker noise or pink noise one description of the origin: 1/f results from a random walk of the inter-event time between successive pulses if there is memory in the process, introduces 1/f power spectrum in the fluctuations Example: 1/f Noise Origin take a real-world amplifier; it's mostly stable, but its gain may drift over time due to many factors can you list some? even if not, you know that this can happen over a time interval of minutes or even hours, the amplifier is mostly stable; gain at 2 pm probably similar to what the gain was at 1 pm over a time interval of days to months, you can't say the same thing; amplifier gain in February could varying significantly from its gain in January the gain might have drifted during a months operation, it random walked from its gain value in January to its gain value in February over a long period (i.e. low frequency); it has larger variation the randomness (variations) seen at low frequency is greater than at high frequency 4

5 1/f Noise is not Exactly f 1 systems that follow 1/f have additional scaling constants that can't be determined from first principles may have frequency spectral features superposed (bumps and wiggles) in general, the slope is pretty much = 1 An Example from Physics Research noise characteristics of carbon nanotubes as conductors single walled carbon nanotubes conducting along 1D slope = 1.06 ± 0.02 what causes this noise? fluctuations of atom locations electromigration of defects it's not really known! 5

6 1/f and White Noise Compared in Time Domain simulated white noise simulated 1/f noise Glossary: Gaussian Noise and White Noise Gaussian noise means Gaussian distribution of noise amplitudes in the time domain if successive values are independent of each other, then this is random and produces white noise (flat frequency spectrum) Gaussian noise can also have 1/f spectrum or other spectrum; this implies that successive values in time are related to each other (somehow) you can make this by taking white Gaussian noise and passing it through a filter white noise can also be non-gaussian; amplitude distribution in the time domain is different 6