NoiseNoise A definition (not mine) Electrical Noise S. Oberholzer and E. Bieri (Basel) T. Kontos and C. Hoffmann (Basel) A. Hansen and B-R Choi (Lund and Basel) ¾ Electrical noise is defined as any undesirable electrical energy (?) T. Akazaki and H. Takayanagi (NTT) E.V. Sukhorukov and D. Loss (Basel) C. Beenakker (Leiden) and M. Büttiker (Geneva) T. Heinzel and K. Ensslin (ETHZ) M. Henny, T. Hoss, C. Strunk (Basel) H. Birk (Philips Research and Basel)
Effect of noise on a signal. (a) Without noise (b) With noise
¾ we like it though (even have a whole conference on it)
National Center on Nanoscience Swiss National Science Foundation 1 2
Noise may have been added to by ... Introduction: Noise is widespread
Noise (audio, sound, HiFi, encodimg, MPEG) Electrical Noise Noise (industrial, pollution)
Noise (in electronic circuits)
Noise (images, video, encoding)
Noise (environment, pollution) it may be in the “source” signal from the start Noise (radio) it may have been introduced by the electronics Noise (economic => theory of pricing with fluctuating it may have been added to by the envirnoment source terms)
it may have been generated in your computer Noise (astronomy, big-bang, cosmic background)
Thermal noise for the latter, e.g. sampling noise Quantum noise
Neuronal noise
Standard quantum limit and more ... 3 4 Introduction: Noise is widespread Introduction (Wikipedia)
Noise (audio, sound, HiFi, encodimg, MPEG) Electronic Noise (from Wikipedia) Noise (industrial, pollution)
Noise (in electronic circuits)
Noise (images, video, encoding) Electronic noise exists in any electronic circuit as a result of random variations in current or voltage caused by the random movement of the Noise (environment, pollution) electrons carrying the current as they are jolted around by thermal energy.energy
Noise (radio) The lower the temperature the lower is this thermal noise.
Noise (economic => theory of pricing with fluctuating source terms) This same phenomenon limits the minimum signal level that any radio receiver can usefully respond to, because there will always be a small but significant Noise (astronomy, big-bang, cosmic background) amount of thermal noise arising in its input circuits. This is why radio telescopes, Noise figure which search for very low levels of signal from stars, use front-end ciruits, usually mounted on the aerial dish, cooled in liquid nitrogen to a very low Shot noise Î focus on Noise in temperature. Electronic Circuits Thermal noise Electronic Circuits
Quantum noise
Neuronal noise
Standard quantum limit and more ... 5 6
Introduction (Wikipedia) Introduction: spectra of a „small“ signal
Electronic noise is often measured in μV/√Hz, a term that derives from the fact that doubling the bandwidth of the measurement doubles the power level measured, but voltage is proportional to the square root of power. Integrated circuit devices, such as op-amps commonly quote equivalent input noise level in these terms (at room temperature). Î white noise
7 8 Introduction (Wikipedia) Introduction (Wikipedia)
Thermal Noise (from Wikipedia) Shot Noise also from Wikipedia Johnson-Nyquist noise (thermal noise, Johnson noise, or Nyquist noise) is the noise generated by the equilibrium fluctuations of the electric current inside an electrical conductor, which happens regardless of any applied voltage, due Shot noise is a type of noise that occurs when the finite number of particles that to the random thermal motion of the charge carriers (the electrons). carry energy, such as electrons in an electronic circuit or photons in an optical device, gives rise to detectable statistical fluctuations in a measurement. It is It was first measured by J. B. Johnson at Bell Labs in 1928. He described important in electronics, telecommunications and fundamental physics. his findings to H. Nyquist, also at Bell Labs, who was able to explain the results by deriving a fluctuation-dissipation relationship. The strength of this noise increases with the average magnitude of the current or Thermal noise is to be distinguished from shot noise, which consists of intensity of the light. Often, however, as the signal increases more rapidly as the additional current fluctuations that occur when a voltage is applied and a average signal becomes stronger, shot noise often is only a problem with small macroscopic current starts to flow. For the general case, the above definition currents and light intensities. applies to charge carriers in any type of conducting medium (e.g. ions in an electrolyte).
Nyquist Schottky 9 10
Introduction (Wikipedia) Introduction (Wikipedia)
Shot Noise also from Wikipedia Shot Noise also from Wikipedia
Shot noise is a type of noise that occurs when the finite number of particles that Shot noise is a type of noise that occurs when the finite number of particles that carry energy, such as electrons in an electronic circuit or photons in an optical carry energy, such as electrons in an electronic circuit or photons in an optical device, gives rise to detectable statistical fluctuations in a measurement. It is device, gives rise to detectable statistical fluctuations in a measurement. It is important in electronics, telecommunications and fundamental physics. important in electronics, telecommunications and fundamental physics.
The strength of this noise increases with the average magnitude of the current or The strength of this noise increases with the average magnitude of the current or intensity of the light. Often, however, as the signal increases more rapidly as the intensity of the light. Often, however, as the signal increases more rapidly as the average signal becomes stronger, shot noise often is only a problem with small average signal becomes stronger, shot noise often is only a problem with small currents and light intensities. currents and light intensities.
what would the author likely like to say? input by a reader: for example, as the rms number fluctuations grow slower than the average “Shot noise and thermal noise have long been believed to be the results of distinct number (i.e. like the square of N in a Poissonian process), noise is only a mechanisms but they aren't. Thermal noise is shot noise due to internal problem for small numbers (i.e. small signals). diffusion currents in electronic devices… Shot noise is shot noise. The main result is illustrated in these measured and theoretical results for noise in subthreshold MOS transistors ….”
11 12 Introduction (Shot Noise on Wikipedia) Introduction (Shot Noise on Wikipedia)
Shot noise in electronic devices consists of random fluctuations of the electric Shot noise is a type of noise that occurs when the finite number of particles current in an electrical conductor, which are caused by the fact that the current is that carry energy, such as electrons in an electronic circuit or photons in an optical carried by discrete charges (electrons). device, gives rise to detectable statistical fluctuations in a measurement. It is important in electronics, telecommunications and fundamental physics. Shot noise is to be distinguished from current fluctuations in equilibrium (Johnson- Nyquist noise). I thought shot noise was to do with electrons passing over a junction between two different conducting materials? … I’ll go else where and check... [email protected] Shot noise is a Poisson process and the charge carriers which make up the current will follow a Poisson distribution. The strength of the current fluctuations can I think that's called a flicker noise... [email protected] be expressed by giving the variance Hi, 2 2 Δ I := ()I − I You are absolutely right, and this page needs some editing. I think I or someone with more detailed knowledge should do this. of the current I, where is the average ("macroscopic") current. Briefly speaking, the granular nature of charge is not responsible for shot noise by itself. Shot noise occurs when charge carriers must cross a junction. However, the value measured in this way depends on the frequency range of So for instance, a copper wite carrying current will exhibit only thermal noise but not shot fluctuations which is measured (bandwidth): The measured variance of the noise, while a semiconductor junction will exhibit both. current grows linearly with bandwidth. Therefore, a more fundamental quantity is the noise power, which is essentially obtained by dividing through the bandwidth Flicker noise is an altogether different phenomenon, and while there are several theories (and, therefore, has the SI units Ampere squared divided by Hertz). and models regarding flicker noise, there is no universally accepted physical model to explain it yet (perhaps a Nobel Prize waiting for someone), despite and perhaps owing to its universal nature. It is seen in phenomena from microscopic scale to the astronomical scale. Vivek vivkr.at.yahoo.com 13 14
Vacuum Tube: its implication to flicker noise Vacuum Tube: a comment
"The triode was invented by Lee de Forest in 1907, and soon afterwards the first amplifiers were built. By 1921 the 'thermionic tube' amplifiers were so developed that C.A. Hartmann made the first courage experiment to verify Schottky's formula for ths shot noice spectral density. Hartmann's attempt failed (that’s not fully true, though), and it was finally J.B. taken from R.E. Burgess et al., British Journal of Appl. Phys., Vol. 6 (1955) Johnson who successfully measured the predicted white noise spectrum. However, Johnson also measured an unexpected 'flicker noise' at low frequency... and shortly thereafter W. Schottly tried to provide a theoretical explanation. Schottky's explanation was based on the physics of electron transport inside the vacumm tube, but in the years that followed Johnson's discovery of flicker noise, it was found that this strange noise appeared again and again in many different electrical deviced."
15 16 Introduction (Flicker Noise on Wikipedia) Introduction (Flicker Noise on Wikipedia)
Pink noise, also known as 1/f noise, is a signal or process with a frequency spectrum such that the power spectral density is proportional to the reciprocal of Some researchers describe it is as being ubiquitous. It should be noted that at the frequency. Sometimes pronounced as one over f noise, it is also called flicker high enough frequencies 1/f noise is never dominant. noise. In other words, it is a sound that falls off steadily into the higher frequencies, 1/f Noise in Electronic Devices instead of producing all frequencies equally. It occurs in many other fields of study. 1/f Noise in Ecological Systems 1/f noise occurs in many physical, biological and economic systems. In physical 1/f Noise in Network Traffic systems it is present in some meteorological data series, the electromagnetic 1/f and 1/f2 Noise in Financial Data radiation output of some astronomical bodies, and in most electronic devices. In 1/f Noise in Heartbeat biological systems it is present in heart beat rhythms and the statistics of DNA 1/f Noise in DNA Sequences 1/f Noise in Meteorology and Oceanography sequences. In financial systems it is often referred to as a long memory effect. 1/f Noise in Neuro Systems A pioneering researcher in the field of 1/f noise in electronic devices was 1/f Noise in Astronomy Aldert van der Ziel. 1/f Noise in Human Coordination 1/f Noise in Cognition, Psychology and Psychiatry 1/f Noise in Music and Speech 1/f and 1/f2 Noise in Granular Flow 1/f Noise in Magnetic Systems 1/f Noise in Traffic Flow 1/f Noise in Geophysical Records 1/f Noise in Chemical Systems 1/f Noise in Number Systems 1/f Noise in Written Language 1/f Noise at Phase Transition
17 18
Introduction (Flicker Noise on Wikipedia) Commercial Noise Sources for everyone “Experiments on human subjects have been done for reducing blood pressure simply by staring at a flickering candle flame. Would the candle flame not be a rich source of 1/f "noise". I know that it is soothing to me when my blood pressure drops, so perhaps that is the effect of the candle flame and music. I think it might also be true that if a person was placed in an environment where there was no 1/f present whatsoever (to any of the five senses), that person might go stark, raving mad ?”
Patent 5657640. Filed: Sep 14, 1995; Issued: Aug 19, 1997 "An air conditioner for exchanging heat from a refrigerant to the outside and adjusting at least one of room temperature and humidity to desired temperature and humidity and having a 1/f fluctuation function for controlling an air supplying means for supplying conditioned air to the room so as to vary the air supplying amount in multiple levels corresponding to a designated reference air amount and irregularly. A reference air amount of the air supplying means can be designated and there is an air amount control for controlling the air supplying means corresponding to the designated reference air amount, and a fluctuation width designates a fluctuation width (volume of the air amount) corresponding to the air amount designated. When the 1/f fluctuation function is employed, the fundamental functions (coolness, warmness, and so forth) of the air conditioner are improved. In addition, the noise of the air conditioner is reduced and the comfort of the user is improved. "
19 20 Commercial Noise Sources for everyone Commercial Noise Sources for everyone
Î Pure White Noise Î sample again
21 22
Noise: a Nuisance Understanding Noise in Measurements
Noise as something that masks wanted signals… Generally the aim is to maximize the signal to noise ratio by minimizing the noise, maximizing the signal or a combination of both.
Noise can be added to a signal at various stages – ¾ the measured parameter themselves can be noisy ¾ the sensor can add noise in the sensing process ¾ noise can be "picked up" in the cabling ¾ there can be noise in the input stages of the measuring device ¾ if the signal is being digitized there can be various sampling noises added ¾ … With all these potential noise sources, it is surprising it is possible to measure anything!
“The topic of noise is a big one, but in practice you may never need to deal with the problem because the equipment you are using is effective in removing the noise or there is no significant noise present in you measurement. The two major noise sources are 50/60 Hz pickup and radio frequency interference - understanding these may be all you need to know.”
Cartoon by Rand Kruback, Agilent Technologies. 23 24 Understanding Noise in Electronics Understanding Noise in Measurements
Noise sources (intrinsic and extrinsic): Power EM emission Thermal Noise The random movement of electrons in a material induces small temperature dependent currents in a conductor. The resulting voltage is dependent on the resistance value according to Johnson's noise equation:
Crosstalk System noise Solutions: Keep circuit series resistance as low as necessary (till thermal noise is about half other sources if possible) Signals EM emission In & Out Minimize the measuring device's bandwidth by input filtering and / or software Crosstalk filtering Ground/power noise
Shielding this and the following pages have been taken from CAPGO: http://www.capgo.com Capgo is about automatic data acquisition and provides industry and research professionals, solutions based on measurement, monitoring, data acquisition, data logging and data loggers. 25 26
Understanding Noise in Measurements Understanding Noise in Measurements
Noise sources (intrinsic and extrinsic): Noise sources (intrinsic and extrinsic):
Input Noise (of amplifier) Sampling Noise The input circuit of a measuring device can introduce significant noise. This noise Today, signals are digitized and therefore subject to sampling errors, i.e. resolution is has a number of components: thermal noise, excess current noise, flicker due to a finite number of digits and due to finite time resolution. noise (1/f) and shot noise. The latter two are characteristics of semiconductors There may also be a problem, if the sampler has not a proper input filter. with the flicker noise generally the most significant.
Solutions: Solutions: ¾ Use well specified measuring device ¾ make use of the full range ¾ Minimize bandwidth by post input filtering - in hardware or software ¾ buy a better ADC … ¾ Where the sampling speed is high compared to the low frequency noise, the ¾ don’t forget a low-pass filter on the input noise can sometimes be treated as a rapid zero drift. Take zero readings between measurements and subtract the zero drift. ¾ use a lock-in (avoid zero-frequency)
27 28 Understanding Noise in Measurements Understanding Noise in Measurements
Noise sources (intrinsic and extrinsic): Noise sources (intrinsic and extrinsic):
Thermal Gradients Dielectric Absorption Temperature gradients in conductors of different materials can induce small No insulator, other than a vacuum, is perfect. When measuring very small currents thermoelectric voltages by the Peltier effect. an often neglected source of this error (say < pico Amp), the charge storage effects of the insulators in the system can is resistors - some of which have very high Peltier voltages (150µV/°C). become significant. This effect only occurs with high impedance (>100 MΩ) sensors such as pH electrodes and then only when the sensor output changes relatively quickly. Strictly, dialectric absorption is more of a signal distortion than Solutions: superimposed noise, however the effect can amplify the impact of other noise sources. ¾ Keep same metals in positive and negative leads ¾ Keep leads together so they are exposed to same temperatures Solutions: ¾ Minimize temperature gradients within measuring device ¾ Use low absorption insulators such as polypropylene, polystyrene, Teflon. ¾ Minimize magnitude of voltage changes to less than a few volts.
29 30
Understanding Noise in Measurements Understanding Noise in Measurements
Noise sources (intrinsic and extrinsic): Noise sources (intrinsic and extrinsic):
Audiophonic Noise Electric Field Coupling Physical vibration of the measuring system can induce measurable noise by Varying electric fields such as from a power cable can be coupled into a measuring several different mechanisms. The most common of these is the piezoelectric circuit by the "stray" capacitance between the two circuits. This capacitance is voltage generated by stress on ceramic capacitors constructed from materials with usually very small - 0.1 to 20 pF, but it is sufficient to introduce voltage noise into high dielectric constants. The problem arises when these capacitors are use in the high impedance circuits. signal path of sensitive amplifiers. A second source of audiophonic noise is the strain gauge effect with resistors, Solutions: especially surface mount types. Server stress may lead to ±0.5% or more variation with stress. ¾ Use a shielded cable connected to Ground at one end, preferably at the measuring device end. Solutions: ¾ Increase the distance between the electric field source and the measuring circuit. ¾ Avoid high value ceramic capacitors in input circuits. ¾ Lower the measuring circuit impedance as much accuracy considerations allow. ¾ Avoid component stress by restricting and damping vibration of circuit boards. ¾ Place susceptible components such that the net effect cancels out. This is most easily done with resistors in bridge or divider networks.
31 32 Understanding Noise in Measurements Understanding Noise in Measurements
Noise sources (intrinsic and extrinsic): Noise sources (intrinsic and extrinsic):
Inductive Coupling Radio-Frequency Coupling Varying magnetic fields will induce a varying current into a measuring circuit. These Radio frequency coupling can be a difficult noise source to identify. The are far more troublesome than electrically coupled voltages because the current measurement circuit acts as an antenna to receive radio energy from radio tends to flow regardless of the circuit impedance and magnetic shielding is very transmitters (including mobile phones and radars). The inputs of many solid state expensive. circuits act like rectifiers to radio frequencies. The result is often an fluctuating DC The magnitude of the induced current is proportional to the magnetic field strength, input voltage. the rate at which it changes and and the area of pickup loop in the measuring circuit. Solutions: ¾ If possible locate system away from sources of radio frequency energy such as Solutions: radio transmitters, microwave systems, radar and inductive heaters. ¾ Minimize measurement circuit area. ¾ Use shielded cable, grounding the shield to the case of the measuring device ¾ Use twisted pair cables. and capacitively coupling it via a 100nF capacitor to the case of the sensor. The ¾ Use mu-metal shielding if affordable. reason for capacitively coupling at one end is to avoid ground loops. ¾ Place measurement circuit away from magnetic fields. ¾ Place ferrite beads over the signal wires including signal grounds, at both ends of the sensor cable. ¾ Avoid moving or vibrating magnetic materials near measurement circuit. ¾ Employ low pass RC or LC filters in the measuring device. ¾ Tied down cables to prevent vibration.
33 34
Understanding Noise in Measurements Understanding Noise in Measurements
Noise sources (intrinsic and extrinsic): Noise sources (intrinsic and extrinsic):
Ground Loops Common Mode Rejection Noise Ground Loops are sufficiently common a problem that they are dealt in detail on a Put simply, common mode voltages can cause the equivalent to a zero shift in the separate page. Most commonly, ground loops induce 50/60 Hz and related input circuit of a measuring device. The resulting noise added to the signal will tend harmonics into a system. These can sometimes be filtered out. However ground to reflect the form of the common mode voltage. loops can also induce unstable DC errors. Solutions: Solutions: ¾ Minimize common mode voltage, particularly AC voltages. ¾ Avoid ground loops. ¾ Select a measuring device with high CMRR ¾ Use measuring devices with electrically isolated inputs . ¾ Use a distributed measurement system.
35 36 Understanding Noise in Measurements At the end of the day Î insight at the limit
Noise sources (intrinsic and extrinsic):
positive cross-correlations Cable Noise due to inter-edge-state scattering negative cross-correlations: The cable can exhibit piezoelectric charge generation when mechanically moved or antibunching of single state stressed. The result is a most noticeable with high impedance sensors. A similar but less well known cable induce noise is the triboelectric effect. Again, the result is a most noticeable with high impedance sensors. Electrical leakage currents in the cable can also inject noise currents into a measurement circuit. This can be particularly troublesome when the voltage between the cable conductors exceeds a volt and the circuit impedance is greater than say 100 kΩ.
Solutions: ¾ Avoid high impedance sensors ¾ Minimize the length of cable between sensor and measuring device. ¾ Minimize the cable movement opportunities - e.g. wind and vibration. ¾ Use cable with quality insulating material (Teflon is excellent). S. Oberholzer et al., Phys. Rev. Lett. 96, 046804 (2006)
37 38
Noise basics Noise basics Consider n carriers of charge e moving with a velocity v through a sample of length L. The current I through the sample is: nev I = L The (mean-square) fluctuation of this current is given by the total differential
³ ne´2 ³ev ´2 δ2I := (I(t) I )2 = δ2v + δ2n h − h i i L L where the two terms are added in quadrature since they are (assumed to be) statistically uncorrelated (?) Two mechanisms contribute to the total noise: F velocity fluctuations, e.g. thermal noise F number fluctuations, e.g. shot noise excess or ’1/ f ’ noise Thermal noise and shot noise are both white noise sources, i.e. power per unit bandwidth is constant: dP dδ2V noise = const or noise = const df df
whereas for 1/ f noise
dP 1 noise = df f α
(typically α =0.5 ...2) 39 40 Signal and Noise 1/f-Noise
Frequency dependence of noise
Low frequency ~ 1 / f α v2 = .Δf – example: temperature (0.1 Hz) , pressure f f α (1 Hz), acoustics (10 -- 100 Hz)
High frequency ~ constant = white noise – example: shot noise, Johnson noise, spontaneous emission noise 1/f noise is present in all conduction phenomena. Physical origins are multiple. It is Total noise depends strongly on signal freq negligible for conductors, resistors. It is weak in bipolar junction transistors and strong for – worst at DC, best in white noise region Noise amplitude MOS transistors. 1/f noise power is increasing as frequency decreases. 1/f noise power is constant in each 1/f noise frequency decade (i.e. from 0 to 1 Hz, 10 to 100Hz, 100MHz to 1Ghz)
log(Vnoise) 1/f noise
Circuit bandwidth White noise Thermal noise Depending on circuit bandwidth, 0 1/f noise may or may not be contributing 0.1 1 10 100 1kHz log( f ) 41 42
1/f-Noise 1/f-Noise questions
α if 1 / f with 0 < α < 2 , the power-spectral density diverges!
43 44 1/f-Noise questions Fundamental Limits are set by
what is the correlation function ? thermal fluctuations (in equlibrium)
how much memory content and where is the memory ? non-equlibrium fluctuations: excess energy allows in general for additional stochastic is 1/f noise present at equlibrium ? channels Î leads to excess noise, i.e. more noise (there are exceptions though: negative excess noise) is 1/f a stationary process ? (can it be distinguished from drift at low frequencies ? quantum fluctuations
is it Gaussian ?
45 46
Quantum-Fluctuations Standard Quantum Limit (example)
What does the state ψ tell an experimentalist ? Determination of momentum p of particle by measuring its position: ψ τ not much, because only if is an eigenstate of the measurement apparatus measure first x1(t=0) and later x2(t= ) can we predict the outcome of a measurement. Δ the first measurement with an „inaccuracy“ of the apparatus of m1 yields an uncertainty in Δ = Δ in general, need to measure many times to get some information, i.e. the mean
m 2 2 2 this is not a good way to get to know the past, right! Δp = ()()()Δx + Δx+ + Δx τ m1 m2 there is an „optimal way“ to measure bound by quantum mechanical uncertainties: m Δp ≥ ()()Δx 2 + Δx 2 „standard quantum limit“ τ m1 + τ 2 m 2 ⎛ τ ⎞⎛ 1 ⎞ ΔxΔp ≥ / 2 the meaning of this is: Δp ≥ ()Δx + h ⎜ ⎟ h m1 ⎜ ⎟⎜ Δ ⎟ ⎝ 2m ⎠⎝ xm1 ⎠ Δx Δp ≥ / 2 measure perturb h minimizing, yields: m τ backaction Δp = h Δx = h Δ Δ ≥ SQL 2τ SQL 2m x perturb pmeasure h / 2 δ ΔqΔI ≥ h / 2 implication for electronic noise: Î 2 ∝ ω I h 47 48 Standard Quantum Limit (example) The „standard“ derivation of
SQL of the measure energy E of a harmonic oscillator:
Δ = ω ESQL h 0 E
if << ω a) thermal 2 kT h 0 ΔE −1/ 2 SQL = ⎛ 1 + 1 ⎞ ⎜ ω / kT ⎟ b) shot noise E ⎝ 2 eh 0 −1⎠ ω h 0 if kT >> ω kT h 0
this is essentially the whole story of the quantum limit, i.e. if ω >> kT / h
but it‘s not, in the opposite – the classical limit (there is shot noise in addition)
within SQL, one can see this (i.e. the transition to the classical limit) as a cut-off that sets in for long measuring time (i.e. low frequency) by the interaction with the bath. Eventually, this leads to an uncertainty in the variance of the momentum scaling with kT
Î current noise will then become frequency independent and proportional to kT!
49 50
Noise: minimal formalism 1st Derivation of Termal Noise l Wiener-Khintchine Theorem
assume diffusive transport in a metallic wire δI(t) := I(t) − I L Be R the electrical resistor, i.e. a metallic wire with length L and cross-section A, having charge carrier density N and momentum relaxation τ. The voltage- current relation reads: for the current I(t), or better for the fluctuations of the current δI around its mean value, we have accordingly: U = R I = RAj = RAeNv · Z Z where j is the current density and v the (mean) velocity of all particles (Drude ∞ iωt ∞ picture), both along the wire direction. SI (ω)=2 e− δI( + t)δI( ) dt =4 cos(ωt) δI( + t)δI( ) dt h · · i 0 h · · i −∞ Because,
if the current-current correlation function CI (t):= δI( + t)δI( ) decays with 1 X h · · i v = v the characteristic time τ, the noise will be white for frequencies ω << 1/τ. NAL j Moreover, SI will then be given by: we have Z Re X X ∞ U = vj =: Uj SI (0) = 2 CI (t)dt L ∞ Because velocities of different particles are uncorrelated, the square fluctuations 2 2 Exercise: derive it yourself + in what other context iis this used in Cond. Matter Physics ? just sum up, i.e. δ v = NAL δ vj . 51 h i 52 1st Derivation of Termal Noise Thermal Noise
We use the Wiener-Khintchine theorem and start from the velocity correlation function for δv: R 2 t /τ kT t /τ Cv(t):= δvj ( )δvj ( + t) = δv e−| | = e−| | h · · i h j i m Using the theorem i+ Z kT ∞ 4kT τ S (ω)=4 e t/τ cos(ωt)= v m − m 1+(ωτ)2 i- 0 temperature Θ Adding up for the final quantity SU (ω)yields
µ ¶2 1 2 1 Thermal has a normal distribution of amplitude with time. Re 4kT τ m v = k T SU (ω)= (NAL) x B L · m 1+(ωτ)2 2 2
2 Bandwidth limiter and by noting that e Nτ/m is the Drude conductivity σD and σDA/L =1/R, we arrive at the desired result: R G=1 1 SU (ω)=4kTR 4kTR · 1+(ωτ)2 ∼ 2 = v 4kTR.BWnoise = tot SU (0) 4kTR J. B. Johnson and H. Nyquist (1928)
53 54
Johnson-Nyquist Noise 2nd Derivation of Termal Noise (Nyquist 1928) = P12 SU (0) 4kTR J. B. Johnson and H. Nyquist (1928)
In early times, this equation was used to determine the absolute zero by extrapolating the data points to zero noise. TT ° By fixing the temperature scale at 0 and 100 C to the Celsius scale, R1=R R2=R Boltzmann‘s constant could be determined as well.
P21 I1 = I1 U1 / 2R P = U 2 / 4R U1 12 1 P12 P R 12 2 P12 = P21 R1 (for all frequencies seperately) P21
55 56 2nd Derivation of Termal Noise (Nyquist 1928) 2nd Derivation of Termal Noise (Nyquist 1928) ideal transmission line with Z = L' / C ' = R This energy has been supplied by the two sources R1 and R2 during the time R SSR τt = L/c, so that the power delivered by each resistor within ∆f is equal to 1 c P1,2 = E∆f Gedankenexperiment: After equilibrium has been established, let us close the 2 L switch. The energy which was moving in and out is now trapped. so that the spectral power density SP (f)reads:
In order to calculate it we need to know the eigenmodes. If L is the length, the 1 hf SP (f)= hf + wavelength λ has to obey nλ/2=L, so that the spectrum is given by: 2 ehf/kT 1 − cn 2 f = Recalling P12 = U1 /4R from before leads to SU (ω)=4RSP (ω), hence: n 2L 4R~ω SU (ω)=2R~ω + e~ω/kT 1 According to Planck, each mode carries the energy − whichintheclassicallimitkT >> ~ω leads to the desired result: 1 hf hf + n n hf /kT 2 e n 1 SU (ω)=4kTR − f The total trapped energy within the frequency window ∆ is obtained by mul- and in the quantum limit kT << ~ω to: tiplying with the number of modes ∆f2L/c: µ ¶ SU (ω)=2R~ω 2L 1 hf E = ∆f hf + ∆f c 2 ehf/kT 1 − 57 58
Derivation of Shot Noise 1st Derivation of Shot Noise
t tk
The current is described by a random sequences of pulses, each pulse carrying charge e.Hence, X I(t)= eδ(t Tk) − k
The auto-correlation function CI (t)ofI(t)isgivenby:
CI (t)=eIδ(t)
so that by using the Wiener-Khintchine Theorem, we immediately find:
SI (0) = 2eI
Shot noise is white, as long as the assumption of a delta function is correct, that is as long as ω is smaller than the inverse of the pulse width. There is no further assumption. If pulses overlap, that does not matter in this calculation. It may matter, if the “channel” has some maximal capacity, given e.g. by Pauli’s principle.
59 60 2nd Derivation of Shot Noise 2nd Derivation of Shot Noise
t t tk t tk t tk tk
At a given frequency ω := 2πn/T ,pulsesatdifferent instances T contribute The current is described by a random sequences of pulses, each pulse carrying n k with a cos(ω t φ )termwithequalamplitude2e/T and random phases φ . charge e. At not too large frequenciy, each pulse occurring at t = T can be n k k k Hence, the intensities− of each pulse can be added. described by a delta function: 2 The intensity δi n per pulse and at frequency with index n is then given by: h i i(t Tk)=eδ(t Tk) 2 − − 2 4e 1 δi n = We are using the discrete Fourier series to express the pulse with T much larger h i T 2 2 than the pulse width. Then, Let r be the number of pulses per unit time. Then, during time T , the total intensity (of all pulses) at frequency with index n amounts to: e 2e X∞ i(t)= + cos(2πnt/T) T T 2 0 2 2e 2eI δi n,T = r = ,whereI is the mean current h i T 2 T The first part, the DC component, only adds to the mean current and can be omitted for the fluctuations. Hence, This has to be multiplied with the number of different frequency allowed by the input filter of the measurement setup. If this frequency span is ∆f,then µ ¶ 2e X∞ 2πn ∆n = ∆fT, so that we obtain as the final result: δi(t Tk)= cos (t Tk − T T − 0 2 δi ∆f =2eI∆f h i 61 62
Schottky Exercise
Schottky
Ipp
Calculate the expected spectral current noise density for so-called random telegraph noise, which is usually associated with two-level fluctuators.
Assume that there are only two current values, separated by Ipp and that the correlations decay with an exponential at times τ I 2 τ⋅ the outcome is something like: S ∝ pp I 1+ ()ωτ 2 is this super- or sub-Poissonian at low frequencies ?
63 64 Another look at hermal and shot noise Scattering Approach (PART 1B)
The emergence of equilibrium noise can also be understood as caused by un- certainties in the occupation of the eigenstates of the system. In this picture here, add page 7-11,12 of my handwritten notes coherence is implicitly assumed. Consider a single electron state at energy Ei in thermal equilibrium. This state on „Messen am Limit“ is occupied with probability p: 1 Noise in Mesoscopic Physics pi = f(Ei)= 1 exp(Ei/kT )+1 (add also as an example negative excess noise) The random variable ni(t), which is 1 if the state i is occupied and zero other- wise, fluctuates. Because n2 = n , we obtain: i i .... 2 2 2 δ ni = n ni = pi(1 pi) h i h i i − h i − .... Since fluctuations at different energies are uncorrelated, the (partial number) fluctuations sum up and yield a term proportional to Z f(E)(1 f(E))dE = kT −
Hence, thermal noise is given by kT
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Summary
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