Experiment 1: Johnson Noise and Shot Noise

Transcription

1 Experiment 1: Johnson Noise and Shot Noise Ulrich Heintz Brown University 2/4/2016 Ulrich Heintz - PHYS 1560 Lecture 2 1

2 Lecture schedule Date Thu, Jan 28 Tue, Feb 2 Thu, Feb 4 Tue, Feb 9 Thu, Feb 11 Tue, Feb 16 Thu, Feb 25 Tue, Mar 1 Tue, Mar 22 Tue, Apr 5 Topic Introductory meeting Safety training Lab 1: Johnson noise and shot noise Lab 2: Tunneling in thin film superconductors Measurement and Error Parameter Estimation Lab 3: Electron paramagnetic resonance Lab 4: Pulsed nuclear magnetic resonance Lab 5: Doppler free spectroscopy Lab 6: Lasers and holography Note: I may modify this schedule as needed during the semester 2/4/2016 Ulrich Heintz - PHYS 1560 Lecture 2 2

3 Lab schedule Dates Teams A & C Teams B & D Lab reports due Feb 2-15 Feb 16-Mar 1 Mar 2-15 Mar 16-Apr 5 Mar 27-Apr 4 Apr 6-19 Apr 20-May 3 Johnson noise and shot noise Tunneling in thin film superconductors Pulsed nuclear magnetic resonance Electron paramagnetic resonance Spring break Lasers and holography Doppler free spectroscopy Tunneling in thin film superconductors Johnson noise and shot noise Electron paramagnetic resonance Pulsed nuclear magnetic resonance Doppler free spectroscopy Feb 22 Mar 7 Mar 21 Apr 11 Apr 25 Lasers and holography May 9 Please also note the updated dates for the lab sessions, taking into account that we loose Tuesday, Feb 22 to spring weekend. This shifts the end date for Teams A&C 2/4/2016 Ulrich Heintz - PHYS 1560 Lecture 2 3

4 Sources of electronic noise Most prevalent noise in electronic circuits is 1/f noise Equal power in every decade of frequency Sometimes called pink noise In a resistor this noise arises from resistance fluctuations over time It depends on the details of the construction of the resistor Present in many systems around us Fundamental sources of noise Johnson noise Shot noise These are small but irreducible sources of noise which arise from very fundamental physics This goal of this experiment is to reveal this connection by measuring fundamental physical constants from this noise 2/4/2016 Ulrich Heintz - PHYS 1560 Lecture 2 4

5 Examples of 1/f noise Noise in electronic circuits Speed of ocean currents Flow of sand in an hour glass Traffic flow on expressways Yearly flow of the Nile over the last 2000 years Loudness of a piece of classical music 2/4/2016 Ulrich Heintz - PHYS 1560 Lecture 2 5

6 Johnson noise First measured by J.B. Johnson in 1928 Explained by Nyquist, also in 1928 Thermal motion of electrons gives rise to small random currents in an electronic circuit which gives rise to a potential difference across a resistor and power dissipation. Characteristic properties J.B.Johnson, Phys. Rev. 32, 97 H. Nyquist, Phys. Rev. 32, 110 Independent of external voltage this noise is present even when there is no external voltage applied to the circuit Independent of frequency Johnson noise is white noise, ie it has equal power in all equal frequency intervals Temperature dependent this noise increases with increasing temperature Universal function of resistance and temperature only this noise does not depend on the material or the geometry of the circuit element 2/4/2016 Ulrich Heintz - PHYS 1560 Lecture 2 6

7 Johnson noise The amplitude of the Johnson noise voltage at any particular time is random Sampled many times, the Johnson noise voltage populates a Gaussian bell curve Johnson noise sets a lower limit on the noise in any circuit 2/4/2016 Ulrich Heintz - PHYS 1560 Lecture 2 7

8 Nyquist equation Nyquist considered a circuit consisting of 2 resistors R connected by a transmission line Johnson noise gives rise to oscillations in this circuit Every mode of oscillation contributes energy kt to the circuit In a frequency interval dν the power is dp = ktdν The mean square voltage in the circuit is dv 2 = di 2 2R 2 = 4RdP = 4RkTdν 2/4/2016 Ulrich Heintz - PHYS 1560 Lecture 2 8

9 Measure Boltzmann s Constant! Measure mean square voltage dv 2 = 4RkTdν Plot dv 2 versus 4RTdν dv 2 slope = k 4RTdν 2/4/2016 Ulrich Heintz - PHYS 1560 Lecture 2 9

10 Measure absolute zero temperature! Measure mean square voltage dv 2 = 4RkTdν Plot dv 2 vs T dv 2 Intercept = T 0 0 T 2/4/2016 Ulrich Heintz - PHYS 1560 Lecture 2 10

11 Practical details Johnson noise is very small need amplification to measure it Resistor with Johnson noise V J Input capacitance C I of amplifier Input resistance R I of amplifier A 10k resistor at room temperature Has an open-circuit rms voltage of 1.3 V over a bandwidth of 10 khz Capacitance of coax cable C c In dv 2 = di 2 2R 2 replace R with impedance Z of circuit 1 Z = 1 R + i2πνc I J Therefore dv 2 = 4R J kt 1 + 2πνR J C I 2 dν 2/4/2016 Ulrich Heintz - PHYS 1560 Lecture 2 11

12 g(ν) More practical details The formula dv 2 = 4R J kt 1 + 2πνR J C I 2 dν holds only for hν kt Restrict frequency range that is passed to amplifier Output of amplifier is dv o 2 = g ν 2 Amplifier noise 4R J kt 1 + 2πνR J C I 2 dν + dv A 2 (ν) g(v) is the (frequency dependent) gain of the amplifier Integrate over all frequencies V 2 g ν 2 o = 4R J kt 2 dν + V A πνR J C I ln ν/ν 0 Note the Logarithm! 2/4/2016 Ulrich Heintz - PHYS 1560 Lecture 2 12

13 Now put it all together V o 2 = 4 R J k T 0 g ν πν R J C I 2 dν + V A 2 Stuff you have to measure Given (manual) Do this integral numerically = G V o 2 Don t forget to estimate your uncertainties! Fit a straight line Slope = k 4R J TG 2/4/2016 Ulrich Heintz - PHYS 1560 Lecture 2 13

14 Shot noise W. Schottky, Ann. Phys. (Leipzig) 57, 541 (1918) First described by Schottky in 1918 Statistical fluctuations in the number of electrons give rise to random fluctuations in the current Characteristic properties Only present when a current is flowing Only important for small currents statistical fluctuations become negligible for large currents because the number of electrons involved becomes very large Independent of frequency shot noise is white noise Does not depend on temperature nor the material or the geometry of the circuit element 2/4/2016 Ulrich Heintz - PHYS 1560 Lecture 2 14

15 Statistical fluctuations If certain events occur with a fixed probability per unit time p and are independent of each other the number of such events which occur in a given time interval follows a Poisson distribution The average number of events observed is μ = pt The actual number of events observed varies randomly from one time interval to the next. The probability to observe n events is P P n; μ = μn n! e μ The RMS of the observed counts n is equal to μ 2/4/2016 Ulrich Heintz - PHYS 1560 Lecture 2 15

16 Poisson distribution 2/4/2016 Ulrich Heintz - PHYS 1560 Lecture 2 16

17 Statistical fluctuations in electric current If a conductor carries a current I ave we expect to observe n e = I avet e electrons to pass a given point in the conductor in a time interval T The number of electrons that pass in any given time interval is subject to fluctuations and is described by a Poisson distribution The rms of the electron counts is I ave T e The mean square current should therefore be I ave Mean squared current di 2 = 2eI ave dν 2/4/2016 Ulrich Heintz - PHYS 1560 Lecture 2 17

18 Measure the elementary charge! Measure mean square voltage dv 2 = di 2 R 2 = 2eI ave R 2 dν Plot dv 2 versus 2I ave R 2 dν dv 2 slope = e 2I ave R 2 dν 2/4/2016 Ulrich Heintz - PHYS 1560 Lecture 2 18

19 Measurement of shot noise Shot noise is small Measured over a bandwidth of 10kHz A current I ave = 1A has a shot noise of 57 na (= %) A current I ave = 1 A has a shot noise of 57 pa (=0.006%) A current I ave = 1pA has a shot noise of 57 fa (=6%) Measure voltage across resistor R 1 dv 2 = di 2 R 1 2 = 2eI ave R 1 2 dν Amplify and integrate amplifier output over all frequencies V o 2 = 2eI ave R 2 0 g ν 2 dν + V A 2 2/4/2016 Ulrich Heintz - PHYS 1560 Lecture 2 19

20 Put it together V o 2 = 2e I ave R 2 0 g ν 2 dν + V A 2 Stuff you have to measure Measure gain g ν Do this integral numerically = G V o 2 Fit a straight line Slope = e Don t forget to estimate your uncertainties! 2I ave R 2 G 2/4/2016 Ulrich Heintz - PHYS 1560 Lecture 2 20