1 Shot Noise - 123.physics.ucdavis.edu

1 1 Shot History and BackgroundShot Noise is due to the corpuscular nature of transport. In 1918, Walter Schottky discoveredShot Noise in tubes and developed Schottky's theorem. Shot Noise isalwaysassociated withdirect current flow. In fact, it is required that there be dc current flow or there isnoShotnoise. Electrical currents do not flow uniformly and do not vary smoothly in time like thestandard water flow analogy. Current flow is not continuous, but results from the motion ofcharged particles ( electrons and/or holes) which are discrete and independent. At some(supposedly small, presumed microscopic) level, currents vary in unpredictable ways. It isthis unpredictable variation that is you could \observe" carriers passing a point in a conductor for some time interval youwould nd that a \few" more or less carriers would pass in one time interval versus the is impossible to predict the motion of individual electrons, but it is possible to calculate theaverage net velocity of an ensemble of electrons, or the average number of electrons driftingpast a particular point per time interval.

2 The variation about the mean value or average ofthese quantities is the Noise . In order to \see" Shot Noise , the carriers must be constrainedto flow past in one direction only. The carrier entering the \observation" point must do soas a purely random event and independent of any other carrier crossing this point. If thecarriers are not constrained in this manner then the resultant thermal Noise will dominateand the Shot Noise will not be \seen". A physical system where this constraint holds is apnjunction. The passage of each carrier across the depletion region of the junction is a randomevent, and because of the energy barrier the carrier may travel in only one direction. Sincethe events are random and independent, Poisson statistics describe this try and nd the statistics of this process will require a physical model to analyze,therefore we will consider anLCtank Derivation of Shot NoiseAs an illustration think about anLCtank circuit being charged through an ideal switch (suchas an idealpnjunction diode) from a battery with a voltageV.

3 Now the switch is capableof turning on and o in such a short time interval that only single electrons pass through tothe tank circuit. Therefore the current pulse is negligible. Also the switch randomly turnson and o such that the current pulses are independent and uncorrelated. Then we can1approximate the current flowing into the tank circuit as a spike of current or delta functionI(t)=Xjq (t tj);(1)where thetj's are the random arrival times of the electrons. What we know abouttjis thaton average there should be I=qof them per unit of time following the de nition of , what does one of these current pulses do to theLCcircuit? A short pulse ofcurrent is not going to go through the inductor, so it must end up charging the will produce a rapid change in the voltage on the capacitor whose magnitude is V=q=C :(2)If initially theLCcircuit contained no energy ( voltage and current identically zero),the rst pulse of current would start the circuit oscillating with a voltage amplitude of V.

4 Subsequent pulses of current would arrive at unpredictable times within the period ofoscillation, so that some pulses might increase the amplitude of the oscillation and othersmight decrease the amplitude. Let us write the voltage onCasV(t)=<hVaexp it=pLC i;(3)whereVais the complex-valued amplitude of the oscillation. Then the e ect of the arrivalof a current pulse is to translateVain the complex plane by a distance Vbut in a randomdirection. Thus,Vawill execute arandom walkin the complex plane. For the presentpurposes, the signi cant feature of a random walk is that the average value ofVais zero, butthe actual value is almost never zero. That is, the spikes in the current through the switchwill keep theLCtank circuit excited to some level. This is a characteristic feature of you are familiar with the theory of random walks, you will have noticed thatjVajcangrow without limit.

5 This unphysical result is due to our having neglected any possibility ofback-action of theLCcircuit on the ideal switch/battery system. However this is necessaryfor the physics of Shot Noise since the electrons must be restricted to travel in one directiononly and not retrace their path. If the expectation of the electron was equally likely to goeither direction then Thermal Noise would lets us suppose that the switch will close for a long time and open for a shorttime, so that the current pulses are of a long duration compared to the resonant frequencyof theLCcircuit. Such pulses are not going to excite much of an oscillation in that , there is a signi cant di erence in the properties of the Shot Noise , depending upon theduration of the current pulses. This is clearly a property of the Noise -producing component,2rather than of theLCtank circuit.

6 The underlying concept is that the Noise is distributedover a spectrum of frequencies, and the form of the distribution function, ornoise spectrumis the key physical switch that has this property is apnjunction diode. It is well known thatsemiconductor diodes exhibit Shot Noise . This is because the built-in potential across thedepletion layer of thepnjunction is high enough to prevent the majority carriers fromreturning once they cross the junction. The transit time across the depletion region is thekey time constant for the diode and the carrier arrival is an independent and random will examine the mathematical machinery by which one evaluates the Noise spectrum,and then apply it to nd Shot and Thermal Noise . The mathematical object which allows usto characterize the duration of the current pulse is called theautocorrelation functionand isde ned byRI(t0) = limT!

7 11 TZT=2 T=2I(t)I(t+t0)dt :(4)The Wiener-Khintchine theorem states that the Noise spectrum is the Fourier transform ofthe autocorrelation function:SI(f)=2Z1 1RI(t0)e i2 ft0dt0;(5)whereSI(f) is the one-sided power spectral density (PSD) and physically for this case is themean-square current fluctuation in a unity bandwidth,SI(f)=i2= f. These de nitionsfollow from the facts that only real, positive frequencies are used in circuit analysis and thatfor a Noise process the mean is always zero so that the variance is equal to the mean-squarevalue. The two-sided spectral density is an even function of frequency so thatS(f)=S( f)which leads to the fact thatS(f)=2S(f), forf 0 and provides the factor of 2 in theFourier transform (5).Now, we apply (4) to (1) to nd the autocorrelation of delta-function current (t0) = limT!

8 1q2 TXkXk0ZT=2 T=2 (t tk) (t tk0+t0)dt ;= limT!1q2 TXkXk0 (tk tk0+t0);(6)where the properties of the delta function are used to evaluate the integral. Now, consider theterms in the double summation above. In the case where the summation indices arek=k0which means the arrival times are equaltk=tk0,wejusthave (t0), and if there areNvalues3oftksuch that T=2<tk<T=2, these terms will contributeN (t0) to the autocorrelationfunction. Fortk6=tk0, the delta functions will occur at randomly distributed, nonzero valuesoft0. We argue that, with suitable averaging, the contributions from these delta functionsto the Fourier transform in (5) will vanish. (Note, however, that entire textbooks have beenwritten on the details that are hidden in the phrase \suitable averaging.") So, the part ofthe autocorrelation function which remains is given byRI(t0)=q I (t0);(7)wherewehaveusedN=T= I=qwith Ibeing the dc the Fourier transform (5): (Ff (t)g$1 ); we nd Schottky's theoremSI(f)=2q I:(8)The spectrum is uniform and extends to all frequencies.

9 This kind of spectrum is calledwhiteand many textbooks use the symbolSI(0) to mean no frequency , let us consider the case for the current pulses being of signi cant duration. Inparticular, supposeI(t)=Xkqs(t tk);(9)wheres(t) is a square current pulse of duration , as illustrated in Figure 1. The auto-1/ s t( )0tt t 1/*s s Figure 1: A square current pulse of duration and its autocorrelation functioncorrelation function can be found by a similar argument to that which we made for thedelta-function case earlier and leads toRI(t0)=q Is(t) s(t+t0);(10)4wheres s0is the autocorrelation function ofs, and is shown in Figure 1. The Fouriertransform of a single triangle is:F(1 jt0j= :jt0j 0:jt0j> )$ sin( f ) f 2:When the pulses repeat, as is our case, then the transform is scaled by 1=.

10 Atthispointweneed to remember that the Fourier transform of a repeating series of pulses with constantperiod and constant pulse-width, leads to a discrete spectrum with an average or dc valueof Iand harmonics atn= ;n=1;2; :::. There would beno Noise producedfrom dc out to1= . This is clearly wrong! The key to nding the right answer is that the electrons arriveat random times ( random periods) and with random transit times ( random ).Then we should take the Fourier transform of all the possible cases and ensemble averagethe results. However it is easier to ensemble average the 's! and then apply the Fouriertransform. This is equivalent since the Fourier transform is a linear operation. Now applying(5) yieldsSI(f)=2q I sin( f ) f 2:(11)This equation gives the same result as (8) at low frequencies, but has a cuto atf=1= asshown in Figure 2.