7.7 The Schottky Formula for Shot Noise

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1 110CHAPTER 7. THE WIENER-KHINCHIN THEOREM AND APPLICATIONS 7.7 The Schottky Formula for Shot Noise On p. 51, we found that if one averages τ seconds of steady electron flow of constant current then the variance of the noise that rides on the current, I avg, is: σi 2 = e o τ I avg. In measuring the current we integrate the amount of current that crosses some point over τ seconds and then divide by τ. (Each electron that crosses the point is modeled as a delta function of current with height e o.) This can be considered a filtering operation. The filter that performs this operation has impulse response: { 1 t τ/2 h(t) = τ 0 elsewhere. As the electrons arrive in a perfectly random fashion (by assumption), the noise must be totally uncorrelated. Thus, the noise is white. The PSD of the noise must be a constant S NN (f) = σ 2 N. To work out what the constant is, let us work with the facts we have the variance of the current as measured using a particular filtering operation and the nature of the filtering operation. The PSD of the measurements of the averaged signal must be: H(f) 2 σ 2 N. From Fourier transform 5 on p. 96 and from property 3 on page 92, we find that: H(f) = sin(πτf). πτf Thus, the power spectral density of the averaged current is: sin 2 (πτf) (πτf) 2 σ 2 N. From the definition of the Fourier transform, it is clear that the inverse Fourier transform at 0 is just the integral of the Fourier transform. As the inverse Fourier transform of the PSD is the autocorrelation, and the autocorrelation at zero is just the variance (as long as the expected value of the stochastic process is zero as it is here), the integral of the above function must give the variance if the noise. In Problem 10 of Chapter 6 it is shown that: sin 2 (πf) (πf) 2 df = 1.

2 7.8. JOHNSON NOISE AND THE NYQUIST FORMULA 111 Let us use this to calculate the integral of the PSD. We find that: sin 2 (πτf) (πτf) 2 σ 2 N df u=τf = 1 τ = 1 τ σ2 N. sin 2 (πu) σ 2 (πu) 2 N du Comparing this with the already known value of the variance of the measurement, we find that: σ 2 N = e o I avg Thus the energy in any given band of width f must be e o I avg f. However, if one wants all of the energy in a given band one must consider the energy in the band and in the symmetric negative band. Considering both contributions, we find that in a band of width f, the noise power is: 2e o I avg f. This is known as the Schottky 2 formula. Note that though we treat the arrival of an electron as an event which does not take any time, this is not actually so. Thus the Schottky formula is not completely correct. However, the time that the event takes is generally small enough that the Schottky formula is valid up to very high frequencies. 7.8 Johnson Noise and the Nyquist Formula Shot noise is a phenomenon associated with the discreteness of the electron. Johnson noise 3 or thermal noise (or occasionally Johnson-Nyquist Noise) is noise due to the random motion of the electrons in a resistor. If one considers the free electrons in a resistor as a gas which is reasonable under appropriate conditions, then on the basis of standard statistical mechanics arguments we can say that the average kinetic energy associated with motion in the x direction of a given electron (due to the temperature of the resistor) is: Average Kinetic Energy = 1 2 kt where T is the absolute temperature and k is the Boltzmann constant. (See [16] for a nice presentation of statistical mechanics and the theorem of 2 After Walter Schottky ( ) [4] who predicted it. 3 Johnson noise is named after John Bertrand Johnson ( ) who was one of the early researchers into the nature of noise. Johnson was at Bell Labs at the time his noise research was carried out. He was a contemporary and coworker of Harry Nyquist[15].

3 112CHAPTER 7. THE WIENER-KHINCHIN THEOREM AND APPLICATIONS equipartition of energy.) If temperature is measured in Kelvins, and energy is measured in Joules, then the numerical value of the Boltzmann constant is (approximately): k = J/K. In order to derive the Nyquist 4 formula we consider the following situation 5. We consider a resistor connected to a V volt battery. We assume that the resistor is a cylinder of length L and cross-sectional area A. We assume that current flow is in the x direction. Consider that when an electron moves inside the resistor, its motion is composed of the drift that caused by the electric field impressed by the battery and the thermal motion of the electrons (which is superimposed on the drift). Let us consider the velocity of an electron in the x direction the direction of the electric field impressed upon the resistor by the battery (and the lengthwise coordinate of the resistor), v x. It is clear that: v x = v d + v t where v d is the drift velocity and v t is the x-velocity due to the thermal agitation of the electrons in the electron gas. The electric field inside the resistor is assumed to be constant and must then be V/L. The change in the amount of work done by the electric field due to the random motion is just v t τe 0 V/L where τ is the amount of time that the electron travels with x-velocity v t and e 0 is the charge on the electron. In order to offset this change, the number of electrons being produced by the battery must change. The work done by the battery on r electrons crossing through it is just re 0 V. We find that if r is the number of electrons that the battery acts on that pass through the battery to offset the effects of the thermal noise on the amount of work done on the the electrons in the resistor by the electric field in the resistor, then r satisfies: re 0 V = v t τe 0 V/L. Thus, the compensatory charge that the battery moves that flows in the circuit external to the resistor must be q = re 0 = v t e 0 τ/l. Assume that n electrons per unit volume are free to be part of our electron gas. Then the total number of electrons in the gas is nal. Furthermore assume that we are integrating the current over t 0 seconds. Let τ now be taken to be the mean time between collisions of the electrons. In t 0 seconds each electron should experience t 0 /τ collisions. The total number of steps 4 Named after Harry Nyquist ( ) who in 1927 derived the formula[3, 15]. 5 The presentation here follows [13] closely

4 7.8. JOHNSON NOISE AND THE NYQUIST FORMULA 113 taken by all the electrons should be N = nalt 0 /τ where N is rounded to the nearest integer. Let the total charge movement due to thermal noise be denoted by Q. Let (v t ) i be the i th motion of an electron in the x direction. We find that: Q = N 1 i=0 e o (v t ) i τ/l. We know that: 1 E(v t ) i = 0, 2 me(((v t) i ) 2 ) = 1 2 kt. As the (v t ) i are independent, we find that: E(Q) = 0, E(Q 2 ) = e2 0 L 2 τ 2 NkT/m = e2 0 L 2 nalτ 2 t 0 kt/(mτ). The average current measured is just Q/t 0. Letting the average current be denoted by I, we find that: E(I) = 0, E(I 2 ) = e2 0 L nalτ 2 kt/(mτt 2 0 ) = ne2 0τA 2mL 2kT t 0. As this average noise current is the sum of many IID random variables, the PDF of I must be (approximately) Gaussian. It can be shown that the resistance of a resistor is: R = 2mL/(ne 2 0τA). Combining all of these facts, we find that: E(I 2 ) = 2kT/(Rt 0 ). Let the instantaneous current flowing in the resistor due to thermal noise be denoted by i(t). As the current flow at one instant is independent of the current flow at any other time, the noise current must be white noise. As the operation performed on the current in our calculation is just averaging over t 0 seconds, then, just as in 7.7, the power spectral density of the unaveraged noise current, i(t), must be: S ii (f) = 2kT/R. As the voltage across a resistor is just v(t) = Ri(t), we find that the PSD of the voltage across the resistor is just: S vv (f) = R 2 S ii (f) = 2kT R.

5 114CHAPTER 7. THE WIENER-KHINCHIN THEOREM AND APPLICATIONS If we are interested in the ( voltage related ) energy in a frequency band of width f, then we must consider the contribution both from positive frequencies and negative frequencies. As the power spectral density is an even function of f, the energy in a band of width f is just: Energy = 4kT R f. (7.4) This is the Nyquist formula for the the thermal noise of a resistor, and it is true for any resistor not just a cylindrical one. 7.9 The Random Telegraph Signal Another Low-pass Signal Suppose one has a signal, X(t), that like a telegraph signal always assumes one of two values say ±a. Further suppose that one expects the signal to switch between the two possible values µ times per second in a totally random way. Also assume that the probability of a change occurring in any given time interval is independent of what happened in any other (disjoint) interval. Let Y be the number of sign changes in an interval of time τ. From what we saw in 3.5, it is easy to see that the probability of M N sign changes is just: P (Y = M) = ( N M ) (µτ/n + f(n)) M (1 µτ/n f(n)) N M where f(n) = o(1/n). Here N is the number of ( virtual ) intervals into which we have chosen to break the interval of length τ, and µτ/n + f(n) is the probability of a single change of sign in the interval of length τ/n. (Later we let N which gives us our final answer.) Let us consider the autocorrelation of X(t). Clearly: X(t)X(t + τ) = ±a 2. It is obvious that the sign will be a plus when there have been an even number of sign changes between t and t + τ and it will be negative if there have been an odd number of sign changes between the two times. We find that: E(X(t)X(t + τ)) = a 2 P (Y even) a 2 P (Y odd) = a 2 (P (Y even) P (Y odd)).

Fundamentals of Noise

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