Johnson Noise and Shot Noise: The Determination of the Boltzmann Constant, Absolute Zero Temperature and the Charge of the Electron
MIT Department of Physics
Some kinds of electrical noise that interfere with a measurement may be avoided by proper attention to experimental design such as provisions for adequate shielding or multiple grounds. Johnson noise [1] and shot noise, however, are intrinsic phenomena of all electrical circuits, and their magnitudes are related to important physical constants. The purpose of this experiment is to measure these two fundamental electrical noises. From the measurements, values of the Boltzmann constant, k, and the charge of the electron, e will be derived.
PREPARATORY QUESTIONS To illustrate the dilemma faced by physicists in 1900, consider the highly successful kinetic theory of gases 1. Define the following terms: Johnson noise, shot based on the atomic hypothesis and the principles of noise, RMS voltage, thermal equilibrium, temper- statistical mechanics from which one can derive the ature, Kelvin and centigrade temperature scales, equipartition theorem. The theory showed that the well- entropy, dB. measured gas constant Rg in the equation of state of a 2. What is the Nyquist theory ?? of Johnson noise? mole of a gas at low density, 3. What is the mean square of the fluctuating compo- nent of the current in a photodiode when its average P V = RgT (1) current is Iav ? is related to the number of degrees of freedom of the The goals of the present experiment are: system, 3N , by the equation 1. To measure the properties of Johnson noise in a k(3N) R = = kN (2) variety of conductors and over a substantial range g 3 of temperature and to compare the results with the Nyquist theory; where N is the number of molecules in one mole (Avo- gadro’s number), and k is the Boltzmann constant de- 2. To establish the relation between the Kelvin and fined so that the mean energy per translational degree of centigrade temperature scales, freedom of the molecules in a quantity of gas in thermal 3. To determine from the data values for the Boltz- equilibrium at absolute temperature T is kT/2. At the mann constant, k, and the centigrade temperature turn of the century nobody knew how to measure pre- of absolute zero. cisely either k or N. What was required was either some delicate scheme in which the fundamental granularity of atomic phenomena could be detected and precisely mea- INTRODUCTION sured above the smoothness that results from the huge number of atoms in even the tiniest directly observable “Thermal physics connects the world of object, or a thermodynamic system with a measurable everyday objects, of astronomical objects, analog of gas pressure and a countable number of degrees and of chemical and biological processes with of freedom. the world of molecular, atomic, and electronic The Millikan oil drop experiment of 1910 was a delicate systems. It unites the two parts of our world, scheme by which the quantum of charge was accurately the microscopic and the macroscopic.”[4] measured. It compared the electrical and gravitational forces on individual charged oil droplets so tiny that the By the end of the 19th century the accumulated ev- effect of a change in charge by one or a few elementary idence from chemistry, crystallography, and the kinetic charges could be directly seen and measured through a theory of gases left little doubt about the validity of the microscope. The result was a precise determination of atomic theory of matter, though a few reputable scien- e which could be combined with the accurately known tists still argued strongly against it on the grounds that values of various combinations of the atomic quantities there was no “direct” evidence of the reality of atoms. such as the faraday (Ne), e/m , atomic weights, and the In fact there was no precise measurement yet available gas constant (kN), to obtain precise values of N, k, and of the quantitative relation between atoms and the ob- other atomic quantities. Therefore, a current will not be jects of direct scientific experience such as weights, meter continuous in the mathematical sense, it should exhibit sticks, clocks, and ammeters. a “noise” due to the granularity of charges. Twenty years later Johnson discovered an analog of by one joule causes a very large increment in σ , the gas pressure in an electrical system, namely, the mean magnitude of τ in common circumstances like room tem- square “noise” voltage across a conductor due to ther- perature must be much less than 1. In practical ther- mal agitation of the electrical modes of oscillation which mometry, the Kelvin temperature T is proportional to τ, are coupled to the thermal environment by the charge but its scale is set by defining the Kelvin temperature carriers. Nyquist showed how to relate that mean square of the triple point of water to be exactly 273.16 K. This voltage to the countable number of degrees of freedom puts the ice point of water at 273.15 K and the boiling of electrical oscillations in a transmission line. The only point 100 K higher at 373.15 K. The constant of propor- atomic constant that occurs in Nyquist’s theoretical ex- tionality between fundamental and Kelvin temperatures pression for the Johnson noise voltage is the Boltzmann is the Boltzmann constant, i.e. constant k. A measurement of Johnson noise therefore yields directly an experimental determination of k. states in the solar corona. τ = kT (5)
In classical statistical mechanics, k/2 is the constant 23 1 where k = 1.38066 10− JK− . By a quantum sta- of proportionality between the Kelvin temperature of a × system in thermal equilibrium and the average energy tistical analysis, based on the Schr¨odinger equation, of N per dynamical degree of freedom of the system. Its ul- particles in a box in thermal equilibrium at temperature timate quantum physical significance emerged only with T , one can then show that the mean energy per transla- the development of quantum statistics after 1920 [4]. A tional degree of freedom of a free particle is τ/2 so the total energy of the particles is 3 Nτ = 3 NkT (see [4], p. summary of the modern view is given below. (see Kittel 2 2 and Kroemer for a lucid and complete exposition): 72). Given the quantum statistical definition of τ , and the definition of T in terms of τ and the triple point of wa- Entropy and Temperature ter, one could, in principle, compute k in terms of the atomic constants such as e, me, and h if one could solve A closed system of many particles exists in a number the Schr¨odinger equation for water at its triple point in of distinct quantum states consistent with conservation all its terrible complexity. But that is a hopeless task, constraints of the total energy of the system and the to- so one must turn to empirical determinations of the pro- tal number of particles. For a system with g accessible portionality constant based on experiments that link the states, the fundamental entropy σ is defined by macroscopic and microscopic aspects of the world. A link between the microscopic and macroscopic was σ = ln g (3) reported by Johnson in 1928 [1] in a paper paired in the With the addition of heat the number of states ac- Physical Review with one by H. Nyquist [2] that provided cessible within the limits of energy conservation rises, a rigorous theoretical explanation based on the principles and the entropy increases. An exact enumeration of the of classical thermal physics. Johnson had demonstrated quantum states accessible to a system composed of many experimentally that the mean square of the voltage across non-interacting particles in a box and having some defi- a conductor is proportional to the resistance and abso- nite energy can be derived from an analysis based on the lute temperature of the conductor and does not depend solutions of the Schr¨odinger equation (Ref. [4], p 77). It on any other chemical or physical property of the con- shows that for one mole of a gas at standard tempera- ductor. At first thought, one might expect that the mag- ture and pressure (273K, 760 mm Hg) σ is of the order nitude of Johnson noise must depend in some way on of 1025. The corresponding value of g is of the order of the number and nature of the charge carriers. In fact 25 the huge number e10 ! Nyquist’s theory involves neither e nor N. It yields a Suppose that the total energy U of the system is in- result in agreement with Johnson noise observations and creased slightly by ∆U, perhaps by the addition of heat, a formula for the mean square of the noise voltage which while the volume and number of particles are held con- relates the value of the Boltzmann constant to quantities stant. With the increase in energy more quantum states that can be readily measured by electronic methods and become accessible to the system so the entropy is in- thermometry. creased by ∆σ. The fundamental temperature is defined by NYQUIST’S THEORY OF JOHNSON NOISE
1 ∂σ (4) Two fundamental principles of thermal physics are τ ≡ ∂U used: Ã !N,V The units of τ are evidently the same as those of en- 1. The second law of thermodynamics, which implies ergy. Since an increase in the energy of one mole of gas that between two bodies in thermal equilibrium at Differential the same temperature, in contact with one another Amplifier
but isolated from outside influences, there can be A R R C Vj’
no net flow of heat; B Vj 2. The equipartition theorem of statistical mechanics [4], which can be stated as follows: FIG. 1: Equivalent circuit of the thermal emf across a con- ductor of resistance R connected to a measuring device with “Whenever the hamiltonian of a sys- cables having a capacitance C tem is homogeneous of degree 2 in a canonical momentum component, the thermal average kinetic energy associ- presented to the input of the measuring device (in our ated with that momentum is kT/2, where case the A–input of a low-noise differential preamplifier) T is the Kelvin temperature and k is is a fluctuating voltage with a mean square value Boltzmann’s constant. Further, if the 2 dVj0 = 4Rf kT df (7) hamiltonian is homogeneous of degree 2 in a position coordinate component, the where thermal average potential energy associ- ated with that coordinate will also be R R = (8) kT/2.” f 1 + (2πfCR)2 If the system includes the electromagnetic field, This equation is equivalent to Equation (A9) in Ap- 2 then the Hamiltonian includes the term (E + pendix B and results from AC circuit theory. 2 B )/8π in which E and B are canonical variables Attention was drawn earlier to an analogy between corresponding to the q and p of a harmonic oscil- the mean square of the Johnson noise voltage across a lator for which (with p and q in appropriate units) conductor and the pressure of a gas on the walls of a 2 2 the hamiltonian, is (q + p )/2. container. Both are proportional to kT and the num- ber of degrees of freedom of the system. The big differ- Nyquist’s original presentation of his theory is magni- ence between the two situations is that the number of ficient, please see the Junior Lab e-library for a copy. translational degrees of freedom per mole of gas is the Nyquist invoked the second law of thermodynamics to “unknown” quantity 3N, while the number of oscillation replace the apparently intractable problem of adding up modes within a specified frequency interval in the trans- the average thermal energies in the modes of the elec- mission line invoked by Nyquist in his theory is readily tromagnetic field around a conductor of arbitrary shape calculated from the laws of classical electromagnetism. and composition with an equivalent problem of adding Since the Boltzmann constant is related to the number up the average thermal energies of the readily enumer- of accessible quantum states, one might well ask: ated modes of electrical oscillation of a transmission line shorted at both ends. Each mode is a degree of freedom Where is Planck’s constant, h, which fixes the of the dynamical system consisting of the electromagnetic actual number of accessible states? field constrained by the boundary conditions imposed by the transmission line. According to the equipartition the- The answer is that the Nyquist theorem in its original orem, the average energy of each mode is kT , half electric form, like the classical Rayleigh-Jeans formula for the and half magnetic. spectral distribution of blackbody radiation, is valid only The Nyquist formula for the differential contribution in the range of frequencies where hf << kT , in other 2 words, at frequencies sufficiently low that the minimum dVj to the mean square voltage across a resistor of resis- tance R in the frequency interval df due to the fluctuating excitation energy of the oscillations is small compared to 14 emfs corresponding to the energies of the modes in that kT . At 300K and 100 kHz, kT = 4 10− ergs (0.04 22 × interval is eV) and hf = 6 10− ergs. Thus at room temperature kT is 108 times× the minimum energy of an oscillation mode with∼ a frequency near 100 kHz. The exact quantum 2 dVj = 4RkT df (6) expression for the mean energy ² of each oscillation mode, noted by Nyquist in his paper, is To measure this quantity, or rather its integral over the frequency range of the pass band in the experiment, hf one must connect the resistor to the device by means of ² = (9) ehf/kT 1 cables that have a certain capacitance C which shunts − (short circuits) a portion of the signal, thereby reducing which reduces to ² = kT for hf << kT over the range its RMS voltage. The equivalent circuit is shown in Fig- of frequencies and temperatures encountered in this ex- 2 ure 1. The differential contribution dVj0 to the signal periment. On the other hand, for the electrical noise at the input of a radio telescope operated at liquid he- R lium temperature for measurement of the 2.7K cosmic background radiation at frequencies near the∼ peak of the Shielded A 11 Twisted Pair AC Krohn-Hite black body spectrum, i.e. around 10 Hz. To SW2 Differential 1kHz - 50kHz Digital Ohmmeter Amplifier Bandpass Oscilloscope GND Filter SW1 B or AC
EXPERIMENT Variable Sine-Wave Kay Connected for calibration only! Generator Attenuator In the present experiment you will actually measure an amplified version of a portion of the Johnson noise FIG. 2: Block diagram of the electronic apparatus for mea- power spectrum. The portion is defined by the “pass suring Johnson noise. band” of the measurement chain which is determined by a combination of the gain characteristics of the amplifier be represented as a Fourier series consisting of a sum of si- and the transmission characteristics of the low-pass/high- nusoids with discrete frequencies n/2t, n=1, 2, 3 ...., each pass filters that are included in the measurement chain to with a mean square amplitude equal to the value specified provide an adjustable and sharp control of the pass band. by the equipartition theorem. When the Fourier series is The combined effects of amplification and filtration on squared the cross terms are products of sinusoids with any given input signal can be described by a function of different frequencies, and their average values are zero. frequency called the effective gain and defined by Thus the expectation value of the squared voltage is the sum of the expectation values of the squared amplitudes, and in the limit of closely spaced frequencies as t , V0(f) g(f) = (10) the sum can be replaced by an integral. → ∞ Ã Vi(f) ! 2 Given the linear dependence of VJ on T in Eq. 13, it is evident that one can use the Johnson noise in a resistor where the right side is the ratio of the RMS voltage as a thermometer to measure absolute temperatures. A V out of the bandpass filter to the RMS voltage V of a 0 i temperature scale must be calibrated against two phe- pure sinusoidal signal of frequency f fed into the ampli- nomena that occur at definite and convenient tempera- fier. A critical task in the present experiment is to measure the square of the effective gain as a tures such as the boiling and melting points of water, function of frequency for the particular settings which fix the centigrade scale at 100◦ C and 0◦C, re- of the measurement chain which are used in the spectively. In the present experiment you will take the subsequent measurement of Johnson noise. centigrade calibrations of the laboratory thermometers for granted, and determine the centigrade temperature When the input of the measurement chain is connected of absolute zero as the zero-noise intercept on the nega- across the conductor under study, the contribution dV 2 J tive temperature axis. to the total mean square voltage out of the bandpass filter in a differential frequency interval is PROCEDURE OVERVIEW 2 2 2 dV = [g(f)] dV 0 (11) J j The experiment consists of the following parts: where V 2 is the voltage measured across the resistor. j0 1. Calibration of the measurement chain; We obtain an expression for the measured total mean square voltage by integrating Equation 11 over the range 2 2 2 2. Measurement of VJ0 = VR VS , where VR = RMS of frequencies of the pass band. Thus voltage at the output of the− band-pass filter with the resistor in place; and VS = RMS voltage with
2 the resistor shorted); for various resistors and tem- VJ = 4RkT G (12) peratures;
where the quantity G is defined by 3. From the data, determine the Boltzmann constant;
4. Determine the centigrade temperature of absolute 2 ∞ [g(f)] zero. G 2 df (13) ≡ 0 1 + (2πfCR) Z Figure 2 is a schematic diagram of the apparatus show- The rationale behind this integration is that over any ing the resistor R mounted on the terminals of the alu- given time interval t the meandering function of time that minum box, shielded from electrical interference by an is the instantaneous noise voltage across the resistor can inverted metal beaker, and connected through switches SW1 and SW2 to the measurement chain or the ohm- Measuure the variation of the test signal RMS voltage and meter. The measurement chain consists of a low-noise the gain of the measurement chain as a function of frequency differential amplifier, a band-pass filter, and a digital os- cilloscope. Sinusoidal calibration signals are provided by Select the sinusoid signal of the function generator a function generator. (FG) and adjust its amplitude so the RMS voltage Vi To minimize the problem of electrical interference in out of the FG is 2V as measured on the digital oscillo- the measurement of the low-level noise signals it is es- scope on Channel∼1. Use a tee to simultaneously send the sential that all cables be as short as possible. The two test signal through the Kay attenuator with 60 dB (1000) cables that connect the resistor to the ‘A’ and ‘B’ input of attenuation to the ‘A’ input of the amplifier with the connectors of the differential amplifier should be tightly ‘B’ input grounded. Display this attenuated signal on twisted, as shown, to reduce the flux linkage of stray AC Channel 2 of the oscilloscope. magnetic fields. Without touching the amplitude control of the FG, The digital oscilloscope emits a variable magnetic field measure and record the RMS voltage of the FG signal from its beam-control coil which will have a devastating at several frequencies over the range that will pass the effect on your measurements unless you keep it far away filter ( 0.5 kHz to 100 kHz). Plot the measured RMS ( 5 feet) from the noise source. Take special care to ∼ ∼ ≥ voltages against the frequencies so as to have a handy avoid this problem when you arrange the components of database of the input RMS voltages required for the de- the measurement chain on the bench. termination of g(f). You should expect to find that the There are various possible choices of settings and pro- RMS voltage varies only slightly over most of the fre- cedure for carrying out measurements that will yield a quency range of interest. Also Measure the RMS volt- value of k. The following recipe works satisfactorily, but ages out of the filter at the same frequencies to define you may well devise a better one. The procedure is based accurately the curve of the square of the gain [g(f)]2 ver- on the assumption that the component of VR not gener- sus f. Plot g2 against f as you go along to check the ated in the resistor can be measured with sufficient ac- consistency and adequacy of your data. curacy by The digital oscilloscope can be set up for these mea- 1. opening switch SW2, surements as follows:
2. unplugging the connections to the ohmmeter and 1. Press ’1’ under the channel ’1’ input. temperature meter, and 2. Select ’COUPLING:AC’; ’Bandwidth limit: ON’. 3. shorting the resistor with switch SW1. The pass band of the Krohn-Hite filter is fixed with 3. Select ’VOLTAGE Measurement: Vrms’. the 3dB High Pass frequency = 1 KHz and the Low Pass 3dB point = 50 KHz. 4. Select ’Frequency Measurement’.
5. Press ’DISPLAY’, and select ’AV 8’ for test sinu- DETAILED PROCEDURE soid measurements, and ’normal’ for noise measure- ments. (Note: You must select ‘normal’ for noise Calibrate the measurement chain measurements since the RMS voltage of the average of n random wave forms approaches zero as n . → ∞ Typical RMS voltages of the Johnson noise signals you On the other hand, the RMS voltage of the aver- will measure are of the order of several microvolts. The age of many wave forms consisting of a constant digital oscilloscope can measure the RMS voltage of both sinusoid plus random noise approaches the RMS periodic and random signals over a dynamic range of voltage of the pure sinusoid.) somewhat more than 103, from several millivolts to sev- eral volts. Thus, with the differential amplifier set to 6. Adjust the digital scope amplitude and sweep- a nominal gain of 1000, the microvolt noise signals will speed controls so that several ( 5-10) cycles of the ∼ be amplified sufficiently to be measured in the millivolt sinusoid appear on the screen. range of the oscilloscope. To determine the overall ampli- fication of the amplifier/filter combination, one can feed To reduce errors of measurement and obtain an error a sinusoidal test signal with an RMS voltage Vi in the assessment, you can use the ‘stop’ and ‘run’ buttons to millivolt range to the ‘A’ AC-Coupled of the PAR ampli- advantage. Press ‘run’ and then ‘stop’, record the Vrms fier (with the ‘B’ input grounded), and measure the RMS and frequency displayed at the bottom of the screen; re- voltage V0 of the output of the filter in the volt range of peat n times (e.g., n=5) at each setting. For each setting the oscilloscope. The gain of the system at the frequency compute the mean Vrms, and the standard error of the σ of the test signal is g(f) = V o/V i. mean (= √n 1 ). − 02 Measure V J for a variety of resistors Measure Johnson noise as a function of temperature
With a 20 KΩ resistor and the amplifier gain set to Measure the Johnson noise in a metal film or wire 1000, the ∼RMS voltage at the output of the bandpass wound resistor over a range of temperatures from that of filter should be in the range of several millivolts. About liquid nitrogen (77 K) to 150◦C. Clip the 30kΩ resis- ∼ ∼ half the RMS voltage is noise generated in the amplifier tor with the thermistor attached to the aligator clips, and itself. Interference pickup may vary. Since all the con- plug the thermistor leads into the color-coded sockets, tributions to the measured RMS voltage are statistically taking care with the delicate wires of the thermister. To uncorrelated, they add in quadrature. To achieve ac- make a temperature measurement, plug the thermome- curate results it is essential to make repeated measure- ter into the color-coded socket on the side of the box. ments with each resistor with the shorting switch across The high temperatures are obtained by inverting the box the conductor alternately opened and closed. The mea- with the mounted test resistor and thermister and plac- sure of the mean square Johnson noise is ing it in the cylindrical oven which contains heater tape and insulation. Connect the heater to the Variac and set the Variac at 40 V. Measure the resistance, the RMS 2 2 2 ∼ VJ0 = VR VS (14) voltages of the Johnson noise and the background as the − temperature rises. For cold measurements, fill the De- where VR and VS are the RMS voltages measured with war flask with liquid nitrogen and place it in the large the shorting switch open and closed, respectively. metal beaker. Then rest the inverted box on the lip of Measure the Johnson noise at room temperature in the metal beaker to assure good electrical shielding. 10 metal film and/or wire-wound resistors with values According to the Nyquist theory the points represent- ∼ 3 6 2 from 10 to 10 ohms. Mount the resistors in the alliga- ing the measured values of VJ0 /4RG plotted against T tor clips projecting from the aluminum test box equipped (◦C degrees) should fall on a straight line with a slope with a single pole single throw (SPST) shorting switch, equal to the Boltzmann constant, and an intercept on a double pole double throw (DPDT) routing switch, and the temperature axis at the centigrade temperature of connections for a thermistor for use in the later temper- absolute zero. Note that if the resistance of the conduc- ature measurement. Cover the resistor and its mounts tor varies significantly with temperature, then G must with a metal beaker to shield the input of the system be evaluated separately at each temperature, i.e. the from electrical interference. After each noise measure- integral of Equation 13 must be evaluated for each sig- ment measure the resistance of the resistor: plug a dig- nificantly different value of the resistance. ital multimeter into the pin jacks on the aluminum box 2 and flip the DPDT switch on the sample holder to the re- 1. Make a plot of VJ0 /4RG against T (in ◦C degrees). sistance measuring position. Before each noise measure- 2. Derive a value and error estimate of k from the ment, be sure to disconnect the multimeter (to avoid in- slope of the temperature curve. troducing extraneous electrical noise) and flip the DPDT switch back to the noise-measurement position. 2 3. Derive a value and error estimate of the centigrade Plot VJ0 /R against R . See how precisely you can temperature of absolute zero. account for the results by a proper choice for the values of k and C. According to equation 12 the value of k can be expressed in terms of measured quantities and G, SHOT NOISE which is a function of R and C: A current source in which the passage of each charge V 2 carrier is a statistically independent event (rather than a k = J0 (15) 4RT G steady flow of many charge carriers) necessarily delivers a “noisy” current, i.e., a current that fluctuates about an The factor G must be recalculated by numerical inte- average value. Fluctuations of this kind are called “shot gration for each new trial value of R and C. In principle, noise”. The magnitude of such fluctuations depends on if you use the correct value of C in your calculations of the magnitude of the charges on the individual carriers. G, then the values of k obtained from Equation 15 should Thus a measurement of the fluctuations should, in prin- cluster around a mean value close to the value for R 0 ciple, yield a measure of the magnitude of the charges. → and should not vary systematically with R. Using the Consider a circuit consisting of a battery, a capacitor best values of k and C derived in this way, you can plot in the form of a photodiode, a resistor, and an inductor, 2 a calculated curve of VJ0 /R against R for comparison connected in series as illustrated in Figure 3a. Illumi- with your data. nation of the photodiode with an incoherent light source Question: What happens as R ? causes electrons to be ejected from the negative electrode → ∞ Light Bulb I(t)
e- i(t)
|e| = Integral from 0 to tau of i(t)dt Photocathode
R F Iav = ke
t - +
τ
FIG. 3: (a) Schematic diagram of a circuit in which the cur- t rent consists of a random sequence of pulses generated by the T passage of photoelectrons between the electrodes of a photo- diode. (b) Schematic representation of the current pulse due FIG. 4: Plot of a fluctuating current against time with a to the passage of one photoelectron. straight line indicating the long-term average current. in a random sequence of events. Each ejected photoelec- The fluctuating current is tron, carrying a charge of magnitude e, is accelerated to the positive electrode, and during its passage between the electrodes it induces an increasing current in the circuit, I(t) = Σkik(t) (16) as shown in Figure 3b. When the electron hits the positive plate the current and its mean square during the time interval T is continues briefly due to the inductance of the circuit and T a damped oscillation ensues. The shape of the current 2 1 2 pulse depends on the initial position, speed and direc- < I >= [Σkik(t)] dt (17) T 0 tion of the photoelectron as well as the electrical char- Z acteristics of the circuit. The integral under the curve The problem is to derive the relation between the mea- is evidently the charge e. If the illumination is strong surable properties of the fluctuating current and e. Al- enough so that many events occur during the duration though the derivation is somewhat complicated (see Ap- of any single electron pulse, then the current will appear pendix C), the result is remarkably simple: within the as in Figure 4, in which the instantaneous current I(t) Iav frequency range 0 < f << e the differential contri- fluctuates about the long-term average current Iav . bution to the mean square of the total fluctuating cur- In this experiment you will measure, as a function of rent from fluctuations in the frequency interval from f to Iav, the mean square voltage of the output of an am- f + df is (see Appendix B) plifier and a band pass filter system whose input is the 2 continuously fluctuating voltage across RF . According d < I >= 2eIavdf (18) to the theory of shot noise a plot of this quantity against the average current should be a straight line with a slope Suppose the fluctuating current flows in a resistor of proportional to e . The problem is to figure out what the resistance RF connected across the input of an amplifier- proportionality factor is. filter combination which has a frequency-dependent gain g(f). During the time interval T the voltage developed across the resistor, IRF , can be represented as a sum THEORY OF SHOT NOISE of Fourier components with frequencies m/2T , where m = 1, 2, 3, ..., plus the zero frequency (DC) component of amplitude I R . Each component emerges from the The integral under the curve of current versus time av F amplifier-filter with an amplitude determined by the gain for any given pulse due to one photoelectron event is e, of the system for that frequency. The mean square of the charge of the electron. If the illumination is con- the sum of Fourier components is the sum of the mean stant and the rate of photoelectric events is very large, squares of the components (because the means of the then the resulting current will be a superposition of many cross terms are all zero). Thus, in the practical limit of such waveforms i (t) initiated at random times T with k k a Fourier sum over closely spaced frequencies, we can ex- a “long term” average rate we will call K, resulting in press the mean square voltage of the fluctuating output a fluctuating current with an average value I = Ke, av signal from the amplifier-filter as the integral as illustrated in Figure 4. The fluctuating component of such a current was called “shot noise” by Schottky in 1919 who likened it to the acoustic noise generated by a V 2 = 2eI R2 [g(f)]2df + V 2 (19) hail of shot striking a target. 0 av F A Z0 2 where VA has been added to represent the constant Photodiode Differential Bandpass Digital contribution’s of the amplifier noise , and Johnson Noise Box Amplifier Filter Oscilloscope variable in Rf to the total mean square voltage, and where the light test-in DC term is omitted because the DC gain of the amplifier- source Multimeter filter is zero. To get a feel for the plausibility of the shot noise for- mula one can imagine that the current in the photodi- Function To oscilloscope when measuring shot noise Generator To "test in" when measuring the gain versus frequency ode circuit is a step function representing the amount of charge ne released in each successive equal time interval of duration τ divided by τ, i.e., the mean current in each FIG. 5: Block diagram of the experimental arrangement for ne measuring shot noise. interval τ . The expectation value of n is < n >= Kτ. We assume there is no statistical correlation between the numbers of events in different intervals. According to Poisson statistics the variance of n (mean square devia- 3. Calculation of the charge of the photoelectrons. tion from the mean) is the mean, i.e.,
CALIBRATION OF THE MEASUREMENT < (n < n >) >2=< n2 > < n >2=< n >= Kτ CHAIN − − (20) It follows that the mean square value of the current Figure 5 is a block diagram of the electronic appara- over time would be tus, and Figure 6 is a diagram of the diode circuit and ne 2 preamplifier. The current I(t) in the photodiode circuit < I2 > = < > τ is converted to a voltage V = IRF at the point indicated e³ 2 ´ in Figure 6 by the operational amplifier with precision = < n2 > τ feedback resistors in the first stage of the preamplifier ³ ´ 2 inside the photodiode box. This voltage is filtered to re- e = < n > + < n >2 move frequencies <100 Hz and is fed to the second stage Ãτ ! where it is amplified by a factor of 10. The output h i ∼ I e signal is further amplified with the differential amplifier, = av + I2 (21) τ ave filtered with the bandpass filter and measured with the which shows that the fluctuating term is proportional RMS voltmeter. The photodiode box has a test input for calibration of to eIav as in the exact expression for the differential con- tribution, Equation 18. the overall gain of the measurement chain as a function Actually the current at any given instant from an il- of the frequency. The typical shot noise RMS voltage luminated photodiode is the sum of the currents due to across the precision resistor RF is of the order of 10 mi- the photoelectrons ejected during the previous brief time crovolts. Since the gain of the circuit in the photodiode box is 10, a gain of 100 in the amplifier will yield a interval. Thus the currents at any two instants separated ∼ 3 in time by less than the duration of the individual pulses total gain of 10 and bring the signal in the pass band of the filter up to the 10 millivolt level that can be readily are not statistically independent. Moreover, the simple ∼ scheme provides no handle on the frequency spectrum of measured by the digital oscilloscope. As in the Johnson the noise which one must take into account in evaluating noise calibration, you can determine the effective gain of the response of the measurement chain. One approach to the amplifier-filter system as a function of frequency by a rigorous solution is presented in Appendix B. Others feeding a millivolt sinusoid signal of measured RMS volt- are possible. age from the function generator into the test input of the phototube box and measuring the RMS voltage of the signal out of the filter. You can use the same band pass SHOT NOISE EXPERIMENTAL PROCEDURE settings of the filter as in the Johnson noise measurement.
The procedure has three parts: Calibration 1. Calibration of the gain of the measurement chain as a function of frequency; Select the sine wave output of the function generator, 2. Measurement of the mean square noise voltage at set the the RMS voltage to 2 mV, and feed the signal the output of the measurement chain as a function directly to the digital oscilloscop∼ e. Without touching the of the average current in the diode circuit as it is amplitude control on the function generator, measure the varied by changing the intensity of illumination. RMS voltage for several frequencies over the range of the R F V = Rf I 476k 10 k Photodiode
- 0.3 µ f - [1] J. B. Johnson, ”Thermal agitation of electricity in con-
+ 1k 2nd stage output ductors”, Phys. Rev. 32, 97 (1928). -45V 476k + [2] H. Nyquist, ”Thermal agitation of electric charge in con-
10k 1k ductors”, Phys. Rev. 32, 110 (1928). "test in" [3] C. Kittel, W. R. Hackleman, and R. J. Donnelly, Amer.
1st stage output J. Phys., 46, 94 (1978). [4] C. Kittel and H. Kroemer, Thermal Physics, Freeman: FIG. 6: Diagram of the photodiode and preamplifier circuit. New York, 1980. [5] Goldman, S. 1948, Frequency Analysis, Modulation and Noise, McGraw-Hill. [6] Horowitz, P. and Hill, W. 1980, The Art of Electronics, filter band pass and plot the result so you have a handy Cambridge:London. data base for the subsequent gain measurements. [7] Reif, F. 1965, Fundamentals of Statistical and Thermal Measure the gain of the measurement chain as a func- Physics, McGraw-Hill:New York, pp. 582-587. tion of frequency. Make sure the photodiode tube voltage [8] Schwartz, M. 1959, Information Transmission, Modula- is switched off so that the tube is an open circuit with tion, and Noise, McGraw-Hill: New York. no photoelectric current. Switch on the amplifier volt- [9] Van Der Ziel, A. 1976, Noise in Measurements, Wiley:New age on the phototube box. Feed test signals of various York , pp. 30-39. SUGGESTED THEORETICAL TOPICS frequencies into the ‘test input’ and measure the RMS voltage at the output of the bandpass filter. Use the sig- The Nyquist theorem. nal averaging feature and the ‘stop’, ‘run’ buttons on the • oscilloscope to enhance the accuracy of your measure- Shot noise theory • ments. Plot the values of g2 as you go along to assess where you need more or less data to define accurately the gain squared integral.
MEASUREMENT OF THE AVERAGE CURRENT AND THE CURRENT NOISE
Remove the cable from the test input and cover the input plug with the cap provided to minimize the pickup of electrical noise from the outside. Set the multi-meter to “volts” on the 200 mV scale and plug it into the “first stage output” to measure the voltage RF I. Leave the rest of the measurement chain just as it was when you calibrated it. Verify that the flashlight bulb is actually working by opening the box and looking. If it doesn’t glow brightly when the voltage is turned up, fix it. Record the RMS voltage and the multimeter read- ing for various settings of the light bulb voltage control. Many repeated measurements at each light intensity will beat down the random errors.
ANALYSIS
2 Plot Vo as a function of the combined quantity
2 ∞ 2 2RF Iav g (f)df (22) Z0 From the slope of this line determine the charge on the electron. APPENDICES
dQ I = (25) A MECHANICAL EXPERIMENT TO dt DETERMINE k 2 We seek an expression in terms of d < VJ >, R, and C for the contribution to the RMS voltage across the Before turning to a detailed consideration of the John- terminals in a narrow frequency range, i.e. son noise experiment, it is amusing to consider the pos- sibility of a mechanical determination of k with a macro- scopic system having one degree of freedom, namely a 2 2 d < V 0 >= d < Q > /C (26) delicate torsion pendulum suspended in a room in ther- J mal equilibrium (i.e. no drafts, etc.) at temperature Consider one Fourier component of the fluctuating T . The degree of freedom is the angular position θ with thermal emf across the resistor, and represent it by the which is associated the potential energy 1 κθ2. Accord- 2 real part of ν = ν expjωt, where j = √ 1. The re- ing to the equipartition theorem (see below), the mean J 0 sulting current is the real part of i = i−expj ωt, the thermal potential energy is 0 charge on the capacitor is the real part of its integral q = (j/ω)i, and the desired output voltage is the real − q j 1 1 part of = ( i). Substituting the expressions for i κ < θ2 >= kT (23) C − ωC 2 2 and q into Equation 24 and canceling the time-dependent terms, we find where κ is the torsion constant of the suspension, and < θ2 > is the mean square angular displacement of the pendulum from the equilibrium orientation. Thus, in j ν0 = (R )i0 (27) principle, by measuring < θ2 > over a time long com- − ωC pared to the period, one can determine k . To judge Solving for i we obtain the relation what this might require in practice, imagine a torsion 0 balance consisting of a tiny mirror (for reflecting a laser beam) suspended by a 0.5 mil tungsten wire 10 feet long. ν0 io = (28) Such a suspension has a torsion constant of the order of R j 3 1 ωC 10− dyne cm rad− . According to Equation 23, at 300 − K the value of < θ2 >1/2, i.e. the RMS value of the an- so gular deflection, would be about 1 arc second. Such an experiment might be possible, but would be exceedingly q j jνJ difficult. ν0 = = ( )i = − (29) J C − ωC ωRC j −
DERIVATION OF THE RMS THERMAL The statistically independent contribution which this VOLTAGE AT THE TERMINALS OF AN RC mode gives to the measured total mean square noise volt- CIRCUIT age is the square of its amplitude which we find by mul- tiplying νJ0 by its complex conjugate: Figure 1 shows the circuit equivalent to the resistor and coaxial cables that are connected to the PAR preampli- 2 2 < νJ > fier for the measurement of Johnson noise. The equiva- < ν0 >= (30) J 1 + (ωRC)2 lent circuit consists of a voltage source of the fluctuating thermal emf VJ in series with an ideal noiseless resistor of Summing all such contributions in the differential fre- resistance R and a capacitor of capacitance C. According quency range df, we obtain to Faraday’s Law, the integral of the electric field around the RC loop is zero, so 2 < dV 0J >= 4Rf kT df (31) Q V = IR + (24) J C where
According to charge conservation (from Ampere’s Law R and Gauss’ Law), the current into the capacitor equals Rf = (32) the rate of change of the charge on the capacitor, so 1 + (2πfRC)2 2 DERIVATION OF THE SHOT NOISE EQUATION obtain the differential contribution d < I0 > in the fre- quency interval df to the mean square of the sum. We Following is an abbreviated version of the shot noise use the well known fact that the mean square of the sum theory given by Goldman (1948). We begin by expressing of all the Fourier components is the sum of the mean the current in the photodiode circuit due to a single event squares of the individual components (the mean values that occurs at time Tk, like that depicted in Figure 3b, of the cross-frequency terms in the squared Fourier series 2 as a Fourier series over a long time interval from 0 to T . are all zero). The quantity d < I0 > is therefore the sum Calling this current pulse i(t Tk) we represent it as a of the mean squares of the individual contributions in the Fourier series: − frequency range df. To evaluate it we first focus atten- tion on the nth Fourier component which we represent by a ∞ 2πnt 2πnt i(t T ) = 0 + a cos + b sin (33) − k 2 n T n T n=1 à ! X 2πnt cn cos φn (38) where à T − !
to which the kth event contributes the quantity 2 T 2e a0 = i(t Tk)dt = (34) T × 0 − T Z 2e 2πn(t T ) cos − k (39) T Ã T ! T 2 2πnt 2e 2πnTk an = i(t Tk) cos dt = cos (35) th 2 T × 0 − T T T The mean square value of the n component is cn; our Z 2 immediate problem is to evaluate the quantity cn. Since the events occur at random times from 0 to T , their T th 2 2πnt 2e 2πnTk contributions to the n component have random phases bn = i(t Tk) sin dt = sin (36) which are distributed uniformly from 0 to 2π. Conse- T × 0 − T T T Z quently, we must add them as vectors. To do this we with i(t Tk) = eδ(t Tk). first group them according to their phase. The expected − − dφ The area under the curve in Figure 3b is the charge e number with phases between φ and φ + dφ is 2π KT . of one electron; it represents the “impulse” of the shot, and is accounted for in the Fourier representation by dφ the lead term in the series whose coefficient is given q = KT (40) 2π by Equation 34. To justify Equations 35 and 36 in the context of the present experiment we note that the Combining this with equations 38 and 39 we find gain of the amplifier-filter system used in this measure- that the average value of the sum of the contributions ment is different from zero only for frequencies such that with phase angles in the range from φ to φ + dφ for the f = n/T << 1/τ. Consequently we can confine our Fourier component of frequency n/T is calculation of the Fourier coefficients to those for which nτ << T . It follows that the cos and sin factors in the integrands of equations 35 and 36 do not vary signifi- dφ 2e 2πnt Ke 2πnt KT cos φ cos φ (41) cantly over the range of t in which i(t Tk) differs from 2π T Ã T − ! π Ã T − ! 0, and that they can therefore be tak−en outside their integrals with their arguments evaluated at the instant We now represent each one of the preliminary sums of the event. In other words, the function representing given by equation 40 by a differential vector in a two the current impulse of a single event acts like a delta- dimensional phase diagram. Added head to tail in order function. Substituting the expressions for a0, an, and bn of increasing phase, these vectors form a closed circular from equations 34, 35, and 36 into 33 we obtain polygon of many sides. If instead of the exact expected numbers of events contributing to each differential vector we use the actual numbers q1, q2, ....., then in general the e 2e ∞ 2πn(t Tk) polygon will not quite close due to statistical fluctuations i(t Tk) = + cos − (37) − T T T in these numbers. The line segment that closes the gap is n=1 X h i the overall vector sum for this particular Fourier compo- We suppose now that many such events pile up to pro- nent. It will have a random direction and a tiny random duce the total current at any given instant. We seek a length that represents the net effect of the fluctuations. way to add the currents due to the individual events to Its x and y components will have zero expectation values, but finite variances (mean square values). Each contri- and it follows that the contribution of the nth Fourier 2 2 2
4e2 (q cos2 φ + q cos2 φ + ...) = T 2 1 1 2 2 2 2π 2 4e KT 2 2e K = cos φdφ d < I2 >= 2 T 2 2π 0 T df = 2e Kdf (44) Z0 T 2e2K = (42) T The mean square of the y components has the same value. And since the vector itself is the hypotenuse of The average current due to the many events is Iav = the right triangle formed by the x and y components, Ke. Thus the final expression for the differential contri- the mean square of its length is the sum of the mean bution to the mean square of the fluctuating component squares of the two components. Thus of the current from the differential frequency interval df is just that given by Equation 18 from which follows Equa- tion 19, the desired formula for the relation between mea- 4e2K < c2 >= (43) sured quantities and e. n T