Unified derivation of Johnson and shot noise expressions
Luca Callegaro
Istituto Nazionale di Ricerca Metrologica (INRIM), Strada delle Cacce, 91-10135 Torino, Italy
共Received 31 October 2005; accepted 20 January 2006兲
Shot noise and Johnson noise in electrical circuits are usually introduced by referring to completely
separate physical models and derivations. We derive Johnson and shot noise expressions from the
same physical model, an ideal tunnel junction, to show the deep connection between the two types
of noise. The derivation uses concepts of quantum mechanics, thermodynamics, and signal
processing. © 2006 American Association of Physics Teachers.
关DOI: 10.1119/1.2174034兴
I. INTRODUCTION
The sources of noise observed in electrical circuits are
usually attributed to Johnson noise and shot noise. Johnson
noise 共from the names of those who did the first exhaustive
experimental observations1 and theoretical explanation2 of
the phenomenon兲 is the random voltage 共or current兲 caused
by thermal fluctuations of the electric charge in conductors.
See Ref. 3 for a pedagogical review. Shot noise is caused by
the discreteness 共quantization兲 of charge carriers and can be
observed when a detector measures a current. Shot noise was
first described by Schottky.4,5
These disparate descriptions of Johnson noise and shot
noise 共see Refs. 6 and 7 for a typical introduction兲 suggest
separate explanations and models of the two noise processes.
The derived quantitative expressions are independent and the
corresponding noise power densities are additive. This independence has been questioned for a long time and in the last
few decades the treatment of electrical conductance as a
quantum-mechanical transmission phenomenon has shown
that both types of noise have the same explanation8 and unified descriptions are now commonplace 共see Refs. 9 and 10
for reviews兲.
A unified picture has not yet appeared in general physics
and engineering courses, possibly because of the difficulties
involved in a rigorous derivation. The aim of this paper is to
show that if rigor is not an issue, expressions for the thermal
and shot noise can be derived from the same physical model.
The derivation invokes several assumptions from various
branches of physics and signal analysis and is a good review
problem for physics or electronic courses.
II. THE MODEL
Consider an ideal model of a physics device, the tunnel
junction. Figure 1 shows a tunnel junction J with a small
insulating gap g 共vacuum, dielectric兲 dividing two metal contacts A and B connected to an ideal voltage generator G with
constant voltage V. The current carriers have an elementary
charge q; for convenience, we will call them electrons. We
let I共t兲 equal the current flowing in the electrical circuit. The
system is isothermal at temperature T.
Because electrons in a circuit are quantum-mechanical
particles, there are finite probabilities per unit time PAB and
PBA that an electron crosses by tunneling through the junction in the A → B or A ← B direction, respectively. The charge
transfer is considered instantaneous. If we know the nature of
the junction, quantum mechanics10 lets us calculate the actual values of PAB and PBA. However, we will assume phe438
Am. J. Phys. 74 共5兲, May 2006
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nomenological values for PAB and PBA dependent on the
junction and its environment, in particular V and T.
We assume that current is not observed continuously, but
in discrete chunks of duration ␶ ; for each chunk only its
average value Ī = 兰␶ I共t兲dt is recorded. We can think of this
observation as given by an integrating analog-to-digital converter with a sampling frequency f s = 1 / ␶. From Nyquist’s
sampling theorem,11 the equivalent bandwidth B for the observation is half the sampling frequency,
B=
fs 1
= .
2 2␶
共1兲
We assume that the sampling time is sufficiently rapid that
only the following events can be observed. 共a兲 Ī = + q / ␶. An
electron has crossed the junction in the A → B direction. Such
an event has a probability PAB␶. 共b兲 Ī = −q / ␶. An electron has
crossed the junction in the A ← B direction. Such an event
has a probability PBA␶. 共c兲 Ī = 0. No electron has crossed the
tunnel barrier. Such event has a probability 1 − 共PAB + PBA兲␶.
Thus, at most only one electron passes the barrier in either
direction.
III. CURRENT
The passage of an electron is a probabilistic event and
therefore Ī is a random variable. The probability distribution
p¯I of Ī is discrete 共only three realizations of Ī are allowed兲.
Because the probability of each realization is given, p¯I is
determined and all moments can be calculated. The average
具I典 is given by
1
具I典 =
兺 Īp¯I
k=−1
=
冉 冊
冉 冊
−q
+q
关PBA␶兴 + 共0兲关1 − 共PAB + PBA兲␶兴 +
关PAB␶兴
␶
␶
= q共PAB − PBA兲.
共2兲
The mean square value of the current is
具I2典 =
冉 冊
−q
␶
2
PBA␶ + 共0兲 +
冉 冊
+q
␶
2
PAB␶ =
q2
共P + PBA兲. 共3兲
␶ AB
Therefore, knowledge of PAB and PBA permits us to derive
the average current and current noise power.
© 2006 American Association of Physics Teachers
438
For T = 0 and V ⫽ 0, Eq. 共10兲 gives the pure shot noise
expression,
兩具I2典兩T=0 = 2q具I典B.
共11兲
For T ⬎ 0 and V → 0, Eq. 共10兲 gives the Johnson noise
expression,
lim 具I2典 =
V→0
4kT
B.
R
共12兲
Fig. 1. Schematic of the model. J is a tunnel junction where contacts A and
B are separated by a gap g, connected to a generator G that supplies a
constant voltage V. The current I共t兲 flows in the junction and T is the temperature of the circuit.
VI. COMMENTS
IV. TRANSITION PROBABILITIES
AND MACROSCOPIC PHYSICS
The average current 具I典 can be written in terms of V and
the macroscopic resistance R of the junction,
具I典 =
V
.
R
共4兲
We can think of the tunneling event as a transition between two states: electron in A or electron in B. The states A
and B have occupation numbers nA and nB. The average
number of transition events per unit time in the A → B direction is PABnA; such events decrease nA and increase nB. In the
same unit time, the number of transition events in the A
← B direction is PBAnB. Because the system is in a steady
state, that is, on the average nA and nB are constant, the
detailed balance condition is
PABnA = PBAnB .
共5兲
The states A and B have energies EA and EB; the energy
difference is EA − EB = qV. Their occupation numbers satisfy
the Boltzmann distribution and, hence,
nA
= e−qV/kT .
nB
We have shown that the same model, the tunnel junction,
gives an expression for the noise whose limits for V = 0 and
T = 0 correspond to Johnson and shot noise, respectively. If
V ⫽ 0 and T ⬎ 0, Eq. 共10兲 shows that Johnson noise and shot
noise are not additive even for the ideal model we have considered.
The main result, Eq. 共10兲, is essentially correct, although
its derivation is far from rigorous.12 A review with a full
quantum-mechanical derivation can be found in Ref. 10.
Nyquist has used a simple thermodynamical argument to
show that Eq. 共12兲 is valid for any electric dipole in thermodynamic equilibrium.2 Equation 共12兲 has been generalized
beyond electric circuits by the fluctuation-dissipation
theorem.13 Instead, the shot noise expression Eq. 共11兲 共giving
the Poissonian shot noise value兲 is strongly related to our
assumptions: mainly that charge carriers must be localized
and independent particles. Typically 共although not strictly
necessarily14兲 localization occurs because of a potential barrier. Correlation among charge carriers can generate subPoissonian or super-Poissonian shot noises 共see, for example,
Ref. 15兲.
共6兲
ACKNOWLEDGMENTS
V. NOISE
If we combine Eqs. 共5兲 and 共6兲, we obtain the relation
between PAB and PBA,
PBA
= e−qV/kT ,
PAB
1
共7兲
and, hence,
具I典 =
V
= q共PAB − PBA兲 = qPAB关1 − e−qV/kT兴,
R
共8兲
q2
q2
共PAB + PBA兲 = PAB关1 + e−qV/kT兴.
␶
␶
共9兲
具I2典 =
If we substitute Eq. 共1兲 into 共9兲, we obtain
具I2典 = 2q
439
V 共1 + e−qV/kT兲
B.
R 共1 − e−qV/kT兲
Am. J. Phys., Vol. 74, No. 5, May 2006
The author warmly thanks W. Bich, G. Mana, G. Brida,
and M. Pisani for fruitful discussions and for reviewing the
manuscript.
共10兲
J. B. Johnson, “Thermal agitation of electricity in conductors,” Phys. Rev.
32, 97–109 共1928兲.
2
H. Nyquist, “Thermal agitation of electric charge in conductors,” Phys.
Rev. 32, 110–113 共1928兲.
3
B. Abbott, D. R. Davis, N. J. Phillips, and K. Eshraghian, “Simple derivation of the thermal noise formula using window-limited Fourier transforms and other conundrums,” IEEE Trans. Educ. 39, 1–13 共1996兲.
4
W. Schottky, “Über spontane stromschwankungen in verschiedenen elektrizittsleitern,” Ann. Phys. 23, 541–567 共1918兲.
5
W. Schottky, “Small-shot effect and flicker effect,” Phys. Rev. 28, 74–
103 共1926兲.
6
C. Kittel, Elementary Statistical Physics 共Dover, New York, 1967兲, Chap.
29, pp. 141–146.
7
S. Engelberg and Y. Bendelac, “Measurement of physical constants using
noise,” IEEE Instrum. Meas. Mag. 6, 49–52 共2003兲.
8
R. Landauer, “Solid-state shot noise,” Phys. Rev. B 47, 16427–16432
共1993兲.
Luca Callegaro
439
9
Y. Imry and R. Landauer, “Conductance viewed as transmission,” Rev.
Mod. Phys. 71, S306–S312 共1999兲.
10
Y. M. Blanter and M. Büttiker, “Shot noise in mesoscopic conductors,”
Phys. Rep. 336, 1–166 共2000兲.
11
H. Nyquist, “Certain factors affecting telegraph speed,” Bell Syst. Tech.
J. 32, 324–352 共1924兲.
12
Some of the simplifications are the following: The sampling procedure
assumes only three possible events; the noise power is identified with the
noncentral moment 具I2典 instead of the variance and hence includes the dc
power; the electron populations obey a Boltzmann instead of a Fermi
distribution; more generally, thermodynamic equilibrium distributions
and detailed balance are assumed for a steady-state dissipative system
共see Refs. 16 and 17兲. These simplifications nevertheless give a correct
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Am. J. Phys., Vol. 74, No. 5, May 2006
result in the low-frequency limit, that is, for f ⬍ 100 GHz at room temperature 共Ref. 3兲.
13
H. B. Callen and T. A. Welton, “Irreversibility and generalized noise,”
Phys. Rev. 83, 34–40 共1951兲.
14
G. Gomila, C. Pennetta, L. Reggiani, M. Sampietro, G. Ferrari, and G.
Bertuccio, “Shot noise in linear macroscopic resistors,” Phys. Rev. Lett.
92, 226601-1–226601-3 共2004兲.
15
Y. M. Blanter and M. Büttiker, “Transition from sub-Poissonian to superPoissonian shot noise in resonant quantum wells,” Phys. Rev. B 59,
10217–10226 共1999兲.
16
M. J. Klein, “Principle of detailed balance,” Phys. Rev. 97, 1446–1447
共1955兲.
17
R. M. L. Evans, “Detailed balance has a counterpart in non-equilibrium
steady states,” J. Phys. A 38, 293–313 共2005兲.
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