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1. Introduction

When one tries to measure a weak signal, a lower limit set by the spontaneous fluctuation in

the current, voltage, and other physical variables of the system under test is always encountered.

These spontaneous fluctuations are referred to as noise. Noise is a significant problem in science

and engineering because it places an ultimate sensitivity on all measurements. In this class, the

mathematical methods, physical origins and characteristics of noise in various devices and systems

will be studied.

Chapter 2 provides a mathematical method for treating noise. Two complimentary averaging

methods, time averaging and ensemble averaging, and the experimental realization of the two

averaging processes are described. The Fourier analysis, specifically the Wiener-Khintchin theorem,

is introduced to calculate the power spectrum and autocorrelation function of a noisy waveform.

A noisy waveform often consists of discrete random pulse trains. Various statistical distribution

functions of random pulse trains (i.e., binomial, Poisson, sub-Poisson, super-Poisson, and Gaussian)

are derived and the effect of random deletion of pulses (partition noise) is discussed using the

Burgess variance theorem. A method for calculating the power spectrum and autocorrelation

function of random pulse trains using the Carson theorem is presented.

Chapter 3 describes how to calculate the noise figure of linear networks. A noisy electrical

network can be represented by a noise-free network with external noise generators. If such a network

has two ports (input and output), the noise figure is often used as a figure of merit expressing the

inherent noisiness of the circuit. The technique is also useful to express the noise properties of

optical systems such as laser and parametric amplifier systems.

Chapter 4 discusses the physical origins of two types of intrinsic noise: thermal noise and

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quantum noise. Thermal noise of a resistor is calculated by microscopic theory using a Brownian

particle model or by macroscopic theory using the equipartition theorem. The latter approach,

developed by Nyquist, is very general and easily extended to include quantum noise (i.e., the

effect of zero-point fluctuation). The expression for the power spectrum of an open circuit voltage

fluctuation, including both thermal and quantum noise, is called a generalized Nyquist formula.

The noise characteristics of a mesoscopic conductor on an extremely small scale is enlightening

with regard to the physical origins of thermal and quantum noise. The transition from mesoscopic

conductor to macroscopic conductor will be discussed in some detail.

Chapter 5 treats the noise of semiconductor pn junctions diodes. Random diffusive motion

and generation-recombination event of carriers contribute to the noise of a pn junction. A pn-

junction diode driven by a low-impedance, constant-voltage source has a shot-noise-limited current

fluctuation, but a junction voltage fluctuation is suppressed. A pn-junction diode driven by a high-

impedance, constant-current source has a shot-noise-limited voltage fluctuation, but a junction

current fluctuation is suppressed. A collective and single charge Coulomb blockade effect in a pn

junction is discussed as the principle of sub-Poisson light generation.

Chapter 6 discusses the noise of a bipolar transistor using the method of a noisy linear network.

Chapter 7 describes the mathematical models and physical mechanisms of 1/f noise. In prac-

tically all electronic and optical devices, the excess noise obeying the inverse frequency power law

exists in addition to intrinsic thermal and quantum noise. This 1/f noise places a serious limit on

the sensitivity of precision measurements at a very low frequency.

Chapter 8 treats the noise of a tunnel diode. While a macroscopic tunnel diode driven by

a constant voltage source has a shot-noise-limited current fluctuation, a mesoscopic tunnel diode

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features various sub-shot-noise behaviors due to single charge Coulomb blockade effect.

Chapter 9 describes the noise of negative conductance oscillators which generate a coherent

electromagnetic field at various frequencies. A laser is also described as a negative conductance

oscillator. A negative conductance oscillator consists of a frequency selective element which provides

positive feedback and a gain element which provides signal amplification and nonlinearity. A general

electrical circuit theory (van der Pol oscillator model) is presented to calculate the amplitude and

phase noise of the coherent radiation. The difference between the cavity internal field noise and

output field noise, and the difference between a free-running oscillator and an injection-locked

oscillator, are pointed out.

Chapter 10 provides a brief overview of the noise of parametric amplifiers. The fundamental

limit of phase-insensitive and phase-sensitive linear amplifiers are discussed. A degenerate para-

metric amplifier is introduced as a noise-free, phase-sensitive amplifier.

Chapter 11 provides a brief overview of the noise in optical communication systems. The

signal-to-noise ratio and bit error rate of optical-preamplifier and repeater-amplifier systems are

calculated and compared with a direct detection scheme. The concept of signal regeneration for

a digital system and the advantage of using optical amplifiers are discussed. The thermal and

quantum limits of communication systems are elucidated using the channel capacity.

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2. Mathematical Methods

2.1 Time Averaging vs. Ensemble Averaging

Noise is a stochastic process of a randomly varying function of time and thus is only statis-

tically characterized. One cannot argue a single event at a certain time; one can only discuss the

averaged quantity over a certain time interval (time average) or many identical systems (ensemble

average). Let us consider N identical systems which produce noisy waveforms x(i)(t), as shown in

Fig. 2-1.

x(1)(t)

x(2)(t)

x(N)

(t)

time average

ensemble average

t

t

t

Figure 2-1

One can define the following time-averaged quantities for the i-th member of the ensemble:

x(i)(t) = limT

1

T T

2

T2

x(i)(t)dt ,

(mean = first-order time average) (2.1)

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x(i)(t)2 = lim

T

1

T

T2

T2

x(i)(t)

2dt ,

(mean square = second-order time average) (2.2)

(i)x () x(i)(t)x(i)(t + ) = limT

1

T

T2

T2

x(i)(t)x(i)(t + )dt .

(autocorrelation function) (2.3)

These time-averaged quantities directly correspond to analog measurements using a simple inte-

grator (low-pass filter) for the measurement of the mean (2.1), square-law detector followed by an

integrator for the measurement of the mean square (2.2) and delayed correlator followed by an

integrator for the measurement of the autocorrelation function. Alternatively, one can measure the

noisy waveform x(i)(t) using an analog-to-digital (A/D) coverter and sampling circuit and then

calculate these quantities using a computer (digital measurements).

One can also define the following ensemble-averaged quantities for all members of the ensemble

at a certain time:

x(t1) = limN 1N

Ni=1

x(i)(t1) =

x1p1(x1, t1)dx1 ,

(mean = first-order ensemble average) (2.4)

x(t1)2 = limN

1

N

Ni=1

x(i)(t1)

2=

x21p1(x1, t1)dx1 ,

(mean square = second-order ensemble average) (2.5)

x(t1)x(t2) = limN

1

N

N

i=1

x(i)(t1)x(i)(t2) (2.6)

=

x1x2p2(x1, t1; x2, t2)dx1dx2 .

(covariance = second-order joint moment)

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Here, x1 = x(t1), x2 = x(t2), p1(x1, t1) is the first-order probability density function, and

p2(x1, t1; x2, t2) is the second-order joint probability density function. p1(x1, t1)dx1 is the prob-

ability that x is found in the range between x1 and x1 + dx1 at a time t1 and p2(x1, t1; x2, t2)dx1dx2

is the probability that x is found in the range between x1 and x1 + dx1 at a time t1 and also in the

range between x2 and x2 + dx2 at a different time t2.

If x(t1) and x(t1)2 are independent of the time t1 and if x(t1)x(t2) is dependent only on the

time difference = t2 t1, such a noise process is called a statistically-stationary process. For a

statistically-nonstationary process, the above is not true. In such a case, the concept of ensemble

averaging is still valid, but the concept of time averaging fails. An ensemble average is a convenient

theoretical concept since it is directly related to the probability density functions. On the other

hand, time averaging is more directly related to real experiments. One cannot prepare an infinite

number of identical systems in a real situation. Theoretical predictions based on ensemble averaging

are equivalent to the experimental measurement of time averaging when, and only when, the system

is a so-called ergodic ensemble. One can loosely say that ensemble averaging and time averaging

are identical for a statistically-stationary system, but are different for a statistically-nonstationary

system.

2.2 Fourier Analysis

When x(t) is absolutely integrable, i.e.,

|x(t)|dt < , (2.7)

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the Fourier transform of x(t) exists and is defined by

X(j) =

x(t)ejtdt . (2.8)

The inverse transform is given by

x(t) =1

2

X(j)ejtd . (2.9)

This is proven by taking the integral in (2.9) over the limits N (instead of), substituting for

X(j) from (2.8), and interchanging the order of integration to obtain

12

X(j)ejtd = 1

lim

N

x(t) sin{N(t

t)}(t t) dt

=

x(t)(t

t)dt

= x(t) . (2.10)

When x(t) is a real function of time, as it always is the case for an observable waveform, the

real part of X(j) is an even function of and the imaginary part is an odd function of [i.e.,

X(j) = X(j)].

When x(t) is a statistically-stationary process, condition (2.7) is not satisfied and thus the

Fourier transform does not exist. The total energy of the noisy waveform x(t) is infinite, but

the noise power (energy flow per second) can be finite. In any practical noise measurement, a

measurement time interval T is finite and the energy of such a gated function xT(t), defined by

xT(t) =

x(t)|t|

T

2

0 |t| > T2, (2.11)

is also finite. The Fourier transform of such a gated function does exist.

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If x1(t) and x2(t) have Fourier transforms X1(j) and X2(j), one obtains

x1(t)x

2(t)dt =

1

2

X1(j)X

2 (j)d .

(Parseval theorem) (2.12)

This relation is known as the Parseval theorem and is proven by substituting the inverse transform

ofx2(t) into the LHS of (2.12), interchanging the order of integration, and identifying the resulting

integral about t as X1(j). If one uses x1(t) = xT(t + ) and x2(t) = xT(t) in (2.12), one obtains

xT(t + )xT(t)dt =1

2

|XT(j)|2eid . (2.13)

When = 0, (2.13) is reduced to

[xT(t)]2dt =

1

2

|XT(j)|2d .

(Energy theorem) (2.14)

The physical interpretation ofXT(j) and |XT(j)|2 is now clear from the above relations. XT(j)

is the (complex) amplitude of the ejt component in a gated function xT(t) and |XT(j)|2 is the

energy density of xT(t) with units of energy per Hz. Equation (2.14) is the total energy of xT(t)

and increases linearly with T for a statistically-stationary process.

The average power of xT(t), defined by

limT

1

T

[xT(t)]2dt = lim

T

1

2

0

2|XT(j)|2T

d , (2.15)

is a constant and universal quantity. If ensemble averaging is taken for a gated function xT(t) in

(2.15), the order of limT

and

0d can be interchanged. The power spectral density is defined as

Sx() = limT

2|XT(j)|2T

.

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(unilateral power spectral density) (2.16)

When a statistically-stationary noisy waveform x(t) is input into an RF spectrum analyzer, the

above (unilateral) power spectral density is displayed on the screen, usually with units of power

dBm = 10log10 P(mW), as shown in Fig. 2-2. The noisy waveform x(t) beats against a strong

local oscillator wave at a frequency in a mixer and only the low-frequency part of the beat

signal is detected by using a low-pass filter with a bandwidth f, where f is termed a resolution

bandwidth. The detected noise power is thus due to the frequency component at of the noisy

waveform x(t). An RF spectrum analyzer measures the noise powers of a fixed frequency for N

wavepackets with a duration of T = 1/f and displays the averaged noise power. This is essentialy

the ensemble average procedure. The averaging time T N determines a video bandwidth f /N. If

one divides this averaged noise power by a resolution bandwidth f, one can obtain the (ensemble

averaged) power spectral density Sx().

Local Oscillator wave A sin t

Noisy waveform x(t)

f = : resolution bandwidth1

T

= : video bandwidth1

TN

f =1

T

1

Np(i)i

TT TTTT

N

t

mixer

~

low-pass filter signal averagert

noisepower

frequency

envelope detector

Figure 2-2

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When = 0 in (2.13), one can also divide both sides of (2.13) by T, take an ensemble average,

and take a limit as T to obtain

limT

1T

xT(t + )xT(t)dt = lim

T

12

0

2|X(j)|2T

cos d . (2.17)

The LHS of this expression is the (ensemble averaged) autocorrelation function x(). Using (2.16)

in the RHS of this expression, one obtains

x() =1

2

0

Sx()cos d . (2.18)

The inverse relation of this expression is

Sx() = 4

0

x()cos d .

(Wiener-Khintchine theorem) (2.19)

Equation (2.19) is proven by using (2.18) for x() in the RHS of (2.19), interchanging the order

of the integral, and identifying the resulting integral about as a Dirac delta function ( ).

Equations (2.18) and (2.19) constitute the Wiener-Khintchine theorem and indicate that 2 x()

and Sx() are the Fourier transform pairs.

If a noisy waveform x(t) is a statistically-stationary process, as shown in Fig. 2-3, with an

exponentially decaying autocorrelation function

x() = x(0) exp

||

1

, (2.20)

where x(0) = x2 and 1 is a relaxation time constant (systems memory time). Substituting (2.20)

into (2.19), one obtains the unilateral power spectral density

Sx() =4x(0)11 + 221

. (2.21)

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statistically-stationarynoisy waveform

statistically-nonstationarynoisy waveform

x(t)

t = 0

y(t) = x(t')dt'0

t

t = 0

y(t)2

= Dyt

t

t

Figure 2-3

The spectrum is Lorentzian with a cut-off frequency of 1/1 and the low-frequency spectral density

is Sx( = 0) = 4x(0)1. The autocorrelation function and the unilateral power spectrum are

shown in Fig. 2-4.

x( )S

x( )4

01

x() x( )0

1

0.5

0 1-1 /1 10-2 10-1 101

10-2

10-1

1

1

Autocorrelation Function Unilateral Power Spectrum

Figure 2-4

A time-integrated function y(t) =t0 x(t

) dt

of a statistically-stationary process x(t

) goes

through a random walk diffusion, as shown in Fig. 2-3. This is called a Wiener-Levy process and

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is an example of a statistically-nonstationary process. A gated function is defined by

y(t) =

t0 x(t

) dt (0 t T)

0 (otherwise)

. (2.22)

The corresponding autocorrelation function becomes a function of and T:

y(, T) =1

T

T||0

y(t + )y(t) dt

=T

2

1 ||

T

2Dy . (2.23)

In order to derive the second line of (2.23), y(t) was used as a cumulative process and thus

y(t + )y(t) = [y(t) + y(t + )]y(t) = y(t)2 = Dyt, where Dy is a diffusion constant. The unilat-

eral power spectral density is given by

Sy(, T) = 4

T0

y(, T) cos() d

=4Dy2

1 sin(T)

T

. (2.24)

The correlation time is now proportional to the measurement time interval T and a finite mea-

surement time T prevents the divergence of the power spectral density at = 0, as shown in

Fig. 2-5.

2.3 Random Pulse TrainsBinomial, Poisson, and Gaussian Distributions

A noisy waveform x(t) often consists of a very large number of random, discrete events and

are represented by

x(t) =Kk=1

akf(t tk) . (2.25)

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0-1 1/

y(, T) /1

2DyT

1

0.5

Sx( , T) / 24DyT2

10-2

10-1

1

10-2 10-1 1 10

Autocorrelation Function Unilateral Power Spectrum

Figure 2-5

One assumes the pulse amplitude ak and the time of event tk are random variables, but that the

pulse-shape function f(t) is a fixed function, as shown in Fig. 2-6. In a real physical situation, f(t)

is determined, for example, by the relaxation time of a system or the transit time of a carrier, and

thus it is a reasonable assumption that f(t) is a fixed function.

a1

a2

aka4

a3

t1 t2 tkt4t3t

measurement time interval T = N

Figure 2-6

Suppose the average rate of such random pulse events per second is and the measurement

time interval T is divided into N = T time slots with a duration . The probability p of pulse

emission in each time slot is , which, by making N large enough, can be much smaller than one.

In such a case, the probability of more than two pulses being emitted in the same slot is negligible.

If a time slot duration is much longer than a systems correlation time (memory time), each pulse

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emission occurs independently (that is, the pulse emission at a certain time is not influenced

by the previous events of the system). In such a case, the probability of finding K events in the

measurement time interval T (out of N trials) is given by the binomial distribution:

WN(K) =N!

K!(N K)!pKqNK , (2.26)

where q = 1 p is the probability of zero pulse emission. Since this is the expansion coefficient of

(p + q)N, the binomial distribution WN(K) satisfies the normalization condition,

N

K=0

WN(K) = (p + q)N = 1 . (2.27)

The mean of the number of pulses is calculated by

KN

K=0

KWN(K) =N

K=0

N!

K!(N K)!p

p(pK)

qNK

=

p

p

(p + q)N

= pN . (2.28)

The mean-square is

K2 N

K=0

K2WN(K) =N

K=0

N!

K!(N K)!

p

p

2(pK)

qNK

=

p

p

2(p + q)N

= (pN)2 + pqN . (2.29)

The variance of the number of pulses is then given by

K2 K2 K2 = N p(1 p) . (2.30)

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We start with a fixed (constant) number of potential events N = T / and introduce random

deletion with the probability q = 1 p. The variance of the output event (2.30) is a rather general

result for such a stochastic process and is referred to as partition noise. In some cases, the

number of potential events N itself fluctuates. In such a case, the variance of the number of output

events is given by

K2 = N2p2 + N p(1 p) ,

(Burgess variance theorem)

(2.31)

where N2 = N2 N2. The above relation is known as the Burgess variance theorem and is

proven by replacing WN(K) in (2.28) and (2.29) by W(N)WN(K), where W(N) is the probability

distribution function for potential events N. The first term on the RHS of (2.31) indicates that

the fluctuation of the initial number of events is suppressed by a loss probability p. The second

term, on the other hand, indicates the new noise term (partition noise).

In the limit ofp 1 and N 1, one obtains the following approximate relations:

(1 p)NK

ep(NK)

epN

, (2.32)

lnN!

(N K)!

d

dNln N!

K ln NK . (2.33)

Using these relations in the binomial distribution (2.26), the approximate expression for WN(K)

is obtained as

W(p)N (K) =

(pN)K

K!epN =

KK

K!eK .

(Poisson distribution) (2.34)

This is the Poisson distribution. The variance of the output events K2 can be calculated as

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follows:

K2 K

K=0

K(K 1)W(p)N (K) = K2

K=2

KK2

(K

2)!eK = K

2

K2 K2 K2 = K = N p . (2.35)

The result is, of course, obtained directly from (2.30) by letting p 0, but keeping Np finite. The

same variance is also obtained from (2.31) in the limit p 0. Therefore, the Poisson distribution is

a quite general distribution for a random-point process. Whenever a random-deletion process with

a very large attenuation rate is imposed, the final statistics always obey the Poisson distribution,

irrespective of an initial distribution, as shown in Fig. 2-7.

However, p cannot be always made much smaller than one for a given average rate of random

pulse events per second because any physical system has a finite memory time m. If the time

slot = TN is made smaller than the memory time m of the system, each event can no longer

be considered a statistically-independent process. Therefore, there is a lower limit for which, in

turn, places a lower limit on p = for a given average rate .

The other important distribution function is obtained when Npq 1, i.e., N 1 and p and q

are not too small. In such a case, the approximate expression for WN(K) is obtained as

W(G)N (K) =

12K2

exp

(KK)2

2K2

.

(Gaussian distribution)

(2.36)

The mean K and variance K2 are given by (2.28) and (2.30) for the binomial distribution. When

p 1, but the mean K = N p is much greater than one, the Poisson distribution is reduced

to the Gaussian distribution. [The Gaussian distribution (2.36) is obtained from the binomial

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K2

N2

N2

=

Poisson

1

0

0 1

total noise

attenuated original noise

partition noise

K2

N2

1

0

0

0

1

N2

= 2

Super-Poissontotal noise


partition noise

K2

N2

1

0

0 1

N2

= 12

Sub-Poisson

total noise


partition noise

T

T

T

Figure 2-7

distribution (2.26) by considering WN(K) as a continuous function of K, expanding ln[WN(K)]

around the averaged point K by a Taylor series expansion and truncating to second order.]

2.4 Random Pulse TrainsSpectrum and Autocorrelation Function

The Fourier transform of (2.25) is

X(j) = F(j)Kk=1

ak ejtk . (2.37)

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The unilateral power spectral density for such random pulse trains is defined by

Sx() limT

2|X(j)|2T

= limT

2|F(j)|2T

K

k,m=1

akam exp[j(tk tm)] . (2.38)

The summation in (2.38) over k and m can be split into the summation for k = m and k = m,

Sx() = limT

2|F(j)|2T

Kk=1

a2k +k=m

akam exp[j(tk tm)]

= 2a2|F(j)|2 + 4x(t)2() .

(Carsons theorem) (2.39)

Here, = limT

KT is the average rate of the pulse emission and a

2 = limT

1K

Kk=1

a2k is the mean-

square of the pulse amplitude. The mean of the noisy waveform x(t) is

x(t) = a

f(t)dt , (2.40)

and a = limT

1

K

Kk=1

ak is the mean of the pulse amplitude. The term containing the delta func-

tion represents the dc contribution to the power spectral density and is obtained by calculating

exp[j(tk tm)] using completely random distributions of tk and tm, identifying F( = 0) with

f(t)dt, and replacing limT

2sin2(T/2)2T with (). For a symmetric distribution of ak about

zero, the second term of (2.39) is zero because a is zero. However, when ak are not symmetrically

distributed about zero, the dc term appears in the power spectral density.

From the Wiener-Khintchine theorem, the autocorrelation function is calculated as

x() =a2

0|F(j)|2 cos d + 2x(t)2

0()cos d

= a2

f(t)f(t + )dt + x(t)2

, (2.41)

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where the Parseval theorem (2.12) is used to derive the second line. When = 0, one obtains

x(t)2 x(t)2 = a2

[f(t)]2dt

=a2

0

|F(j)|2d

(Campbells theorem of mean square) , (2.42)

where the energy theorem (2.14) is used to derive the second line.

2.5 Current Noise of a Vacuum Tube

Let us consider a fluctuating current i(t) caused by random emission of electrons from the

cathode of a vacuum tube, as shown in Fig. 2-8. Emission of electrons and subsequent travel

of the electrons induce the surface charges on the cathode and the anode. There is a relaxation

current flowing into a circuit to build up such a surface charge at the electrode surfaces and to

restore the steady state of the two electrodes. This relaxation current constitutes the fluctuating

current. In the vacuum tube case, the pulse energy (integrated relaxation current pulse) is equal

to q (constant) and the pulse shape f(t) is determined by an electron transit velocity between the

cathode and the anode. The pulse emission event follows the Poisson distribution with an average

rate of emission = limT

KT . Therefore, the average current is i(t) = q = I.

If one assumes f(t) is an exponentially decaying function with a time constant 1, the autocor-

relation function is calculated from (2.41) as

i() =qI

21e |

|1 + I2 . (2.43)

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space charge region

electron multiplication region

t

t

t

Poisson-point-process

sub-Poisson-process

reflected electron

super-Poisson-process

added electron

Figure 2-8

The unilateral power spectral density is calculated from (2.39) as

Si() =2qI

1 + 221+ 4I2() .

(Schottky formula) (2.44)

As shown in Fig. 2-9, the low-frequency power spectral density ( 11 ) is equal to the full shot

noise 2qI and the spectral shape is Lorentzian with a cut-off frequency of 11 due to the finite

duration of a relaxation current pulse. The dc component of the spectrum is caused by the flat

response of the autocorrelation function I2. On the other hand, the Lorentzian noise spectrum

stems from an exponentially decaying autocorrelation function centered at = 0.

The physical origin for the full shot noise of the current spectrum (2.44) is the fact that the

emission of each electron in such a thermal-limited vacuum tube is statistically independent and thus

obeys a Poisson-point process. This is not always true. In a vacuum tube of a space-charge limited

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0 1 2 3 4 50

1

1 20

2I

q

Si( ) 2qI

1 -1-2-3-4

1

12I /q

i()qI2( (1

1

Figure 2-9

regime, there is a strong electron-electron Coulomb interaction in the space-charge region near

the cathode. A potential minimum produced by the space charge is modulated by the fluctuating

emission rate and provides negative feedback to regulate the effective electron emission rate, as

shown in Fig. 2-8. In other words, each electron emission event is no longer statistically independent

but, rather, anti-bunched. Consequently, the circuit current fluctuation is less than the full shot

noise. This shot noise suppression due to negative feedback effect is a rather common feature in

various electronic systems; examples will be seen in the following chapters, in which either the

Coulomb interaction or the Pauli exclusion principle (or both) play a role in the negative feedback

mechanism to suppress shot noise.

On the other hand, if randomly emitted electrons are amplified by an electron multiplier, the

initial Poissonian electron stream is transformed to the bunched electron stream, as shown in

Fig. 2-8. Consequently, the circuit current fluctuation is more than the full shot noise. This excess

noise is also a rather common feature in various amplifier systems, i.e., photomultipliers, avalanche

photodiodes, and laser amplifiers.

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3. Noise figure of linear networks

A noisy electrical network can be represented by a noise-free network with external noise

generators. The magnitude of the external generator is expressed either by an equivalent noise

resistance or an equivalent noise temperature. If such a network has four terminals or two (input

and output) ports, the noise figure is often used as a figure of merit expressing the inherent noisiness

of the circuit. The technique is also useful to express the noise properties of photonic systems such

as lasers and optical parametric amplifiers.

3.1 Two-terminal networks Thevenin equivalent circuit

A noisy two-terminal network with impedance Z() = R()+jX() generates the open circuit

voltage fluctuation v(t) as shown in Fig. 3-1(a). The two equivalent circuits based on Thevenins

theorem[1] are shown in Fig. 3-1(b) and (c). One is the noise-free network with impedance Z()

v(t)Z() Z() Y() i(t)

(a) (b) (c)

v(t)

Figure 3-1: (a) A noisy two-terminal network. (b) Thevenin equivalent circuit with an external

voltage source. (c) Thevenin equivalent circuit with an external current source.

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in series with a voltage generator v(t). The spectral density of v(t) is expressed by

Sv() = 4kBRn , (3.1)

where is the absolute temperature and Rn is the equivalent noise resistance. The other is the

noise-free network with admittance Y() = G() +j B() in parallel with a current generator i(t).

The spectral density of i(t) is expressed by

Si() = 4kBGn , (3.2)

when Gn is the equivalent noise conductance.

If the network is linear and passive, and there is no net energy flow, i.e. the circuit is at thermal

equilibrium condition, then Rn = R() and Gn = G() = R()/[R()2 + X()2]. The noise in

this case is equal to Johnson-Nyquist thermal noise. However, in nonlinear active circuits, or in

non-equilibrium condition, these equalities generally do not hold.

When the electron temperature is different from the lattice temperature, which is the case for

hot electron devices, it is convenient to express (3.1) and (3.2) in the alternative forms:

Sv() = 4kBnR , (3.3)

Si() = 4kBnG , (3.4)

where n is the equivalent noise temperature. In circuits containing shot noise sources as primary

noise sources, it is convenient to use the expression

Si() = 2qI , (3.5)

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where I is the terminal current and is the shot noise suppression factor. If some smoothing

mechanisms are dominant, for instance due to space charge effect, becomes smaller than unity.

If there is no smoothing mechanism in the system, the power spectral density is full-shot noise

( = 1).

Consider the parallel RC circuit shown in Fig. 3-2(a). The thermal noise of the register is

represented by the parallel current source i(t) with the spectral density of Si() = 4kB/R. The

series voltage source v(t) in the Thevenin equivalent circuit shown in Fig. 3-2(b) has the spectral

density of

Sv() =R2

1 + 2(CR)2Si() . (3.6)

The frequency dependent power spectral density (3.6) is due to the impedance of the capacitor.

i(t)

(a) (b)

R C R C

vn

Figure 3-2: (a) A parallel RC circuit with thermal noise current source. (b) The Thevenin equivalent

circuit with thermal noise voltage source.

Using the Wiener-Khintchine theorem, we obtain the mean-square value of the voltage generator

v(t)2 = 12

0

Sv()d = kBC

. (3.7)

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The mean-square energy stored in the parallel RC circuit is thus equal to

1

2Cv(t)2 =

1

2kB . (3.8)

This is an example of the equipartition theorem of statistical mechanics [2]. That is, if the system

energy is of the form of quadratic dependence on generalized coordinate, i.e. the voltage in this case,

the average energy of the system under thermal equilibrium condition is equal to 12kB per degree

of freedom. Notice that the noise energy is independent of the resistance R while the magnitude

and the bandwidth of the noise spectrum are dependent on R.

3.2 Four-terminal networks

A network with two pairs of terminals, input and output ports, is known as a four-terminal or

two-port network. For a noiseless four-terminal network, the currents and voltages at the terminals

are related to each other in terms of the impedance matrix Z or the admittance matrix Y as follows:

V1

V2

=

Z11 Z12

Z21 Z22

I1

I2

, (3.9)

I1I2

=

Y11 Y12

Y21 Y22

V1

V2

. (3.10)

The subscripts 1 and 2 refer to the input and output ports, respectively, and the sign convention is

that currents flowing into the network are positive, as shown in Fig. 3-3. The upper case letters I

and V indicate the Fourier transforms of the current and voltage, which are in general dependent

on frequency.

A noisy four-terminal network is represented by an extension of Thevenins theorem[1]. In

Fig. 3-4(a), a series noise voltage generator appears at each port. Some degree of correlation may

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v1(t)

i1(t) i2(t)

v2(t)

Figure 3-3: A noiseless four-terminal network.

exist between these generators since the same physical mechanism may be responsible, at least in

part, for the open circuit voltage fluctuations. The dual of Fig. 3-4(a) is shown in Fig. 3-4(b), in

v1(t)

i1(t)

Z

vn1(t)

v2(t)

i2(t)

vn2(t)

v1(t)

i1(t)

Y v2(t)

i2(t)

in1(t) in2(t)

(a) (b)

Figure 3-4: (a) Thevenin equivalent circuit with two external series voltage generators. (b) Thevenin

equivalent circuit with two external parallel current generators.

which the internal noise is represented by parallel current generators.

The current-voltage relation of a noisy four-terminal network becomes

V1 + Vn1

V2 + Vn2

=

Z11 Z12

Z21 Z22

I1

I2

, (3.11)

I1 + In1

I2 + In2

=

Y11 Y12

Y21 Y22

V1

V2

. (3.12)

It is often more convenient to refer both external generators to the input port. The equivalent

circuit shown in Fig. 3-5 has a series voltage generator and a parallel current generator at the

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input, for which the current-voltage characteristic is expressed by the relation

I1 + Ina

I2

=

Y11 Y12

Y21 Y22

V1 + Vna

V2

. (3.13)

Comparing (3.12) and (3.13), the Fourier transform of the new voltage generator vna(t) should be

v1(t)

i1(t)

Y v2(t)

i2(t)

ina(t)

vna(t)

Figure 3-5: The equivalent circuit of a noisy two-port with external current and voltage generators

in the input port.

related to that of the current generator in2(t):

Vna = In2Y21

. (3.14)

The Fourier transform of the new current generator ina(t) is

Ina = In1 Y11Y21

In2 . (3.15)

The arrangement of Fig. 3-5 is particularly convenient for calculating the noise figure of the

two-port network. However, this equivalent circuit is valid only for calculating the noise in the

output port. It does not give the correct description of the noise in the input port. This can be

easily seen by the fact that Ina is not equal to In1.

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3.3 Noise figure of a linear two-port

The noise figure of a two-port is defined by

F =total output noise power per unit bandwidth

output noise power per unit bandwidth due to input termination

at a specific frequency and temperature. A signal input is(t) is transferred to the output through

an input admittance Ys and noisy two-port network, as shown in Fig. 3-6. The noise in the input

Yina

(t)

vna(t)

ins

(t)Ys

is

Figure 3-6: A circuit for calculating the noise figure of a two-port network.

admittance Ys and the noisy two-port are independent, and so the noise figure of the whole system

can be expressed as

F =|Ins + Ina + YsVna|2

|Ins|2

= 1 +Sia()

Sis()+ |Ys|2 Sva()

Sis()+ 2Re(ivY

s )

Sia() Sva()

1/2Sis()

, (3.16)

where Sia(), Sva() and Sis() are the power spectra ofina(t), vna(t) and ins(t), respectively, and

iv is the normalized cross-correlation spectral density between ina(t) and vna(t).

The power spectral densities can be expressed as

Sia() = 4kBGni , (3.17)

Sva() = 4kB/Gnv , (3.18)

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Sis() = 4kBGs . (3.19)

Here Gni and Gnv are the equivalent noise conductances, which are not necessarily actual conduc-

tances of the two-port network. On the other hand, Gs = Re(Ys) is the actual source conductance.

The current generator ina(t) is split into two, one part of which is uncorrelated with vna(t) and the

other part is fully correlated with vna(t). Therefore we obtain

Ina = Inb + YcVna , (3.20)

where Yc is the correlation admittance of ina(t) and vna(t). Since InbVna = 0, we have

iv InaV

na|Ina|2 |Vna|2

1/2 = Yc|Vna|2|Ina|2

1/2=

YcGniGnv

. (3.21)

The noise figure of (3.16) is now rewritten as

F = 1 +GniGs

+(Gs + Gc)

2 + (Bs + Bc)2 (G2c + B2c )

GnvGs, (3.22)

where Gc and Bc are the real and imaginary parts of the correlation admittance Yc. This expression

is easily transformed into the form

F = F0 +(Gs Gso)2 + (Bs Bso)2

GnvGs. (3.23)

Here, F0 = 1 +2

Gnv(Gso + Gc) is the minimum noise figure achieved when the source admittance

satisfies the following matching condition:

Gs = Gso = (GnvGni B2c )1/2 , (3.24)

Bs = Bso = Bc . (3.25)

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The conditions (3.24) and (3.25) for the source conductance and susceptance are referred to as

noise tuning or noise matching. The four parameters Fo, Gso, Bso and Gnv completely characterize

the noise fluctuations of the two-port network.

3.4 Noise figure of amplifiers in cascade

One of the important applications of the equivalent circuit discussed in the previous section

is the overall noise figure of the system when several amplifiers are connected in cascade as shown

in Fig. 3-7. Suppose each amplifier in the cascade is connected to a matched load, i.e. the output

F1, G1ina1(t)

vna1(t)

ins(t)Ysis(t)F2, G2 loadina2(t)

vna2(t)

Figure 3-7: The equivalent circuit of a cascade amplifier system.

and input admittances of adjoining amplifiers are equal. The noise figure of the whole system in

such a case can be written as

F =|Isn + Ina1 + YsVna1|2

|Ins|2+|Ina2 + Y1Vna2|2

G1|Ins|2+|Ina3 + Y2Vna3|2

G1G2|Ins|2

+ . (3.26)

Here Yi and Gi are the input admittance and power gain of the i-th amplifier. If Fi is the noise

figure of the i-th amplifier defined by (3.16), the overall noise figure is

F = F1 + (F2 1)/G1 + (F3 1)/G1G2 + . (3.27)

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Equation (3.27) is known as Friisss formula[3]. The expression indicates, that the noise figure of

the cascade amplifier system is essentially determined by the noise figure of the first stage if the

power gain of the first stage is sufficiently high. It is important to use a low-noise amplifier in the

first stage in order to realize a small overall noise figure.

References

[1] L. Thevenin, Comptes Rend. Acad. Scie. Paris, 97, 159 (1883).

[2] R. Lief, Fundamentals of statistical and thermal physics (McGraw-Hill, New York, 1965).

[3] H. T. Friiss, Proc. IRE, 32, 419 (1944).

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4. Thermal Noise and Quantum Noise

There are two types of intrinsic noise in every physical system: thermal noise and quantum

noise. These two types of noise cannot be eliminated even when a system is perfectly constructed

and operated. Thermal noise is a dominant noise source at high temperatures and/or low frequen-

cies, while quantum noise is dominant at low temperatures and/or high frequencies. A conductor is

the simplest physical system which produces these two types of intrinsic noise. The intrinsic noise

of macroscopic and mesoscopic conductors will be discussed in this chapter.

A conductor in thermal equilibrium with its surroundings shows, at its terminals, open-circuit

voltage (or short-circuit current) fluctuations, as shown in Fig. 4-1. Thermal equilibrium noise

R

R

v

i

Sv( ) = 4k B R Si( ) =4kB

R

Figure 4-1:

was first observed by M. B. Johnson in 1927. He discovered that the noise power spectral density

is independent of both the material the conductor is made of and the measurement frequency,

and is determined only by the temperature and electrical resistance. The open-circuit voltage

noise spectral density is Sv() = 4kBR and the short-circuit current noise spectral density is

Si() = 4kB/R. This noise is referred to as thermal noise and is the most fundamental and

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important noise in electronic devices.

The physical origin of thermal noise in a macroscopic conductor is a Mrandom-walk of

thermally-fluctuated electrons. An electron undergoes a Brownian motion via collisions with the

lattices in a conductor. The statistical properties of such a Brownian particle were studied first

by A. Einstein twenty years before Johnsons observation of thermal noise. The electrons in a

conductor are thermally energetic via collisions with the lattice and travel randomly. The electron

velocity fluctuation is a statistically-stationary process. However, the mean-square displacement

of an electron increases in proportion to the observation time. The electron position fluctuation is

a statistically-nonstationary process. Such a microscopic approach can indeed explain Johnsons

observation.

Nyquist employed a completely different approach to the problem. He introduced the concept

of a mode for the system by using a transmission line cavity terminated by two conductors. He

then applied the equipartition theorem of thermodynamics to these transmission line modes. In

this way he could explain Johnsons observation without going into the details of a microscopic

electron transport process. Nyquists approach is very general and is easily extended to include

quantum noise, which is important in a high-frequency and low-temperature case.

In order to introduce quantum noise to a conductor, one must assume the quantization pos-

tulate. The current and voltage of a LC circuit are a pair of conjugate observables just like the

position and momentum of a particle or a mechanical harmonic oscillator. One cannot determine

these two conjugate observables simultaneously with arbitrary accuracy due to the Heisenberg

uncertainty principle. These quantum uncertainties in the current and voltage have their micro-

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scopic origins in that the momentum (corresponding to the short circuit current) and position

(corresponding to the open circuit voltage) of an electron in a conductor cannot be determined

simultaneously.

A mesoscopic conductor does not impose any scattering on electrons and an electron travels as

a coherent wave in such a small-scale device. Nevertheless, there is still a finite electrical resistance

(quantum unit of resistance RQ =h2e2 ) due to the Pauli exclusion principle. It is truly surprising to

find that the formula for thermal noise and quantum noise of a macroscopic conductor with many

scatterings still hold for such a mesoscopic conductor.

4.1 Microscopic Theory of Thermal Noise Einsteins Approach

4.1-1 Mean Free-Time and Mobility

u(t)

collision

f1f2f3 E E E E E E E

u(t) : average driftvelocity

V

collision with the lattice

t

Figure 4-2:

Consider a one-dimensional conductor under an applied voltage (Fig. 4-2). We assume that an

electron is accelerated by the uniform electric field between collisions with the lattice and that an

3


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electron velocity returns to zero at every collision event. This is not true in a real collision process

in a conductor, but the conclusion one obtains using this assumption is essentially same as that

obtained by more realistic collision models. Since the electron drift velocity in this model is given

by u(t) = t = qEm

t during the time between collisions f, the displacement between two collisions

is

x(f) =

22f =

qE

2m2f . (4.1)

After K collisions with the lattice, the total displacement is qE2m 2f K. Therefore, the mean drift

velocity u is given by

u total displacementtotal time

=

qE2mK

2f

Kf=

q2f2mf

E . (4.2)

Here, f and 2f are the mean free time and mean-square free time. The mobility is defined by

u = E. Thus, one obtains

=q2f

2mf. (4.3)

As shown in Fig 4-2, f is randomly distributed around its mean value f. The probability

pi(m, ) that the ith electron experiences exactly m collisions in a time interval [0, ] obeys a Poisson

distribution if the probability of electron collision with the lattice is independent of the electron

drift velocity which is a reasonable assumption for a weak field; thus it is a statistically independent

process. Strictly speaking, the electron collision with a lattice obeys a binomial distribution with

a very small collision probability starting from a very large number of (potential) collision events,

which is well approximated by a Poisson distribution. Thus, one obtains

pi(m, ) =(i)m

m!ei , (4.4)

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where i is the mean rate for collision per second. The probability for a free time f lying between

fi and fi + dfi is equal to the joint probability of zero collisions in a time interval [0, fi] and

one collision in a time interval [fi, fi + dfi]. Thus,

q(fi)dfi = pi(0, fi) pi(1, dfi)

= i eifi dfi , (4.5)

where eidfi 1 is used and i eifi is considered a probability density function by which we

can calculate the mean free time and mean-square free time,

fi = i

0

fi eifi dfi = 1

i, (4.6)

2fi = i

0

2fi eifi dfi =

2

2i= 2fi

2 . (4.7)

As seen from (4.6) and (4.7), fi does not obey a Poisson distribution, but, rather, a geometrical

distribution. Figure 4-3 compares pi(m, ) and qi(fi). If N electrons behave in a similar but

0

0.5

1

0 10

0.5

1

0 1 2 3 4 5

Poisson distribution

m

Geometical distribution

i fi i

2 i p i m,qi fi / i

Figure 4-3:

independent way, the Maddition theorem of a Poisson process is applied and the total probability

of collision p(m, ) still obeys a Poisson distribution with the mean rate =Ni=1

i =Nf

. The

(collective) mean free time f is defined by f =N

.

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Using (4.7) in (4.3), the mobility is uniquely related to the mean free time,

=qfm

. (4.8)

According to the collision model, any departure of the drift velocity u(t) = u(t)u from its mean

value decays with a time constant f =mq

.

4.1-2 Velocity and Current Fluctuations

In a thermal equilibrium condition (zero applied voltage), the mean drift velocity of an electron

is zero, u = 0. However, an electron acquires a non-zero momentum when it collides with a lattice

which decays toward the mean value of u = 0 with a time constant f. If its momenta before and

after a collision are p1 and p2, respectively, the collision imparts a momentum change (p2 p1) in

an infinitesimal time. The equation of motion is then

du(t)

dt= u(t)

f+

(p2 p1)(t)m

. (4.9)

The drift velocity u(t) is kicked randomly by the second part of the RHS of (4.9) and is simulta-

neously damped by the first part of the RHS of (4.9). The Fourier analysis of (4.9) requires the

introduction of a gated function uT(t) = u(t) for T2 t T2 and 0 for otherwise because u(t) is

a statistically-stationary process. The Fourier transform of this gated function leads to

UT(j) =

q

(p2 p1)1 +jf

. (4.10)

From the Carson theorem, the power spectral density of the velocity fluctuation is given by

Su() = 2(p2 p1)2 (/q)2

1 + 22f, (4.11)

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where = limT

K

T=

1

fis the mean rate of collisions, K is the number of collision events, and

(p2 p1)2 = limT

1

K

K

k=1

(p2k p1k)2 is the mean square of the momentum change per collision.

Since momenta before and after a collision are statistically independent,

(p1 p2)2 = 2p2

= 2 limT

1

K

Kk=1

p2k

. (4.12)

For a thermal equilibrium condition, the mean square of the electron momentum is determined by

the equipartition theorem:

p2

2m

=1

2

kB (one

dimensional case) . (4.13)

From (4.11)(4.13), one obtains

Su() =4kB/q

1 + 22f, (4.14)

where (4.8) is used.

From the Wiener-Khintchine theorem, one can calculate the autocorrelation function

u() =1

2

0

Su()cos d

=kB

qfexp

||

f

. (4.15)

Therefore, the mean square value of u(t) is given by u(0) =kBqf

. Figures 4-4(a) and (b) show

Su() and u().

The velocity fluctuation u(t) of the electron produces a short-circuited current fluctuation i(t) =

qu(t)/L in an external circuit, where L is the length of the conductor. Since the current fluctuation

of each electron is additive, the total current fluctuation power spectral density is given by

Si() = Su()q2

L2ALn , (4.16)

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where A is a cross-sectional area of the conductor and n is the electron density. If one uses (4.14)

and the expression for electrical resistance, R = LA

= LnqA

, in (4.16), one obtains

Si() =4kB/R

1 + 22f. (4.17)

Therefore, the short-circuited current fluctuation power spectral density at low frequency is 4kB/R,

which is exactly what Johnson observed experimentally.

1

0.5

0 1-1101

1

Su( ) /4kB

q

f

u )q f

k

f

10-1

10-110

-2

10-2

Figure 4-4:

4.1-3 Position and Charge Fluctuations

The integral of a statistically-stationary process which shows a white power spectral density

is called a Wiener-Levy process. The surface charge induced by the fluctuating short-circuited

current across a conductor is

q(t) =

t0

i(t)dt . (4.18)

Consider again the one-dimensional conductor shown in Fig. 4-2. An electron transit over a free

path f between collisions gives rise to a surface charge on the electrode equal to qfL

. If there

are m = t independent events in a time duration t , then the mean-square value of the surface

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charge q(t) is calculated as

q(t)2 = q2(t)2fL2

=2kB

Rt , (4.19)

where = nAL/f is the mean rate of collisions, 2f = v

2T

2f = 2kB f/q is the mean-square

free path, and v2T = kB/m is the mean-square thermal velocity; therefore, the charge fluctuation

is not stationary, but, rather, diffuses with time. The diffusion coefficient, defined by q(t)2 = 2Dt,

is given by

Dq =kB

R=

1

4Si( 0) . (4.20)

Since q(t) is a cumulative process, one can write

q(t + ) = q(t) + q(t, ) , (4.21)

where q(t, ) is the charge fluctuation added between times t and (t + ). Since the correlation

time of the additive fluctuations is very short (f ), the two processes in the RHS of (4.21) are

uncorrelated; therefore, the covariance function is equal to the mean square value:

q(t + )q(t) = q(t)2 + q(t)q(t, ) q(t)2 . (4.22)

The autocorrelation function must be redefined in such a manner that the observation time remains

finite rather than infinite and the ordinate is set to zero outside the observation time interval. Thus,

the autocorrelation function is a function of the time delay and the observation time T:

q(, T) =1

T

T||0

q(t + )q(t)dt

= 1T

T||0

2kBR

t dt

=kB

RT

1 ||

T

2. (4.23)

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The power spectral density is obtained by the Wiener-Khintchine theorem as:

Sq(, T) = 4 T

0q(, T)cos d

=8kB

2R

1 sin(T)

T

. (4.24)

When T 1, the power spectral density is proportional to 2, which is characteristic of a

Wiener-Levy process. Figures 4-5(a)(c) show the real-time function, autocorrelation function,

and power spectral density of the surface charge q(t). The integral of an electron velocity u(t)

t

1

1 10

q(t)

0

random walk diffusion

=

0.1

S , T)q

0-1 1

1

0.5

, T)q.

q(t)2 2kB

Rt

3R

4kB T2

-2

-2

-1

R

k T

Figure 4-5:

is an electron displacement x(t), which is also a Wiener-Levy process. By performing the same

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calculation, one obtains the (position) diffusion coefficient,

Dx =kB

q=

1

4Su( 0) .

(Einsteins relation) (4.25)

4.2 Macroscopic Theory of Thermal Noise Nyquists Approach

Nyquists treatment of thermal noise appeared very soon after Johnsons observation and the

result is often referred to as Nyquists Law. Nyquist initially considered two electrical conductors,

R1 and R2, connected in parallel, as shown in Fig. 4-6. The open-circuit voltage noise source v1

associated with conductor R1 produces a current fluctuation in the circuit, leading to an absorbed

power R2v21/(R1 + R2)2 by conductor R2. A similar flow of absorbed power, R1v22/(R1 + R2)

2,

exists from R2 to R1. Since the two conductors are at the same temperature, the power flow in

each direction must be exactly the same and cancel each other out; otherwise, the second law of

thermodynamics would be violated. The second law of thermodynamics states that it is impossible

to take heat from one reservoir to another reservoir at an equilibrium temperature. In order to

satisfy this exact cancellation of power flow from R1 to R2, and vice versa, the open circuit voltage

noise v2i should be proportional to the electrical resistance Ri. This exact cancellation of power flow

must hold not only for the total power, but also for the power exchanged in any frequency band;

otherwise, the second law of thermodynamics would be violated simply by inserting a frequency

filter between the resistors. In other words, the power spectrum Sv() of the voltage fluctuations

should be independent of the detailed structure and material of the conductor and should be a

universal function ofR, , and (angular) frequency . Specifically, if the conductors R1 and R2 are

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at different temperatures, 1 and 2, the net heat flow should be proportional to the temperature

difference 12 and thus the power spectrum Sv() is proportional to the temperature . Nyquist

R1conductor 1

conductor 2 2vv1

R2

R2

R1 + R22 v1

2

)(

R

R1 + R22 v

2

)(1

2

Figure 4-6:

then considered a long, lossless transmission line terminated at either end by conductors R1 and

R2 (Fig. 4-7). This lossless LC transmission line has a characteristic impedance Z0 =LC

which

is equal to the end terminal conductance R1 = R2 = R and the wave velocity v =1LC

, where

L and C are the inductance and capacitance of the transmission line per unit length. The power

delivered to the transmission line from R1 or R2 in a frequency interval d/2 is given by

dP =1

4RSv()

d

2. (4.26)

A time duration in which this noise power travels in the transmission line is t = v

, where is the

length of the transmission line. Hence, the total stored energy in the transmission line in the same

frequency band is

dE = dP t 2 = 2Rv

Sv()d

2. (4.27)

Suppose the transmission line is suddenly short-circuited by closing the two switches across R1

and R2 and, therefore, the energy density (4.27) on the transmission line is trapped as standing

waves. The resonant frequencies of these standing waves are N =v2 N, where N is a positive

12


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1=

v

2l

2=

v

2l2~

3

=v

2l3~

N=

v

2lN~

R 1

2v

R2

l

v1

Figure 4-7:

integer. The number of standing wave modes in the frequency interval d2 is given by

m =

d

2

v

2

=

d v

. (4.28)

As the transmission line length goes to infinity, the number of degrees of freedom (DOF) of the

system (given by the number of the standing wave modes) also goes to infinity. Therefore, it is

permissible to invoke the equipartition theorem to determine the total energy in the transmission

line at thermal equilibrium. The average energy per mode (with two DOFs) is kB and thus the

stored energy in the frequency band d2 is given by

dE = mkB = dv

kB . (4.29)

13


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From 4.27) and 4.29), the power spectrum is obtained as

Sv() = 4kBR . (4.30)

As expected from a thermodynamic argument, Sv() is proportional to R and . If an angular

frequency becomes greater than kB/h, the average thermal energy per DOF is calculated as

h nth =h

exphkB

1

, (4.31)

where the photon number distribution function Pn obeys the geometrical distribution

Pn =nnth

(1 + nth)n+1= exp

nh

kB

1 exp

h

kB

, (4.32)

and thus the mean thermal photon number is

nth n=0

nPn =1

exphkB

1

. (4.33)

Therefore, the total energy in the frequency band d2 is

dE = mhnth =d

v h

exphkB

1

. (4.34)

Comparing this result with (4.27), the power spectral density of the voltage noise is found to be

Sv() =4hR

exphkB

1

. (4.35)

In (4.35), the quantized energy h of a photon is taken into consideration. However, the full

quantum mechanical analysis of the problem suggests that it is still insufficient since the zero-point

fluctuation is not included.

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4.3 Quantum Noise

Consider a lossless LC circuit in order to derive the zero-point fluctuation. The resonant

frequency of an LC circuit is given by 0 =1LC

. One now models each transmission line cavity

mode in Fig. 4-7 by such a lumped LC resonant circuit. If the current flowing in the inductance L

is denoted by i(t) and the voltage across the capacitance C is denoted by v(t), the Kirchhoff laws

are represented by

Cdv(t)

dt= i(t) , (4.36)

L

di(t)

dt = v(t) . (4.37)

These equations can be expressed in terms of the normalized voltage, q(t) Cv(t), and the

normalized current, p(t) Li(t):dq(t)

dt=

1

Lp(t) , (4.38)

dp(t)

dt= 1

Cq(t) . (4.39)

The total energy stored in the lossless LC circuit is given by

H =1

2Li2 +

1

2Cv2 =

p2

2L+

q2

2C. (4.40)

Suppose one interprets the above expression in terms of an analogy with a mechanical harmonic

oscillator with a mass m and oscillation frequency 0 =1LC

. One can then identify the inductance

L corresponding to the mass m, the capacitance C corresponding to the inverse of the spring

constant 1k

= 1m2

0

, and p and q corresponding to the momentum and position, respectively. If one

considers 4.40) as a Hamiltonian function of the system, the classical Hamilton equations for the

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position q and momentum p are written as

dq

dt H

p=

1

Lp , (4.41)

dp

dt H

q= 1

Cq , (4.42)

which are identical in form to the Kirchhoff equations (4.37) and (4.38) for the voltage q and

current p. Hence, one can conclude that the voltage and current in a lossless LC circuit

are a pair of conjugate observables, just like the position and momentum of a mechanical har-

monic oscillator. Quantum mechanically, conjugate observables q and p must satisfy the following

commutation relation,

[q, p] qp pq = ih , (4.43)

where q and p are no longer complex numbers (c-numbers); rather, they are quantum mechanical

operators (q-numbers). Using the Schwartz inequality, one can derive the following Heisenberg

uncertainty principle for the product of the variances of q and p:

q2

p2

h2

4 . (4.44)

The (non-Hermitian) creation (a) and annihilation (a) operators are defined by:

a =1

2h0L(0Lq + ip) , (4.45)

a =1

2h0L(0Lq ip) . (4.46)

The creation operator a is a Hermitian conjugate of the annihilation operator a. The Hamiltonian

function (4.40) and the commutation relation (4.43) are rewritten as

H = h0

aa +

1

2

, (4.47)

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[a, a] = 1 . (4.48)

If the LC circuit is not excited at all, the quantum mechanical state of such an unexcited LC circuit

is referred to as the ground state, or the vacuum state. The vacuum state is mathematically

defined by

a|0 = 0 . (4.49)

One cannot annihilate a photon from the vacuum state because there is no photon in the vacuum

state, which is the meaning of this expression. On the other hand, the operation of the creation

operator on the vacuum state creates a single-photon state:

a|0 = 1 . (4.50)

The mean and mean-square of the voltage and current of the vacuum state are calculated by

taking the ensemble average of the respective operators:

q 0|q|0 = 0|

h0C

2(a + a)|0 = 0 , (4.51)

q2 0|q2|0 = 0| h0C2

(a2 + a2 + aa + aa)|0 = h02

C , (4.52)

p 0|p|0 = 0|i

h0L

2(a a)|0 = 0 , (4.53)

p2 0|p2|0 = 0| h0L2

(a2 a2 + aa + aa)|0 = h02

L . (4.54)

Here, the orthogonality relations between photon number eigenstates such as 0|1 = 0|2 = 0 are

used. The variances of the voltage and current satisfy the minimum uncertainty product,

q2 q2 q2 = h02 C

p2 p2 p2 = h02 L q2 p2 =

h2

4. (4.55)

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Notice that both voltage and current have intrinsic fluctuations (variances) even at absolute zero

temperatures. The expectation value of the zero-point energy is given by

H 0|H|0 = 0|h0

aa +1

2

|0 = h0

2. (4.56)

If one adds this zero-point fluctuation contribution to (4.35), one obtains the full quantum

mechanical expression for an open-circuit voltage fluctuation spectral density:

Sv() = 2hR coth

h

2kB

. (4.57)

As shown in Fig. 4-8, Sv() is reduced to the thermal noise value (4kBR) in the high-temperature

limit (kB h) and is reduced to the quantum noise value (2hR) in the low-temperature limit

(kB h). Equation (4.57), including the quantum mechanical zero-point fluctuation, is referred

to as the generalized Nyquist noise.

1

10

1 10

102

10-1

10-1

10-2 102

Thermal Noise limit

Sv() / 4k BR

Quantum Noise Limit

h

2kB

Figure 4-8:

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4.4 Thermal and Quantum Noise of a Mesoscopic System

If the size of a conducting wire becomes smaller and smaller, an electron can eventually transit

from one electrode to the other without a collision with the lattice. Such a small-scale conductor is

called a mesoscopic conductor. A collision-free electron transport in a mesoscopic conductor is

termed a ballistic transport, while a conventional electron transport accompanying many collisions

with the lattice in a macroscopic conductor is termed a dissipative transport. Since an electron

propagates as a coherent wave in such a mesoscopic conductor, various coherent electron wave

interferometric devices can be constructed. Even though such an electron wave is not scattered by

the lattice, it is known that the terminal current of such a mesoscopic conductor still features noise.

Consider a mesoscopic conductor with two terminal electrodes connected to an external constant

V

electrode electrodemesoscopic conductor

electronenergy

coherent electron wave

kN =2

LN

1

2

1 2 qV

Figure 4-9:

voltage source, as shown in Fig. 4-9. The electron distributions in the two terminal electrodes

obey the thermal equilibrium Fermi-Dirac distributions. The Fermi-level difference is determined

by the external voltage. An electron wavepacket with a certain energy is emitted from electrode 1

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and absorbed by electrode 2, which contributes to an external current. As shown in Fig. 4-9, the

longitudinal wavenumber of such a coherent electron wave is quantized by the boundary condition:

kN =2

LN (N = positive integer) . (4.58)

The mode density per energy is given by

dN

dE=

dN

dkN

dkNdE

=L

2hvN, (4.59)

where vN =ddkN

is the electron group velocity. The total number of independent modes in the

energy range 1 2 is dNdE (1 2). Each mode can accommodate a single electron due to the

Pauli exclusion principle and has a transit time tN =LvN

between the two electrodes. Therefore,

the average current is

I = qdN

dE(1 2) 1

tN=

q2

hV , (4.60)

where 1 2 = qV (V = external bias voltage) is used. A mesoscopic conductor has a finite

conductance GQ =q2

h(quantum unit of conductance) that is not due to scattering with the lattice

(dissipation), but due to the Pauli exclusion principle.

An alternative method to derive the quantum unit of conductance GQ is to count the number

of independent temporal modes (rather than Fourier modes). As shown in Fig. 4-10, the

inverse Fourier transform of a bandpass-filtered spectrum centered at a frequency 0 and bandwidth

(FWHM) = 12h

is a Nyquist function fN =sin(t)(t) . A sequence of Nyquist functions

displaced by = 1 reproduces any bandwidth-limited function, and thus one can consider the

arrival rate of independent temporal modes as 1

= . If one assigns a single electron per

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independent temporal mode, the current is

I =q

=

q2

hV . (4.61)

The same result is obtained as in 4.60). If a mesoscopic conductor supports only one transverse

mode, two spin-degenerate states can still be accommodated for the single transverse mode and

thus the total conductance is 2q2

hinstead of q

2

h. If a system is at an absolute zero temperature,

Spectrum

0 E=

1

Nyquist

function

12 t

Figure 4-10:

as shown in Fig. 4-11(a), the state above the Fermi energy is completely empty and the state

below the Fermi energy is completely occupied. The electron wavepackets (temporal modes) with

center frequencies lying between the two Fermi energies are emitted from electrode 1 and travel

toward electrode 2. The electron wavepackets with energies outside of this region do not contribute

to a current. This electron transport is a deterministic process and thus there is no dc current

noise in the external circuit. However, if one measures the current noise at a finite frequency, the

noise is produced by the beating between the electron wavepacket emitted from one electrode in

the energy below the Fermi energy 1 and the empty electron wavepacket (vacuum state) emitted

from the same electrode in the energy above the Fermi energy 1. By increasing the measurement

21


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frequency, additional occupied and empty electron wavepackets contribute to the beat noise and

the noise power thus increases linearly in proportion to . This is the origin of the quantum noise

which is dominant in a low-temperature/high-frequency region, as shown in Fig. 4-12.

If a system is at a finite temperature, as shown in Fig. 4-11(b), the electron occupancy near

the Fermi energy becomes probabilistic. The electron wavepackets with energies close to 1 and

2 may or may not be emitted from electrode 1 and absorbed by electrode 2. This stochasticity

introduces another current noise in the external circuit. By increasing the temperature, the partial

occupancy of electron states increases and the noise power thus increases linearly in proportion to

. This is the origin of the thermal noise which is dominant in a high-temperature/low-frequency

region, as shown in Fig. 4-12.

beating

empty packets

occupiedpackets beating

(a) = 0 (Quantum Noise) (b) = 0 (Thermal Noise)

statistical occupancy

2

1

2

1

Figure 4-11:

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1

10

1 10

10

101010 10

Thermal Noise limit

Quantum Noise

h

2kB

S ( )RQ

4kB

Figure 4-12:

References

1. Mobility and diffusion constant of electrons in conductors:

W. Shockley, Electrons and Holes in Semiconductors, Van Nostrand, New York (1963).

2. Langevin equation and fluctuation-dissipation theory of a Brownian particle:

M. Sargent III, M. O. Scully and W. E. Lamb, Jr., Laser Physics (Addison-Wesley,

Reading, Mass. 1974).

3. Thermal noise:

M. B. Johnson, Nature 119, 50 (1927); Phys. Rev. 32, 97 (1928).

H. Nyquist, Phys. Rev. 32, 110 (1928).

4. Quantum zero-point fluctuations of a loss less LC circuit:

23


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W. H. Louisell, Radiation and Noise in Quantum Electronics, McGraw-Hill Book Co.,

New York (1964); Quantum Statistical Properties of Radiation, John Wiley & Sons, New

York (1973).

5. Generalized Nyquist formula:

H. B. Caller and T. A. Welton, Phys. Rev. 83, 34 (1951).

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Contents

5. Inherent Noise of pn Junction Diodes 2

5.1 pn Junction Diodes Under Constant Voltage Operation . . . . . . . . . . . . . . . . 4

5.1.1 Current-Voltage and Capacitance-Voltage Characteristics of a pn Junction . . 4

5.1.2 Thermal Diffusion Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

5.1.3 Generation-Recombination Noise . . . . . . . . . . . . . . . . . . . . . . . . . 13

5.1.4 Total Current Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5.1.5 Short Diode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5.2 pn Junction Diodes Under Constant Current Operation . . . . . . . . . . . . . . . . 21

5.2.1 Effect of a Finite Source Resistance . . . . . . . . . . . . . . . . . . . . . . . 21

5.2.2 A pn Junction Diode With Negligible Backward Thermionic Emission . . . . 23

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5. Inherent Noise of pn Junction Diodes

Shockleys 1949 paper heralded a new era in the history of semiconductor device physics.

Basic physical processes of a junction diode and a junction transistor were presented in this 1949

Bell Syst. Tech. J. paper. In the same issue, the first report appeared on the noise of a point

contact transistor. The observed noise figure was 50-70 dB above the intrinsic noise limit! It took

almost 30 years to suppress this excess noise (mainly due to 1/f noise and surface recombination

noise) and to obtain a noise figure very close to the theoretical limit (= 0 dB). This intrinsic noise

of a transistor is determined by the thermal noise in the bulk resistive region and the shot noise in

the junction region. In this chapter we will study the inherent noise of pn junction diodes, which

set a fundamental limit on the noise performance of various semiconductor devices.

There are two distinct bias conditions for a junction diode: constant voltage operation and

constant current operation. The former is obtained when the junction (differential) resistance Rd

is much larger than the source resistance RS, and the latter is obtained in the opposite limit.

There are two types of junction diodes: a macroscopic junction and a mesoscopic junction. An

electrostatic energy for a single electron q2/2C (where C is a junction capacitance) is much smaller

than the thermal energy kB in a macroscopic junction and the opposite is true for a mesoscopic

junction. A junction diode features markedly different noise characteristics in such different bias

conditions and operational regimes.

Consider a pn junction diode biased by a constant voltage source with a source resistance RS,

as shown in Fig. 5-1. If the source resistance RS is much smaller than a diode differential resistance

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defined by

Rd

dI

dV

1

, (5.1)

where I and V are the junction current and the junction voltage, then the junction voltage V is

p

n

Rs

v

RdCdep CdifRsis i

Figure 5-1:

pinned by the source. There is no fluctuation in the junction voltage V due to the fast relaxation

time (CRS) of an external circuit, but there is a fluctuation in the junction current I. This bias

condition is referred to as constant voltage operation. Standard theoretical studies on the noise

characteristics of a pn junction diode have considered this mode of operation; therefore, our analysis

also starts with this bias condition (Section 5.1).

On the other hand, some experimental studies on the noise of a pn junction diode have consid-

ered the opposite bias condition; that is, the source resistance RS is much larger than the differential

resistance Rd of the diode. In such a case there is no fluctuation in the junction current I due to the

slow relaxation time (CRS) of an external circuit, but there is a fluctuation in the junction voltage

V. This bias condition is referred to as constant current operation. This mode of operation will

be analyzed in Section 5.2.

There are two types of pn junctions which feature drastically different noise characteristics:

macroscopic junctions and mesoscopic junctions. When a single-electron charging energy (q2/2C)

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(where C is the junction capacitance) is much smaller than the thermal characteristic energy

kB, the behavior of each individual electron does not affect the junction dynamics. This is a

macroscopic junction limit. On the other hand, when (q2/2C) is much greater than kB, a so-called

(single-electron) Coulomb blockade effect emerges and a previous single-electron event determines

the junction dynamics at a later time. This is a mesoscopic junction limit.

5.1 pn Junction Diodes Under Constant Voltage Operation

5.1.1 Current-Voltage and Capacitance-Voltage Characteristics of a pn Junction

The noise characteristics of a p+-N heterojunction with a heavily p-doped narrow bandgap

material and lightly n-doped wide bandgap material will be studied, rather than a conventional

p-n homojunction, because this specific junction structure is used in various semiconductor devices

(i.e., a double-heterostructure semiconductor laser and a heterojunction bipolar transistor). The

extension of the following analysis to a pn homojunction is straightforward.

The band diagram of a p+

-N heterojunction diode at a zero bias condition (V = 0) and a

forward bias condition (V > 0) are shown in Fig. 5-2. The built-in potential VD is divided into the

potentials in the p+- and N-layers:

VDp =VDK

, (5.2)

VDn = VD

1 1

K

, (5.3)

where

K = 1 +1(N

A1 N+D1)2(N

+D2 NA2)

. (5.4)

4


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N-layer

Ecq

V

electron gas

+

+

+

+

+

+

x

+

+

x

electron gas

+

hole gas hole gas

x = -d x = 0

(v = 0)

x = -d x = 0

(v > 0)

EVq

VDn

VDp

VDn - V2

VDp - V1

F

p+

- layer

Fp

Fn

Figure 5-2:

Here, 1, NA1, and ND1 are the dielectric constant, acceptor concentration, and donar concentration

of the p+-layer, and 2, NA2, and ND2 are those of the N-layer. If a p+-N diode satisfies 1 > 2

and NA1 N+D1 N+D2 NA2, then K 1. Consequently, the built-in potential VD is mainly

supported in the N-layer, i.e., VDn VD and VDp 0. The transmitted electron flux from theN-layer to the p+-layer across the potential barrier height VDn should be equal to the transmitted

electron flux from the p+-layer to the N-layer across the potential barrier Ec/q because there is

no net current at V = 0.

When a forward bias (V > 0) is applied, only the potential barrier seen by the electrons in the

N-layer decreases to VDn V2 VD V, where the applied voltage supported in the N-layer is

V2 = V

1 1K

V. The electron (minority carrier) density at the edge of the depletion layer

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(x = 0) in the p+-layer is given by

np = XnN0 expVD V

VT = np0 expV

VT , (5.5)where VT =

kBq is the thermal voltage and

np0 = XnN0 exp

VD

VT

, (5.6)

is the thermal equilibrium electron density in the p+-layer. X is the transmission coefficient of

an electron at the heterojunction interface and nN0 is the electron (majority carrier) density at

the edge of the depletion layer (t =

d) in the N-layer which is equal to the thermal equilibrium

electron density in the N-layer.

The excess electron density 5.5) at x = 0 diffuses towards x = W, where a p-side metal contact

is located. The distribution of the excess electron density n(x, t) obeys

tn(x, t) = n(x, t) np0

n 1

q

xin(x, t) , (5.7)

where n is the electron (minority-carrier) lifetime and, since there is no electric field in the neutral

p+-layer, the current in(x, t) is carried only by a diffusion component

in(x, t) = qDn x

n(x, t) . (5.8)

Here, Dn is the electron diffusion constant. Solving (5.7) and (5.8) with the boundary conditions,

np =

np0 expVVT

at x = 0

np0 at x

Ln

, (5.9)

the steady-state solution for n(x) is now given by

np(x) = np0 + (np np0)ex/Ln , (5.10)

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where Ln =

Dnn is the electron diffusion length. The junction current density is determined by

the diffusion current (5.8) at x = 0:

i in(x = 0) = qDnLn

(np np0) = qDnnp0Ln

eVVT 1

. (5.11)

The total current I = Ai vs. the junction voltage V is plotted in Fig. 5-3, where A is a cross-

sectional area.

10

-10

-2 -1

0 1 2

5

-5

ILn

AqDnnp0

V/VT

Figure 5-3:

The differential resistance Rd, defined bydIdV

1

, is approximately given by VT/I under a

strong forward bias condition. The diffusion capacitance Cdif of the diode is defined as the voltage

derivative of the (total) minority carrier charge:

Cdif ddV

Q(minority carrier) = Ad

dV

q

0[np(x) np0]dx

= AqLnnp0

VTeVVT

IVT

n . (5.12)

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The CR time constant characterized by the differential resistance Rd and the diffusion capacitance

Cdif is equal to the electron (minority carrier) lifetime n.

The depletion-layer capacitance Cdep of the diode is defined as the voltage derivative of the

(total) space charge:

Cdep ddV

Q(space charge)

=2

WdepA

=

q2ND2

2(VD V) A (5.13)

Here, Wdep =

22qND2

(VD V) is the depletion layer width in the N-layer. The capacitance con-

tributed by the depletion layer in the p+-layer is neglected. The CR time constant characterized

by the differential resistance Rd and the depletion layer capacitance Cdep is equal to the thermionic

emission time te, which we will discuss later in this chapter.

It will be shown that the thermionic emission time te = CdepRd is a key parameter for deter-

mining the noise characteristics of a pn junction diode under weak forward bias, while the minority-

carrier lifetime n = CdifRd is a key parameter for determining the noise characteristics of a pn

junction diode under strong forward bias. The junction capacitance of a pn junction is determined

by the depletion-layer capacitance under weak forward bias and by the diffusion capacitance under

strong forward bias.

5.1.2 Thermal Diffusion Noise

When a pn junction is biased by a constant voltage source, the electron (minority carrier)

densities at x = 0 (edge of the depletion layer) and x = W (p-side metal contact) are constantly

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fixed to be np0 eV/VT and np0, respectively. The electron density fluctuates, however, between

x = 0 and x = W due to microscopic electron motion induced by thermal agitation and electron

generation and recombination processes. In order to keep the boundary conditions at x = 0 and

x = W and to restore the steady-state electron distribution, the relaxation current pulse flows in

the entire p+-region between x = 0 and x = W. This relaxation current inside the p+-region results

in the departure from charge neutrality of this region, which induces the external circuit current.

If an electron makes a transit over a small distance f between collisions with the lattice, an

instantaneous current q(t) flows at the two locations x = x

and x = x

+ f, as shown in Fig. 5-4.

This instantaneous current creates the departure from the steady-state electron distribution (5.10)

and triggers the relaxation current in the entire p+-region between x = 0 and x = W, which,

after a reasonably short time, restores the steady-state electron distribution (5.10). The electron

distribution deviation n

(x, t) = n(x, t) np0(x), which results in such a relaxation current in the

entire region, satisfies the diffusion equation (5.7) and the boundary conditions n

= 0 at x = 0

and x = W. The Fourier transform of the diffusion equation (5.7) is given by

2

x2N

(j) =1

L2N

(j) , (5.14)

where N

(j) is the Fourier transform of n

(x, t) and

L2 =L2n

1 +jn. (5.15)

The Fourier transform of the relaxation currents i

1(t) at x = x

and i

2(t) at x = x

+ f are

expressed in terms of the Fourier-transformed electron density deviations N

1(j) at x = x

and

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q(t)

x = 0 x = Wx' x' + l f

initial current

relaxation current

diffusion current direct return current

0

N' ( j)

N' 1

N' 2

Wx

Figure 5-4:

N

2(j) at x = x

+ f:

I

1(j) = qDnN

(j)

x

x=x = k1N1(j) , (5.16)I

2(j) = qDnN

(j)

x x=x+f = k2N

2(j) , (5.17)

where

k1 =qDn

Lcoth

x

L

, (5.18)

k2 =qDn

Lcoth

W x

L

. (5.19)

There are also direct return curre

yamamoto noise

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