Chapter 8

REVIEW OF NOISE PROCESSES AND NOISE CONCEPTS RELEVANT TO MICROWAVE

SEMICONDUCTOR DEVICES

The universality of 1/1 noise transcends the limits of electro­physics. Indeed, 1/1 noise is present in the phase and frequency of all known frequency standards ... It has also been found in the flow rate ofsand in an hourglass, the frequency ofsunspots, the light output of quasars, the flow rate of the Nile over the last 2,000 years, the central Pacific Ocean current velocities at a depth of 3,100 meters, the flux of cars on an expressway, and in the loudness and pitch fluctuations of classical music (Quoted from (Handel, 1980), with permission).

INTRODUCTION

In some of the chapters to follow, we will treat low-noise amplifiers and detectors for microwaves/millimeter waves. The receiver/detector noise often represents one important side to the limitations posed on a microwave system, once one has chosen a suitable source for the transmitter. The transmitter has important characteristics such as available power and close-to-carrier noise. These were discussed in earlier chapters. Often, improvements to the sensitiv­ity ofthe receiver or detector, which is determined by its noise properties, may be even more effective in terms of optimizing total system performance. The present chapter reviews some of the basic noise concepts and processes which will be useful in understanding and characterizing such devices. For further details we refer to the excellent recent book, (Van der Ziel, 1986), and the annotated reprint collection by Gupta (1977).

THERMAL NOISE - NOISE FIGURE AND EQUIVALENT NOISE TEMPERATURE

The concepts which are used for characterizing receiver noise performance go back to the well-known Nyquist theorem, which may be stated as follows.

Nyquist'll Theorem

To find the noise (power, voltage or current) produced by a two-terminal network in thermal equilibrium at temperature T (OK) and having an impedance with real part R, we replace the network by a noiseless resistor R in series with a noise generator of voltage en, see Figure 8.1. Here ((Nyquist, 1928) and (Johnson, 1928)),

e~ = 4kBTRp(f)B (8.1)

S. Yngvesson, Microwave Semiconductor Devices© Kluwer Academic Publishers 1991

20B Microwave Semiconductor Devices

kB = Boltzmann's constant

B = Bandwidth over which noise is measured

and

p(f) = k;T x ehJllt~T _ 1 is the "PLANCK FACTOR" (B.2)

Noise in resistive circuits is directly related to black-body radiation, and Ny­quist's theorem can be derived from Planck's black-body radiation law: This is why p(J) is called the Planck factor.

It is also important to note that the black-body radiation law assumes that the system is in thermal equilibruim at some temperature, T. This situa­tion correctly describes passive microwave loads, transmission lines, etc. Most active devices are not in thermal equilibrium, since they dissipate (or produce) power. The non-thermal equilibrium case will be discussed in later sections in this chapter. At microwave frequencies and T ::: room-temperature,

Then (expand the exponential!)

hi «1 kBT

e~::: 4kBTRB (B.3)

Note that in this approximation, the noise voltage is independent of frequency - white noise. The time-dependence of e,,(t) can be illustrated as in the insert in Figure B.l. The available noise power can be found from the circuit in Figure B.2:

e 2 P" =...!!.. = kBTB

4R (B.4)

Recently, it has become possible to calculate the noise in a bulk piece of semiconductor by numerical simulation (Adams, 1991). Using the Monte Carlo method, individual particles are followed, which go through scattering events in the material, whereupon fluctuations in the current are obtained. One such simulation demonstrated clearly that the current fluctuations in a short section of GaAs follow the Nyquist expression. From (B.4) we can easily derive that

"'2 P" kBTB (B 4 ) '''=R=~ . a

Figure B.1.b. shows that this relation is satisfied by the simulated GaAs data for two temperatures (77K and 300K) (Adams, 1991).

Chapter 8 209

• t

R ............ Noise Less

R .. .. £]('1 a

1000.0

100.0 ["'.,

• Me calculation (300K) r--.

Theory (300K) Ul

N « "-v

N I 0 10.0

.25.

S 1.0 (j) • 0.1

"-"- • "-

"-"-

--Me calculation (77K) "-

"-"-

Theory (77K) "-"-

0.1 1.0 1 0.0

Resistance (kO)

b

"-

1 00.0

Figure 8.1. (a) The noise-equivalent circuit of a resutor i6 a random voltage 60urce (en (t)) in 6erie8 with a noi8e-Ie8' re,i6tor of the same value. The time-dependence o/en(t) i6 illustrated. (b) Monte Carlo calculation ofi"£/B for GaAs resistors of length 0.1-0.t JLm. The ordinate i6 Si(f) (8ee (8.,11)) which i8 equivalent to i"£/B. Courte8y of Dr. J. Adam8 (Adam" 1991).

210 Microwave Semiconductor Devices

R

R

Figure 8.2. The available noise power is obtained when a matched load il connected to the noisy resistor.

Noise Figure

For the definition of the noi .. e figure, we use a standardized test set-up (Fig. S.3):

We then compare the noise power out (Pn)~&T from the circuit in Figure

S.3 with that from the following circuit (Figure S.4), (Pn)~&T'

The noise figure is defined as

) (1) F = (Pn OUT

- (2) (Pn)OUT

(S.5a)

We can calculate (Pn)~&T by noting that the Nyquist theorem gives the available noise power from the input load (R) as kBTB:

Thus,

( ) (2) Pn OUT = kBTB x Go

( ) (1) F = P" OUT

- GokToB

Effective Input Noise TeUlperature

(S.5b)

(S.5c)

The effective input noise temperature, T., of a network (i.e. amplifier, mixer, etc.) is defined as that temperature of a matched input termination for which the output noise power (per unit bandwidth) is twice that which would occur if the input termination were at absolute zero temperature.

Chapter 8

Matched Load, T=290K

~R Amplifier (or Mixer)

Power gain = Go

211

Figure 8.3. Standard configuration for mea.furement of the noi.fe figure of an amplifier.

R

290K

Amplifier Power gain = Go

Noise Free

(2) ------ (P n)out

Figure 8.4. The .ame amplifier configuration a. in Figure 8.3, but lU.uming a noi.e-free amplifier.

We thus compare the networks in Figures 8.5 and 8.6 below:

Ezample 1: Use the definitions of noise temperature and noise figure to derive the relationship between these concepts.

212

R

(at T = OaK)

Microwave Semiconductor Devices

Amplifier

Figure 8.5. An amplifier with a matched input termination at T = 0 Kelvin.

(2) (1) --(P nJout = 2 x (P n)out

Figure 8.6. The same amplifier as in Figure 8.5., ezcept that the temperature of the input termination has been increased to a temperature of T. Kelvin, which doubles the output noise power.

Chapter 8

R

(atT=OOK)

Amplifier Power gain = Go Noise Figure = F

213

Figure 8.7. An amplifier with a matched input termination at a temperature of 0 Kelvin, and 6pecified gain and noile figure.

First, use Rat T = OOK (see Figure 8.7).

We find (with To = 2900 K)

(P,,)OUT = FGolcBToB - GolcBToB = (F - 1)GolcBToB (8.6)

The fird term is the output noise of the amplifier, gain Go, noise figure F' ,

The 6econd term has to be subtracted because in the definition of noise figure, a matched termination at To (2900 K) is included. We subtract the noise this would produce since in our example the load is at OOK (i.e. zero noise).

The equivalent noise temperature, Til is now obtained if we increase the temperature of the input load until (P,,)OUT double6. The (extra) noise due to the input load thus equals the value of (P,,)OUT above

(8.7)

i.e. T. = (F -1)To (8.8)

This expression relates the two definitions (noise temperature and noise figure).

Cascaded Networks (Frii.' formula)

Figure 8.8 shows several cascaded networks (two-ports) with given noise figure and gain for each network. One can easily derive an expression for the

214 Microwave Semiconductor Devices

Fl' F2• -----0---- Fn· G l G2 Gn

R(To) > > > (0 CD -{).. (0

Figure 8.8. A cascaded arrangement of n matched amplifiers.

kBToB kBToB --- --,..,. Transmission Line

....,

R > Char. Imp. = Zo ~ .(> Temp. = To

(To) Loss =L

R(To)

R=Zp " R=Zp

Figure 8.9. A transmission line with matched loads at each end.

noise figure or noise temperature of the entire cascaded network, by using the above concepts. These expressions are (Friis, 1944):

F2 - 1 Fs - 1 Fn - 1 F1 ... n = FI + -G + -G G + ... + G G G (8.9)

1 1 2 1 2'" n

Chapter 8 215

and T. 2 T."

T. , ... n = Tel + -G + ... + G G G 1 1 2'" ,,-1

(8.10)

These relations express the fact that the noise figure (or noise temperature) of the first "stage" yields the dominant contribution to the system noise figure, particularly if its gain is large.

A special case is when all networks are identical. For this case, we can define the noi6e mea6ure

F-1 M=l_-b (8.11)

This equation can be derived from (8.9) by summing the geometric series. It is useful if one wants to estimate the actual system noise figure (= M) which is obtained if one has available a number of identical amplifiers, for which the gain is not large enough to make the contributions of the following stages to the system noise figure negligible.

Ezample 2: Noise temperature of a transmission line.

Analyze the network in Figure 8.9.

Since there is thermal equilibrium, the right resistor must receive as much noise power as it delivers, i.e. kBToB.

The right resistor receives noise from (1) the left resistor and (2) the transmission line

(1) The line has a loss factor

L = Pi" - Pout

(i.e. > 1 for loss)

The power delivered by the left resistor is attenuated by 1/ L; it is

pel) = ~kBToB L

(8.12)

(8.13)

(2) Call the noise power produced by the line p(2) = (P,,)L, Also call the noise temperature of the line To,. As stated above, the right resistor must receive as much noise power as it delivers:

(8.14)

Use the definition of noise figure (8.5c) on the transmission line and (8.14):

Fl = kBToB (8.15) GkBToB

216

since

Further, using (8.8)

Microwave Semiconductor Device,

(T.)l = (L - 1) To

(8.16)

(8.17)

(8.18)

Eq. (8.18) can be generalized if the line is at some other temperature, Tl:

(T.)l = (L - l)Tl (8.19)

This expression also applies to a general matched attenuator with loss =L.

Variation of Noise Figure with Source Admittance

In all relations derived so far in this chapter, we have assumed that the source is matched to the input of the network. It is often a good approxi­mation to make this assumption, but especially with transistor amplifiers, we must consider the influence of the source admittance on the noise figure. We generalize the definition of noise figure so that it is still

F = 1 NOISE INTERNAL TO NETWORK + NOISE FROM TERMINATION

(8.20)

(The noise from the termination will now not be lcBToB anymore, of course, if it is not matched.)

We represent our two-port network as shown in Figure 8.10 (IRE Sub­committee on Noise, 1960).

Here, e(t) and i(t) represent internal noise sources in the two-port. The two­port equations yield:

{ V1 = AV2 + B12 + E

11 = CV2 +D12 +1 (8.21 )

If the input source (with internal admittance, Ys, and a real part = G s) is included, the circuit is as shown in Figure 8.11.

The noise figure for this circuit becomes

F - 1 Ii + Ysel 2 - + ·2

IS (8.22)

In general, i and Yse are partially correlated, and this must be taken into account when performing the operations indicated. The result can be stated in the following concise form:

F=Fmin+ ~IYs-YoI2 (8.23)

where the following four noi6e parameter6 have been defined:

Chapter 8

11 e(t)[E]

12

I Noise-

V1 Source-Free

I(t) [I] Two-Port

Figure 8.10. Equivalent circuit of a typical transistor amplifier.

E

Noise Free Two-Port

I

Figure 8.11. Inclu.ion of an input .ource in the circuit of Figure 8.10.

Fmin = Optimum noise figure.

217

V2

Yo = Optimum source admittance for which Fmin is obtained (real and imagi­nary parts).

218 Microwave Semiconductor Devices

R". = A resistance parameter which specifies how sensitive the noise figure is to changes in Ys away from Yo.

Typically, Yo corresponds to a quite large reflection coefficient (vs 50n) for MESFET amplifiers.

In order to determine the four "noise parameters" one measures F for a number of values of Ys = Gs + jB and then fits the data to equation (8.23) (Lane, 1969; Mitawa and Katoh, 1979).

SHOT NOISE

Shot noise is due to the fluctuations in particle current in any device with an average (DC) current flow. The concept was first applied to electron emis­sion from the cathode in a vacuum tube (Schottky, 1918). Any current consists of discrete particles (electrons or holes). If we have N particles per second, then there will be a statistical fluctuation in N, with a standard deviation ~v'N.

N _ IDC. - , (8.24) e

Thus, the noise current magnitude is

liNI ~ e v'N ~ e JI~e (8.25)

=? i~ ~ eIDc

(Since N is per second, BW ~ 1 Hz)

Closer investigation shows that an extra factor of two must be included, as discussed below:

(8.26)

For a more rigorous look at shot noise, let's assume that we are recording the current i(t) (with its fluctuations) over a finite time interval, T. Expand i(t) in a Fourier series (see e.g. (Marcuse, 1970)):

00

i(t) = L Cm e;(21r/T)ml (8.27) m=-oo

The Fourier coefficients must be random variables, i.e. with random phase, so that

< Cm > = 0 for m f:. 0

Co = IDC, and

< CmC~ > =< ICm l2 > 6mn

Also, < i(t) > = Co = IDe

(8.28)

(8.29)

(8.30)

(8.31)

Chapter 8 219

The symbol <> denotes that an ensemble average is taken. Since the current is real, i(t) = i· (t), from which follows that

00 L < CmCn > e;(2 ... /T)(m-n)1

m=-oon=-oo 00

(8.32)

m=-oo

As for thermal noise, we measure the noise current in a finite bandwidth, say from w(m = a) to w(m = b) and the bandwidth is

From (8.32) it follows that

1 B = -(b - a)

T

b

< i 2 >B= 2 L < ICml >2 m=a.

The fact that C_ m = C:,. gives rise to the factor of two.

(8.33)

(8.34)

To calculate the Fourier coefficients, we assume that the current due to each electron can be represented by a 6-function

(8.35)

" where t" is the time at which the lI'th electron p/l.Sses the plane at which we measure i(t). Now,

(8.36)

and

< ICm l2 > = ;: E L ([e;(2"'/T)m(T.-T~)]) " I'

e2 e2

=T2Ll=NT2

(8.37)

" All terms in the sum are 1 since the events at T" are independent. N is the number of events, i.e. the number of electrons observed in the time T. This makes it clear that

eN IDe = - (8.38)

T Finally, combining (8.33), (8.34), (8.37) and (8.38) we find the desired result quoted earlier:

(8.26)

220 Microwave Semiconductor Device6

If the device has an internal resistance = R we find a noise voltage

e~ = R2i~ ... The available noise power = ~, i.e.

1 (PN)SHOT-NOISE = 2' x R x e x Inc x B

DIFFUSION NOISE

(8.39)

(8.40)

The expressions for noise in a semiconductor device have had to be ex­tended beyond the standard concepts of Nyquist and shot noise due to the special conditions in many devices. We will discuss some of these extended concepts and expressions in this section (Van der Ziel, 1986).

We first need to define the spectral intensity, S,,(I), of a random variable, X(t):

(8.41)

The quantity X(t)X(t + 8) is the autocorrelation-function, and for the important example of white noise, it has a value of:

X(t)X(t + 8) = A X 6(8) (8.42)

Therefore, S,,(I) = 2A for this case, which is frequency-independent as it should be. Another example is the noise voltage from a Nyquist source, which has a spectral intensity

(Compare (8.3)!) (8.43)

We will be especially interested in the noise produced by carriers which move due to diffusion. Thus we introduce a slab of (uniform) semiconductor, with a concentration of carriers, n, and a concentration gradient, ~:. We sub­divide the slab into smaller "boxes" which have dimensions az, ay and az, as shown in Figure 8.12 below. Each of the boxes is labeled (Ie, I, m), with an index for each of the three coordinate directions.

Next, we want to model the motion of individual electrons by assuming that there is a probability = a X at that an electron will jump from the box (Ie, I, m) to any ofthe adjacent boxes during the time interval at- allindividual jumps are also assumed to be independent. If n(k, I, m) is the density of electrons in box (k, l, m), then we can set up equations for the particle current between this box and box (Ie + 1, I, m) as follows:

wl.Hl = an(Ie, I, m)azayaz (8.44)

Chapter 8 221

~: Box (k + 1,1, m)

Box(k,I, m) Electron densny is n(k,I, m)

z

Figure 8.12. A section of a bulk semiconductor, with a particle gradient 1J..n/1J..z in the z-direction. The particle flow in and out of the incremental boz designated by (1J..z, 1J..y, 1J..z) is calculated in the tezt.

and WII+l,1I = an(k + 1, I, m)1J..z1J..y1J..z =

= a[n(k,l, m) + (an) 1J..z]1J..z1J..y1J..z az Is,I,m

(8.45)

The net particle current becomes

W = -a (::) 1J..z 21J..y1J..z ",',m.

(8.46)

The quantity w/1J..y1J..z is the particle current per unit cross-sectional area, and should be independent of the size of 1J..y and 1J..z. Therefore, this quan­tity is a constant for a given concentration gradient in the z-direction, and consequently a x 1J..z2 must also be a constant. We will call the constant D,.., I.e.

(8.47)

and

w = -D,.. (::) 1J..y1J..z h,I,m

(8.48)

It thus becomes clear that D,.. is the diffusion constant (for electrons).

222 Microwave Semiconductor Devices

Particle currents in both the forward and the reverse directions contribute to the noise. In each direction we have a given average particle current con­sisting of a finite number of particles, and the situation is therefore equivalent to the one we encountered in the derivation of the shot-noise expression. From (8.26) we can find the spectral intensity for the shot noise current:

S, = 2eIDc

If we apply this to the particle current, w, we find that

S,. (I) = 2 W Ia,Ia+l + 2WAo+l,/o

We can express WIa,1o+1 as

Thus, S",(I) = 4Dnn(k,l,m)AyAz/Az

(8.49)

(8.50)

(8.51)

(8.52)

We can convert the noise in the particle current to noise in the electrical current by multiplying by e2 , i.e.

(8.53)

Let us now compare the spectral intensity derived from particle random "jumps" as given above, with the situation for a uniform material in thermal equilibrium. In this case, we expect the Nyquist relation to be valid. We can indeed find the Nyquist noise formula by using the Einstein relation (1.24a), which is valid for thermal equilibrium conditions:

which leads to

S'(I) = 4kBT[e~n(z)lAyAz/Az = 4kT/AR

where AR is the resistance of the box,

AR= Az q~n(z) AyAz '-v--'

IT

(8.54)

(8.55)

(8.56)

Equation (8.55) is the equivalent of (8.43) above, i.e. we have obtained the Nyquist noise formula, provided that the Einstein relation is obeyed.

Our main objective is to find a noise expression which is valid in the sat­urated portion of the channel in a MESFET (see Chapter 10), where electrons are moving essentially at the saturated velocity, and may have a fairly high electron temperature. It is clear that the Nyquist relation and the Einstein relation do not apply. However, as discussed in greater detail in, for example,

Chapter 8 223

van der Ziel (1986), (8.53) still applies to this case. If we change from using the current to using the current density, and also use (8.42) (assuming that elec­trons in different positions (z) are uncorrelated), we find the auto-correlation function to be:

j .. (z)j:(z') = 4e2 (D,,: n) B x c5(z - z') (8.57)

Here, A is the cross-sectional area, and B the bandwidth. In using this equa­tion, we will not let the c5-function go to the limit, but instead assume a smallest element in the z-direction of ~z, so that the value of the c5-function for z = z' will be 11 ~z. Equation (8.57) is on the form which we will make use of in Chapter 10 for discussing noise in MESFETs. We will refer to the type of noise represented by the generalized noise spectral density in (8.53), or the auto-correlation function in (8.57), as "diffusion noise".

FLICKER NOISE, OR I/F-NOISE

A large number of physical phenomena exhibit low-frequency fluctuations which have a power per unit bandwidth which varies with frequency as 11 fa, where a is close to one. Usually, this noise is therefore described as "11 f­noise". It was first identified in electron tubes, and was at that time termed "flicker noise" (Johnson, 1925; Schottky, 1926). Typically, 11 f-noise is im­portant in the frequency range from mqch less than 1 Hz to kHz or MHz, depending on the device or the material (in one case, measurements showed a Ilf-dependence down to 10-7 Hz!). As the frequency is increased, eventu­ally thermal (Nyquist) noise will take over, and a constant noise power will be observed versus frequency. The transition between the two noise processes identifies a "corner frequency". Clearly, there must also be a low frequency limit for which the III-noise becomes independent of frequency, in order for the integrated noise power to be finite. There are actually a number of physical noise processes which exhibit III-character, as we shall see in our discussion below.

The simplest component which shows III-noise is a resistor. A bulk piece of semiconductor also has III-noise. Based on measurements on a number of materials, Hooge(1969, 1976; also see Hooge et al., 1981) postulated the following empirical relationship:

(8.58)

Here, Si is the spectral intensity of the 111 noise current, 1 and N are averages of the current and the number of charge carriers in the device, respectively, while aH is a constant which Hooge found to be 2 x 10-3 • This law appears to apply to a number of fairly large devices, but recent research has indicated that the same expression may be used with values of the constant which differ depending on the type of material, and the quality of the material. Smaller

224 Microwave Semiconductor Device"

values for the Hooge parameter, aN, are generally obtained in small devices. Evidence is also mounting which shows that there is a universal limit for how small aN can be in a given material, based on the theory for "quantum 1/ f­noise" (Handel, 1975, 1980). These topics will be reviewed below.

Since 1/ I-noise exists in a device as simple as a resistor, it is appropriate to start the discussion with this case. A resistor of length L, containing N electrons, and with a mobility 1', has a resistance:

L2 R=­

ql'N (8.59)

Fluctuations in the resistance can come about because of fluctuations in either I' or N in this expression (Van der Ziel, 1986), i.e. we have either number­fluctuation noi"e, or mobility fluctuation noise. In general, both may occur, and thus (if the the two noise processes are independent):

(8.60)

If we first look at number fluctuation noise, this may for example be due to electrons being randomly absorbed and reemitted by traps in the semiconduc­tor, either at the surface, or in the bulk. This idea is reasonable, since empirical studies of devices have shown decreased 1/1 noise when for example the den­sity of surface traps was decreased by appropriate processing. In analogy with (8.41), we can find the spectral density of the number fluctuations as:

(8.61)

We have assumed a characteristic time constant, T, for the process by which electrons get trapped and reemitted by the traps. The spectral intensity shows a frequency-dependence which is constant up a frequency of about l/T, and then drops off proportional to 1/ p. Intuitively, we then realize that if a num­ber of different traps exist, with an appropriate distribution of time constants, Ti, we could simulate a frequency dependence which is close to 1//. We may ascribe a probability distribution, g(T), which governs the probability with which a particular time constant occurs. For the particular case of

l/T g(T) - for TO < T < Tl

- In(Tt!To) (8.62)

and g( T) = 0 otherwise

one finds that the spectral intensity varies as 1/1 over the frequency range from l/n to l/To:

SN(f) _ aNi (.N")2 - (.N")2Ilnh/To)

(8.63)

By assuming that (8.64)

Chapter 8 225

Klaasen (1971) found that

SN(!) _ Si(f) _ al f3 (N)2 = (1)z - IN' where al = In(Tdro) (8.65)

This relation is of the same form as (8.58), which was empirically found by Hooge. It thus appears plausible that carrier number fluctuations due to traps may be one physical mechanism behind the observed 1/ I-noise. One will have to investigate in a particular case, whether a sufficiently wide range of time constants may be obtained, of course, and we will return to this question in conjunction with MESFETs in Chapter 10.

It has also been shown that mobility fluctuations may result in 1/ I-noise. The carrier mobility will in general show random variations due to the scat­tering processes which the carriers undergo. This problem is probably best treated in the frame work of Handel's quantum 1/1 noise theory (Handel, 1975; 1980), which was applied to the semiconductor case by Van der Ziel et al. (1985).

QuantuDl 1/1 Noise

Any charged particle, which is accelerated or decelerated, will radiate broadband electromagnetic radiation, so-called "bremsstrahlung". In semi­conductors, charge carriers constantly experience random accelerations and decelerations, which should thus give rise to radiation of electromagnetic noise. The probability of bremstrahlung emission for low-energy particles is indepen­dent of the photon energy. The noise power emitted per unit bandwidth is proportional to the number of quanta, N, which are emitted within that band­width per second. If we compare two equal bandwidth intervals at frequencies It and 12, respectively, where It < 12, we will find that It/h more photons are emitted in the interval near h. This follows from the energy independence ofthe emission probability, plus the fact that the photon energy is hf. It then also follows that the noise power per unit bandwidth is proportional to 1/1.

The Hooge parameter can be derived in terms of the average change in velocity for the scattering events the electrons go through, as follows:

_ 4a (.Il1/Z ) aH- - --3'11" c2

(8.66)

Here, a is the fine-structure constant, well-known from atomic physics, with a value of 1/137. For an elastic scattering process, one can show that .Il1/2 = 21/2

(Van der Ziel et al., 1985). Since 1/z = 3lcT /m·, we find the following equation for the case of elastic scattering:

4a 6lcT aH.=- x--

3'11" m·c2 (8.67)

226 Microwave Semiconductor Devices

Table 8.1. Representative values of the Hooge-parameter (aH) for several devices at T = 300 K. (from (Van del Ziel et al., 1985)), @1985 IEEE.

Device Experintental Theoretical aH aH

Silicon BJT, n+pn aH. < 1.1 X 10-1 8.8 X 10-8 < aH. < 8.3 X 10-1

Silicon BJT, p+np aNn = 6 x 10-8 aHn = 2.2 x 10-8

Silicon MOSFET, p-channel 3 x 10-1 - 9 X 10-7 8.8 X 10-8 - 8.3 X 10-7

Silicon MOSFET, n-channel 2.9 x 10-6 2.2 X 10-8

Silicon JFET, n-channel 2.5 X 10-8 2.2 X 10-8

NEe GaAs MESFET 2 x 10-4 7.1 X 10-6

GaAs MESFET, MBE 5.6 x 10-5 7.1 X 10-6

For a given material, the strength of the l/f-noise is then inversely proportional to the effective mass. aH is also proportional to the temperature. This result immediately agrees, qualitatively at least, with the well-known fact that silicon devices have much lower 1/ i-noise than GaAs devices - note the much smaller effective mass for GaAs. It also leads to the conclusion that there is a trade-off between device speed and 1/ i-noise, since it is largely the small effective mass (and the high mobility which follows from it) which gives GaAs devices their high speed compared with silicon devices. Van der Ziel et al. (1985) compared experimentally measured and predicted Hooge-parameters for devices made from both GaAs and silicon, and some of these data are given in Table 8.1. There is agreement for a few devices, while when there is disagreement, the experimental value is always largest. There is also some controversy about the temperature-dependence of the noise (Van der Ziel, 1986). Handel's quantum 1/ i noise theory thus is basically consistent with experimental data, if we assume that in the cases for which the experimental Hooge-parameter is larger than what the theory predicts, other processes which are also of 1/ i-type dominate. These other processes could be associated with surface or bulk traps, etc., as discussed earlier in this section. Only in a few cases does the measured 1/ i-noise get as low as the predicted quantum 1/ i noise limit. Generally, this happens for very high quality material, such as is grown by MBE. At the present state of materials growth, it is thus impossible to test the theory accurately. Future work will undoubtedly clarify these issues further, but it

Chapter 8 227

appears fruitful at the present time to regard the quantum 1/1 noise as a lower limit for 1/ I-noise in general. In most practical devices other factors still are more important in determining the actual low frequency noise level. We discuss the occurrence of 1/ I-noise in two-terminal devices in Chapter 6, and 1/ I-noise in three-terminal devices in Chapters 10-12.

REFERENCES

ADAMS, J.G. (1991). "Numerical Modeling of Noise in GaAs Semiconduc­tor Devices Using the Monte Carlo Method," Ph.D. Thesis, University of Massachusetts, Amherst, MA, February 1991.

FRIIS, H.T. (1944). "Noise Figures of Radio Receivers," Proc. IRE, 32, 419.

GUPTA, M.S.(1977). "Electrical Noise: Fundamentals and Sources," IEEE Press, New York.

HANDEL, P.H. (1975). "1/1 Noise, An Infrared Phenomenon," Phys. Rev. Lett., 34, 1492.

___ , (1980). "Quantum Approach to 1/1 Noise," Phys. Rev. A, 22, 795.

HOOGE, F.N. (1969). "1/1 Noise is No Surface Effect," Phys. Lett., A-29, 139.

---, (1976). "1/1 Noise," Physica, 83B, 9.

--, KLEINPENNING, T.G.M., and VANDAMME, L.K.J. (1981). "Ex-perimental Studies of 1/ I-Noise," Rep. Progr. PhYII., 44, 479.

IRE Subcommittee on Noise (1960). "Representation of Noise in Linear Two­ports," Proc. IRE, 48, 69.

JOHNSON, J.B. (1925). "The Schottky Effect in Low-Frequency Circuits," Phys. Rev., 26, 71.

--_, (1928). "Thermal Agitation of Electricity in Conductors," Phys. Rev., 32,97.

KLAASEN, F.M. (1971). "Characterization of Low 1/1 Noise in MOS Tran­sistors," IEEE Tran". Electron Devices, ED-18, 887.

LANE, R.Q. (1969). "The Determination of Device Noise Parameters," Proc. IEEE, 57, 1461.

MITAMA, M. and KATOH, H. (1979). "An Improved Computational Method for Noise Parameter Measurement," IEEE Tran". Microwave Theory Techn., MTT-27, 612.

NYQUIST, H. (1928). "Thermal Agitation of Electric Charge in Conductors," PhYIl. Rev., 32, 110.

228 Microwave Semiconductor Devices

SCHOTTKY, W. (1918). "Uber Spontane Stromschwankungen in Verschiede­nen Elektrizitatsleitern," Ann. Phys., 57, 541.

SCHOTTKY, W. (1926). Phys. Rev., 28, 74.

VAN DER ZIEL, A., HANDEL, P.H., ZHU, X., and DUH, K.H. (1985). "A Theory of the Hooge Parameters of Solid-State Devices," IEEE Trans. Elec­tron Devices, ED-32, 667.

___ , (1986). "Noise in Solid State Devices and Circuits," John Wiley, New York.

FURTHER READING

FUKUI, H. (1981). "Low-Noise Microwave Transistors and Amplifiers," IEEE Press, New York.

GUPTA, M.S. (1977). Cited above.

VAN der ZIEL, A. (1986). Cited Above.

___ , (1988). "Unified Presentation of 1/! Noise in Electronic Devices: Fun­damentall/! Noise Sources," Proc. IEEE, 76, 233.