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Q NBS TECHNICAL NOTE 1173

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U.S. DEPARTMENT OF COMMERCE/National Bureau of Standards

Three Guises

of

Generation-Recombination

Noise

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IGO

. U5753

1173

1983
NATIONAL BUREAU OF STANDARDS

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IJATIOKAI. BTJKEAO; OP STAITDRRDE Three Guises may n ij3

of Generation-Recombination Noise ^

^i- 53

Julius Cohen

Center for Radiation Research

National Measurement Laboratory

National Bureau of Standards

Washington, DC 20234

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U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued April 1 983
National Bureau of Standards Technical Note 1 1 73

Natl. Bur. Stand. (U.S.), Tech. Note 11 73, 12 pages (Apr. 1983)

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THREE GUISES OF GENERATION-RECOMBINATION NOISE

Julius Cohen

Radiometric Physics Division

Center for Radiation Research

It is shown that the noise in a zero-biased junction

may be just a manifestation of the normally-occurring

generation-recombination process, rather than shot noise,

as is usually presumed. In addition, an attempt is made

to clarify some noise mechanisms in semiconductors by

addressing mathematical interpretation and terminology.

In particular, for a biased homogeneous material at low

frequencies, where the relevant transport mechanism is

drift, a shot-like expression of the g-r noise equation

is derived. For a zero-biased junction at low frequencies,

where the relevant transport mechanism is diffusion, a pure shot-like expression of the g-r noise equation and

an equivalent thermal (Nyquist) expression is derived.

In both the homogeneous and the junction cases, however,

the true noise remains generation-recombination noise,

i.e., the origin of the noise is the fluctuations in

the rates of generation and recombination of free carriers.

Key words: generation-recombination; junction; noise;

semiconductors.

This paper has two objectives: 1) to clarify some noise

mechanisms in semiconductors by addressing mathematical interpretation

and terminology, and 2) to show how the normally occurring generationrecombination process can alone account for the noise in a zero-bias

junction, i.e., without making the usual presumption of another

noise mechanism or origin.

That generation-recombination noise is relatively unfamiliar,

belies its importance. Indeed, g-r noise is one of the primal

noises in semiconductor devices, including photon detectors, and

oftentimes it is predominant. G-r noise originates in thermally or

optically-stimulated electronic interband transitions or transitions

1
between impurity levels, traps, or recombination centers and one of

the bands, either conduction or valence. Associated with these

transitions are fluctuations in the numbers of free carriers and in

their lifetimes, thus giving rise to the g-r noise. The mathematical

expressions for g-r noise depend on the details of the number of

energy levels, the energies corresponding to these levels, the

electron population, and the occupancy of states. Formulas for

specific models have been derived by van Vliet and co-workers. ' '^'^'^

An interesting, but sometimes deceptive aspect of g-r noise is

that it may appear in several guises, thus obscuring its true identity;

hence origin. This paper shows how two basically different guises

of g-r noise--a shot form and a thermal (Nyquist) form--may be

easily derived from an explicit g-r noise formula, and discusses the

significance of the various forms.

As an example, the common case of an extrinsic semiconductor

such as germanium or silicon containing both donors and acceptors,

one of which is predominant; each of which exceeds in number the

number of free carriers , will be treated here. The general formula

for the extrinsic g-r noise spectrum S (co) is given by

S^^^(a)) = 4 !

T + (ojt)"

(1)

where

Af

a- AN^ , is a cpmplicated statistical factor, '^

IT
and

oj = 2^f, f is the electrical frequency, i^ is the mean square noise

current, Af is the noise-equivalent band pass, I is the (steady) direct

current, t is the lifetime of the free carriers, and N is the number of

free carriers.

For the particular case being treated here, Eq. (1) reduces to

S^^'^co) = 4 I 2 \WW^'] ' ^^^

At low frequencies, as assumed here for simplification, where

a3T«.l , the term in brackets ;^ t Hence Eq. (2) may be re-written as Oj

S No)) = 4 IH/N. (3)

Because N = gx, where g is the generation rate, the quantity t/N in Eq.

(3), is just the inverse of the generation rate. (Although the exact

noise spectrum formula for other models may not be identical to the

above, they are expected generally to be similar.)

The free carriers, on being generated, will undergo some form of

transport. Although the transport process itself may contribute a noise

in addition to the g-r noise, "transport noise" will be assumed to be

negligible. For a bulk material, the zero-bias case is physically

uninteresting, as g-r noise apparently is not observed unless a net

current is passed, and the preponderant noise mechanism in a passive

homogeneous material is thermal agitation noise of the free carriers,

i.e., Johnson noise. For the useful case of a bulk material with bias,

3
the transport mechanism of concern is drift. In pure drift,

collisions between the free carriers and the lattice are assumed to

be negligible.

For the case of pure drift, Ohm's law is applicable; viz.,

J = (N/u)eyE , (4)

where J is current density, u is volume, e is electron charge,

y is mobility, and E is electric field. Expressing N as a function

of I and substituting in Eq. (3) gives

S(i) (oj) = 4el (t/tj) , (5)

where x^ = l^ yV is the carrier drift time across the region I between

electrodes, and V is voltage. (This result is consistent with a

similar derivation made for a more general case. )

Equation (5) resembles that for the classical shot noise of a

thermionic diode; viz.,

S(^*^(a)) = 2 e I 3 . (6)

When a thermionic diode is operated in the saturated emission

current mode, full (or pure) shot noise is obtained and 3=1. When

the diode is operated in the space-charge-limited mode, partial

shot noise is obtained and 3 < 1. Hence, Eq. (5) is the shot-like

form of the g-r noise equation, but it should not be construed as

meaning that the g-r noise is shot noise. The origin of g-r noise is

the fluctuations in the rates of generation and recombination of the

carriers; the origin of true shot noise is the corpuscular nature of

electrons, and the noise is associated with emission or conduction.

4
Equation (5) relates to extrinsic g-r noise in a biased, bulk

semiconductor. Now a zero-biased pn junction will be treated: A

pure shot-like form of the g-r noise equation will be derived and

shown to have a mathematically equivalent thermal (Nyquist) form.

In the bulk regions, which are the regions adjacent to the

depletion region (i.e., the junction region proper) the electric

field is of negligible strength. Hence, drift is inconsequential

and diffusion is the relevant transport mechanism. Of the free

carriers in the bulk regions, only minority carriers within a diffusion

length L of the junction region are able to contribute to the

junction noise, and the appropriate time of transport is the lifetime

T. Those carriers reaching the depletion region boundary are swept

across the junction region (by the built-in field), and "instantaneously"

collected with virtually no recombination having occurred.

It follows from this model that

I ^c Ne/T . (7)

Substituting the above equation into Eq. (3), one obtains

S^^'^o)) = 4el , (8)

which is the pure shot-like form of the g-r noise equation for a

zero-biased junction. Like the previously-treated homogeneous case,

the true noise is generation-recombination noise, not shot noise,

because the origin of the noise is the fluctuations in the generation

and recombination rates of minority carriers, not the discreteness

of the electron. Broadly interpreted, however, this noise could be

thought of as a shot noise in the sense that the number (concentration)

5
of carriers fluctuates.

It is axiomatic that the magnitude of the available noise power

of a system in thermal equilibrium must be given by Nyquist's theorem--

regardless of the particular noise mechanism; i.e.,

P = k T Af . (9)

But,

P = i' R/4 . (10)

Thus, from (8), (9), and (10),

4 e I Af = 4 k T Af/R .

' (11)

Equation (11) should not be construed to mean that the shot noise

is thermal noise. It merely expresses a mathematical (numerical)

equivalence.

The present model ascribes the noise in a zero-biased junction to

fluctuations in the number of free minority carriers. In this respect

o it conforms to established theory. However, where earlier models

g either presume the existence of shot noise, or otherwise presume

that the fluctuations of concern are associated with transport,^

our model makes no presumption. Instead, it argues that the fluctuations responsible for the junction noise originate in processes

known to be occurring; namely, generation and recombination. But

the present model does not preclude the possibility of a true shot

noise in addition, especially if the junction is forward biased.*

The confusion about noise is abundant. One cause is

*With reverse bias, the depletion region widens, thus increasing the

generation within; recombination is negligible by comparison, and the

noise is essentially "generation noise," given by Eq. (8) with the

factor 4 replaced by 2.
misinterpretation of mathematical formulas ;^0 another is ambiguous

terminology. At this juncture it is appropriate to discuss briefly

the terminology used herein, which conforms with common use.

The physical agents of generation-recombination noise are heat

and/or light. These cause fluctuations in the rates of generation

and recombination of carriers, which is manifested as noise. Thus,

this noise is appropriately termed generation-recombination noise. The

carriers' property which fluctuates is their number (concentration).

The physical agent of thermal noise is heat, and the carrier

property which fluctuates (owing to thermal agitation) is velocity.

Obviously this noise is temperature dependent and its formula is given

by the Nyquist formulation. Other common names for thermal noise are

Johnson noise, Johnson-Nyquist noise, and Brownian/Brownian movement

noise.

The physical basis of pure shot noise is per se the corpuscular

nature of the particles and the carrier property which fluctuates is

1

2

number (concentration). Pure shot noise (all carriers are free) is

independent of temperature and dependent only on current.

Basing terminology on the origin of the fluctuations, as is done

here, should reduce possible ambiguity.
REFERENCES

1. K. M. van VI let and J. R. Fassett, in Fluctuation Phenomena in

Solids , R. E. Burgess, Ed. (Academic, New York, 1965).

2. J. E. Hill and K. M. van Vliet, J. Appl . Phys. 29, 177 (1958).

3. M. B. Colligan and K. M. van Vliet, Phys. Rev. 171, 881 (1968).

4. K. M. van Vliet, Appl. Opt. 6, 1145 (1957).

5. This situation is typical of cooled Ge and Si detectors,

for example.

6. P. R. Bratt, in Semiconductors and Semimetals , Vol. 12, R. K.

Willardson and A. C. Beer, Eds. (Academic, New York, 1977).

7. D. Long, in Optical and Infrared Detectors , R. J. Keyes, Ed.

(Springer-Verlag, Berlin, 1980).

8. R. L. Petritz, Proc. I.R.E. 40, 1440(1952).

9. V. Weisskopf, N.S.R.C, MIT Rad. Lab. Series, No. 133, May 14,

1943, or for an outline, see H. C. Torrey and C. A. Whitmer,

Crystal Rectifiers (McGraw-Hill, New York, 1950).

10. Early investigators of noise deceived themselves, through

mathematical deduction, into believing that shot noise in

thermionic diodes was thermal noise (as in conductors). Eventually.

physics (logic) prevailed! [See, e.g., D. A. Bell, J. Instn.

Elec, Eng. 82_, 522 (1938), and its appended Discussion.] But

shortly, the specious reasoning was carried over to semiconductors, it being proposed by Weisskopf that both thermal

and shot noises be designated shot noise. [See reference 9.]

Obfuscation of this sort persists yet. Thus, to cite a recent

example: for an unbiased junction-- "The Johnson noise current
arising in the junction ..." [W. Swindell, in Applied

Optics and Optical Engineering , Vol. VIII, R. R. Shannon and

J. C. Wyant, Eds. (Academic, New York, 1980.] It is not always

clear whether the fallacies, real or apparent, arise because of

deficiencies in mathematics, analysis, conceptualization, terminology, or exposition, but the result is the same--confusion.

11. H. Nyquist, Phys. Rev. 32_, 110 (July, 1928).

12. W. Schottky, Ann. Phys. 57, 541 (1918).
NBS-n4A IREV. 2-8C) U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

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1. PUBLICATION OR

REPORT NO.

NBS TN 1173

2. Performing Organ. Report No. 3. Publication Date

April 1983

4. TITLE AND SUBTITLE

Three Guises of Generation-Recombination Noise

5. AUTHOR(S)

Julius Cohen

6. PERFORMING ORGANIZATION (If joint or other thon NBS. see instructions)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

Final

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State. ZIP)

Same as item 6.

10. SUPPLEMENTARY NOTES

I I Document describes a computer program; SF-185, FlPS Software Summary, Is attached.

11. ABSTRACT (A 200-word or less factual summary of most si gnificant information. If document includes a significant

bi bliography or literature survey, mention it herej

It is shown that the noise in a zero-biased junction may be just a manifestation

of the normally-occurring generation-recombination process, rather than shot

noise, as is usually presumed. In addition, an attempt is made to clarify some noise mechanisms in semiconductors by addressing mathematical interpretation

and terminology. In particular, for a biased homogeneous material at low

frequencies, where the relevant transport mechanism is drift, a shot-like

expression of the g-r noise equation is derived. For a zero-biased junction

at low frequencies, where the relevant transport mechanism is diffusion, a pure

shot-like expression of the g-r noise equation and an equivalent thermal (Nyquist)

expression is derived. In both the homogeneous and the junction cases, however,

the true noise remains generation-recombination noise, i.e., the origin of the

noise is the fluctuations in the rates of generation and recombination of free

carriers.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

generation-recombination; junction; noise; semiconductors.

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