Electrical 1f noise related to carrier transport

Solid-StareHecrronicsVol. 36,No. 10,pp.

1477-1480, 1993

0038-I IO]/93 s6.00 + 0.00 Copyright 0 1993 Pcrgamon Press Ltd

Printed in Great Britain. All rights reserved

ELECTRICAL 1,” NOISE RELATED CARRIER TRANSPORT ANDRZEJ Instytut

TO

M. ZAKLIKIEWICZ

Technologii Elektronowej, 02-668 Warsxawa, Poland

(Received 27 October 1992; in revised form 26 January 1993) Abstract-A short survey of many years of l/fnoise investigation is given underlining the main, commonly accepted theories. A modified formula of the Hooge expression is presented. This formula describes electrical I/fnoise with the accuracy depending on the factor a, related to the carrier transport conditions. The need for further investigation of l/f noise signals in the time domain is pointed out.

1.

INTRODUCTION

l/f noise has been observed and interpreted since 1925[1]. Later, in semiconductor times, many trials were made to explain the l/f noise phenomenon. Three turning points can be recorded in this matter. The first is McWhorter’s investigation in 1955 of slow surface states activation[2]. As the second we can recognize Hooge’s paper “l/f noise is no surface effect”[3], published in 1969. The third is the concept of Handel (1975), who tried to interpret l/_fnoise on the base of quantum mechanics[4]. Many investigations and a great number of models and theories appeared as a result of the subsequent idea: if one cannot describe l/f noise by means of one universal relation, it seems suitable to try to solve this problem for a single semiconductor device or for a particular phenomenon, e.g. charge trapping. It is difficult to conclude that these investigations have achieved successful results. However, some of these investigations can be useful as a base to formulate the equation, describing phenomenologically electrical l/f noise. In the last years, the papers of Van der Ziel[S] and Van Vliet[6] have contributed to a very profound knowledge of l/f noise. However, the author presents here his own short review, which should help appreciate the idea of the proposed equation of l/f noise.

DIFFICULTIESIN FORMULATING A UNIQUE I// NOISE EXPRESSION

2.

All electronic materials and devices investigated up to now show l/f noise during d.c. current flow. This kind of noise is even more spread and it concerns various physical phenomena with some distinguished stochastical properties. Considering nonelectrical phenomena, e.g. traffic current investigation on an expressway, made by Musha and Higuchi[7], seems to be suitable as a demonstration of the author’s concept of carrier transport related l/f noise.

Computer processing of recording results, made at proper time intervals, distinctly showed the existence of the l/f effect related to car number fluctuation at a tixed place on the expressway. The power spectral density of these fluctuations was derived under the following two assumptions: (a) cars on an expressway are Brownian particicles with random motion, which cause diffusion in an ordinary sense; and (b) their local drift velocity u is a linearly decreasing function of the local car concentration: u = u,

( > l-“,

n,

where v, is the drift velocity when the number of cars n approaches zero and the constant n, is usually much larger than the mean concentration. The equation of continuity was transformed into a Burgers equation:

where a = 2v,/n,, D is the effective diffusion coefficient and t’ and x’ are new coordinates of time t and distance x. The Burgers nonlinear equation (the second term at the left-hand side is nonlinear) has a turbulence solution whose power spectral density with respect to the wave number is proportional to the reciprocal of the wave number. In the original frame of reference (t, x) the power spectrum in the wave number was transformed into a power spectrum in the frequency. The final result: S(w) = 0:/w,

(3)

is applicable only to hydrodynamical scheme of the cars current, i.e. for suitably low frequencies. According to McWhorter’s theory, l/f noise in semiconductors is caused by carriers tunneling between the inside of a semiconductor and traps present at the surface oxide layer. The spectral

1478

ANDRZEJM. ZAKLIKIEWICZ

density of noise related to this effect can be expressed

as[8]: - - (~'

S ( f ) = 4a:

dt/t

t

J.2 1 + co2t 2 i n ( t 2 / t l )

4a 2 - co ln(t 2/tl ) (arctg cot2 - arctg cot l),

It is difficult to present a new appreciation of quantum l / f noise theory. Even a discussion on top level presented in [6] does not support it unanimously. A well grounded statement that it is difficult to find a satisfactory experimental verification of Handel's theory has been presented in [10].

(4) 3. THE PROPOSED FORMULA FOR

where a is the amplitude of a stochastic process, t i , t2 are the minimum and maximum values of common time constants of tunneling and g - r processes. In the range of l/z2 ,~ co ,~ l/'c t we have a 1If type spectrum:

Taking into account the current knowledge of l / f noise and starting from Van der Ziel's[5] and Van Vliet's[6] papers and Musha's[7] experiments, it is possible to formulate an equation, describing electrical Ill noise with higher accuracy:

a2

S( f ) = ~rql~f - ',

S ( f ) = fln(z2/z l).

(5)

The tunneling effect can explain l / f noise not only in semiconductors with a surface oxide layer but also in grained structures (grain boundaries) and for interface contacts. In many eases, especially with high quality semiconductor technologies, such an interpretation of the 1/f noise mechanism cannot be accepted. In the 60's Hooge made experiments with many samples and various materials and geometries. He stated that 1/f noise was not an exclusive surface effect. Very often it is the interior of semiconductor which is responsible for 1If noise generation rather than its surface. Hooge formulated a semiempirical equation for homogeneous semiconductor samples, expressing the spectral density of resistance fluctuations:

sR(f) R2

sAf) = --~

~a

= ~-],

(6)

where N is the number of free carriers in the sample and I is the direct current flowing through the sample. In the beginning it was assumed that ~n has an approximately constant value of 2 × 10 -3. Later experiments[9,10] showed that =a can have values in a very large range (10-10-6). In the last years one can observe a growing interest in quantum l / f noise theory, proposed by Handel[4], who derived the following equations: S(f)=

4~Av 2 el 3nc2 z f '

I[f NOISE

(9)

where ~T is the constant related to carrier transport conditions, I is the d.c. current flowing through the considered object and q is the charge of electron. It has been stated that the d.c. current exponent// is equal to 2 or nearly 2 for many different investigated elements. However very different values o f / / have been also observed[1 i,12] and even a/3 = 0 ease would be appropriate to explain l / f type resistance fluctuations observed in thermodynamic equilibrium conditions[13]. According to the author's observations, the exponent of frequency f should be equal to - 1 or very close to - 1 . All deviations from the 1/f rule are caused by other kinds of noise, especially by burst noise[14]. The sum of l / f noise and burst noise can give a 1If "/noise, where y is in the range from nearly 0 up to above 2. Figure 1 is given as the illustration of these possibilities. The noise spectra with y > 2 have been observed in capacitance (electret) microphon circuits[15], but this was recognized as an effect of flickering. Unfortunately we still do not have a more precise definition of 1If noise. The author proposes here to use the term " l / f noise" only for noise having f - t 0 type spectra allowing only some small deviations.

togs

(7)

where: 40cAr2 3~C 2 = ~H"

(8)

This second relation is known as Handel's equation[5]. In the above c is the velocity of light, ~ is the fine structure constant of a charge congiomerate[5], Av = v2 - v~ is the vectorial change in velocity along the electron path dl2 and t = 2d~2/(v2 + vt ).

tog f Fig. 1. Burst noise and l/f noise representing three different levels of spectral density.

Electrical l/f noise related to carrier transport

The author’s study of various points of view, presented in many papers and his own observations and investigations allow him to state that the I/fnoise level resulting from the cur coefficient value depends on many factors determining carrier transport. In semiconductors we can specify these factors as the number of carriers N, their mobility p, the velocity of carriers v and even more important carrier velocity changes Au, the crystal lattice structure and some imperfections of this lattice, material resistivity and finally the trap (g-r centers) density. Probably these are not all the factors determining carrier transport conditions with regard to their influence on probability distribution of appearance of separated electrons at the fixed test point (plane). The distinction of N in eqn (6) describing the l/f noise spectral density seems to be unsatisfactory grounded. It could be eventually considered as ‘IN’ part with q and exponent 6 having various values. The inversely proportional relation between the noise spectral density S and the carrier number N has only in a few cases been confirmed experimentally[9]. The elementary noise event is related to the transport of single carriers and therefore the application of the electron charge q to eqn (9) seems to be well based. 4. THE I/f NOISE LAW

The l/J noise and also other kinds of observed noise spectra demand treatment in the wider perspective than simple physical and mathematical problem. As we know, no real physical process can produce a white noise spectrum in an unlimited frequency range. We know also how to describe the limitation of the upper frequency for thermal, shot and g-r noise. On the contrary, too little attention was given <

1479

to low frequency boundaries of noise spectra. Currently developed semiconductor devices have shown a lower and lower l/‘noise level. Therefore the question arises how far this lowering is possible. The general problem of very low frequency noise is illustrated in Fig. 2. The cases (a) and (b) are observed in many investigated samples under d.c. current flow condition. Figure 2(c) presents the case when a sample is not supplied by any kind of external energy except heat. Very good agreement between experiments and theory is here observed in a large frequency range. However what we can say about thermal noise close to f = 0 or what kind of noise exists at the frequency pointed as A in Fig. 2(b)? Is it the sum of thermal, shot and l/f noise? The ideal spectrum without l/f noise at d.c. current flow conditions is shown in Fig. 2(d). Several mathematical methods may be applied to derive a noise spectral density inversely proportional to frequency. We refer to the short presentations (McWhorter, Musha) and results (Handel) included in this paper. Further progress in carrier transport theory should give some facilitations and improvement in the theoretical description of I/f fluctuations. One can state the necessity of further investigations of noise in the time domain. Such investigations were conducted many years ago mainly to solve l/f noise stationarity problems[ 16-241. Nearly all experiments show a lack or very serious limitation of stationarity[ 16,21,22,24]. Variance fluctuations of l/f noise have been observed and determined[l7-191. The disagreement between the results of these investigations can likely be explained by too short measurement times in relation to the low frequency components of the noise signal. The amplitude limitation of commonly used measurement sets could also introduce an additional error.

s I/f

yJ

Thermal

Shot

Thermal noise

noise

..

4 -..

I..

“7 7

A (b)

(a1

S

If..-... Shot noise

Thermal

jP ?

I

0

?

noise

Thermal

noise

...

ii

... \

*.* w

(cl

f

_

0

(d)

f

Fig. 2. Noise spectral density vs frequency (both in linear scale): (a,b) observed in electronic devices; (c,d) according to theoretical equations.

1480

ANDRZEJ M. ZAKLIKIEWICZ

1.0

Distinguishing in the general equation such parameters as the number of carders N or the carrier mobility # requires additional information about investigated semiconductor device, material, structure and sample dimensions. Presently it can be even considered as a success if it is possible to formulate a satisfactory equation for the l / f noise level in a specific device, e.g. a Schottky diode. Elaboration of l / f noise in the strict sense and the criteria of its stationarity could be very helpful in any further investigation.

O O. O. >-.I

rf O. I.U

-> 0.1

0.01

-3

I

I

-2

-i

Acknowledgements--I am indebted to Professor F. N. Hooge for his criticism which helped the correction and improvement of this paper. I would like to thank Professor J. Marciak-Kozlowska and Professor J. F. Kolodziejski for discussion of some considered problems and--in the case of Professor Kolodziejski--for his help in the preparation of the text. Thanks are also given to Professor F. M. Klaassen for his kind assistence in the course of elaboration of this paper.

I

;

I

3

(d-doll6 Fig. 3. The probability amplitude distribution of noise processes. Dashed lines--experimentally stated deviation from normal distribution. The amplitude distribution of l / f noise is well approximated by the normal (Gaussian) distribution equation[25]: P(d) = a ( 2 ~1 7 ~ e x p l

], (d ~-~ - do)el

(10)

where do is the expected value of noise amplitude d(t) and 0.2 is the variance of d(t). Normal distribution by itself predicts very high amplitudes with very little probability P(d), thus appearing very rarely. In addition m a n y experiments (including that of Musha) show (in the case of l / f fluctuations) the existence of appreciate deviations from the normal distribution (dashed line in Fig. 3). Thus the probability of appearing of very high noise amplitudes should be even higher. 5. C O N C L U S I O N S

The variety of I l l noise kinds and sources can be observed and should be considered. I l l noise in semiconductors and semiconductor devices is mainly dependent on the carrier transport conditions. The electrical I / f noise in semiconductors and semiconductor devices cannot be expressed by means of more detailed equations than given here (9). The survey of results of selected investigations gives a reason of this statement.

REFERENCES

1. J. A. Johnson, Phys. Rev. 26, (1925). 2. A. L. McWhorter, l/f Noise and Related Surface Effects in Germanium. MIT Lincoln Lab., Rep. 80 (1955). 3. F. N. Hooge, Phys. Lett. 29A, 139 (1969). 4. P. H. Handel, Phys. Rev. Lett. 34, 1492 (1975). 5. A. Van der Ziel, Proc. IEEE 76, 233 (1988). 6. C. M. Van Vliet, Solid-St. Electron. 34, 1 (1991). 7. T. Musha and H. Higuchi, Jap. J. appl. Phys. 15, 1271 (1976). 8. A. Ambrozy, Electronic Noise, p. 117. Akademiai Kiado, Budapest (1982). 9. F. N. Hooge, Proc. lOth Int. Conf. on Noise in Physical Systems, p. 375, Budapest (1989). 10. F. N. Hooge, in Noise in Physical Systems and I/f Fluctuations (Edited by T. Musha), p. 7. Ohmsha (1991). I 1. E. Kuzma, J. Stawarz and A. Zaklikiewicz, in Elektronika Polprzewodnikow (Edited by ITE PAN, Warszawa), p. 321 (1967) (in Polish). 12. A. Zaklikiewicz, 1TE Reports (Prate ITE), No. 58 (1970) (in Polish). 13. R. F. Voss and J. Clarke, Phys. Rev, B 13, 556 (1976). 14. A. M. Zaklikiewicz, Solid-St. Electron. 24, 1 (1981). 15. W. Denda, E. Stolarski and A. M. Zaklikiewicz, ITE Reports (Prace ITE), No. 4, 91 (1984) (in Polish). 16. J. J. Brophy, Phys. Rev. 166, 827 (1968). 17. J. J. Brophy, J. appl. Phys. 40, 3551 (1969). 18. J. J. Brophy, J. appl. Phys. 41, 2913 (1970). 19. W. E. Purcell, J. appl. Phys. 43, 2890 (1972). 20. F. N. Hooge, Physica 42, 331 (1969). 21. R. A. Dell Jr, M. Epstein and C. R. Kannewur, J. appl. Phys. 44, 472 (1972). 22. J. L. Tandon and H. R. Bilger, J. appl. Phys. 47, 1697 (1976). 23. R. F. Voss, Am. Phys. Soc. 40, 913 (1978). 24. M. S. Keshner, Proc. IEEE 70, 212 (1982). 25. J. S. Bendat and A. G. Piersol, Random Data: Analysis and Measurement Procedures. Wiley, New York (1971).