Hooge_expStudies1byFnoise

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Rep. Prog. Phys., Vol. 44, 1981. Printed in Great Britain

Experimental studies on llf noiseF N Hooge, T G M Kleinpenning and L K J VandammeDepartment of Electrical Engineering, Eindhoven University of Technology, Eindhoven,The Netherlands

AbstractExperimental studies on l/f noise ar e reviewed with emphasis o n experiments th at may bedecisive in finding the correct theoretical m odel fo r this type of noise. Th e experimentalresults are confronted with two theories: McWhorters surface state theory and Clarkeand Vosss theory of local tem pera ture fluctuations. Th e applicability of either theo ryturns ou t to be very limited. Th e validity of an empirical relation is investigated. Itsapplication to electronic devices proves rather successful. Experiments show th at l/fnoise obeying the empirical relation, which we shall call a: noise, is a fluctuation in th atpa rt of th e mobility th at is due to lattice scattering.This review was received in December 1980.

0034-4885/81/050479+54 $06.50 981 The Institute of Physics31


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480

Contents

F N Hooge, T G M Kleinpenning and L K J Vandamme

1. Introduction1.1. Spectrum and correlation function1.2. llfnoise2.1. The experiment of Voss and Clarke2.2. AC excitation of l/f noise3.1. Two-electrode arrangements3.2. Four-electrode arrangements3.3. Point contacts4.1. Accuracy of C1,f4.2. Some conditions for reliable a values4.3. 01 values5.1. The McWhorter model5.2. Discontinuous films5.3. The McWhorter model and a noise6. Temperature fluctuations7. Fluctuations in number or in mobility?

2. llfnoise due to fluctuations in the conductance

3. Configurations for measurement

4. Results for 01

5 . Surface states and l/f noise

7.1. l/f noise in therma l voltage7.2. l/fnoise in Hall voltage7.3. I/fn ois e an d lattice scattering7.4. llfnoise of hot electrons7.5. E mpirical llfn ois e source term8. Lattice scattering causes 01 noise9. Non-ohm ic junctions9.1. Single-injection space-charge-limited-current iode9.2. p-n diode9.3. Schottky barrier diode10. MOS ransistors10.1. Oxide trap model10.2. a-noise model10.3. Comparison of the two models

References11. Summary and conclusions

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Experimental studies on llf noise 48 1

1. IntroductionGenerally speaking, electrical noise is a well-understood phen om enon . Its theory is welldeveloped in the more abstract fields of thermodynamics and statistical mechanics aswell as in the field of models that show what the electrons are doing in specific cases.M uch experimental work o n electronic devices sup po rts these theories an d models. Inmo st cases the theory helps us to d ecide whether such devices ca n be improved by lower-ing the noise. Fo r instance, generation-recombination noise ca n be reduced-at leastin principle-by using cleaner preparation techniques to avoid trapping centres. On theothe r han d, thermal noise can not be reduced by whatever method of preparation is used,based a s it is o n fundam ental physical laws, an d it is therefore unavoidable.In this well-ordered field of noise research, with agreement between experiment andtheory almost everywhere, there is one notorious exception: I f noise. This type ofnoise manifests itself as fluctuations in electrical conductan ce. I t often causes trou bleat low frequencies, let us say below 1 kHz, since its spectral density is inversely propor-tional to the frequency. Th e noise has been name d l/f after this spectrum. I n this reviewwe restrict ourselves to l/f noise in electrical condu ctance, althou gh l/f fluctuations ar emet with in geological, physiological and musical phenomena (Mandelbrot 1977). Somepeople have speculated on the com mo n origin of all l/f phenom ena. W e shall no t followthem but concentrate on conductance noise.Th e electrical llfn oi se occurs ab un da ntly in contacts, films, resistors a nd in nearly allsemico nduc tor devices. In spite of such omnipresence an d half a century of research o nllfn ois e, there is no accepted the ory explaining such noise.Many models have been proposed, but each has its own specific experimental factswith which it is a t variance. There is no t even agreement a s to whether all observedelectrical l/f noise belongs to the same physical phenom enon . Perha ps there are severaltypes of If n o is e, requiring different theories to explain all the experimental facts. Th emain controversy is whether the noise is caused by generation-recombination processesbetween the conduction ban d a nd trap s at the surface, o r by some as yet no t understoodfluctuation in the lattice scattering o ccurring in the bulk . Th e first possibility is supp ort edby a simple an d very reason able theory, the second by experimental facts. I n conse-quence the simple question as to whether l/f noise is avoidable is not to be answeredunanimously a t present. Yes, avoid traps , w ill be the an swer of those wh o believe in ageneration-recombination mechanism. No, not for a given number of electrons, sayothers who think it is fundamental to the scattering of electrons. In this phase of l/fresearch the em phasis mu st lie on simple experiments tha t directly decide whether certaintheoretical p roposals ar e right o r wrong. This explains why experimental is the firstword in the title of this review. But in orde r to avo id an unsystematic p resentation ofunconnected experimental facts we have grouped the facts around theoretical problems.W e shall first sketch the application of noise theory to the ot he r types of noise. Th enwe shall see where this leads to in the case of Ilfnoise.1. . Spectrum and correlation func tionWhen a quant i ty X shows noise we may write

X ( t )= ( X )+ A X ( t )


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482 F N Hooge, T G M Kleinpenning and L K J Vandammewhere ( ( A X ( t ) } z ) is constant for stationary noise. Th e symbol ( ) means averagingover a lon g enough time interval. Now A X ( t ) may be written in a Fourier series :

A X ( t )=Eat exp ( j 2 5 ~ h t ) a t * exp ( - j 2 n h t ) (1.2)where the coefficients ai ar e fluctuating amplitudes. Th e noise componen t measured at afrequ enc yh is then

(AX,) =O (1 3)1.4)

If one measures in a final bandwidth A A the Fourier components inside the bandwidthad d up quadratically. Th e noise measured in a unit bandwidth is called the spectraldensity. This spectral density S x ( f ) s closely related to the autocorrelation functionrpx(t), rpx(t) describes how, on the average, a deviation A X ( t0 ) will decay. Th e au to-correlation function is defined by

( ( AXZ)~)2(aiaz *).

p x ( t ) = ( A X ( t o ) A X ( t o + t ) ) . (1 5 )Th e relation between qx(t) nd Sx(f) is given by the Wiener-Khintchine relations:

S x ( f ) = 4 1 qx( t )COS 2rff dtrpx(t)= 1, S x ( f ) OS 2 7 t d.6 (1.6)(1.7)

This theory can be found in any introductory book on noise, e.g. MacDonald (1962),van der Ziel (1959, 1970) or Wax (1954). Th e Wiener-Khintchine relations (1 -6) and(1 .7 ) are all we need here. They are used when making a physical model f or a n observednoise. Th e differential equation describing the transitions in th e proposed physicalmodel gives the correlation function. Its Fourier trans form should be in agreement withthe observed noise spectrum, with its shape as well as with magnitude. Th e simplestcase, also the one most often met, is the exponential correlation function, correspondingto th e Lorentzian spectrum. If, on the average, a deviation AX decays according to-dAX/dt = AXIT

then the correlation function iscpx(t)= A X ( to + t ) >=


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Experimental studies on llf noise 483The linear equation (1 .8) is a very general one. It does not say anything abo ut whatthe electrons are doing, but only that, whatever they do, they do it independently. So,with the slight deviations that we are dealing with, this equation will appear quite often.In mo st cases we then find the Lorentzian spectrum according to equation (1.10). A t

low frequencies f ~ g) the spectrum is white, a nd at high frequencies ( ~ T B) it is l /f2.There a re three well-understood types of noise th at a re frequently seen in noise studies.These are thermal noise, shot noise and generation-recombination noise. Th e third oneis also of theoretical interest for llfn ois e models.(a) Thermal noise originates from the thermal motion of the charge carriers. Inequilibrium situations this motion has a n average energy of 4kT. Th e relaxation time isextremely fast, T N 10-12 s, so that at all attainable frequencies we measure a whitespectrum with Sv = 4kTR (1.13)or

SI= 4kTG. (1.14)(6 ) Sh ot noise is also a white iioise in the usual frequency range. It is found when acurrent of discrete particles leaves a cathod e or passes a potential barrier. Its spectraldensity is given by

SI= 2eI. (1.15)This spectrum is white si n ce fn 4 1 at all attainable frequencies because of the very sh orttransit time TI of the electrons.(c) Generation-recombination noise is of particular interest to us, since some l/fnoise theories are variations on this theme. Consider a semiconductor with a number ofidentical tra p levels. A fraction of them will be occupied by electrons. Since there is acontinuing trapping and detrapping between the traps and the conduction band (or val-ence band) the number of trapped electrons, and therefore also the number of free elec-trons, will fluctuate. Th e transitions between tra ps and b and are described by equation(1 .8). The generation-recombination spectrum of the conduction electrons is given by

471-t 277f ) 2S, f) = ((An')) - -- (1.16)

1.2. l/f noiseAfter this sh ort presentation of some elements of noise theory we can now ap pro ach theproblem of ll fn oi se . This review will discuss its physical origin a t length. But in orderto keep the introduction short we now simply accept the experimental fact that theconductance of a semiconductor fluctuates with a l/fspectrum. T he condu ctance fluctua-tions of an ohmic sam ple can be measured as voltage fluctuations when a constant currentis passed throu gh the sample o r as current fluctuations when the voltage dr op across thesample is kept co nstan t:(1.17)

Clif is a number which is a measure of the relative noise of the sample. It is independ entof the measuring conditions, such as current or voltage. It enables us also to directlycom pare measurements t ha t were mad e in different frequency ranges and reported in theliterature.It is a well-established experimental fact that, in many cases, spectra have been


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484 F N Hooge, T G M Kleinpenning and L K J Vundammemeasured with Scocf-l.0. Often the exponent is - 1.02 0.1 over six or m ore decades offrequency. Spectra can be measured in the range 0.1 H z t o 100 kH z without any diffi-culty. In a few cases measu remen ts have been extended dow n to 10-6 Hz. I t must berealised th at the spectrum cann ot be exactly l/f in the whole range 0 < CO. In the firstplace, S- oc when f- 0. Also, the integral of S-, CO at both frequency limits f 0 andf CO. Therefore, if the noise is stationary the spectrum must flatten below a certain lowfrequency an d it must become steeper above a certain high frequency. Neither of thesefrequency limits has ever been observed. Two solutions are possible: ( U ) the noise is no tstationary, o r ( b )we must measure a t still lower frequencies. Rega rding the last point itmus t be realised that it is difficult to obtain conclusive experimental results. Since 10-6 Hzcorresponds to measuring times of the order of a month, extending the measuring timeto a century adds only some 20 to JS df when the spectrum was originally measuredover 12 decades from H z to 106 Hz. A n exact llfsp ec trum also makes it impossibleto find the correlation function by Fourier transformation of the spectrum. It is true tha t,because of these mathematical arguments, exact llf noise cannot exist. But the import-ance of this statem ent should no t be over-estimated. All difficulties occur a t f = O o rf CO. Because of the logarithmic dependencies there are no practical difficulties at verylow or very high frequencies. So a spectrum experimentally indistinguishable from llfcan exist over man y decades. When the spectrum is ab ou t l/f in a limited frequency rangethe correlation function goes approximately with In t in a limited time interval. Thiscorresponds t o a relaxation of A X according to the differential equation

with B A X B l. (1 18)A Xdt- = A exp (BAX )F or symm etry reasons this suggests something like

- d A X - s i n h C A Xd t CT with (1.19)as the equation analogous to equation (1 .8).No physical model has been inspired by these purely mathematical considerations.Th e usual way to make a model is to consider the llfs pectr um as the summation of alarge number of Lorentzian spectra. If the Lorentzian spectra have relaxation timesbetween 71 and 7 2 an d if their statistical weights are propo rtional to 7-1

71< 7 < 7 21 1In 72/71 7g ( r ) dr= -___ - d r (1.20)then a llfspe ctrum is foun d in the frequency range 7 2 - 1 to 71-1

(1.21)The equation can be approximated as follows :

< W2>Sx )=___ -In 7 2 / 7 1 f1/2n72


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Experim ental studies on llf noise 485We thus find a l/f spectrum over a wide frequency range. Th e integral of the approxi-mated Sx(f) over this range is ((AX)2). Th e white and the l l f 2 branch hardly contributeto the total integral fromf=O to f= 00 if T z B ~ .Here we must make a remark that is important when comparing model calculationswith experimentally observed spectra. Th e approxim ation (1 .23) in the lifregion showsthat the noise is very insensitive with regard to TI and 7 2 . Only In 72/71 appears in theequation. Therefore one need no t know 7 1 and 7 2 to calculate the spectrum. Even thevaguest idea abou t their ratio is sufficient. If the model has serious errors in the absolutevalues of 7 nd 7 2 they will no t show up in the calculated spectrum. Numerical agreementbetween the calculated and observed spectrum-which is always measured in a limitedfrequency range including neither 1/71 no r l/.rz-can no t be considered as a definiteproof of the correctness of the model. If the model does not give absolute values for7 nd 7 2 one usually takes In 72/71 N 10 (see, fo r exam ple, 45.2). If it is desired t o expressthe distribution in the correlation function then we must take

In this type of theory the l/f noise results from the summation of a number of linearprocesses, all obeying equation (1.8). Th e amplitude distribution is then Gaussian. Inorder t o avoid t he somew hat artificial set of linear relaxation processes one might try t odevelop non-linear theories. In such theories the amplitude distribution need no t neces-sarily be Gaussian. Therefore it is worthw hile investigating whether l/f noise has aGaussian am plitude distribution. All experimental evidence suggests th at if the distribu-tion is non-Gaussian a t all, it is so with only minor deviations from the Gaussian charac-ter. Very accurate measurements have been don e by Voss (1978) who fo un d l/f noisewith a perfect Gaussian am plitude distribution. In his measurements such large ampli-tudes were measured that their relative probability was 10-7 of the probability a t the topof the distribution curve. In some cases Voss foun d slight deviations from the G aussiancurve. Two explanations are possible.(i) Gaussian an d non-Gaussian llfn ois e exist. This would mean tha t there are severaltypes of l/f noise.(ii) Vosss non-Gaussian l/f noise could be contaminated with some other noise.Burst noise, especially, is highly effective in spo iling a G aussian distribution. In this typeof noise the current jump s a t random between two c onstant, nearly equ al levels.This brings us back to the problem Is there only one type of l/fno ise ? There isstrong experimental evidence tha t a t least the greater part of the llfn ois e spectra meas-ured have the same physical origin. In 1969 Hoog e proposed an empirical relation fo rthe l / f noise in homogeneous samples :

(1.26)

in which N is the total num ber of charge carriers an d a s a dimensionless constant with avalue of about 2 x 10-3. Cl/f was introduced into relation (1.17) to normalise resultsmeasured a t different currents o r at different frequencies. Fu rth er normalisation ispossible assuming that each electron has its own independent contribution to the noise.Then the absolute noise of the sample must be divided by the number of electrons, andfor the relative noise we findClif= a / N . (1 .27)


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486 F N Hooge, T G M Kleinpenning and L K J VandammeThe surprising result was that 01 turned ou t to be a constant. There was no theoreticalmodel behind this result, only the idea that whatever the electrons do, they do it inde-pendently. In this way of thinking N in the denom inator is a trivial factor. It certainlydoes not mean that the llffluctuations must necessarily be fluctuations in number.

Th e validity of relation (1 .26), the accuracy of a tc, will be discussed extensively inthis review. But we have introduced a here already to create some order in the vastam ou nt of llfd at a. Noise intensities will be expressed as a: values according to equa tion(1.26).Application of equation (1 .26) to hom ogeneous samples often gives a values of ab ou t2 x 10-3. n inhomogeneous situations equation (1 -2 6)can be applied to volume elements.Inte gra tion procedures-treated in $3--make it possible to express experim ental resultsin 01 values. Com plicated electronic devices can also be treated in this way (see $99 an d 10).We then find expressions for the current or voltage fluctuations from which a can bedetermined . In all cases where direct o r indirect app lication of equation (1 .26) yields01 values of ab ou t 2 x 10-3 we are obviously dealing with the same type of llfnoise. Thisl/f noise will be called a-type llf noise. In $8 we repo rt on experimental results thatsuggest th at 01 noise is a fluctuation in the lattice scattering of the electrons. In $5 anothertype of l/fn oi se will be treated, i.e. McW horters generation-recombination type.So at the end of this introduction we d o no t claim a priori tha t all Ilfnoise must be01 type. We shall use the empirical relation (1.26) to order experimental data. If, inthe end, such an ordering procedure proves to be successful, the summarising of theproperties of a noise is easy. This gives then th e conditions th at have to be met by atheoretical model still to be made.

2. l f noise due to fluctuations in the conductanceIt has been demo nstrated only lately tha t it is the conductance th at fluctuates with a l/fspectrum. Before the decisive experim ent by Voss an d C larke (1976a, b) this idea wasgenerally assumed t o be correct. Some aut ho rs were already studying the next question,whether l/f conductance fluctuations were caused by fluctuations in the number of thefree charge carriers or by fluctuations in their mobilities. The general acceptance ofconduc tance fluctuations was based o n the following simple observations. When aconstant current is flowing through an ohmic sample, I/fnoise is found in the voltageacross the sample with a spectral density proportional to 1 2 ; when the voltage is keptconstant the l lfcu rre nt noise is proportional t o V2. However, there were some worryingexperimental facts. Wh en such simple experiments were performed w ith AC currentwith frequency W O noise was sometimes found in frequency regions around 2w0 andhigher harmonics an d, most disturbing, a t very low frequencies near 0 Hz. These extranoises were more than proportional t o the norm al llfnoise. Therefore it was thought th atthe extra noises an d the llfn ois e itself were generated by non-linear mechanisms. Is thellfn ois e a kind of turbulence in the curre nt? Such questions bring us to the more generalproblem. Does the current generate the noise or does the current only serve to measurethe already existing conduc tance noise ?We shall now first present Voss and Clarkes demonstration of the conductancefluctuations. Knowing then t ha t the l lfn ois e is in the conductance we shall discuss thevarious AC current experiments. We shall try to remove the paradoxes, showing th at theseAC experiments do not provide serious arguments against a conductance model for thel/f noise.


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Experimental studies on llf noise 487

% -U -% -i t )w v

2.1. The experiment of Voss and ClarkeWhen a resistor R is in equilibrium a t a temperature T we always find voltage fluctuationsacross the resistor

S V ( f ) =4kTR. (2 .1)This is called Nyqu ist noise, Joh nso n noise or therm al noise. Relation ( 2 .1 ) has beenvery well established, bo th theoretically a nd experimentally. If desired eq ua tion (2 .1 )can be interpreted as the Brownian movement of the electrons with kT kinetic energyper degree of freedom. If a current flows thr ou gh th e resistor, there is n o longe r equilib-rium an d noise additional to ( 2 .1 ) may be foun d, e.g. l/f noise. But in equilibrium wehave Nyquist noise and nothing else.The Voss and Clarke experiment could be described as the measurement of l/f noisein the Ny quist noise. W hen R and T are constants SV is found according to (2 .1) : nol/f noise is observed. If R o r T fluctuates with a l/f spectrum then-also accord ing to(2.1)-we find lif noise in S v ( f ) if the m easuring time is no t infinitely long. If themeasuring time is infinitely long-which should be the case if we really inten d to measurethe average value of (AV)Z-then of cou rse T and R in (2.1) are also values averagedover an infinitely long time, so that no noise other than Nyquist noise will be observedin (AV)z. By saying that there is l/f noise in the Nyquist noise, we do not mean tocorrect relation (2 .1) (which would be against thermodynam ics) but th at l/f noise in Rwill show up in the thermal noise.When studying 1, noise in the thermal noise we should be careful not to confusenoise fro m different sources. Since we measure the pro du ct RT, Sv ( f ) will be a directmeasure of temp erature fluctuations. Tem perature fluctuations will also influence SV( )since R depends on T. But under well-chosen conditions it will be possible to measureequilibrium fluctuations in R as fluctuations in V.The measurement is performed as follows (see figure 1). The equilibrium voltagefluctuations V(t) of the resistor R are amplified. Th e white thermal noise passes a ba nd -pass filter with lower frequency limitfi and higher frequency limit f h . Th e signal is thensquared a nd averaged over a time 7 hich results in a signal P ( t ) . Then a F ourie r analysisof P ( t ) s mad e, which gives a l/f spectrum .W e express V ( t ) s a Fou rier series

where ( A )-I is the length of the truncated time sample.According to (2 .1) and (1 .4)


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488 F N Hooge, T G M Kleinpenning and L K J VandammeThe squaring of (2.2 ) leads toV 2 ( t ) = X (AnAn*+An*An)+ZZAd,* exp [2nj(fn-fm) t l

?k TL nb+XZAn*Ant exp [- n j fn - m) ( m f n ) . (2 .4)Here we have omitted terms with sum frequencies because, in the end, they will make nocontribution due t o the following averaging process over the relatively long time T.

n m

Th e signal V ( t ) s averaged according to

with1P ( t ) = - V ( t ) t

n l A f $ 1 / T $ A7

The contribution of the first term on the right-hand side of (2 .4) is called P I . ForT+ 00 we find

= 4 k T R ( f h f i). (2 .6)If 7 is large b ut n ot infinite P l ( t ) shows fluctuations because R shows fluctuations:

(2 .8)1S P ~ = C I / ~[4kTRCfh-f i>Iz.fThe second and third term on the right-hand side of ( 2 . 4 ) give rise to white spectralcontributions P z . Straightforward calculations show that the lif part P I is greater thanthe white part Pa at frequencies below f c :

(2.91c = c1lf ( f h - f i ) cl f h .Relation (2.9) shows why it is so difficult to measure Il fn oi se in thermal noise. Oneneeds very noisy samples. It is rare to find Cllf values higher than 10-7. The high-frequency limit fh must be less than the reciprocal R C time of the m easuring equipment.A corner frequency f, less tha n 1 Hz makes the measurement of the l / f part of thespectrum difficult.This method of measuring l/f noise in samples in equilibrium was designed by Vossand Clarke, who performed successful measurements on semiconducting InSb films andon metal films of N b. Similar measurements on carbon samples were later done byBeck and Spruit (1978) who then obtained relation (2.9), which also agreed with theearlier measurements of Voss and Clarke.2.2. A C excitation of l / f noiseSince Voss and Clarke's experiment we know th at lifn ois e is a fluctuation in the resist-ance. Ou r discussion here of the AC effects is based on the knowledge we have now.Part of the A C effects had been observed before the Voss and Clarke experiment wasdone. Some a uth ors gave interpretations of their experimental results th at differ there-


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Expe rimental studies on lif noise 489fore fro m mo dern interpretations. We can accept all experimental results, b ut no t allinterp retations originally given. Ohms law states directly what will happen w hen anAC current with frequency f I flows through a sample in which the resistance shows l/ffluctuations :

SI =PCl / f / f . (2.10)An alternating voltage will be found a t l , accompanied a t bo th sides by noise according toS d f i fi A f )= t VAC^ CiiflAf. (2.11)

F o r obvious reasons this noise is called l / A noise. (In this connec tion A does not meanbandw idth.) There is perfect agreement between the calculated an d observed ratio ofthe magnitude of the l/Af noise and the magnitude of the llfnoise when measured withDC current (Lorteije and Hoppenbrouwers 1971).

6is the observed spectrum. Th e

f -Figure 2. Noise excited by an alternating current. The broken curvescales are logarithmic.

What is much more puzzling is that in some samples a l/f noise also appeared atAC excitation. We then have the situation shown in figure 2. T he observed spectrum 2:can be interpreted as a l/f spectrum and two l/Af spectra, the lower frequency branchshowing spectrum folding. Th e noise intensity of the l/f spectrum is proportional toVAC^ with 2 < n < 4 . Roughly speaking, the intensity is several per cent of the intensityof the l / A spectrum.Montagnon discovered Ac-generated low-frequency noise in 1954. It has furtherbeen studied by Sutcliffe (1972) who established th at the spectrum was 11M any auth ors thou ght th at this Ac-generated l / f spectrum disagreed with resistancefluctuations as th e source of 1 noise. In 1977 van H elvoort and Beck proposed a modelin a paper t ha t also summarised th e older literature. This model in its simplest form is

a resistor with its resistance depending o n the direction of the current. A n applied


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490 F N Hooge, T G M Kleinpenning and L K J Vandanzmesinusoidal voltage will lead to a superposition of a sinusoidal current on a D C componentZDC, proportional to VAC,ZDC will give lif noise, proportional to I D $ , and hence toVAC^. In a somewhat more complicated form we take

(2.12)=Ro exp (aV) .If aV is not too large series expansion yields

Th e second term between th e brackets yieldsavo2cos2 2xf t = (++3 cos 4 T f l ) .Vo2-_RO Ro

Thus there is a constant term(2.13)

The l/fnoise, being proportional to Z D C ~ ,will now show a V A C 4 dependence. If weconsider one asymmetrical resistor, it must be strongly asymmetric in order to producea large ZDC th at can account for the observed Ilfnoise. This is the case for a gold contacton p-type germanium.An othe r possibility is given by grainy samples, fo r instance carbo n resistors. Contac tsbetween the grains, if rectifying, will be 5 0 % of the exp (aV) type and 50% of theexp (- aV) type. The I-V characteristic will be symmetrical and fairly linear. But thenoise con tributio ns of the individual contacts add up. Th us this seemingly ohm ic resistorproduces strong llfnoise on A C excitation. Such models can also explain the appearanceof l/f noise aro un d the higher harmonics of the excitation frequency. Jones and Francis(1975a, b) studied correlation effects a t different frequencies in the Ilf ba nd s an d betweenl/f an d l/Af bands. They, too, concluded from these measurements tha t l / f noise isa resistance fluctuation.

3. Configurations for measurementConductance fluctuations are easily probed by passing a constant current through apair of driver electrodes and measuring the voltage fluctuations across a pair of sensorelectrodes. I n such a four-electrode configuration sensors an d drivers are separate. Th enoise SV across the sensors is proportional to the square of the current through thedrivers, but only in very special cases will S V = I ~ S E , here S R is the spectral powerdensity in the fluctuations of the resistance between the sensors. Four-electrode arrange-ments are used to minimise the influence of the noise generated at the current contacts.When the sensors are placed so that the average voltage across them is zero, then theso-called transverse noise will be measu red. In this case, the measured transverse noisedoes not change when the current source Z t the drivers is replaced by a voltage sourcepassing an average current I through the drivers. The transverse noise arrangem ent isvery convenient when using DC pre-amplifiers to measure low-frequency spectra. If onecan make good noiseless contacts then two-electrode arrangements are attractive formeasuring noise in low conductance samples by applying a constant voltage across thesample and measuring the current spectrum S I . Two-electrode configurations have the


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Experimental studies on llf noise 49 1advantage of an unam biguous representation of the noise S v / V 2= S I / I ~ = R / R ~ =G/G2= CI/j/fm(nvefff)-l. F o r hom ogen eous samples subm itted to uniform fields the effectivevolume v e f f equals the volume of the sample. I n non-u niform fields Deft is smaller th an thesample volume. In ord er to interpret experimentally obtained values of SV in terms of cygeneral formulae are needed for the calculation of the resistance R and the resistancefluctuations SR for different electrode configurations. Th e measured noise in two- andfour-electrode arrangements always consists of the thermal noise and a conductionnoise term. Th e thermal noise equals 4kTR with R the resistance between the sensors.Th e conduction noise is proportional to the square of the bias curren t I. We shall nowconsider only noise due to conductivity fluctuations.3.1. Two-electrode arrangementsThe general formula for the resistance between an electrode pair on a sample is basedon the relation for the total dissipated energy in the sample with a bias current I . Thetotal dissipated energy is given by

RIz=S [ pJ2 dv (3.11and the integral must be carried out over the whole conductor except the electrodesbecause ideal contacts ar e assumed. Som e local increase in resistivity leads to (i) a nincrease in the dissipated energy, an d (ii) a small change in the local cu rre nt density J .No w the correlation function for the conductance must be introduced for thecalculation of ~ v ( T ) ,he correlation function fo r the voltage. Butterweck (1975) simpli-fied the calculation of ~ v ( T )y assuming 6 p uncorrelated in space except at distancesshorter than a c orrelation length. Th e correlation length is small in com parison withcontact dimensions. J2 does not change substantially on displacement over a correlationlength. F o r the resistivity fluctuation apzlnf is introduced so that (1.26) will result inhomo geneous situations. He re p is the resistivity an d n is the carrier density. Th e generalrelation for the power density in the voltage for a two-electrode arrangement thenbecomes

When the resistivity is homogeneous and the current density is not, an effective volumemay be introduced by Clif=a/nveff . Using equations ( 3. 1) an d (3 .2 ) the effective volumefor a two-electrode configuration is written as

Th e integrals must be taken over the whole sample except the electrodes. T he effectivevolume is at m ost a s large as the sample volume.Any deviation of the homogeneous field in uniform samples will lead to an increasein R and SV. Then Sv/V2 is no longer determined by the total number of free chargecarriers in the sample. Th e noise would seem to be con centrated in an effective volumewith higher current density.3.2. Four-electrode arrangementsVandamme and van Bokhoven (1977) have given a general relation fo r the noise voltage


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492 F N Hooge, T G M Kleinpenning and L K J Vandammebetween arbitrarily shaped and placed sensors when a constant current is applied toarbitrarily shaped an d placed drivers :

Th e integral must be taken over the whole sample except the electrodes. Relation (3 .4 )is quite similar to equation (3. 2) except that J4 s replaced by ( J g ( which is the sq uareof the scalar product of the current density J caused by the cu rrent I through the driversand adjoint cu rrent density j. n a thoug ht experiment the adjoint current is the currenttha t flows when the curren t source has been switched from the drivers t o the sensors. Aproof for equ ation ( 3. 4) was given by van Bok hoven (1978). This general relation w asinspired by the sensitivity theorem in electrical networks. F o r practical app lications ofthe equations one has to solve boundary value problems in order to find the requiredJ and J . In general this problem will be to o difficult to solve analytically. Then theapplication of an electrical network giving a discretisation of the Laplace equation leadsto a tractable problem. Th e sensitivity of a change in voltage across a sensor pair due toa slight change in a resistor in th e kth bran ch is given by 6Vi8R = k & / I ,where the currentand ad joint current in the kth branc h a re denoted by ik an d fk. A number of numericalresults are given by Vandamme and van Bokhoven (1977) and Vandamme andde Kuijper (1979). The adjoint current app roa ch of equation (3 -4 )provides a possibilityof finding the areas of high and low contribution to th e noise. Th e noise measured acrosssmall sensors stems from the neighbourhood around the sensors where J j s large.Quite often one can make some qualitative statements about SV by sketching the currentlines J and j,ed by physical intuitio n. Regions of high current density contribute heavilyin equation (3. 4) if at least the densities J and J are no t perpendicular. W hen the sensorsand drivers coincide, then J = I and the four-terminal equation (3.4) reduces to thetwo-terminal equation (3.2). If we assume the sample to be homogeneous also in thestatistical properties of the conductivity fluctuations, then p and ap2inf can be put infront of the integrals in the a bove equations.3.3. Point contactsIf one ca n m ake reliable contacts, point contacts a re eminently suitable for noise measure-men ts because of their very sma ll effective volume. A good approximation to the equa-tions of the noise of a point contact can easily be derived by a simple treatment of thecontact between two large spheres (Bell 1960, Hooge and Hoppenbrouwers 1969a)(see figure 3). Th e contac t are a is a circle with rad ius a. Th e equipotential surfaces ar eassumed to be spherical an d concentric aro un d the centre of the contact area. We considera shell between th e equipotentials a t distances x and x + d x from the centre. Th e shell ishomogeneous, also in the current density, so tha t equation (1 .26) applies:

Fo r the resistance in on e contact mem ber we find


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Experimental studies on llf noise 493c e c - - - -/ - --

/ // /

ConductorFigure 3. Simplified model of a point contact between two conducting spheres. The broken curvesrepresent hemispherical equipotential surfaces.Th e noise in one contact m ember is

By analogy to equ ation (1 .26) this is expressed as

F or a com plete contact with two contact mem bers we find from SR= 2SR, and R = 2R1

This result also follows directly from equations (3.1) and (3.2) when J ( x )= 1/2nx2 issubstituted.The equipotentials close to the contact area are hemiellipsoidals rather than hemi-spheres. Vandamm e (1976a) calculated the noise using the two-electrode equation (3 . 2 )an d the more realistic equipotentials. His results give a firmer theoretical basis to theresult obtained from the simple model of figure 3. The two models give different resultsfor SE as a function of a and for R as a function of a. However, approximation (3.9)giving the relation between SR and R is surprisingly good.In experimental work one often uses the contact between two crossed cylindricalbars. The resistance can then easily be varied by a factor of 20. When there are nocomplications one finds that Clif is proportional to R3. This permits the calculation of01 by means of equation (3.9). In this way a: of metals has been determined, which canscarcely be done in a no the r way. W ith semiconductors there is still an oth er possibility,the use of an ohmic contact between a large semiconductor sample and a thin metalwire. Th e metal wire, a good condu ctor, contributes neither to th e resistance no r to thenoise. Then a: can be calculated from equation (3 .8 ) .


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494 F N Hooge, T G M Kleinpenning and L K J Vundumme4. Results for a4.1. Accuracy of C1/jWhen l/f noise is measured it is usually accompanied by other noise which we shall callhere back grou nd noise. Th e spectral density of the l / f noise will be denoted by S, thatof the background noise by S b . When S + S b is of the same order of magnitude as Sbthe accuracy in S will be low. Th e background noise S b consists of the pre-amplifierinput noise 4kTRe and the thermal noise 4 k T R of the sample where Re is the equivalentnoise resistance. When th e filtered and squared noise is passed throu gh a low-pass filterto give a running-time average we estimate tha t th e detection limit for reliable a: valuesis S / s b > o . 5 . When the background noise is dominated by the pre-amplifier noise across-correlation measuring set-up as described by Storm (1978) is preferable. Then thefiltered and multiplied noise of the two channels with inputs in parallel is passed throughan integrator. Th e estimate for the detection limit is S / S b 2 0.1 for samples with aresistance greater than the equivalent noise resistance of the system. The equivalentnoise resistance of a cross-correlation set-up can be 100 times lower than the Re of thepre-amplifiers used in each channe l. Using equation (1 .26) for homogeneous samplessubmitted to uniform fields we find

S E P d UE29PS


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Experimental studies on lif noise 495local increases in th e electric field, leading to non -uniform fields on a microscopic scale.In these cases the experimental results can no t be interpreted by e quation (1 .26). An othercondition f or o btaining reliable 01 values is tha t the sam ples are provided w ith low-noiseohm ic contacts with a low specific con tact resistivity ( 2 cm2). If one were to calculatean a: value for samples with po or con tacts the result would have nothing to d o with thetrue a: of the material. For near-intrinsic materials both electron and hole mobilityfluctuations must be taken into account. Then a: can be calculated from the followingequation :

where an and a, are the conductivities of electrons and holes, respectively, n and p arethe free carrier concentrations an d U is the volume of the sam ple. Cl/f is inversely prop or -tional to the total number of carriers th at dom inate the conductivity. This can be th econcentration of minority carriers as observed in MnO by Kleinpenning (1976a).Surface treatments have a strong influence on the l/f noise (Leuenberger 1967,Hanafi an d van d er Ziel 1978). This is still no t fully und erstood. It may be caused byaccum ulation layers and surface charges in the oxide on the samp le. It always results inlow 01 values.With po int co ntacts serious difficulties may arise from oxide f i l m between the contac tmembers. Such a film may determine the resistance an d the noise of the conta ct. If itdoes then the contact is called film-dominated ;otherwise it is constriction-dom inated inwhich case equation (3.9) applies.If, in the m ode l of figure 3, a film with co nstant resistivity Pfilm an d constant thicknesst is present, two situations ar e possible (Vand amm e 1974a).

Constriction-dominated Film-dominatedcontact contactUpbulk% tp f i lm upbulk< tpfilmRcca-l Rcca-2CiinK R3 Ciif ccR

~

Reliable a: values can be found from crossed-bar experiments only if Clif remainsproportional to R3 during a variation in R by a t least a factor of 10. It is also possiblethat a m ultispot contact is formed. Th e mechanical contact area then contains manysmall conducting spots. If there are k such spots equation (3 .9) becomes

If we had just used equation (3.9) the result would have been a much t oo high 01 value.O n the other hand, with the help of equation (4. 4) the num ber of spots and their averagediameter can be determined when 01=2 x is used (Ortmans and Vandamme 1976).Clarke and Voss (1974) investigated Bi films in order to see whether the noise wasinversely proportional to the number of atom s or to the num ber of free carriers. In Bithere is a great difference between the num ber of atoms a nd free carriers. Th eir resultsgave no sup po rt for relation (1.26). However, they overlooked the fact that the holeconcentration at room temperature is about 3 x 1019 cm-3 instead of 1017 cm-3 which isthe value at 77 K. Vandamm e an d K edzia (1979) obtained from crossed-bar experimentsa: values for Bi of 3 x 10-3, using for the hole concentration 1.5 x 1019 cm-3 as follow ed

32


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496 F N Hooge, T G M Kleinpenning and L K J Vandammefrom the Hall effect measurements. Vandammes dat a on manganin an d bismuth showth at (i) a:-type llfn ois e is no t a temperature-induced noise, and (ii) relation (1 .26 ) holdsfor Bi an d nianganin. One may wonder whether a: differs in the solid and liquid phase.Therefore Kedzia and Vandamme (1978) investigated the lif noise in liquid and solidgallium. Fr om the experimental results they concluded tha t a for solid, liquid an d under-cooled gallium is ab ou t 2 x 10-3 and that a does not greatly depend on the temperaturein the range 77-600 K. Stroeken and Kleinpenning (1976) found for mercury at roomtemperature an a value of abo ut 2 x 10-3. May an d Aniagyei (1974) did no t succeed inmeasuring an 01 lower than 20 on a tungsten wire. Vandamm es (1976b) results inincandescent tungsten filaments were in agreemen t with their results. However, hisexperiments on tungsten crossed bars resulted in a: values of ab ou t 10-3. The largeam ou nt of noise in thin wires is due to multispot contacts a t the clamping contacts of thefilament. Stroeken and Kleinpenning (1976) demonstrated tha t Si an d G e a t 300 an d77 K show a: values of about 2 x even if the samples were plastically deformed.Although t he mobilities decrease with increasing de formation 01 is practically independentof the deformation. Fro m point contact experiments it was found that a: has no appreci-able pressure dependence in the range from 0 to 25 kba r (Vandamme 1976b).4.3 . a: valuesThe survey made by Hooge (1969) of the data published on llfnoise did not provideabsolute evidence th at relation (1 .26) is correct for comp ound semiconductors. Thereforefurthe r study was made of 111-V com pou nds by Vandam me (1974b). Th e averagesof the experimentally obtained results ar e summ arised in the table below.

Concentration fromHall measurements (cm-3) cc T W

n-type InSb 1.6 x lO I 41 6 x 1016p-type InSb 1 .2 x 1OI6p-type GaSb 1 5 x 1017n-type GaA s 2.3 x 1016p-type GaAs 2.3 x 1016n-type GaP 2.9 x loL6

1 . 3 ~0-3 773 . 4 ~0-3 3007 x 10-3 772 x 10-3 773x10-3 3006x10-3 3003 .4 ~ 1 0 -3 3 0 09x10-3 300

It follows from these experiments that (i) a: values for 111-V co m po un ds are approxi-mately 2 x 10-3 and (ii) n does not greatly depend on temperature. Studies of pointcontacts of ten metals showed th at their a: values were about 10-3 (Hoppenbrouwers andHoo ge 1970). Th e tempe rature coefficient of the resistance and the total num ber of atomsin a resistor play a central pa rt in the temperature-noise model of Clarke an d Voss (1974)(see $6). They fo un d experimental evidence for their theoretical prediction th at manganinwith a negligibily low temperature coefficient will show no llfn ois e. However, from m an-ganin point contacts Vanda mm e (1976b) obtained a: values of 0.7 x 10-3 using a n electronconcentration of 3.5 x 1022 cm-3 in the calculation. The conclusion from all theseexperiments is that equation (1.26) describes the l/f noise for many materials, such asmetals, solid and liquid, and semiconductors, n- a n d p-type. Th e numerical value of ais about 2 x 10-3. Because of the many com plications t ha t may occu r it is not surprisingthat the spreading in a: values is so high: about a factor of three from the average. This


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Experimental studies on Ilf noise 497inaccuracy makes it impossible t o determine whether a really is a constant. a may differfor different materials. 01 may be slightly temperature-dependent. All this is true as longas lattice scattering prevails. In 8 we shall see th at other scattering mechanisms reduceCY. values considerably.

5. Surface states and noise5.1. The M c Whorter modelOne of the m ajor difficulties in unde rstanding llfn ois e is the inability of simple physicalmodels t o give a l/f spectrum in a natural way. A simple model that does give a l/fspectrum was proposed by M cW horter (1955,1957). Th e M cW horter model has remainedthe most accepted model for llfnoise in MOST. Trapping of charge carriers in traps locateda t a distance from th e semiconductor-oxide interface is considered as the noise source.The McWhorter school of thought believes in (i) the surface origin of l/f noise, and(ii) fluctuation of num ber instead of fluctuation i n the mobility of th e free charge carriers.Surface states in a bro ad sense are defined as a ny electronic state which is localised a t thesurface, i.e. its wavefunction has a maximum amplitude at or near the surface andvanishes at a sufficiently great d istance from the surface. I n this respect states in th eoxide near the interface or at the interface and in the accumulation or inversion-layerregion of the silicon can be classified as surface states. Th e terms slow an d fastsurface states were used in early investigations of states at the Si/SiOz interface. Th e faststates are conjectured to interact either directly, or through tunnelling, with the siliconbulk an d therefore must be located a t or at least very near th e interface. Charge exchangebetween the silicon and the slow states requires a very long period of time.The orientation dependence of interface states correlates with the density of theavailable bonds per unit are a on the corresponding crystal orientations. Surface statesare thus often attributed to some unsaturated or dangling bonds at the silicon surface.Some physical models for interface states predict the absence of surface states on anoxidised silicon surface. Oth er models assume a stron g correlation between interfacestates and oxide charges or between surface states an d disorder o r misfits. There is nosingle method by which the distribution of interface states over the energy gap can bemeasured throughout the band gap. A complete picture can only be had by combiningresults obtained with two or m ore measuring techniques. Th e peaks in the interfacestate density near the band edges have often been questioned by, for example, Boudry(1973) and Declerck et a1 (1973). Th e bias dependence of the noise in MOST is oftenexplained by the energy distribution of th e interface states in the gap. O nly tra ps within afew kT of the Ferm i level, which is bias-depend ent, a re effective in generating noise. Th isexplanation for the bias dependence of the noise from Pai (1978), for example, seemsdoubtful, considering the apparent peaks in the surface states.In a letter entitled Evidence of the surface origin of th e llfno ise, ah a nd W ielsher(1966) presented correlations between th e noise, the lossy part of the gate impedance dueto carrier recombination an d the interface states of the MOST. However, when we analysetheir results between 20 Hz and 100 kHz a t a constant gate voltage we must concludethat the observed noise is not I/$ For example, at VG= 9 V their spectrum between20 Hz and 100 kHz is proportional to l/f2. This misleading situation was the start formuch experimental and theoretical work on the relation between l/fand the surfacestates (see, for example, Hsu et al 1968, Klaassen 1971, Broux et a1 1975). Since l/fspectra were observed in a frequency range of ten decades or more, the summation


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498 F N Hooge, T G M Kleinpenning and L K J Vandammeapproach of equation (1.21) requires that the 1/r distribution of time constants holdsgood over an extremely large range and tha t the physical processes at different areindependent. Th e values of the limiting frequencies are n ot th at im po rtan t in view of thediscussion given after relation (1.23 ). This wide range of time constants can arise froma thermally activated process or a tunnelling process.In MOST it is usually assumed that the time constant dispersion is caused by quantum-mechanical tunnelling of carriers from the Si/SiOz interface to traps located inside theoxide, This McW horte r model has been used in theoretical calculations, fo r exampleby Christensson et aZ(1968), Leventha l(1968), Berz (1970) an d Hsu (1970). They all gotmore or less the same expressions for the noise. F u and Sah (1972) suggested that directtunnelling of free carriers, either from the conduction band or the valence band into th eoxide traps, is an unlikely mechanism. Instead, they proposed a two-step process inwhich the free carriers first communicate with the fast surface states at the interface andthen tun nel int o the oxide traps elastically. Th e addition of a n intermediate state givesessentially the same expression for the noise. Th e dominant low-frequency noise comesfrom the fluctuation in occupancy of the oxide traps. If the electrons in an element of theinterface do no t interact with one specific oxide tra p bu t with several ones having differentT , then the result is a simple generation-recombination spectrum with an effective rwhich is close to the smallest r since

Only if each element of the interface has its dominant trap with a specific r are the trapsindependent. Th e noise is then a summ ation of generation-recombination spectra, eachwith its own r . The simplest form of the McWhorter model with a continuous set ofwith statistical weight l / r s as follows. It is assumed first that in the oxide layer onthe semiconductor there are electron traps with a constant concentration through thewhole layer. Th e second assumption is th at th e probability of penetration into the oxidelayer decreases exponen tially with distance from the interface. In this case the r aredetermined by quantum-mechanical tunnelling or overlap of the wavefunctions. Th eprobability of penetration is proportional t o 1 / r and to exp (-x/X). Therefore r becomesT = 7 exp ( x / h ) ( 5 -2 )

where h is a characteristic decay length of the wavefunction with an order of magnitudeof 1 A. Th e traps fa r from the interface have large r yielding the low-frequency part ofthe spectrum. Th e distribution function g(T) becomes

which is required t o obtain a l/f spectrum.

5.2. DiscontinuousfilmsDiscontinuous metal films show extremely high l/f noise. Th e high noise intensity isconnected with th e special type of electrical conduction in such films, which is by tunnel-ling. A n extensive study of discontinuous films, their conductance an d their noise hasbeen published in two papers by Celasco et al(l978). Section 5.2 is a summary of thesepapers, especially with regard to the llfnoise.


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Experimental studies on llf noise 499Th e films consist of metal islands evaporated on t o an insulator. Th e main conduc tionmechanism is the tunnelling of electrons fro m island to island. Th e strongest experi-mental s up po rt for this model is tha t the conductivity decreased very steeply from ab ou tinfinity t o very low values during th e evapo ration process when a critical thickness was

reached as in a percolation process. Th e tunnelling process 1 in figure 4 gives the con-duction. Th e potential barrier height is Ec - F , he distance between the conduction b andof the insulator an d the Fermi level. Th e position of the conduc tion band is determinedby the electron affinities of the metal an d th e insulator a nd also by the occupancy of thetrapping centres. Th e origin of the noise is in the fluctuation of the occupancy of the traps.The mean number of trapped electrons nt is, of course, given by Fermi statistics, butthere will be a kind of generation-recombina tion noise because of tunnelling processesbetween m etal an d traps, such as process 2 in figure 4. Th e fluctuation Ant in the number

I I1 fF

l nsui ator

Figure 4. Model of a discontinuous metal film.of trapped electrons changes th e charge Q of the insulating layer and thereby the potentialenergy Ec. This barrier fluctuation causes the tunnelling to fluctuate. Th e noise origin-ating in process 2 causes the noise measured in process 1.Since this noise is McW horters generation-recombina tion noise its spectrum is l/f,Not only does the shape of the spectrum agree with the model but the intensity, too,agrees well with reasonable numerical estimates for the concentration of the trappingcentres.

Apart from the two tunnelling processes there are thermionic transitions from ametal island into the conduction band of the insulator and from there either to the tra pso r to an oth er metal island. There will be generation-recombination noise generated bythe transitions between conduction ban d an d traps. In the original papers it is theoretic-ally and experimentally demonstrated th at the thermionic processes are of no importanc ebelow temperatures of 400 K. In this summary we shall therefore neglect thermionicprocesses from th e start.


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500 I:N Hooge, T G M Kleinpenning and L K J VandammeTh e conductance of the structure of figure 4 follows from the current eq uation givenby Simmons (1964):

Joc(Ec - F) exp [-A(Ec - E F ) ~ / ~ ]E c-EF+e v ) exp [ -A(Ec -EF + eV)1/2] (5.4)where Vis the applied voltage over the barrier and

A = 4~(2m)1/2 -ld 5 ' 5 )where d is the thickness of the barrier. I n this pa rt we shall om it the non-essentialproportionality constants which are complicated products of fundamental constantsand dimensions of barriers and islands. Th e conductance G follows from dJ/dV witheV


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Experimental studies on Ilf noise 50 15.3. The M c Whorter model and a noiseThe strongest objection to the M cWhorter model as a possible explanation of 01 noise isthat a noise is a bulk effect. a noise is experimentally fou nd to be inversely proportionalto the total num ber of free charge carriers. This excludes surface effects as the source ofa noise. Th e noise density is no t related t o the geometrical shape an d th e surface area,bu t depends only on the volume of the bulk. A clear demonstration of the bulk hypothesisis given by the l/f noise experiments on continuous gold films by Hooge andHoppenbrouwers (1969b). Here the l/f noise density of films with different thicknessesdepends purely on the volume of th e films.There are a number of l/fnoise experiments where the results are interpreted as abulk effect. Several examples ar e presented in this review (993, 7 and 9) : l l fnoise insolid-state injection diodes, in p-n diodes, in thermo-EMF, in Hall EM F, n two- an d four -probe configurations, and so on. In all these cases the llfnoise is calculated on thebasis of formula (1 .26). Th e agreement between experimental a nd calculated results isstriking. There are inore experiments in which the M cW horter m odel fails to explain theresults. These experiments ar e described in 97, which discusses the problem of whetherthe fluctuations a re in the concentration o r in the mobility.The l/f noise has been investigated in a number of devices, where the conductionoccurs close to the surface. Here the llfn ois e density depends on the quality of the sur-face. Some people concluded therefore th at McWhorters model prevails. Other peoplecame to the conclusion that the condition of the surface affects the free-carrier densityprofile, the electric-field strength profile and the scattering probability of the carriers.Since the l/f noise depends on these quantities it is possible that the surface indirectlyaffects the llfnois e. Examples a re given in 98 (thin Bi films) an d in $10 (MOSTransistor).Ano ther illustration was given by Kleinpenning (198Qa). H e discussed the llfn ois e of ashort p+-n diode in terms of equation (1-26). The magnitude of the ILfnoise is deter-mined by the concentration of minority carriers in the base, which is related to thecontact recombination velocity of the injected minority carriers at the collecting contact.So the contact recombination velocity indirectly determines the I/ noise density.Van der Ziel (1979) suggested in a review paper that grain boundaries in semi-conduc tors are a source of l/f noise. However, a high intensity of l/f noise du e t o grainboundaries c an also be interpreted as a volume effect. Ar ou nd the point of contact ofthe grains the electric current shows constriction effects, which lead to high electric-fieldstrengths a nd to h igh noise levels (93).

6. Temperature fluctuationsHooge and Hoppenbrouwers (1969b) measured the l/f noise in continuous gold films.The results obey the empirical relation (1.26) with 01=2.4 x 10-3. They concluded thatthe magnitude of llfnoise in gold is of the same order as in semiconductors with thesame numb er of free carriers. Voss and Clarke (1976b) studied resistance fluctuations inthin metal films. They found general agreement with equation (1 ,26 ) with two exceptions,viz. bism uth and m anganin . These exceptions ar e discussed in this review in 994 an d 8.Voss an d C larke interpreted the ob served resistance fluctuations in terms of equilibriumfluctuations in temperature that modulate the film resistance R . Their reasoning is asfollows. According t o the statistical mechanics of th e equilibrium state the average of


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502 F N Hooge, T G M Kleinpenning and L K J Vandarnmethe energy fluctuations 6E of a system in con tact with a heat ba th is given by

( ( 8 E ) z )= kT2Cv (6.1)where k is the Boltzmann constant, T i s the temperature and Cv is the h eat capacity of thesystem. Since energy fluctuations ar e connec ted with tem pera ture fluctuations, 6E= Cv8T,we obtain

((6 T)2) = kT2/Cv. (6.2)Th e mean square resistance fluctuation of a metal film in contact with a heat bath is thengiven by

{ 6R)) = dRjd T) 6 T)) = y2R2k/Cv (6 .3)where y = d In R/d In T. For most metals at room temperature ycz 1, NE N A andC v = 3 NAk where N A is the number of atoms in the film. So the mean square of therelative resistance fluctuations is((SR/R)2)- ~ / ~ N A E/3N. (6 4)A vital problem is now the frequency dependence of the spectral density of the fluctua-tions in R. The spectral density has to be of the magnitude given by equation (1.26)with a l/f dependence over many decades of frequencies. Th e tempe rature fluctuationmodel is encouraging with respect to the m agnitude. This is based on eq uations (1 . I ) ,(1 .26) and(6.4) .As to the power spectrum, a large number of papers have been published on thissubject. The main difficulty here is to obtain an explicit ljfregion over many decadesof frequencies. At best a l / f o r a l/f-like spectrum can be obtained for a very restrictedfrequency range. W e shall present som e results based on the Langevin diffusion equationof temperature fluctuations given by

1at cva T(P t , - D V ~ T ( Y ,)=~-. ( P , )

where T(r, t ) s the temperature at spot Y at time t , D s the thermal diffusivity, cv is thespecific heat, and F(v , t ) is the random Langevin source function associated with thetransport fluctuations in the system. Th e quantity F(v, t ) is uncorrelated in space andtime. Applying equation (6.5) to a system yields the p ower spectrum for the temperatu refluctuations. Th e frequency dependence of this spectrum depends bot h on the geo-metrical dimensions of the system and on the boundary conditions governing the heatflow between the system and su rrounding space. In this review we restrict ourselves tometal films with length 1, width i v an d thickness t , I > w > t . We shall present spectra fordifferent boundary conditions.F or a completely isolated film there is no heat flow between the film and the surround -ings, so ((6R)Z) =O. Let the ends of the film (x= 0, x= Z) be connected with supply wires,so that the system is thermally homogeneous in the x direction, and let the heat flowbetween film an d the rest of the surroundings be negligible. I n this case Kleinpenning(1976b), Liu (1977) and Ketchen and Clarke (1978) found S ~( j )x f-1 /2 o r f< f i= D / d 2and S ~ ( f ) a z f - 3 / 2or f> fi . Normalisation of the spectrum is achieved by usings d f = ( ( W 2 ) . ( 6 . 6 )Th e result is plotted as curve D in figure 5.


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Experimental studies on llf noise 503

Figure 5. Calculated noise spectra of relative resistance fluctuations in metal films using N A = N , a=2 x 10-3 y = 1 and ( l / ~ ) ~ =03. Curve A: according to equation (1.26). Curve B : according toVoss and Clarke s semi-empirical formula, equation (6.7). Here, f o = f i = D / n P . Curve C :according to equation (6.8) with f o = 1/2VT. Curve D: one-dimensional system, thermallyhomogeneous in the x direction, length I, f o = D / n P .Let the film be embedded in a medium, so tha t film and medium can be consideredas a thermally homo geneous system. In this situation Voss and Clarke (1976b) found

f


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504 F N Hooge, T G M Kleinpenning and L K J Vandammea pure surface character. Th e integral Jt f) f s determined by the bulk volume, bu tthe frequency behaviour of S f) s determined by the geometrical shape.F or metal films with good thermal co ntact with the sub strate where the heat exchangebetween film and substrate is dominant, we find

where .r=RthCv is the heat relaxation time, and Rth is the thermal resistance betweenfilm and subs tra te . Equat ion (6 .8) holds for T < I ~ / ~ D .he result is plotted in figure 5.Let us compare the experimental results from the gold films as obtained by Hoogeand Hoppenbrouwers (1969b) and Kleinpenning (1976b) with equation (6 ,7 ). Fo r thesefilms we have I = 800 pm, w = 10 pm , y = 0.5 and D = 1.2 cmz s-1, so that equation (6.7)predicts SR f ) /R2= 7 x lO-3/fN in th e frequency range fi= 60 Hz and f w= 400 kH z.Experimentally, i t is found that S~ (f )/ R 2 = 2 .4x lO-3/fN in th e 10 H z to 30 kH z range,which agrees with the semi-empirical Voss-Clarke formula equation (6.7). However,the heat relaxation time of the Au films was experimentally found to be a bo ut T = 0.3 ps.Since T


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Experimental studies on l l f noise 505(v) For temperature-induced resistance fluctuations there should be a correlationbetween the noise in different parts of the films at low frequencies fiN D j d 2 . Suchspatial correlation effects are observed by Voss an d C larke (1976b) an d C larke an d H siang(1976) o n thin m etal films with po or therm al con tact between film an d substrate a nd in a

frequency region aro un df i. However, in the semiconductor n-GaA s no such correlationeffects could be observed by Kleinpenning (1976b).(vi) Eberhard and Horn (1978) have investigated the temperature dependence of I/noise in metal films o n fused silicon an d sapph ire substrates. I n contrast with bo thequation (1.26) and Voss and Clarke's formula they found a rather strong temperatureand substrate dependence of SB(f)/R2. Neither of the two models can adequatelydescribe the data presented by Eberhard and H or n. Possibly the results may be ascribedto a combination of temperature-induced resistance fluctuations on the one side anda noise o n the other.(vii) Van V liet and M eh ta (198 1) have treated th e problem of temperature fluctuationsin a theoretical way. They have shown th at no l/fnoise will arise fro m heat diffusionin a system, unless very specific noise sources are proposed whose physical origin isuntenable.In a discussion of temp erature fluctuations Kleinpenning (1976b) concluded th at thereis a real danger of confusing 01 noise an d temperatu re-induced resistance fluctuations. Insemiconductors the temperature-induced fluctuations are always negligible with respectto a noise. In m etals the temperature-induced fluctuations are generally greater tha na noise at frequencies ne ar th e thermal relaxation time a nd are negligible with respect tothe a noise a t frequencies f ar away fro m this relaxation time (see figure 5 ) . We concludethat a noise in semiconductors cannot be caused by tempe rature fluctuations. Fu rth er,it is most improbable that temperature fluctuations are the origin of l/fnoise in metals.They give only a l/f-like contribution a t frequencies a rou nd the reciprocal thermal timeconstant.

7. Fluctuations in number or in mobility?Th e conductivity u of an extrinsic semiconductor is given by

u =q p n (7.11where q is the elementary charge, p is the m obility an d n is the free-carrier density. Fr omequation (7.1) it is obvious that conductivity fluctuations can be caused by mobilityfluctuations Sp or by density fluctuations Sn. So we have the possibilitiesSu=(do/dn) Sn=qp8n (7 2)6 o= d old p) 6p = q n8p. (7 .3)

We shall discuss both types of fluctuation for a semiconductor in therm al equilibrium.Hoo ge (1972). He suggested tha t the mobility of a free charge carrier fluctuated asTh e assumption of mobility fluctuations as the origin of l/f noise was introduced by

Spip. = .if. (7 4)

s,/p2= 4 f N (7.5)For homogeneous samples with N free carriers, equation (7 .4 ) becomes


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506 F N Hooge, T G M Kleinpenning and L K J Vandammeassuming that the mobilities of the free charge carriers fluctuate independently of eachother.To discriminate between number fluctuations SN and mobility fluctuations Sp, wehave to study fluctuations in a q uantity X ( p , N ) which depends on p and N in differentways. F or number fluctuations we have

SX,=(dX/dN) SN SxN= NdXldN)2 S N I N ~ (7 6)and f or mobility fluctuations

SX,=(dX/dp) Sp Sxp= s ilp2* (7.7)So we have to investigate the l/f noise of a quantity Xwith 1NdX/dN( # pdX/dpL(. Theconductance is certainly not such a qu antity; here G - pN and thus NdG/dN= pdG ldp.Quantities obeying the inequality are, for example, the thermal voltage and the Hallvoltage of extrinsic semiconductors, the conductivity a t high electric-field strengths (ho tcarriers) and the conductivity of heavily doped semiconductors. In this section we shalltreat successively the a noise in these quantities.7.1. l/f noise in thermal voltageThe treatment of this subject is based on a paper by K leinpenning (1974). For an n-typenon-degenerate semiconductor the thermal voltage Vth is proportional to the energydistance between the Fermi level an d the m ean cond uction level in the conduction band(see figure 6)whereand

Vth=(EF+ I C ) ATIqT (7 8 )(7.9)

(7. IO)

CO= SO& G ( E ) e 1 G(E)de)-'E F = kT In (Nc/N ).

t- - - - - - - -

Figure 6. Energy diagram of a semiconductor.


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Experimental studies on llf noise 507Here N c is the effective num ber of states in the cond uction ba nd, N is the number of freeelectrons, A T is the applied temperature differences, G ( E ) e is the conductance of elec-trons with energy between E a n d & + d e , and G=J, G ( E ) E is the conductance. Forlattice scattering we have ( E k ) = 2 k T . Conductance fluctuations 6 G = J 8G(e) d e lead tofluctuations in k 6 k = J r ( E - ( & ) ( E ) de/(G). (7.11)If the conductance fluctuations are due t o num ber fluctuations, we have

~ E F = ( k T / ( N ) ) 6N SQ = ( kT )2 S N(N ) - 2 . (7 12)Th e imp ortant result is tha t number fluctuations lead to 6 ~ = 0tf


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508 F N Hooge, T G A4 Kleinpenning and L K J VandammeI I I I II

qV , /kTFigure 7. l l fnoise of thermo-EMF with applied voltage. Dots: experimental results of n-Si. Broken curve,calculated results for number fluctuations, full curve fo r mobility fluctuations.

A t the same time Kleinpenning (1974) investigated the llf no ise in th e thermal voltageof intrinsic non-degenerate semiconductors. Here th e therm al voltage is

(7.18)with Gn and G p he electron and hole conductances, and

Vthn= (E F + ~ k ) T/qT vthP= - Eg- F + Ek ) A T/qT. (7.19)Here E g is the band gap and Ek' is the mean conduction level in the valence band (seefigure 6). Let us define a quantity /3 according to equation (7.15), where G=Gn+ Gp.Now the question is, what is the magnitude of p for number and mobility fluctuations,respectively ?

Fo r num ber fluctuations we have the produ ct N P is constant, since a shifting of theFermi level will affect both the number of electrons N and holes P . ConsequentlyS N / N = - P/P.Hence we find6 Vthn= 6 VthP= - kA T / q ) S N / N GGnlGn= - Gp/Gp=6N/Nand 6G/G= [(G' -Gp)/G]SNjN.

Applying these results to equations (7.15), (7.18) and (7.19) we obtain for latticescattering Ek= k = 2 k T )

(7 20)GnGp(Eg/kT+4)-G2 2 2b(Eg/kT+4)- b+1)'= ( G(Gn-Gp) ) = ( b2-1where b =Gn/Gp=pn/pp s the mobility ratio.For mobility fluctuations the free charge carrier mobilities are assumed to fluctuate


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Experimental studies on llf noise 509independently. Consequently (6pn 6 ,up)=0 and (6Gn 6 G p ) =0 . Now we obtain for /Iapproximately P=(E g/kT+ 4)2 2b2(b+ 1)-2(b2+ 1)-1. (7 .21)Experimental results obtained from intrinsic germanium at room temperature yieldsP E 150. Using b=2 and Eg/ kT= 26 and equation (7 .20 ) we obtain P 2 : 1400. Equation(7 .21) yields P = 160. So the mobility fluctuation hypothesis is here also supported bythe experimental results.7.2 . I l fno ise in Hall voltageAn additional sup port fo r the mobility fluctuation hypothesis is obtained fro m investiga-tions on llf noise in the Hall voltage. Recently Vaes an d Kleinpenning (1977) and

1 I I I I 1 II

Figure 8. llfnoise in Hall voltage. Dots: experimental results o f n-Ge. Broken line, calculated result fornumber fluctuations, full line for mob ility fluctuations.Kleinpenning (1980b) have investigated the influence of magnetic induction B on thellfvoltag e noise density between two small circular H all electrodes in terms of th e ratioy(B)=S(B)/S(O) see figure 8). Here S ( B ) and S(0) are the llfvoltage noise densitieswith and without magnetic induction when the applied voltage across the line-shapedouter electrodes is constant, an d the frequency fixed. The ratio y was calculated fo r twocases: l/f fluctuations are due to mobility fluctuations ( y p ) or to free-carrier densityfluctuations ( y ~ ) .In these calculations a n n-type non-degenerate homogeneous isotrope semiconductorwith spherical constant-energy surfaces has been considered. The re is a direct currentdensity J x in the x direction an d a m agnetic induction B in the z direction. The currentdensities in the x and the y directions ( x _ L y l z ) t point r at time t ar e given by

Jx@, t ) = A ( r , t ) E , t ) - D ( r , t ) E&, t ) (7 .22)J d y , t > = D ( r , >E&, t)+A(r, ) E&, t ) (7 .23 )


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F N Hooge, T G M Kleinpenning and L K J Vandamme10withwhere div J ( r , t )= 0

U ( , r , t )=qp(&, , t ) 2 ~ / 3 k T ) (E, r , t ) .

(7.24)(7.25)

(7.26)(7.27)

Here U(&, r , t ) is the conductivity, p ( ~ ,, t ) s the mobility and n(e , r , t ) is the carrierdensity with energy E at point r at t ime t . E( r , t ) s t he electric-field strength. To obtainthe voltage fluctuations between two points in the xy plane, fluctuations around thesteady state have to be considered. These fluctuations are assumed t o be caused by eithermobility or density fluctuations. So the quantities A and D fluctuate and as a result thequantities J and E do also. Th e relation between the fluctuations in A, D, J and E =-grad can be fou nd from equations (7.22)-(7.24). Th e result isA div grad 8rp(r, t )= E grad 8A(r , t )+ [ EA grad 6D(r, t)lz. (7.28)

Making a Fourier transform in the frequency-wavevector representation, equation(7.28 ) yields(7.29)

rp(k, w is the Fourier transform of Srp(r, t ) . The fluctuation in the open-circuit voltagebetween two point electrodes a t r1 an d YZ for a two-dimensional condu ctor ist(r1, re, w)= rp(k, U [exp ( jk .r l ) - ex p(j k-r z)] dk. (7 .30)

In order to find the spectral density of v(r1, rz, U the properties of the a-noise sourceshave to be known.As the calculations of the ratios yp and Y N are rather complicated, we shall restrictourselves to giving the final results. A detailed description is given in the paper by Vaesand Kleinpenning (1977). If the l/f noise can be ascribed to mobility fluctuations,according to equations (7 .4 ) and (7. 5) the result of yp for lattice scattering is

where A(B) and D(B) are the quantities of equations (7.25) and (7.26) at magneticinduction B, and U(&), p(e) and n(E) are the averages of the conductivity O ( E , Y, t ) ,mobility p(e, r , t ) and free-carrier density n(E, r , t ) , respectively. At low magneticinductions ( ~ H B


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Experimental studies on Ilf noise 511At low magnetic inductions this reduces to

yN(B)=l +2pH2B2. (7.34)From equations (7.32 ) and (7.34) i t follows that at low B the difference between y l L ( B )and ~ N ( B )s too slight to enable us to discriminate between mobility and numberfluctuations. Using equations (7.31) and (7.3 3) the ratios y p and y~ at high B can becalculated numerically. In figure 8 the results of y a (full line) and Y N (broken line) havebeen plotted for 0< ~ H B1. Experimental data obtained from n-type germanium at300 K with p =20 2 cm and p~ = 3300 cm2 V-l s-1 are also plotted in figure 8. Here thedots represent the mean values of all measurements and the vertical lines indicate thestandard deviation. Since the experimental da ta follow the line for mobility fluctuations,the conclusion is that these results give additional support to the claim that a noise iscaused by mobility fluctuations.7.3. l l f noise and lattice scatteringThere is still another piece of evidence in favour of mobility fluctuations. Th is is thecase of l/f noise in the conductance of heavily doped semiconductors where the mobilityis determined by a mixture of lattice scattering and impurity scattering. Hoo ge andVandamine (1978) have experimentally found the relation

S G ( f >/G2 (aifN> P/p1att)2 (7.35)where p-I= pimp-' and platt and p i m p are the mobilities that would be foundif only lattice or impurity scattering were present. The result will be treated in d etailin $8. But here we shall use the numerical results for the discussion of the 6p-6Nproblem. By analogy with equations (7 .6 ) an d (7.7 ) we write

(7.36)= X(p~ a t t , ) K Pimpplatt(pimp+p~att)-'] N= p Nso that mobility fluctuations lead to

and number fluctuations toS G / G ~ gT s N / N 2= h ( N ) a/fN.X N )

(7.37)

(7.38)If the mobility does not depend on the nu mb er of free carriers then we have a reductionfactor h(N )= 1. If, according to Hilsum (1974), the relation between p and N is givenby p = platt [1+ ( N / N O > ~ / ~ ] - ~ ,hen we find + ,


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512 F N Hooge, T G M Kleinpenning and L K J Vandamme

Figure 9. CL as a function of p at 300 K. CLO and po are the values found in samples not affected by impurityscattering or surface scattering. Circles: experimental, p-type Ge. Squares : experimental,n-type GaAs. Dots: experimental, thin Bi films. Full line: theory, equation (8.12). Brokenline: theory from equations (8.13) and (8.14).ent. We mak e two calculations assuming mobility fluctuations and n umb er fluctuationsrespectively. Then the calculated results will be compared with experimen tal data .Assuming mobility fluctuations the current density J ( t ) a t time t is

J( t> ~ p [ E ( x ,) ,p o k t ) l t > (7.39)where E ( x , t ) is th e electric-field strength a t point x O


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Experimental studies on llf noise 513Combining this result with equations 7.41) nd 7.42) ields

(po dJldpo)2 a.V2"= (E dJ/dE)2 flv 7.43)

According to Jacoboni and Reggiani 1979) he relation between p, po and E can beapproximated by7.44)

with ,i3 of the order of unity. W ith equations 7.39) nd 7.44)we find po d J /dpo=E dJ/dE, so that equation 7.43) educes toSv = a.V2/fN. 7.45)

If we assumed that l/f fluctuations were caused by number fluctuations, we would havethe following equation : J(t) = q pCL[E(x, t> ln(x, t ) E(x , t ) . 7.46)F o r the open-circuit fluctuations the following applies :and thus A J ( t ) = O = ( d J / d E ) SE(x , t ) + ( dJ / dn) 6n(x, t ) 7.47)

Following the same procedure as in the paragraph above we find7.48)

7.49)where p( E ) is given by equation 7.44) nd pd E) is the differential mobility defined by

Hence, nu mb er fluctuations would lead to noise voltage densities th at a re larger by afactor p2(E)/pd2(E) co mpared to the noise densities fro m mobility fluctuations. There-fore the llfnoise study of hot electrons also gives a possibility of discriminating between6 p and 6N.Since the experiments have to be performed at high electric-field strengths, Jouleheating of the samples creates serious problems. T o avoid this heating problem, sampleswith a hemispherical geometry were investigated. O n the other ha nd , the noise calcula-tions for such a geometry are complicated. T he details of these calculations are om ittedhere. Th e noise is measured as a function of the voltage across the sample. T he experi-mental data agree well with the calculated curve for mobility fluctuations and are faraway from the curve for numb er fluctuations. As an illustration let us com pare the resultsa t the voltage where the current is one-half of what it would have been if the sam ple wereohmic. T he noise calculated for num ber fluctuations is then a hundred times higher thanth at calculated for mobility fluctuations. T he expe rimental results lie within a factor oftwo aro un d the mobility curve. Th e conclusion is that a. noise of hot carriers can bedescribed in terms of low-field mobility fluctuations characterised by equation (1 .26).

d[p(E) E l P .

7.5. Empirical llf noise source termIn order to describe a. noise in systems where the electric-field strength and the free-carrier density are non-homogeneous, we have to transform equation 1.26) nto a l/f


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514 F N Hooge, T G M Kleinpenning and L K J Vandamnzenoise source term. He re we shall derive such a source term which describes the l/fnoise as a result of fluctuations in the lattice scattering. Fr om experimental dat a i t canbe concluded that l/f fluctuations in the conductivity are spatially uncorrelated. Inprinciple there will be a physical limit with respect to the distance at which the fluctua-tions are spatially correlated. In practice this distance turns o ut to be s ho rt comparedwith all characteristic dimensions of condu ctors an d their electric contacts. Hence, thecross-correlation spectral density in the conductivity U can be written as

So(r , r , f ) = [cxu~(r)ifut(r)]( r - r ) (7.50)where n(r) is the free-carrier density at spot P, u r) is the conductivity and 6 s the Diracdelta function . Fro m l/noise in thermo-EMF it is concluded th at the l/f fluctuationsin th e condu ction ba nd are energetically uncorrelated. So the cross-correlation spectraldensity of the l/f fluctuations in the conductivity of non-degenerate semiconductors is

6(v- ) 6(E- E)uZ(r,E )So(r , r, E , E, f ) ~fa@,E ) (7.51)where n(r, E ) is the density of free carriers with energy E a t spo t r , and u(r, E ) is theconductivity of carriers with energy E. Th e delta functions indicate tha t the fluctuationsa t r and r f r are uncorrelated, and so are the fluctuations a t the energies E a nd E # E .Consequently, th e electron conductivity an d the hole conductivity fluctuate uncorrelatedly,which is confirmed by experiments on thermal voltage of intrinsic germanium. Klein-penning (1978) obtained

6(r- r ) (7.52)3n/8) au2(r)S&, r , f ) = / / w S,(r, r, E, E , f ) d E d E =0for lattice scattering. T he factor 3n /8 is the result of the averaging procedure. This factorcan be avoided by using (8/3n-)a in (7 .51 ) for the description of the noise in a sub-band.

8. Lattice scattering causes U noiseAfter it had been demonstrated that 01 noise is a fluctuation of the mobility, furtherexperiments were done in order to investigate the nature of the mobility fluctuations inmore detail. A n obvious question is th en : does l/f noise depend on the kind of scatter-ing ? Th e simplest picture explaining the finite value of the m obility is that each electron,moving around in a Brownian motion at an average thermal velocity, is accelerated bya n applied electric field until i t collides with som e scattering centre. After the collisionsthere is no mem ory of the previous motion. In this way electrons o btain a limited driftvelocity prop ortional to the field E , which leads to the definition of mobility

( U ) = ( U d r i f t ) E (8 .1)Simple mechanical considerations lead to

p =qr/m (8 2where 7 s the average time that the electron is accelerated between two collisions. Th ereciprocal o f this collision time is proportional to t he probability t ha t the electron will be


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Experimental tudies on llf noise 515scattered. Hence, if two independen t scattering mechanisms ar e simultaneously active,we shall, for the effective collision time T, ind

l/T= 1/T1+ 1/ Q .As a direct consequencel / p = l / P l + 1/p2.

Of the many types of scattering three are important in semiconductors and metals :lattice scattering, impurity scattering and surface scattering.Lattice scattering very often dominates other m echanisms. Th e electrons are thenscattered by p ho no ns of the lattice vibrations, either optical o r acoustical.Electrons can a lso be scattered at electrically charged defects in the lattice. Th isso-called impurity scattering dominates in highly doped semiconductors, especially atlow temperatures. In thin layers nearly inelastic scattering occurs a t the surface. Thissurface scattering dominates when the layers are thinner than A the free path betweentwo collisions in a thick sam ple of the sam e material.If there are two scattering mechanisms, each with its own noise, then the observednoise in the effective p will be

as follows from differentiating equations (8.4). Because of (1 .26) this leads to2

al ( 012.We shall show that all experiments can be described very well by assuming llfnoise inthe lattice scattering and very little o r no llf no ise in any other type of scattering. Inother words, we shall demo nstrate tha t it is correct t o take

Hence (8.10)The first demonstration of the correctness of (8.10) was the measurement of cy. on semi-conductors with lattice scattering and impurity scattering (Hooge and Vandamme 1978).Th e noise was measured on ohm ic metal-semiconductor point contacts. Three p-typeGe crystals were used with p=3.0 x 10-2 Q cm, 3.7 x 10-3 Q cm and 4.5 x Q cm.Th e mobility in these samp les is lower tha n platt, the value fo un d in high-ohmic m aterialbecause of the additional impurity scattering. A few point contacts of each crystal wereinvestigated. Th e results ar e presented in figure 9. These experimental results, togetherwith results from n-type G aA s with p =2.7 x 10-3 Q cm, agree with relation (8. IO)represented by the full line. F o r a further discussion see Weissman (1980) an dKleinpenning (1981b). A large reduction of the mobility by the influence of impurityscattering will only occur in highly doped m aterial. T he carrier concen tration will thenbe high, which makes it necessary to use very small samples. Even then t he noise is low.This is why in all previous measurements of a ow-doped material was chosen, wherelattice scattering always prevailed, resulting in a alues close to


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516 F N Hooge, T G A4 Kleinpenning and L K J VandammeAnother demonstration of lattice scattering as the source of lif noise is given bynoise measurements on thin bismuth films (Hooge et a1 1979). The influence of surfacescattering is often described by a relation f or free pa th X

l / X = l/Xo+k/t (8.11)where ho is the bulk value for the free path found in an infinitely thick sample, t is thethickness of the sample, and k is a constant with an order of magnitude of unity. Thesurface influence is representable here by an effective free path t / k . Since (8.11) is verysimilar to (8 .4) the result for a will be

01= (p/po)2010 (8.12)by analogy with (8. 10 ). The subscript refers to bulk values. An advantage of equation(8.12 ) is that it does not depend on k, he numerical value of which has not been agreedupon. Equa tion (8. 12 ) gives the full line in figure 9. Since equation (8.12) is only areasonable guess, one may ask whether it is good enough to permit calculations of thenoise. Therefore th e calculations were also made fo r a model in which the influence of0 was taken into account. Th e angle 0 is the angle between the velocity of the electronan d its distance to the surface. Averaging over all values of 0 and all values of thedistance t o th e surface gives a relation for h as a function of t much more complicatedthan equation (8.11) :

x - l e x p ( - x ) d x rXoF. (8.13)Differentiating (8.13) gives an expression for the noise:

(8.14)From (8.1 3) and (8.14) follows the dependence of a on X and thereby of a: o n p. Thisdependence has been plotted as the broken line in figure 9. Figure 9 shows tha t whenp N po the value of a depends quadratically on p and th at (8.12) is a good approxim ation.When p


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Experimental studies on llf noise 517width w and thickness t . The carrier concentration strongly depends on x the distancefrom the surface (0


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518 F N Hooge, G M Kleinpenning and L K J Vandamme(ii) p-ii junc tion diode. He re we have a semiconductor structure in which one partis p-type and the other n-type.(iii) Schottky barrie r diode. He re we have a semiconductor m aterial in contac t witha metal. Th e metal-semiconductor junc tion has rectifying properties

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