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On the noise spectra of semi-conductor noise and of flicker effect
Physica XVI, no 4
April 1950
ON THE NOISE SPECTRA OF SEMI-CONDUCTOR NOISE AND OF FLICKER EFFECT b y A. V A N D E R Z I E L Department of Physics, University of British Columbia, Vancouver, B.C. Canada
Summary An a t t e m p t is made to explain the fact t h a t in flicker effect and in excess noise in semi-conductors the noise inte.nsity is inversely proportional to the frequency in a wide frequency range. I n section 1. a few general theorems about the F o u r i e r analysis of fluctuating quantities are mentioned and it is shown t h a t it is impossible for the noise i n t e n s i t y to vary inversely proportional to the frequency in the whole interval 0 < [ < oo. I t is then shown in section 2. t h a t a noise i n t e n s i t y which is inversely proportional to the frequency in a wide range can be obtained if a distribution function of correlation times is introduced. I n sections 3. a n d 4. it is shown t h a t such a distribution function should be expected from the various mechanisms involved. I n section 3. the various sources of noise in a semi-conductor are mentioned and it is shown t h a t the "shot effect", discussed b y B r i l l o u i n , Bernamont and G i s o 1 f is not sufficient to account for all the noise. A few other mechanisms, viz. flicker effect and resistance fluctuations due to the motion of foreign atoms or lattice distortions, are then introduced which give the required distribution of correlation times. I n section 4. flicker effect is discussed and it is shown t h a t M a c f a r l a n e ' s theory is equivalent to the one discussed in section 2. I t is then suggested t h a t flicker effect in oxide-coated cathodes might not primarily be due to the surface of the coating b u t much more due to what happens inside it. I n an Appendix it is shown t h a t B e rn a m o n t's and G i s o 1 f's t r e a t m e n t of shot noise are equivalent.
Introduction. A c c o r d i n g t o e x p e r i m e n t a l e v i d e n c e t h e noise spect r a of excess n o i s e i n s e m i - c o n d u c t o r s a n d of f l i c k e r effect i n r a d i o t u b e s (at l e a s t i n t h o s e h a v i n g o x i d e - c o a t e d c a t h o d e s ) h a v e i n c o m m o n t h a t t h e n o i s e i n t e n s i t y is a p p r o x i m a t e l y p r o p o r t i o n a l to t h e s q u a r e of t h e d.c. c u r r e n t I a n d i n v e r s e l y p r o p o r t i o n a l t o t h e freq u e n c y [ i n a w i d e f r e q u e n c y r a n g e 1) 2) s) 4) 5) e). I t is n o t v e r y diffic u l t t o a c c o u n t for t h e 1-21aw, b u t m o s t t h e o r i e s g i v e t h e w r o n g - -
3 5 9
- -
~60
A. VAN D E R ZIEL
frequency dependence, viz. a noise intensity which is constant for low frequencies and which varies inversely proportional to the square of the frequency for higher frequencies. It is the aim of this paper to offer an explanation for this I//-law.
l. A / e w general theorems. In our discussion we need a few general theorems 1) 7), which follow from the Fourier analysis of fluctuating quantities: 1. If the fluctuating q u a n t i t y X(t) is defined for 0 < t < t o and if x,, is the Fourier component of frequency co,, = 2~n/t o = 2n/,,, then :
x,,.2 = F(/) A~ = 4A/f~ ° X(t) X(t + w) cos o~,,wdw
(1)
(we shall drop the s u f f i x , later) where A~ -- 1/to, or:
F(/) = 4 f~" X(t) X(t +iw) cos ~m' dw = 4X(t) 2f ~ c(w) cos ore' dw.
(2)
c(w) is the " n o r m a l i z e d " correlation function:
c(w) =: X(t) X(t + w)/X(t) 2,
(3)
c(w) = I for w = 0 (normalisation), c(-.- w) = c(w), c(w) is independent of t and c(w) = 0 if [w[ 2)> z, where z is the correlation time. In the case of fluctuating quantities which are caused b y a large n u m b e r of independent events occuring at random, z is the duration of the event (e.g. the transit time of an electron in a radio tube) ; in the case of fluctuations involving decay problems ~ measures the average life of the decay. 2) As the spectral distribution function F(/) is completely determined b y the correlation function X(t) X(t + w), it is dangerous to arrive at definite conclusions about the mechanism causing the noise merely from the spectral distribution. Even if a mechanism gives the right spectral distribution it does not necessarily follow t h a t it is the right mechanism ; all mechanisms which give the same correlation function are equally likely. It is also not certain t h a t a single mechanism is involved, it m a y well be t h a t several mechanisms play an i m p o r t a n t part, perhaps even in different parts of the spectrum. 3) F(/) cannot satisfv the condition :
F(/) = a// for 0 < / < oo
(4)
where a is a constant. For according to a well-known Fourier theo-
ON T H E N O I S E SPECTR& OF S E M I - C O N D U C T O R NOISE
361
rein the relation (2) can be reversed as (compare also B e r n amontl)):
X(t) X(t + w) = f~o F(/) cos 2~/ze, d/
(5)
X(t) X(t + w) is a bounded continuous function and therefore the integral in (5) should converge for all values of w. B u t if (4) is subs t i t u t e d into (5). we obtain an integral which diverges at / = 0 for all values of w and at / = oo for w = 0. In order to make the integral in (5) convergent, F(/) has to satisfy the restrictions:
a) F(/) varies slower t h a n 1//for v e r y low frequencies, b) F(/) varies faster t h a n 1//for v e r y high frequencies. This means t h a t the l//-law cannot hold in the whole frequency range; unfortunately, it does not tell us at which frequencies deviations from this law occur. It would be extremely helpfnl if these frequency regions could be determined experimentally, for if F([) is known for all frequencies then
X(t) X(t + w) is known too.
2. Explanation o/ the I/Haw. If the correlation function represents an exponential decay of average life 3, as might be expected in the cases of flicker effect and semi-conductor noise (see below): c(w) =
e - ' ~ '~
(6)
and hence after (2):
F(/) = const. ~ (1 + oJ2 r ~ )-l.
(7)
F(/) is then independent of frequency if ~oz ~. 1 and is proportional t o / - 2 if 0)3 ~ I. This is in m a r k e d contrast to the 1//-law; other types of correlation functions give similar results. It is difficult to find a single event which gives rise to a correlation function such t h a t the 1//-law is the result of it. In order to solve this problem we turn to the t h e o r y of dielectric losses in solid dielectrics, where a similar problem has been solved recently. If the power factor tan ~ is expressed in terms of a single relaxation time 3, one finds: tan ~ = const, oJz (1 + co2r2)-1
(8)
Most solid dielectrics, however, have a power factor which is practically independent of frequency in a very wide f r e q u e n c y range. This
362
A. VAN DER ZlEL
has been a c c o u n t e d for b y introducing a wide distribution of relaxation times. If: dP = g(x) dr; (f~' g(r) dr =: 1) (9) is the probability for a relaxation time between r and (r + dr), one obtains in stead of (8) : t a n 6 ---- const. [~o o r (1 + co2z~)-1 g(r) dr (I0) B y a proper choice of g(r) one can always obtain agreement between t h e o r y and experiment a n d as there are strong a r g u m e n t s in favour of such a wide distribution of relaxation times this t h e o r y is now generally accepted (for an excellent survey compare Dr. G e v e r s' thesis 8)). If we now introduce into our problem a wide distribution of correlation times r in stead of the single correlation time of (7), the same beneficial effect m i g h t be expected. Let again (9) represent the probability of a correlation time between r and (r + dr), then we find for F(]) in stead of (7) :
F([) = 4 X ( t ) 2 f ~ r(1 + ofl~) -1 g(r) dr
(I I)
B y a proper choice of g(r) one can always obtain agreement between t h e o r y and experiment. One might of course question w h a t this " a g r e e m e n t " means; for even if one h a d started with the wrong correlation function one can always choose g(r) such t h a t F([) has the right frequency dependence. One can a t t r i b u t e physical m e a n i n g to the whole procedure only if sound arguments in favour of such a distribution function exist. It will be shown below t h a t this actually is the case. We obtain exactly the l//-law b y introducing the following (normalized) distribution function: g(r) dr ---- [ln (r2/rl] -1 dr/r for z~ < ~ < r2. / / g(r) dr = 0 outside t h a t interval.
(12)
Introducing (12) into (11) we obtain :
F(]) = 4 X(/)2[ln (r2/rl)] - t . (tan -1 o~ 2 - - t a n -1 o~rl)/~o
(13)
which is independent of frequency if cot2 ~ 1, varies as 1/o~ in the region 1/r 2 < oJ < 1/z I and varies as 1/co2 if o r l.>> 1. B y a proper choice of r I and r 2 one can extend the 1//-region as far as is necessary. At v e r y low a n d at v e r y high frequencies the conditions a) and b) for F(]) are satisfied b y (I 3).
ON THE NOISE SPECTRA OF SEMI-CONL, UCTOR NO ISE
363
In o r d e r to test the above t h e o r y one has to find deviations from the 1//-law. One would e x p e c t zt a n d Tz to d e p e n d upon t e m p e r a t u r e . This m e a n s t h a t one should investigate especially the shape of the spectral d i s t r i b u t i o n of the noise for different t e m p e r a t u r e s . In this respect it is interesting to m e n t i o n the m e a s u r e m e n t s of B e r n a m o n t 1) on thin m e t a l layers carrying a d.c. current. F o r one p a r t i c u l a r sample he found a transition from a 1//-law to a 1//Z-law at 72 kc (at room t e m p e r a t u r e ) . T h e a b o v e distribution function (I 2) is a v e r y reasonable or~e. F o r e x a m p l e consider the case t h a t the r e l a x a t i o n times r can be describ e d with the help of ilactivation energies" E such t h a t : 3 = % e~/kr (31 = 30 e~l/kr; 32 = %e~2/~r).
(14)
A r a t h e r n a r r o w distribution in E m a y thus give rise to a wide dist r i b u t i o n in 3, because k T is a v e r y small q u a n t i t y at room t e m p e r a ture. Moreover, (14) shows t h a t t h e relaxation time has a lower limit z 0 as required b y 112). Finally, if we substitute (14) into (12) we o b t a i n the following distribution function in E :
g(E) dE = dE/(E 2 - El) for E 1 < E < E2~ / g(E) dE = 0 outside t h a t interval,
(12a)
where E 2 and E t a r e the energies corresponding to 32 and 3 I. 31 = ~0 if E,. = 0 ; there is no highest limit for 32 b u t as one would not e x p e c t e x t r e m e l y large a c t i v a t i o n energies, it is unlikely t h a t 32 = co (this would m e a n a ~iolation of condition a) for F(/)). I t m a y be, however, t h a t 32 has to be m e a s u r e d in minutes or even hours. O b v i o u s l y a n y function g(E) which has a width (E 2 - - E l ) , a r e a s o n a b l y flat p a r t for E l < E < E 2 and a more or less rapid decrease for E < E 1 and E 3> E 2 will produce the same results *). If the e x p e r i m e n t a l results indicate a slightly different d e p e n d e n c e of the noise u p o n frequency, one has to alter the distribution function g(E) dE accordingly.
3. Types o/noi.~e in a semi-con;t,,'tor. The following types of noise m a y occur in a semi-conductor: *) The whole d i s c u s s i o n is v e r y s i m i l a r to the discussion on the p o w e r factor ot solid d i e l e c t r i c s s). I f the d i s t r i b u t i o n f u n c t i o n (12) is i n t r o d u c e d i n t o (10), t a n ~ b e c o m e s i n d e p e n d e n t of f r e q u e n c y in the region I/T~ < co < I[T~. T h i s does n o t m e a n t h a t the s a m e mechanism is i n v o l v e d in the t w o eases, i t o n l y m e a n s t h a t t h e s a m e methods c a n b e applied.
364
A. VAN D E R Z I E L
a) T h e r m a l noise. Due to fluctuations in the velocity of the e l e c t r o n s in the senti-conductor the net current t h r o u g h a n y cross-section is not zero b u t will show r a n d o m fluctuations. T h e F o u r i e r c o m p o n e n t s of this fluctuating current are given b y N yq u i s t's t h e o r e m 8): _
4kTA/
R
h~
k T " (e h m r
I)--1
(15)
F o r metallic conductors (except for v e r y thin m e t a l layers) this is the only t y p e of noise present, for semi-conductors this is the case only for zero d.c. current. b) S h o t noise. As the n u m b e r of free electrons in the semic o n d u c t o r shows r a n d o m fluctuations (the electrons can either exist in the b o u n d or in the free s t a t e ; the n u m b e r of electrons in b o t h states are in t h e r m a l equilibrium), an excess a m o u n t of noise o v e r and above t h e r m a l noise will occur if the semi-conductor is c a r r y i n g a d.c. current I. This effect is negligible for metal wires, e x c e p t perhaps v e r y thin ones, b u t might be quite i m p o r t a n t for semiconductors. Assuming t h a t the creation of a free electron in the s e m i - c o n d u c t o r is an i n d e p e n d e n t e v e n t occuring at random, one can follow three m e t h o d s of calculation of the Fourier c o m p o n e n t s of this noise current : l) F r o m the velocity distribution function of the electrons as it is modified b y the d.c. current (B r i 11 o u i n J0)). 2) F r o m the fact t h a t the fluctuation in the n u m b e r of free electrons causes a fluctuation in the resistance (B e r n a m o n t 1)). 3) The motion of the free electrons between their creation a n d their subsequent c a p t u r e gives rise to current pulses (G i s o 1 f 11)). The first two m e t h o d s give identical results; t h a t this also is the case for the third m e t h o d is shown in the Appendix. Assuming t h a t all free electrons have the same life-time ~1, we obtain b v G i s o I f's method :
= 2elzJ/'(z;l~2)"
Fsin/ L
1/2/] 2
(o~zt/2)
J
(16)
(compare Appendix); e denotes the electronic charge, r 2 is the t i m e which it will take a free electron to drift from the negative e l e c t r o d e
ON THE NOISE SPECTRA
OF SEMI-CONDUCTOR
NOISE
365
to the positive one. T h e above result holds for 31 ~ % if 31 > 32 we have instead:
r si~ < ' 3 : ) ] ~. i2 -- 2:-M/L (w32/2) / '
(16a)
in between a transition occurs from (1 6) to (1 6a). 32 depends upon the d.c. current I so t h a t in (16) the noise is in fact proportional to 12 (compare Appendix). (16) and (16a) show t h a t the u p p e r limit for this t y p e of noise is just the true shot noise of a s a t u r a t e d diode carrying a d.c. current f ~). It seems t h a t in m a n y cases the a m o u n t of noise which is actually measured is far above this limit, so t h a t this mechanism c a n n o t explain all excess noise. T h e exact a m o u n t of noise depends u p o n the ratio (31/32). For a c r y s t a l diode 32 is the time which is necessary to cross the barrier layer, this time is so small t h a t 31 >~ 32, so t h a t full shot effect occurs *). Note t h a t z~ is tile life-time of the free electron, which will in general be n m c h larger t h a n the time between two collisions with the lattice. There has b e e n quite a discussion in recent literature about the question w h e t h e r or not t h e r m a l noise and shot noise have the same n a t u r e and origin. Those who gave an affirmative answer to this question were misled b y the fact t h a t certain models of a conductor, which were showing true shot effect, gave N y q u i s t's formula. B u t as N y q u i s t ' s result is i n d e p e n d e n t of a n y model (as it follows directly from the second law of t h e r m o d y n a m i c s ) it is comp l e t e l y false to conclude from it t h a t t h e r m a l noise and shot noise h a v e the same origin. We see now t h a t shot noise in diodes is equivalent to a certain t y p e of excess noise in semi-conductors. :) F l i c k e r effect. F o r the noise of t y p e b ) it was assumed t h a t the creation of a free electron was an i n d e p e n d e n t event occuring at random. As the free electrons originate from i m p u r i t y centres in the semi-conductor, one would expect this to be the case if the i m p u r i t y atoms occur as isolated atoms in the lattice. If, however, these impurities occur in clusters (e.g. at the grain boundaries in p o l y - c r y s t a l l i n e substances) an e x t r a a m o u n t of noise will be generated, especially at low frequencies. For let 33 be the average time b e t w e e n the delivery of two subsequent free electrons f:'om the same *) I n c r y s t a l d i o d e s excess noise is the e n o r m o u s a m o u n t of noise o v e r a n d a b o v e t h e r m a l noise a n d full s h o t effect.
366
A. VAN DER ZlEL
cluster. For high frequencies such that a~z3 >/~ 1 this will hardly have any effect, as the electrons still appear to be delivered in single events occuring a t random. But for low frequencies such that coz3 <~ 1 it is just as if the cluster delivers a number of electrons sinmltaneously so that (16) and (16a) have to be multiplied b y a large factor which increases with decreasing frequency. As the clusters will decay b y a diffusion process one would expect that the noise would become independent of frequency if coz4 <~. 1, where ~4 is the average life-time of the cluster. As diffusion is a rather slow process one would expect rather large time constants here. As this effect is similar to the flicker effect of cathodes in radio-tubes s), we refer to the next section where a fuller discussion will be given. d) L o c a l fluctuations in conductivity. These m a y occur due to the diffusion of foreign atoms or lattice distortions through the lattice (place exchange). Due to this effect the electrons will move in a rather slowly fluctuating electric field and this is equivalent to a fluctuating conductivity. This mechanism m a y be a very important one in many cases. One would again expect rather long time constants for this process. It is interesting to mention that it was suggested recently that the excess noise in crystaldiodes might be due to fluctuations in the contact potential due to the motion of ions in the contact surface. This is the same phenomenon as discussed here ~). e) S p o n t a n e o u s fluctuations in temperature. These fluctuations in temperature 13) give rise to spontaneous fluctuations in resistance; if the conductor carries a d.c. current these resistance fluctuations give in turn rise to a noise e.m.f.. Though this effect is certainly present, it seems to be unable to account for the enormous amounts of noise observed *.) Moreover, though it accounts for the long time constants observed, it will give rise to a l//2-1aw at higher frequencies as there is only a single time constant involved in the mechanism. It nfight be though that fluctuations of local origin will give more noise. Mechanisms b), c), d) and e) all give excess noise over and above *) T h e s a m e e f f e c t s h o u l d c o n t r i b u t e t o t h e flicker e f f e c t in d i o d e s w i t h t h o r i a t e d t u n g s t e n or t u n g s t e n f i l a m e n t s . I t c a n be s h o w n , h o w e v e r , t h a t t h i s e f f e c t is u s u a l l y m u c h s m a l l e r t h a n t h e s h o t noise of t h e d i o d e , w h e r e a s t h e flicker e f f e c t is m u c h l a r g e r t h a n s h o t noise a t l o w f r e q u e n c i e s .
ON T H E N O I S E S P E C T R A OF S E M I - C O N D U C T O R N O I S E
367
t h e r m a l noise which is proportional to 12. Experiments show a noise spectrum which varies as 1/[ down to very low frequencies. This requires a wide distribution of correlation times, including rather large ones. Mechanism b) gives rise to a noise spectrum which is constant for cot I ~ 1 and becomes negligible for o~T1 ~ 1. As vt £ To and To is of the order of 10 -4 sec. (compare Appendix), this mechanism cannot cause the bulk of the noise at low frequencies is), though it m a y play a part at h.f.. It would not be difficult to account for a wide distribution in correlation times in this mechanism. Mechanisms c) and d) have in common t h a t t h e y m a y give rise to a wide distribution of correlation times including rather long ones. For the atoms which take part in the diffusion processes will not all have the same activation energy and a large n u m b e r of different processes are possible, so t h a t one would indeed expect such a distribution.
4. Flicker e//ect o~ electron emitting cathodes. S c h o t t k y ~) assumed t h a t the emission depended upon the n u m b e r of foreign atoms present on the emitting surface. As this n u m b e r of foreign atoms can decay b y a diffusion process or b y an evaporation process, one would expect the decay to be an exponential one. This leads to a 1/./2-law at high frequencies, whereas experiments on oxide-coated cathodes indicate a 1//-law *). M a c f a r 1 a n e 14) recently proposed a t h e o r y of flicker effect, based upon S p r o u 1 l's 15) t h e o r y of cathode emission, which can explain the 1//-law. It will be shown below t h a t M a c f a r I a n e's m e t h o d is formally equivalent to the general t h e o r y given in section 2. M a c f a r 1 a n e assumes t h a t the emission occurs f rom small "active specks" which can decay in two ways: a) b y a diffusion process as in S c h o t t k y's theory, b) by drawing current from the speck. Small specks can decay b o t h ways but large specks m a i n l y decay b y process b) according to S p r o u 1 l's theory. Large specks decay m u c h faster t h a n small specks; the larger the speck, the more vio*) Some ex periments on t u n g s t e n and on thoriated tungsten cathodes seem t o indicate a l//2-1aw (compare e.g. S c h o t t k y's paper). I t would be interesting to settle this question definitely by further experiments.
36~
A. VAN DER ZIEL
lently it decays. M a c f a r l a n e t h e n introduces a Gaussian distribution in the size of the specks. If the distribution were a narrow one, all specks would decay at the same rate and S c h o t tk y' s original 1//Z-law would have been obtained ; a wide distribution in size, however, gives a 1//-law at high frequencies as required by experiments. The success is due tot he introduction of a wide distribution in the size of the specks. As the large specks decay so much faster t h a n the small specks, this is equivalent to the introduction of a very wide distribution in the correlation times as presented in section 2 *). M a c f a r I a n e's t h e o r y gives a macroscopic description of the p h e n o m e n o n ; if the problem is investigated from a microscopic point of view, the picture is as follows : The atoms in an active speck do not all have the same work-function, the more neighbours of the right kind an atom has, the lower the work-function will be and thus the larger the probability of decay b y emission of an electron. Moreover, not all atoms of the active speck will have the same probability of diffusion; a distribution in " a c t i v a t i o n energies" is v e r y likely. Therefore, even if M a c f a r 1 a n e's t h e o r y might not be correct in all its details, the actual mechanism will give rise to a wide distribution of correlation times and t h a t is all t h a t is required for the
1//-law. M a c f a r 1 a n e 14) also applied his t h e o r y of flicker effect to the problem of excess noise in semi-conductors. As shown in section 3, this is a very reasonable step. But it is even more reasonable to a p p l y the t h e o r y of excess noise in semi-conductors to the flicker effect of oxide-coated cathodes. For t h o u g h M a c f a r 1 a n e's t h e o r y is u n d o u b t e d l y right for tungsten and thoriated tungsten cathodes (though one might question whether a narrow or wide distribution in the size of the specks has to be introduced there) it is doubtful whether it is the only mechanism in the flicker effect of *) T h e t h e o r y m e a n s a d i s t r i b u t i o n of "r o v e r the i n t e r v a l T, < r < T~ w i t h r~ = 0. M o r e o v e r , n o h.f. l i m i t for the l / l - l a w is f o u n d , w h i c h s e e m s to v i o l a t e c o n d i t i o n b) of s e c t i o n 2. for F([). B o t h r e s u l t s are d u e to a n i d e a l i z a t i o n of the a c t u a l p h e n o m e n o n , w h i c h d o e s n o t o c c u r ill p r a c t i c e . T h e i d e a l i z a t i o n is in t h e d i s t r i b u t i o n futlction of the size of t h e s p e c k s . F o r l a r g e sizes of t h e s p e c k s , t h e n u m b e r of s p e c k s is c e r t a i n l y less t h a n i n d i c a t e d b y a G a u s s i a n d i s t r i b u t i o t h as it is ceLtain t h a t e v e n tile l a r g e s t s p e c k s will be v e r y s m a l l in c o m p a r i s o n to the c a t h o d e a r e a . T h e i n f i n i t e l y s m a l l n u m b e r of i n f i n i t e l y l a r g e s p e c k s , g i v e n b y t h e G a u s s i a n d i s t r i b u t i o n , d e c a y so infinitel3." f a s t t h a t t h e y m a k e I(l) ~ i n f i n i t e l y large.
ON THE NOISE SPECTRA OF SEMI-CONDUCTOR
NOISE
369
oxide cathodes. For the theory pays its main attention to the s u r / a c e of the emitting layer. But it has been found recently 16) that the emission current for an oxide cathode is determined by the conductivity of the coating itself. As the oxide coating is a semi-conductor, its conductivity will show spontaneous fluctuations and it is very reasonable to assume that these fluctuations will show up as fluctuations of the emission current. The measurements indicate that the emission current I is proportional to the conductivity a of the coating, so that I / a is a constant. This means that if R is the resistance of the coating, then the voltage V o - - I R = I d / ( A a ) across the coating is independent of:the emission current. It is reasonable to assume that V0 does not fluctuate, so that one would expect for the current fluctuation AI: ,41 = - - 1 , 4 R / R
or ,4/~ = ( I / R ) 2. (,4R) 2
(17)
As the excess noise in semi-conductors gives rise to a l//-law, it is reasonable to expect the same result for oxide coated cathodes. The above theory offers an explanation for the fact that there is a distinct difference between the flicker effect in oxide-coated cathodes and in tungsten or thoriated tungsten cathodes (quite apart from the possible difference in spectral distribution mentioned before). The measurements seem to indicate that for equal emitting area the oxide-coated cathodes have a much larger flicker effect ; this suggests a different mechanism and that is what the above theory offers *). This does not suggest that the theory of noise in semi-conductors can be applied to the flicker effect of oxide-coated cathodes without any alteration; the above suggestion offers only a new mechanism and it n fight be worth while to develop it further. C o n c l u s i o n s . The discussion in the last two sections shows clearly that there is good evidence in favour of a wide distribution in correlation times and that it is very likely that such a wide distribution *) In a d d i t i o n it offers a v e r y s i m p l e e x p l a n a t i o n for the following p h e n o m e n o n . In a t u b e c o n s i s t i n g of two parallel hot c a t h o d e s h a v i n g a s m a l l d i s t a n c e d and b e i n g k e p t at a u n i f o r m high t e m p e r a t u r e T the flicker effect wil not show up, as the noise is g i v e u b y N y q u i s t's t h e o r e m . As the net c u r r e n t flowing t h r o u g h the t u b e is zero, one would i n d e e d e x p e c t no flicket effect a c c o r d i n g to e q u a t i o n (17). The t h e o r y m i g h t also e x p l a i n the s u p p r e s s i o n of flicker effect iT) which is observed w h e n the diode is s p a c e - c h a r g e d l i m i t e d in steacl of s a t u r a t e d , b u t it is b e y o n d the scope of this p a p e r to d e v e l o p this theory. Physica XVI
24
370
A. VAN DER ZIEL
function can explain the 1//-law. Moreover, it has been shown t h a t it is difficult to separate the various components of excess noise in semi-conductors and oxide-coated cathodes and t h a t it is practically impossible to find out where the fluctuations in resistance end a n d the true flicker effect begins. The author is indebted to Dr. K. S. K n o 1, N.V. Philips' Gloeilampenfabrieken, Eindhoven, The Netherlands and to Dr. A. J. D e k k e r, University of British Columbia, Vancouver, Canada, for stimulating discussions about the above subject and to the Defence Research Board of Canada for a grant in aid and for permission to publish these results. It is a great pleasure, for the a u t h o r to publish this paper as a tribute to Prof. C o s t e r.
Appendix. Proo/ o/ the equivalence o/ Gisol/'s and Bernamont's methods. Let L be the length of a semi-conducting bar having metal electrodes at its ends, let R be the resistance of the bar and I the d.c. current flowing t h r o u g h it, t h e n the d.c. voltage V = IR. If e is the electron charge and u the electron mobility, then the drift velocity is Fu, where F = V/L = I R / L is the electric field strength. R is given as : R -- L2/(euNo), (18) where N Ois the total n u m b e r of electrons in the conductor. As R is inversely proportional to N o we have for fluctuations A N o of No:
,dR----R,dNo/N o or AR 2 ---- (R2/N~0),dN~0;
(19)
the noise e.m.f. AE in series with R is I,dR and the fluctuating current A I flowing through R is IAR/R. Its mean square value is :
,dI 2 = (P/No). (AN~o/No) = I2/No
(20)
for the classical t h e o r y als ,dN~0 =-- N o in this case (this relation holds for semi-conductors but not for metals). As AI(t) 2 is now given b y (20) and as an exponential correlation function with average life z o should be expected, we have, according to (2) and (7), for the Fourier components i of I(t) :
-fi
=
4(I2/No) A/% (1 + W2~o)- '
4 euRF% A/
-- L2(1 + oJ2~o)
(2 !)
This is the noise current according to B e r n a m o n t 1). We now turn to G i s o 1 f's theory. G i s o 1 f 1i) makes the simplifying
ON T H E N O I S E S P E C T R A OF S E M I - C O N D U C T O R N O I S E
371
a s s u m p t i o n t h a t all current pulses have equal length 31 (31 is the time interval between the creation of a free electron a n d its subseq u e n t capture). He t h e m obtains as his final result: Fsin (0~1/2)7 2
-~ = (2euF2i3,/R) A/. L (~---~
J --
Fsin (om/2) J 2
= (2 e # RI2,, A//L2). L ( ~ - ~
(22)
which is G i s o 1 f's formula (20) in slightly different n o t a t i o n (except for a factor 2 which has disappeared in his calculation). If we put 230 = 31, t h e n the two formulae are identical e x c e p t for the f r e q u e n c y dependence (due to the different correlation function used by Gisolf). We shall now prove t h a t 31 a n d 30 h a v e an u p p e r limit 32. F o r as L is the length of the c o n d u c t o r and vg = F,~ = I R u / L t h e drift velocity of the electrons in the direction of the field, it would take an electron an average time 32 = L/v d = L2/(IR,~) to t r a v e l from the one electrode to the o t h e r one if it were not c a p t u r e d before. I n t r o ducing this into (22) we obtain: •7
....
F sin (co31/2)-]~
zl
(23)
This derivation only holds if 31 ~ 32, for otherwise one has to take into account t h a t not all electrons can possibly have the life-time 3~ (those free electrons which are c r e a t e d near the positive electrode arrive at this electrode before t h a t time). If 31 ~> 32, the results become v e r y simple again, as now all electrons e n t e r the semic o n d u c t o r at the negative electrode and leave it at the positive one. We thus obtain in stead of (22): Fsin (o)32/2)1 2
/7 = (2 e I AI) • L ( ~ - ~
J
(23)
which is the formula for true shot effect. It is the merit of O is o 1 f's m e t h o d t h a t it can show the change from excess noise in semi-conductors to true shot effect so nicely. P u t t i n g R = 105 ~2, I = 1 mA, I. = 1 cm and ,~ = 100 cm/sec per unit field strength (Volt/cm), we obtain 30 = 10 -4 sec. Received 26-1-50
372
ON T H E N O I S E S P E C T R A OF S E M I - C O N D U C T O R N O I S E
REFERENCES 1) J. B e r n a m o n t , C . R . , 1 9 8 , 1755 and 2144; 1934;Ann. de Physique T, 71, 1937. 2) H.C. T o r r e y a n d C . A. W h i t m e r , Crystal Rectifiers, Chapt. 6, M c G r a w H i l l Co., New-York, 1948. 3) P . H . M i l l e r J r . , Proc. I. R. E. 3 5 , 2 5 2 , 1947. 4) E. M e y e r andH. Thiede, E . N . T . 1 2 , 2 3 7 , 1935. 5) C . J . C h r i s t e n s e n and G. L. P e a r s o n , B e l l S y s t . Techn. Journ. 15, 197, 1936. 6) W. S c h o t t k y , Phys. Rev. 2 8 , 7 4 , 1926. 7) M.C. W a n g a n d G . E. U h l e n b e c k , Rev. Mod. Phys. 17,326, 1945. 8) M. G e v e r s, Philips Res. Rep. 1, 197, 279, 361, 447, 1946. 9) H. N y q u i s t , Phys. Rev. 32, II0, 1928. I0) L. B r i l l o u i n , Helv. phys. Act. 7 , 4 7 , Supplement, 1934. l l ) J. H. G i s o l f , Physica 15, 825, 1949"). 12) W. S c h o t t k y , Ann. Phys. Lpz., 57, 541, 1918. 13) H . A . v a n d e r V e l d e n , P h . D . Thesis, Utrecht, 1947. 14) G . G . M a c f a r l a n e , Proc. phys. Soc..59, 366, 1947. 15) R . L . S p r o u l l , Phys. Rev. 67, 166, 1946. 16) N . B . H a n n a y , D. M a c . N a i r a n d A . H. W h i t e , Journ. appl. Phys. 20, 669, 1949. 17) W. S c h o t t k y , Physica 4, 175, 1937. The following two papers which are not quoted in this paper are also of great interest for the subject: B. D a v y d o v , J. Phys. U.S.S.R. 4, 335, 1941;B. D a v y d o v andB. Gurevich, J. Phys. U.S.S.R. 7, 138, 1943.
Note added in proo/ Since this paper was written, Mr. W. A. B a i n at the Uuiversity of British Columbia found several cases in which the noise spectrum showed a transition from a 1//-law to a 1/f'-law at higher frequencies. Dr. W. G. S h e p h e r d (U. of Minnesota) told me t h a t similar results have been obtained at the University of Minnesota and at the University of Pennsylvania. Mr. B a i n also found a dependence of the transition frequency upon temperature. Both results are in good accord with the above theory, though Mr. B a i n's results seem to exlude relation (14) between r and T, used in our example in section 2.
*) G i s o 1 f does not seem to have been aware of previous work on this subject. There are a few errors in his duscussion, which can be corrected by following the calculation and the discussion more closely.
April 1950
ON THE NOISE SPECTRA OF SEMI-CONDUCTOR NOISE AND OF FLICKER EFFECT b y A. V A N D E R Z I E L Department of Physics, University of British Columbia, Vancouver, B.C. Canada
Summary An a t t e m p t is made to explain the fact t h a t in flicker effect and in excess noise in semi-conductors the noise inte.nsity is inversely proportional to the frequency in a wide frequency range. I n section 1. a few general theorems about the F o u r i e r analysis of fluctuating quantities are mentioned and it is shown t h a t it is impossible for the noise i n t e n s i t y to vary inversely proportional to the frequency in the whole interval 0 < [ < oo. I t is then shown in section 2. t h a t a noise i n t e n s i t y which is inversely proportional to the frequency in a wide range can be obtained if a distribution function of correlation times is introduced. I n sections 3. a n d 4. it is shown t h a t such a distribution function should be expected from the various mechanisms involved. I n section 3. the various sources of noise in a semi-conductor are mentioned and it is shown t h a t the "shot effect", discussed b y B r i l l o u i n , Bernamont and G i s o 1 f is not sufficient to account for all the noise. A few other mechanisms, viz. flicker effect and resistance fluctuations due to the motion of foreign atoms or lattice distortions, are then introduced which give the required distribution of correlation times. I n section 4. flicker effect is discussed and it is shown t h a t M a c f a r l a n e ' s theory is equivalent to the one discussed in section 2. I t is then suggested t h a t flicker effect in oxide-coated cathodes might not primarily be due to the surface of the coating b u t much more due to what happens inside it. I n an Appendix it is shown t h a t B e rn a m o n t's and G i s o 1 f's t r e a t m e n t of shot noise are equivalent.
Introduction. A c c o r d i n g t o e x p e r i m e n t a l e v i d e n c e t h e noise spect r a of excess n o i s e i n s e m i - c o n d u c t o r s a n d of f l i c k e r effect i n r a d i o t u b e s (at l e a s t i n t h o s e h a v i n g o x i d e - c o a t e d c a t h o d e s ) h a v e i n c o m m o n t h a t t h e n o i s e i n t e n s i t y is a p p r o x i m a t e l y p r o p o r t i o n a l to t h e s q u a r e of t h e d.c. c u r r e n t I a n d i n v e r s e l y p r o p o r t i o n a l t o t h e freq u e n c y [ i n a w i d e f r e q u e n c y r a n g e 1) 2) s) 4) 5) e). I t is n o t v e r y diffic u l t t o a c c o u n t for t h e 1-21aw, b u t m o s t t h e o r i e s g i v e t h e w r o n g - -
3 5 9
- -
~60
A. VAN D E R ZIEL
frequency dependence, viz. a noise intensity which is constant for low frequencies and which varies inversely proportional to the square of the frequency for higher frequencies. It is the aim of this paper to offer an explanation for this I//-law.
l. A / e w general theorems. In our discussion we need a few general theorems 1) 7), which follow from the Fourier analysis of fluctuating quantities: 1. If the fluctuating q u a n t i t y X(t) is defined for 0 < t < t o and if x,, is the Fourier component of frequency co,, = 2~n/t o = 2n/,,, then :
x,,.2 = F(/) A~ = 4A/f~ ° X(t) X(t + w) cos o~,,wdw
(1)
(we shall drop the s u f f i x , later) where A~ -- 1/to, or:
F(/) = 4 f~" X(t) X(t +iw) cos ~m' dw = 4X(t) 2f ~ c(w) cos ore' dw.
(2)
c(w) is the " n o r m a l i z e d " correlation function:
c(w) =: X(t) X(t + w)/X(t) 2,
(3)
c(w) = I for w = 0 (normalisation), c(-.- w) = c(w), c(w) is independent of t and c(w) = 0 if [w[ 2)> z, where z is the correlation time. In the case of fluctuating quantities which are caused b y a large n u m b e r of independent events occuring at random, z is the duration of the event (e.g. the transit time of an electron in a radio tube) ; in the case of fluctuations involving decay problems ~ measures the average life of the decay. 2) As the spectral distribution function F(/) is completely determined b y the correlation function X(t) X(t + w), it is dangerous to arrive at definite conclusions about the mechanism causing the noise merely from the spectral distribution. Even if a mechanism gives the right spectral distribution it does not necessarily follow t h a t it is the right mechanism ; all mechanisms which give the same correlation function are equally likely. It is also not certain t h a t a single mechanism is involved, it m a y well be t h a t several mechanisms play an i m p o r t a n t part, perhaps even in different parts of the spectrum. 3) F(/) cannot satisfv the condition :
F(/) = a// for 0 < / < oo
(4)
where a is a constant. For according to a well-known Fourier theo-
ON T H E N O I S E SPECTR& OF S E M I - C O N D U C T O R NOISE
361
rein the relation (2) can be reversed as (compare also B e r n amontl)):
X(t) X(t + w) = f~o F(/) cos 2~/ze, d/
(5)
X(t) X(t + w) is a bounded continuous function and therefore the integral in (5) should converge for all values of w. B u t if (4) is subs t i t u t e d into (5). we obtain an integral which diverges at / = 0 for all values of w and at / = oo for w = 0. In order to make the integral in (5) convergent, F(/) has to satisfy the restrictions:
a) F(/) varies slower t h a n 1//for v e r y low frequencies, b) F(/) varies faster t h a n 1//for v e r y high frequencies. This means t h a t the l//-law cannot hold in the whole frequency range; unfortunately, it does not tell us at which frequencies deviations from this law occur. It would be extremely helpfnl if these frequency regions could be determined experimentally, for if F([) is known for all frequencies then
X(t) X(t + w) is known too.
2. Explanation o/ the I/Haw. If the correlation function represents an exponential decay of average life 3, as might be expected in the cases of flicker effect and semi-conductor noise (see below): c(w) =
e - ' ~ '~
(6)
and hence after (2):
F(/) = const. ~ (1 + oJ2 r ~ )-l.
(7)
F(/) is then independent of frequency if ~oz ~. 1 and is proportional t o / - 2 if 0)3 ~ I. This is in m a r k e d contrast to the 1//-law; other types of correlation functions give similar results. It is difficult to find a single event which gives rise to a correlation function such t h a t the 1//-law is the result of it. In order to solve this problem we turn to the t h e o r y of dielectric losses in solid dielectrics, where a similar problem has been solved recently. If the power factor tan ~ is expressed in terms of a single relaxation time 3, one finds: tan ~ = const, oJz (1 + co2r2)-1
(8)
Most solid dielectrics, however, have a power factor which is practically independent of frequency in a very wide f r e q u e n c y range. This
362
A. VAN DER ZlEL
has been a c c o u n t e d for b y introducing a wide distribution of relaxation times. If: dP = g(x) dr; (f~' g(r) dr =: 1) (9) is the probability for a relaxation time between r and (r + dr), one obtains in stead of (8) : t a n 6 ---- const. [~o o r (1 + co2z~)-1 g(r) dr (I0) B y a proper choice of g(r) one can always obtain agreement between t h e o r y and experiment a n d as there are strong a r g u m e n t s in favour of such a wide distribution of relaxation times this t h e o r y is now generally accepted (for an excellent survey compare Dr. G e v e r s' thesis 8)). If we now introduce into our problem a wide distribution of correlation times r in stead of the single correlation time of (7), the same beneficial effect m i g h t be expected. Let again (9) represent the probability of a correlation time between r and (r + dr), then we find for F(]) in stead of (7) :
F([) = 4 X ( t ) 2 f ~ r(1 + ofl~) -1 g(r) dr
(I I)
B y a proper choice of g(r) one can always obtain agreement between t h e o r y and experiment. One might of course question w h a t this " a g r e e m e n t " means; for even if one h a d started with the wrong correlation function one can always choose g(r) such t h a t F([) has the right frequency dependence. One can a t t r i b u t e physical m e a n i n g to the whole procedure only if sound arguments in favour of such a distribution function exist. It will be shown below t h a t this actually is the case. We obtain exactly the l//-law b y introducing the following (normalized) distribution function: g(r) dr ---- [ln (r2/rl] -1 dr/r for z~ < ~ < r2. / / g(r) dr = 0 outside t h a t interval.
(12)
Introducing (12) into (11) we obtain :
F(]) = 4 X(/)2[ln (r2/rl)] - t . (tan -1 o~ 2 - - t a n -1 o~rl)/~o
(13)
which is independent of frequency if cot2 ~ 1, varies as 1/o~ in the region 1/r 2 < oJ < 1/z I and varies as 1/co2 if o r l.>> 1. B y a proper choice of r I and r 2 one can extend the 1//-region as far as is necessary. At v e r y low a n d at v e r y high frequencies the conditions a) and b) for F(]) are satisfied b y (I 3).
ON THE NOISE SPECTRA OF SEMI-CONL, UCTOR NO ISE
363
In o r d e r to test the above t h e o r y one has to find deviations from the 1//-law. One would e x p e c t zt a n d Tz to d e p e n d upon t e m p e r a t u r e . This m e a n s t h a t one should investigate especially the shape of the spectral d i s t r i b u t i o n of the noise for different t e m p e r a t u r e s . In this respect it is interesting to m e n t i o n the m e a s u r e m e n t s of B e r n a m o n t 1) on thin m e t a l layers carrying a d.c. current. F o r one p a r t i c u l a r sample he found a transition from a 1//-law to a 1//Z-law at 72 kc (at room t e m p e r a t u r e ) . T h e a b o v e distribution function (I 2) is a v e r y reasonable or~e. F o r e x a m p l e consider the case t h a t the r e l a x a t i o n times r can be describ e d with the help of ilactivation energies" E such t h a t : 3 = % e~/kr (31 = 30 e~l/kr; 32 = %e~2/~r).
(14)
A r a t h e r n a r r o w distribution in E m a y thus give rise to a wide dist r i b u t i o n in 3, because k T is a v e r y small q u a n t i t y at room t e m p e r a ture. Moreover, (14) shows t h a t t h e relaxation time has a lower limit z 0 as required b y 112). Finally, if we substitute (14) into (12) we o b t a i n the following distribution function in E :
g(E) dE = dE/(E 2 - El) for E 1 < E < E2~ / g(E) dE = 0 outside t h a t interval,
(12a)
where E 2 and E t a r e the energies corresponding to 32 and 3 I. 31 = ~0 if E,. = 0 ; there is no highest limit for 32 b u t as one would not e x p e c t e x t r e m e l y large a c t i v a t i o n energies, it is unlikely t h a t 32 = co (this would m e a n a ~iolation of condition a) for F(/)). I t m a y be, however, t h a t 32 has to be m e a s u r e d in minutes or even hours. O b v i o u s l y a n y function g(E) which has a width (E 2 - - E l ) , a r e a s o n a b l y flat p a r t for E l < E < E 2 and a more or less rapid decrease for E < E 1 and E 3> E 2 will produce the same results *). If the e x p e r i m e n t a l results indicate a slightly different d e p e n d e n c e of the noise u p o n frequency, one has to alter the distribution function g(E) dE accordingly.
3. Types o/noi.~e in a semi-con;t,,'tor. The following types of noise m a y occur in a semi-conductor: *) The whole d i s c u s s i o n is v e r y s i m i l a r to the discussion on the p o w e r factor ot solid d i e l e c t r i c s s). I f the d i s t r i b u t i o n f u n c t i o n (12) is i n t r o d u c e d i n t o (10), t a n ~ b e c o m e s i n d e p e n d e n t of f r e q u e n c y in the region I/T~ < co < I[T~. T h i s does n o t m e a n t h a t the s a m e mechanism is i n v o l v e d in the t w o eases, i t o n l y m e a n s t h a t t h e s a m e methods c a n b e applied.
364
A. VAN D E R Z I E L
a) T h e r m a l noise. Due to fluctuations in the velocity of the e l e c t r o n s in the senti-conductor the net current t h r o u g h a n y cross-section is not zero b u t will show r a n d o m fluctuations. T h e F o u r i e r c o m p o n e n t s of this fluctuating current are given b y N yq u i s t's t h e o r e m 8): _
4kTA/
R
h~
k T " (e h m r
I)--1
(15)
F o r metallic conductors (except for v e r y thin m e t a l layers) this is the only t y p e of noise present, for semi-conductors this is the case only for zero d.c. current. b) S h o t noise. As the n u m b e r of free electrons in the semic o n d u c t o r shows r a n d o m fluctuations (the electrons can either exist in the b o u n d or in the free s t a t e ; the n u m b e r of electrons in b o t h states are in t h e r m a l equilibrium), an excess a m o u n t of noise o v e r and above t h e r m a l noise will occur if the semi-conductor is c a r r y i n g a d.c. current I. This effect is negligible for metal wires, e x c e p t perhaps v e r y thin ones, b u t might be quite i m p o r t a n t for semiconductors. Assuming t h a t the creation of a free electron in the s e m i - c o n d u c t o r is an i n d e p e n d e n t e v e n t occuring at random, one can follow three m e t h o d s of calculation of the Fourier c o m p o n e n t s of this noise current : l) F r o m the velocity distribution function of the electrons as it is modified b y the d.c. current (B r i 11 o u i n J0)). 2) F r o m the fact t h a t the fluctuation in the n u m b e r of free electrons causes a fluctuation in the resistance (B e r n a m o n t 1)). 3) The motion of the free electrons between their creation a n d their subsequent c a p t u r e gives rise to current pulses (G i s o 1 f 11)). The first two m e t h o d s give identical results; t h a t this also is the case for the third m e t h o d is shown in the Appendix. Assuming t h a t all free electrons have the same life-time ~1, we obtain b v G i s o I f's method :
= 2elzJ/'(z;l~2)"
Fsin/ L
1/2/] 2
(o~zt/2)
J
(16)
(compare Appendix); e denotes the electronic charge, r 2 is the t i m e which it will take a free electron to drift from the negative e l e c t r o d e
ON THE NOISE SPECTRA
OF SEMI-CONDUCTOR
NOISE
365
to the positive one. T h e above result holds for 31 ~ % if 31 > 32 we have instead:
r si~ < ' 3 : ) ] ~. i2 -- 2:-M/L (w32/2) / '
(16a)
in between a transition occurs from (1 6) to (1 6a). 32 depends upon the d.c. current I so t h a t in (16) the noise is in fact proportional to 12 (compare Appendix). (16) and (16a) show t h a t the u p p e r limit for this t y p e of noise is just the true shot noise of a s a t u r a t e d diode carrying a d.c. current f ~). It seems t h a t in m a n y cases the a m o u n t of noise which is actually measured is far above this limit, so t h a t this mechanism c a n n o t explain all excess noise. T h e exact a m o u n t of noise depends u p o n the ratio (31/32). For a c r y s t a l diode 32 is the time which is necessary to cross the barrier layer, this time is so small t h a t 31 >~ 32, so t h a t full shot effect occurs *). Note t h a t z~ is tile life-time of the free electron, which will in general be n m c h larger t h a n the time between two collisions with the lattice. There has b e e n quite a discussion in recent literature about the question w h e t h e r or not t h e r m a l noise and shot noise have the same n a t u r e and origin. Those who gave an affirmative answer to this question were misled b y the fact t h a t certain models of a conductor, which were showing true shot effect, gave N y q u i s t's formula. B u t as N y q u i s t ' s result is i n d e p e n d e n t of a n y model (as it follows directly from the second law of t h e r m o d y n a m i c s ) it is comp l e t e l y false to conclude from it t h a t t h e r m a l noise and shot noise h a v e the same origin. We see now t h a t shot noise in diodes is equivalent to a certain t y p e of excess noise in semi-conductors. :) F l i c k e r effect. F o r the noise of t y p e b ) it was assumed t h a t the creation of a free electron was an i n d e p e n d e n t event occuring at random. As the free electrons originate from i m p u r i t y centres in the semi-conductor, one would expect this to be the case if the i m p u r i t y atoms occur as isolated atoms in the lattice. If, however, these impurities occur in clusters (e.g. at the grain boundaries in p o l y - c r y s t a l l i n e substances) an e x t r a a m o u n t of noise will be generated, especially at low frequencies. For let 33 be the average time b e t w e e n the delivery of two subsequent free electrons f:'om the same *) I n c r y s t a l d i o d e s excess noise is the e n o r m o u s a m o u n t of noise o v e r a n d a b o v e t h e r m a l noise a n d full s h o t effect.
366
A. VAN DER ZlEL
cluster. For high frequencies such that a~z3 >/~ 1 this will hardly have any effect, as the electrons still appear to be delivered in single events occuring a t random. But for low frequencies such that coz3 <~ 1 it is just as if the cluster delivers a number of electrons sinmltaneously so that (16) and (16a) have to be multiplied b y a large factor which increases with decreasing frequency. As the clusters will decay b y a diffusion process one would expect that the noise would become independent of frequency if coz4 <~. 1, where ~4 is the average life-time of the cluster. As diffusion is a rather slow process one would expect rather large time constants here. As this effect is similar to the flicker effect of cathodes in radio-tubes s), we refer to the next section where a fuller discussion will be given. d) L o c a l fluctuations in conductivity. These m a y occur due to the diffusion of foreign atoms or lattice distortions through the lattice (place exchange). Due to this effect the electrons will move in a rather slowly fluctuating electric field and this is equivalent to a fluctuating conductivity. This mechanism m a y be a very important one in many cases. One would again expect rather long time constants for this process. It is interesting to mention that it was suggested recently that the excess noise in crystaldiodes might be due to fluctuations in the contact potential due to the motion of ions in the contact surface. This is the same phenomenon as discussed here ~). e) S p o n t a n e o u s fluctuations in temperature. These fluctuations in temperature 13) give rise to spontaneous fluctuations in resistance; if the conductor carries a d.c. current these resistance fluctuations give in turn rise to a noise e.m.f.. Though this effect is certainly present, it seems to be unable to account for the enormous amounts of noise observed *.) Moreover, though it accounts for the long time constants observed, it will give rise to a l//2-1aw at higher frequencies as there is only a single time constant involved in the mechanism. It nfight be though that fluctuations of local origin will give more noise. Mechanisms b), c), d) and e) all give excess noise over and above *) T h e s a m e e f f e c t s h o u l d c o n t r i b u t e t o t h e flicker e f f e c t in d i o d e s w i t h t h o r i a t e d t u n g s t e n or t u n g s t e n f i l a m e n t s . I t c a n be s h o w n , h o w e v e r , t h a t t h i s e f f e c t is u s u a l l y m u c h s m a l l e r t h a n t h e s h o t noise of t h e d i o d e , w h e r e a s t h e flicker e f f e c t is m u c h l a r g e r t h a n s h o t noise a t l o w f r e q u e n c i e s .
ON T H E N O I S E S P E C T R A OF S E M I - C O N D U C T O R N O I S E
367
t h e r m a l noise which is proportional to 12. Experiments show a noise spectrum which varies as 1/[ down to very low frequencies. This requires a wide distribution of correlation times, including rather large ones. Mechanism b) gives rise to a noise spectrum which is constant for cot I ~ 1 and becomes negligible for o~T1 ~ 1. As vt £ To and To is of the order of 10 -4 sec. (compare Appendix), this mechanism cannot cause the bulk of the noise at low frequencies is), though it m a y play a part at h.f.. It would not be difficult to account for a wide distribution in correlation times in this mechanism. Mechanisms c) and d) have in common t h a t t h e y m a y give rise to a wide distribution of correlation times including rather long ones. For the atoms which take part in the diffusion processes will not all have the same activation energy and a large n u m b e r of different processes are possible, so t h a t one would indeed expect such a distribution.
4. Flicker e//ect o~ electron emitting cathodes. S c h o t t k y ~) assumed t h a t the emission depended upon the n u m b e r of foreign atoms present on the emitting surface. As this n u m b e r of foreign atoms can decay b y a diffusion process or b y an evaporation process, one would expect the decay to be an exponential one. This leads to a 1/./2-law at high frequencies, whereas experiments on oxide-coated cathodes indicate a 1//-law *). M a c f a r 1 a n e 14) recently proposed a t h e o r y of flicker effect, based upon S p r o u 1 l's 15) t h e o r y of cathode emission, which can explain the 1//-law. It will be shown below t h a t M a c f a r I a n e's m e t h o d is formally equivalent to the general t h e o r y given in section 2. M a c f a r 1 a n e assumes t h a t the emission occurs f rom small "active specks" which can decay in two ways: a) b y a diffusion process as in S c h o t t k y's theory, b) by drawing current from the speck. Small specks can decay b o t h ways but large specks m a i n l y decay b y process b) according to S p r o u 1 l's theory. Large specks decay m u c h faster t h a n small specks; the larger the speck, the more vio*) Some ex periments on t u n g s t e n and on thoriated tungsten cathodes seem t o indicate a l//2-1aw (compare e.g. S c h o t t k y's paper). I t would be interesting to settle this question definitely by further experiments.
36~
A. VAN DER ZIEL
lently it decays. M a c f a r l a n e t h e n introduces a Gaussian distribution in the size of the specks. If the distribution were a narrow one, all specks would decay at the same rate and S c h o t tk y' s original 1//Z-law would have been obtained ; a wide distribution in size, however, gives a 1//-law at high frequencies as required by experiments. The success is due tot he introduction of a wide distribution in the size of the specks. As the large specks decay so much faster t h a n the small specks, this is equivalent to the introduction of a very wide distribution in the correlation times as presented in section 2 *). M a c f a r I a n e's t h e o r y gives a macroscopic description of the p h e n o m e n o n ; if the problem is investigated from a microscopic point of view, the picture is as follows : The atoms in an active speck do not all have the same work-function, the more neighbours of the right kind an atom has, the lower the work-function will be and thus the larger the probability of decay b y emission of an electron. Moreover, not all atoms of the active speck will have the same probability of diffusion; a distribution in " a c t i v a t i o n energies" is v e r y likely. Therefore, even if M a c f a r 1 a n e's t h e o r y might not be correct in all its details, the actual mechanism will give rise to a wide distribution of correlation times and t h a t is all t h a t is required for the
1//-law. M a c f a r 1 a n e 14) also applied his t h e o r y of flicker effect to the problem of excess noise in semi-conductors. As shown in section 3, this is a very reasonable step. But it is even more reasonable to a p p l y the t h e o r y of excess noise in semi-conductors to the flicker effect of oxide-coated cathodes. For t h o u g h M a c f a r 1 a n e's t h e o r y is u n d o u b t e d l y right for tungsten and thoriated tungsten cathodes (though one might question whether a narrow or wide distribution in the size of the specks has to be introduced there) it is doubtful whether it is the only mechanism in the flicker effect of *) T h e t h e o r y m e a n s a d i s t r i b u t i o n of "r o v e r the i n t e r v a l T, < r < T~ w i t h r~ = 0. M o r e o v e r , n o h.f. l i m i t for the l / l - l a w is f o u n d , w h i c h s e e m s to v i o l a t e c o n d i t i o n b) of s e c t i o n 2. for F([). B o t h r e s u l t s are d u e to a n i d e a l i z a t i o n of the a c t u a l p h e n o m e n o n , w h i c h d o e s n o t o c c u r ill p r a c t i c e . T h e i d e a l i z a t i o n is in t h e d i s t r i b u t i o n futlction of the size of t h e s p e c k s . F o r l a r g e sizes of t h e s p e c k s , t h e n u m b e r of s p e c k s is c e r t a i n l y less t h a n i n d i c a t e d b y a G a u s s i a n d i s t r i b u t i o t h as it is ceLtain t h a t e v e n tile l a r g e s t s p e c k s will be v e r y s m a l l in c o m p a r i s o n to the c a t h o d e a r e a . T h e i n f i n i t e l y s m a l l n u m b e r of i n f i n i t e l y l a r g e s p e c k s , g i v e n b y t h e G a u s s i a n d i s t r i b u t i o n , d e c a y so infinitel3." f a s t t h a t t h e y m a k e I(l) ~ i n f i n i t e l y large.
ON THE NOISE SPECTRA OF SEMI-CONDUCTOR
NOISE
369
oxide cathodes. For the theory pays its main attention to the s u r / a c e of the emitting layer. But it has been found recently 16) that the emission current for an oxide cathode is determined by the conductivity of the coating itself. As the oxide coating is a semi-conductor, its conductivity will show spontaneous fluctuations and it is very reasonable to assume that these fluctuations will show up as fluctuations of the emission current. The measurements indicate that the emission current I is proportional to the conductivity a of the coating, so that I / a is a constant. This means that if R is the resistance of the coating, then the voltage V o - - I R = I d / ( A a ) across the coating is independent of:the emission current. It is reasonable to assume that V0 does not fluctuate, so that one would expect for the current fluctuation AI: ,41 = - - 1 , 4 R / R
or ,4/~ = ( I / R ) 2. (,4R) 2
(17)
As the excess noise in semi-conductors gives rise to a l//-law, it is reasonable to expect the same result for oxide coated cathodes. The above theory offers an explanation for the fact that there is a distinct difference between the flicker effect in oxide-coated cathodes and in tungsten or thoriated tungsten cathodes (quite apart from the possible difference in spectral distribution mentioned before). The measurements seem to indicate that for equal emitting area the oxide-coated cathodes have a much larger flicker effect ; this suggests a different mechanism and that is what the above theory offers *). This does not suggest that the theory of noise in semi-conductors can be applied to the flicker effect of oxide-coated cathodes without any alteration; the above suggestion offers only a new mechanism and it n fight be worth while to develop it further. C o n c l u s i o n s . The discussion in the last two sections shows clearly that there is good evidence in favour of a wide distribution in correlation times and that it is very likely that such a wide distribution *) In a d d i t i o n it offers a v e r y s i m p l e e x p l a n a t i o n for the following p h e n o m e n o n . In a t u b e c o n s i s t i n g of two parallel hot c a t h o d e s h a v i n g a s m a l l d i s t a n c e d and b e i n g k e p t at a u n i f o r m high t e m p e r a t u r e T the flicker effect wil not show up, as the noise is g i v e u b y N y q u i s t's t h e o r e m . As the net c u r r e n t flowing t h r o u g h the t u b e is zero, one would i n d e e d e x p e c t no flicket effect a c c o r d i n g to e q u a t i o n (17). The t h e o r y m i g h t also e x p l a i n the s u p p r e s s i o n of flicker effect iT) which is observed w h e n the diode is s p a c e - c h a r g e d l i m i t e d in steacl of s a t u r a t e d , b u t it is b e y o n d the scope of this p a p e r to d e v e l o p this theory. Physica XVI
24
370
A. VAN DER ZIEL
function can explain the 1//-law. Moreover, it has been shown t h a t it is difficult to separate the various components of excess noise in semi-conductors and oxide-coated cathodes and t h a t it is practically impossible to find out where the fluctuations in resistance end a n d the true flicker effect begins. The author is indebted to Dr. K. S. K n o 1, N.V. Philips' Gloeilampenfabrieken, Eindhoven, The Netherlands and to Dr. A. J. D e k k e r, University of British Columbia, Vancouver, Canada, for stimulating discussions about the above subject and to the Defence Research Board of Canada for a grant in aid and for permission to publish these results. It is a great pleasure, for the a u t h o r to publish this paper as a tribute to Prof. C o s t e r.
Appendix. Proo/ o/ the equivalence o/ Gisol/'s and Bernamont's methods. Let L be the length of a semi-conducting bar having metal electrodes at its ends, let R be the resistance of the bar and I the d.c. current flowing t h r o u g h it, t h e n the d.c. voltage V = IR. If e is the electron charge and u the electron mobility, then the drift velocity is Fu, where F = V/L = I R / L is the electric field strength. R is given as : R -- L2/(euNo), (18) where N Ois the total n u m b e r of electrons in the conductor. As R is inversely proportional to N o we have for fluctuations A N o of No:
,dR----R,dNo/N o or AR 2 ---- (R2/N~0),dN~0;
(19)
the noise e.m.f. AE in series with R is I,dR and the fluctuating current A I flowing through R is IAR/R. Its mean square value is :
,dI 2 = (P/No). (AN~o/No) = I2/No
(20)
for the classical t h e o r y als ,dN~0 =-- N o in this case (this relation holds for semi-conductors but not for metals). As AI(t) 2 is now given b y (20) and as an exponential correlation function with average life z o should be expected, we have, according to (2) and (7), for the Fourier components i of I(t) :
-fi
=
4(I2/No) A/% (1 + W2~o)- '
4 euRF% A/
-- L2(1 + oJ2~o)
(2 !)
This is the noise current according to B e r n a m o n t 1). We now turn to G i s o 1 f's theory. G i s o 1 f 1i) makes the simplifying
ON T H E N O I S E S P E C T R A OF S E M I - C O N D U C T O R N O I S E
371
a s s u m p t i o n t h a t all current pulses have equal length 31 (31 is the time interval between the creation of a free electron a n d its subseq u e n t capture). He t h e m obtains as his final result: Fsin (0~1/2)7 2
-~ = (2euF2i3,/R) A/. L (~---~
J --
Fsin (om/2) J 2
= (2 e # RI2,, A//L2). L ( ~ - ~
(22)
which is G i s o 1 f's formula (20) in slightly different n o t a t i o n (except for a factor 2 which has disappeared in his calculation). If we put 230 = 31, t h e n the two formulae are identical e x c e p t for the f r e q u e n c y dependence (due to the different correlation function used by Gisolf). We shall now prove t h a t 31 a n d 30 h a v e an u p p e r limit 32. F o r as L is the length of the c o n d u c t o r and vg = F,~ = I R u / L t h e drift velocity of the electrons in the direction of the field, it would take an electron an average time 32 = L/v d = L2/(IR,~) to t r a v e l from the one electrode to the o t h e r one if it were not c a p t u r e d before. I n t r o ducing this into (22) we obtain: •7
....
F sin (co31/2)-]~
zl
(23)
This derivation only holds if 31 ~ 32, for otherwise one has to take into account t h a t not all electrons can possibly have the life-time 3~ (those free electrons which are c r e a t e d near the positive electrode arrive at this electrode before t h a t time). If 31 ~> 32, the results become v e r y simple again, as now all electrons e n t e r the semic o n d u c t o r at the negative electrode and leave it at the positive one. We thus obtain in stead of (22): Fsin (o)32/2)1 2
/7 = (2 e I AI) • L ( ~ - ~
J
(23)
which is the formula for true shot effect. It is the merit of O is o 1 f's m e t h o d t h a t it can show the change from excess noise in semi-conductors to true shot effect so nicely. P u t t i n g R = 105 ~2, I = 1 mA, I. = 1 cm and ,~ = 100 cm/sec per unit field strength (Volt/cm), we obtain 30 = 10 -4 sec. Received 26-1-50
372
ON T H E N O I S E S P E C T R A OF S E M I - C O N D U C T O R N O I S E
REFERENCES 1) J. B e r n a m o n t , C . R . , 1 9 8 , 1755 and 2144; 1934;Ann. de Physique T, 71, 1937. 2) H.C. T o r r e y a n d C . A. W h i t m e r , Crystal Rectifiers, Chapt. 6, M c G r a w H i l l Co., New-York, 1948. 3) P . H . M i l l e r J r . , Proc. I. R. E. 3 5 , 2 5 2 , 1947. 4) E. M e y e r andH. Thiede, E . N . T . 1 2 , 2 3 7 , 1935. 5) C . J . C h r i s t e n s e n and G. L. P e a r s o n , B e l l S y s t . Techn. Journ. 15, 197, 1936. 6) W. S c h o t t k y , Phys. Rev. 2 8 , 7 4 , 1926. 7) M.C. W a n g a n d G . E. U h l e n b e c k , Rev. Mod. Phys. 17,326, 1945. 8) M. G e v e r s, Philips Res. Rep. 1, 197, 279, 361, 447, 1946. 9) H. N y q u i s t , Phys. Rev. 32, II0, 1928. I0) L. B r i l l o u i n , Helv. phys. Act. 7 , 4 7 , Supplement, 1934. l l ) J. H. G i s o l f , Physica 15, 825, 1949"). 12) W. S c h o t t k y , Ann. Phys. Lpz., 57, 541, 1918. 13) H . A . v a n d e r V e l d e n , P h . D . Thesis, Utrecht, 1947. 14) G . G . M a c f a r l a n e , Proc. phys. Soc..59, 366, 1947. 15) R . L . S p r o u l l , Phys. Rev. 67, 166, 1946. 16) N . B . H a n n a y , D. M a c . N a i r a n d A . H. W h i t e , Journ. appl. Phys. 20, 669, 1949. 17) W. S c h o t t k y , Physica 4, 175, 1937. The following two papers which are not quoted in this paper are also of great interest for the subject: B. D a v y d o v , J. Phys. U.S.S.R. 4, 335, 1941;B. D a v y d o v andB. Gurevich, J. Phys. U.S.S.R. 7, 138, 1943.
Note added in proo/ Since this paper was written, Mr. W. A. B a i n at the Uuiversity of British Columbia found several cases in which the noise spectrum showed a transition from a 1//-law to a 1/f'-law at higher frequencies. Dr. W. G. S h e p h e r d (U. of Minnesota) told me t h a t similar results have been obtained at the University of Minnesota and at the University of Pennsylvania. Mr. B a i n also found a dependence of the transition frequency upon temperature. Both results are in good accord with the above theory, though Mr. B a i n's results seem to exlude relation (14) between r and T, used in our example in section 2.
*) G i s o 1 f does not seem to have been aware of previous work on this subject. There are a few errors in his duscussion, which can be corrected by following the calculation and the discussion more closely.