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Ambipolar transport of carrier density fluctuations in germanium
Hill, J . E . Van Vhet, K.M. 1958
Physica XXIV 709-720
AMBIPOLAR TRANSPORT OF CARRIER DENSITY FLUCTUATIONS IN GERMANIUM*) Department of Electrical Engineering University of Minnesota, Minneapolis, Minnesota
by J. E. H I L L and K. M. VAN V L I E T Abstract
The spectrum of generation-recombination noise in semi-conductors may differ from the usual form at relatively high field strengths or relatively short drift times of the carriers. In contrast to previous opinions it is shown t h a t such a modification occurs only if the ambipolar drift velocity is sufficiently large. Accordingly, the sweeping of ~ r r i e r s through the sample by the electric field has no effect in material with a sino,~ type of carrier or in intrinsic material, but should show up in near-intrinsic ..... "s. Measurements, made to detect this effect, completely confirmed this behavior. ]~lum the "resonances" in the spectra the ambipolar drift time was determined and found to be in good agreement with the calculated values.
Introduction. The noise observed in high resistivity germanium crystals above ~-~ I00 c/s can entirely be attributed to statistical fluctuations in the rate of generation and recombination of holes and electrons, as has been reported before 1). For sufficiently low concentrations of impurity centers, which m a y occur either in the bulk or at the surface, the fluctuations in both carrier densities are almost completely correlated, i.e. An(t) ~. Ap(t). At sufficiently low electric fields the spectrum of the carrier fluctuations is the same as under thermal equilibrium. For larger field strengths an appreciable fraction of the carriers is carried off at the electrodes before recombination occurs. Under these circumstances the noise spectrum m a y be quite different from the low field spectrum. It is the purpose of this paper to discuss this effect and to present experimental results pertaining to this case. The effect of the carrier drift on the noise was first discussed by D a v y d o v and G u r e v i c h 2). Although not stated specifically, it seems that their analysis was concerned with a conductor, containing only a single type of carriers, e.g. electrons. According to their views, the correlation function, which normally has an exponential form, would be zero for times larger than the drift time ,g of the carriers from one electrode to the other electrode (,~ = = L//~nE). This idea is not correct as was shown before by several investigators 3)4)5). The general argument is that in order to maintain space-
*) Work supported by a U. S. Signal Corps Contract. -- 709 -Physica XXIV
710
J.E.
HILL AND K. M. VAN VLIET
charge neutrality, carriers start to move everywhere in the crystal simultaneously, when at some spot a fluctuation in the generation rate has occurred. If the contacts are ohmic, then electrons will move in at the negative electrode, while others are carried off at the positive electrode. This chain continues until most of the extra electrons have recombined after a time of the order of the lifetime T which m a y be much larger than *a. Consequently the correlation function is not affected b y the value of Ta. The "replenishment" of carriers itself also is a statistical process, since at the negative electrode a large reservoir of electrons is available. Therefore, it has been stated before by V a n V l i e t and B l o k 4) and independently by BOer et al. 5) that these injection-extraction fluctuations should give rise to noise of the order of 2eI per cycle in case that the l i f e t i m e , is much larger than the drift time *a of the carriers. This, however, is inconsistant with the picture just given s), since space-charge neutrality will be restored within a time of the order ~r, where ~r is the dielectric relaxation time. The injection-extraction fluctuations, therefore, will be strongly "sp~cecharge suppressed" and, in fact, will not exist in semiconductors. In insulators, like CdS an effect m a y occur at very low light levels. However, in that case the current flow is space-charge limited and a more drastic change of the noise picture is anticipated. We thus conclude that in a semiconductor with a single type of carriers (the minority carriers m a y be trapped and thus immobilized) the noise spectrum does not change in shape and no additional noise effects occur necessarily if we pass from the condition Ta >~ T to *a ~ *, up to such fields where the electrons become appreciably heated and w~ actually enter the pre-breakdown region. In semiconductors involving both electrons and holes the situation is different; this will be investigated in this paper.
2. Theory o~ /luctuations in mixed semiconductors. A very clear picture of arnbipolar motion of carriers in a semiconductor was recently published by S t ~ c k m a n n 7)s) which we will briefly review. In Figure I we will distinguish between three extreme cases. Let at t = 0 extra carriers be generated somewhere in the sample in a region/lx. If the carriers come from donor levels (Figure l a), the positive charge is fixed. Electrons moving out of the slice Ax to the positive electrode are replenished by others to preserve neutrality. Hence, the region of increased conductivity A~ remains at the same spot; the magnitude o f / t ~ decays, of course, with time. If holes and electrons are generated across the band gap in an n-type crystal, the picture changes (Figure l b). The electrons flowing to the right into the lower conductivity region will not produce an extra electron concentration there, since others flow out "simultaneously" (i.e. within a time of the order of the dielectric relaxation time), thus preserving spacecharge neutrality. The holes flowing to the left, however, are mainly neutra-
A M B I P O L A R T R A N S P O R T OF C A R R I E R D E N S I T Y F L U C T U A T I O N S
71 1
lized b y electrons going in the opposite direction. Thus, in the region to the left of the region which at t ---- 0 had an increased conductivity, the electron flow adjusts itself in such a w a y that at each moment An ~ Ap; the slice of increased conductivity, therefore, moves to the negative electrode with approximately the hole mobility. Finally, in an intrinsic semiconductor, (Figure l c) no shift of the concentration disturbance occurs, since the flow of carriers into the lower conductivity regions b y either carrier can be neutralized b y the same type of carrier. (I) Strongly N-type; transitions from donor levels Bn
Bn
t=0
t>0
(2) N-type; hole-electron pair transitions An~ &p Anj Ap
t=O
t>O
(3) Intrinsic ~n,~p
&n, Ap
HI"
tfO
t>O
F i g . 1. C a r r i e r c o n c e n t r a t i o n d i s t u r b a n c e i n : a) a n e x t r i n s i c s e m i c o n d u c t o r i n v o l v i n g d o n o r - c o n d u c t i o n b a n d t r a n s i t i o n s ; b) a n n - t y p e s e m i c o n d u c t o r n o t t o o f a x f r o m i n t r i n s i c s o t h a t h o l e e l e c t r o n p a i r s a r e g e n e r a t e d a c r o s s t h e b a n d g a p ; c) a n i n t r i n s i c c r y s t a l . ( c o m p a r e r e f e r e n c e 7).
This behavior is more rigorously described with the ambipolar transport equation for carriers as given b y v a n R o o s b r o e c k 9) which results from the continuity equations for holes and electrons separately under elimination of the term div E:
O(Ap)/Ot = -- qff= E . V(Ap) -- Ap/.r + D J 2 ( A p ) + g(r, t);
(1)
here . is the lifetime, E the electric field, /za and D= are the ambipolar mobility and diffusion coefficient respectively and g(r, t) is a stochastic source term. The quantities F= and D= are given as *)
#a =
(no -- Po)~n#~ ; #~no + #~Po
Da
=
(no + Po) D~Dn noDs + poD~
(2)
• ) Note t h a t /za and Da are n o t related b y the Einstein relation; other n a m e s for/~= are "group mobility" or "signal mobility". Expressions for/Za and Da in the ease t h a t the dielectric relaxation time is not small compared to "c have been given b y K e i 1 s o n 10) and by S t 8 c k m a n n s).
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j.E.
H I L L AND K. M. VAN V L I E T
In strongly n-type material # a s s / ~ , for strongly p-type material/~a ~ ~ #n and in intrinsic material #a = 0 in accordance with Figure 1. The general solution of (1) is lengthy if we apply proper boundary conditions at the surface. Recently, a solution of (1) neglecting ambipolar effects was obtained 11). Here we shall neglect the diffusion telm since we mainly want to point out the effect of concentration shifts on the noise. We come back to this at the elid of this section. Assuming that the field is in the positive X-direction equation (1) reduces to 0(Ap)/at + q~a E 0(Ap)/0x = -- Ap/~ + g(x, t). (3) If recombination occurs mainly at the surface, t h e n , stands for the dominant surface transient time constant. We will be interested in the fluctuations in the total hole density given b y
AP(t) = foL Ap(x, t) dx.
(4)
R constant current source
C C.
Fig. 2. Geometry of crystals used in the measurements. The limits in the integral depend on the crystal geometry. Let us consider a bridge type sample as was actually used, Figure 2. The noise is measured between probes. Let us assume that the current is driven b y a perfect constant current generator. The voltage fluctuations between the probes are then given by A V 2 = I z AR 9., where R is the resistance between the probes. Hence, the integral in equation (4) extends only over the region (0 - - L ) between the probes. Although in the arrangement of Fig. 2 the noise is physically represented by a voltage generator, it is convenient to also introduce the equivalent short-circuited current generator for the noise between the probes,/lIss ~ = A V2/R 9". Then we find for the relation between /fiBs z and Ap2, or also for the relation between the spectral densities SI([) and S~([) of these respective quantities Sx(]) = Iz [(b + 1)/(bNo + P0)] 2 Sp(]),
(5)
where b =/.tn/#,~, and where No and Po are the total numbers of electrons and holes in the region between the probes.
A M B I P O L A R T R A N S P O R T OF C A R R I E R D E N S I T Y F L U C T U A T I O N S
713
T o find S~(]) we first consider the effect on the noise of fluctuations' generated at t = 0 in a slice between x' and x' + dx'. When we average over an ensemble for which Ap(x', O) = ,4po, the stochastic term drops out in equation (3). The remaining partial differential equation has the general solution p(x, 0 = e
-
(6)
where ~5 (y) is an arbitrary function. We have the condition that/tp(x, 0) = = 6(x -- x') Apodx', where $(x -- x') equals 1/dx' inside the interval (x', x' + dx') and zero outside; it represents a Dirac-delta function in the limit dx' ~ 0. This gives
lip(x, t) -~ Apo e -t/* 6(x --/~a, Et -- x')dx'.
(6a)
This is an obvious result, since the concentration disturbance progresses with the velocity #aE as pointed out in Figure 1. If we take into account diffusion, the concentration disturbance spreads out and the 6-function is replaced b y a more gaussian type of function, as can be found in the same way. Integrating (6a) with respect to x yields.
AP(t) = Apodx' e-t~* if 0 _< /~a,Et + x' < L AP(t) = 0
elsewhere
(7)
The inequalities in equation (7) limit the correlation time in an n-type sample (#a positive) to t <_ (L -- x')/l~E; in a p-type sample (/~a negative) the limit is given b y t ~ -- x'/#aE. These differences correspond to concentration shifts in the positive and negative X-direction respectively. In what follows we shall assume first that the crystal is n-type. Then we obtain for the noise due to the sources between x' and x' + dx', applying the WienerKhintchine theorem: dSp(/) = 4(Apodx')2foCL-~)l~"~ exp (-- u/~) cos cou du.
(8)
The fluctuations in any part dx' are caused b y generation and recombination and b y ambipolar diffusion (see end of this section). Both processes actually mean a transfer of conduction electrons to valence band electrons. Regardiess of which effect prevails, the total variance following from Fermi statistics is: (Ap0dx'F =
porto dx'/(no + Po).
(9)
We can now sum up the contributions from different slices dx'. The sources are statistically independent (at the same instant) if dx' is large compared to the scattering mean free path. As a good approximation it is, however, permissible to integrate (8), which yields 4pon0 f ~ f Ir'-x'll~'"~ Sp(/) = "no + Po dx' J0 exp (--u/-r) cos cou du
(10a)
714
J . E . HILL AND K. M. VAN VLIET
Changing the order of integration of this double integral yields: S p(/) 4 porto [ r4v,~ [ r,-v,.E~, = du dx' e-U~* cos oau du no +
Po Jo
= 4 ponoL no + Po where
fo
Jo
[e-ul* (1 -- U/ra)] cos cou du,
(lOb)
L L(#nno + I~po) -(11) ltZa[E Ino - - P01 t z n l ~ E is the ambipolar drift time. The absolute marks have been added to make the expression also valid for p-type semiconductors as is easily checked by repeating the above derivation for that case. The correlation function given by [] in (10b) is exactly of the form suggested before except for the fact that now the appropriate lifetime Ta enters into the expression. In semiconductors with a single type of carriers or in intrinsic semiconductors ra = oo and the correlation function is always exponential. The spectral density is obtained from (10a) or (10b). The result of the integration is found ill a paper by B u r g e s s lZ); in our case this yields: Sp(/) = (4ponoL'ca/(no + Po))W(cor, (.O'ra) (12) where 02/¢2 exp (--¢/0) W[O, ¢) ---- ~1 + 0 t ~ -------~ 1+ 0 + 1 + 09. [(1--02) cos¢--20sin¢]}. (12a) ~'a - -
-
t~
-
1--02
The low frequency value of W is given by W(O, O) -----('c/'ra)2 (e-'°1. -- 1) + "c/'ra
(13)
Introducing the normalized frequency response F(w'r, coTa) = W(coT, oa'Ca)/W(O, 0 ) w e find from (12), (13), (11)and (5):
SI(/) = 4qI
[~a .2 ] [ ~" + (e- * ' 1 . - 1) ~
b+2+ 1/b ]F(cor, COra)(14) [no/Po-- po/no]
For low field strengths or high values of ~a this expression reduces to a spectrum of the well-known form v/(1 -1- co2"r2). If Ta = r, the deviations are already noticeable; in that case
7
F" b+2W._l/b Sz(/) = (4/e) qI Llno/Po _ po/nolA
([ ~ ~o2~
Finally, in the case that *a ~ • we have F(O, ¢) m 2¢-2(1 -- cos ¢); hence S~r(/)=2qi I b + 2 + l / b ][sin½co-ra] 2 Ino/Po -- po/nol ½co'ra "
].(15)
(16)
Before proceeding we like to point out the connection with other methods employed before. D a v y d o v and G u r e v i c h 2) in their original derivation
AMBIPOLAR TRANSPORT OF CARRIER DENSITY FLUCTUATIONS
715
used the method of addition of elementary current pulses. This method in its simplest form (application of Campbell's or similar theorems) breaks down since the pulses are not independent. This affects the magnitude of the noise. The frequency dependence should be correct if the method is properly applied. According to Figure 1, one could consider as an elementary event the voltage pulse due to a hole-electron pair drifting with the ambipolar velocity #a~. On doing so, one has to take into account the statistical distribution of the lengths of the pulses and the statistical distribution of the origin of the pulses (x' in our notation), noting that the maximum drift length is L - x'. D a v y d o v and G u r e v i c h , although not.dealing with ambipolar transport, carried out this program correctly; accordingly, their frequency dependence is the same as in our case (equation 12a) if we replace their single carrier mobility # by #a. It is curious that in later versions of the "pulse addition method" different results were obtained since either the statistical distribution of the lengths of the pulses was ignored 13) or the difference in drift times for carriers generated at different spots x' was not taken into account 14). The above analysis, which is a collective carrier approach, is simpler because no information is needed about the behavior of the individual carriers. Moreover, it is better justified since the application of the charge neutrality implications to single carriers is highly speculative. I
s(f)
P 162
tO
I0
to~
Frequency (Kc.)
Fig. 3. Plot of equation (16).
We now return to the results given by equations (I 5) and (16). The spectrum of the noise, predicted by (16) is plotted in Figure 3. The striking fact is the appearance of "resonances" at high frequencies. A spectrum like this will never be measured as such, however, since the spectrum analyzer
716
j.E.
HILL AND K. M. VAN VLIET
averages o v e r a certain b a n d w i t h . I n Figure 4, c u r v e I, we p l o t t e d the result t h a t would be m e a s u r e d b y a s p e c t r u m a n a l y z e r with a b a n d w i d t h t h a t is 10% of the center frequency. I t is seen t h a t the higher order peaks average out due to the fact t h a t the b a n d w i d t h increases with increasing c e n t e r f r e q u e n c y of the band. [This is, of course, not necessary if f r e q u e n c y conversion involving n a r r o w low f r e q u e n c y filters is e m p l o y e d b u t usually s p e c t r u m analyzers below 500 kc do not operate this way.] Also p l o t t e d in Figure 4 is the s p e c t r u m o b t a i n e d f r o m e q u a t i o n (15) for r = T=. E v i d e n t l y , the peaks occur at n e a r l y the same frequencies, b u t are m u c h less pronounced. F o r o t h e r values of , / T a all varieties b e t w e e n the two curves in F i g u r e 4 are possible.
16 ~ o z
I: ~'=oo; integrated over 10% bandwidth
X: 1: =1:a
.N_ 0
E
o z
iff 2
~ff3
I
i
0.I
I
A
I I0
Fig. 4. Calculated noise according to equation (16) and equation (15). Curve I: T = co; spectrum of the noise integrated over a bandwidth B = 0.1 f. Curve II: v = va = 20/~ sec. W e shall now point out some limitations of the simple a p p r o a c h given above. Results different from the above ones m a y be o b t a i n e d if the c r y s t a l c u r r e n t is not driven b y a c o n s t a n t c u r r e n t generator. In t h a t case correlation effects occur between the various regions of the crystal as were observed b y M o n t g o m e r y 15). These effects can be c a l c u l a t e d in the same w a y as was done above for the noise. F i n a l l y we shall indicate w h a t modifications follow from inclusion of the diffusion term. L e t us first consider the case t h a t t h e noise is m e a s u r e d between probes which are not close to the e n d . contacts. T h u s end-effects can be ignored. T h e solution of e q u a t i o n (1) is now i n s t e a d of (6a) : Ap(x,
t)
A p ( x ' , O) d x ' (4~Dat)½ exp
r
t
L
r
(x - -
x ' - - #=Et)2 7 4Dat .] "
(17)
AMBIPOLAR TRANSPORT OF CARRIER DENSITY FLUCTUATIONS
717
The maximum spread of a disturbance, located at t = 0 at exaqtly x' is ,-~ (Da(L -- x')/po, E)~ < (DaL/l~o/E)t. In strongly extrinsic material we find from the Einstein relations that this spread is small compared to the distance between the probes if E L >> kT/q. This is always the case and consequently no influence of the diffusion is expected in extrinsic material. In intrinsic material the diffusion can have an appreciable effect. The noise resulting from (17) can be found in the same manner as before, or, in a more sophisticated approach, with the Smoluchowski method of probability-after effects. This results in: Sl(/)
x
=
412( no )(bpo+Po) Po no + Po bno + Po
foodu f; cLvf
dx'
2
cos wu L(4~Dau)~ exp
×
[
u r
4£)-au
a " (18)
For E sufficiently large, this expression should give the same result as equation (14). In an intrinsic semiconductor with/za = 0 and Dar >~ L 2 (18) reduces to a MacFarlane type spectrum. In such a semiconductor the response time is determined b y diffusion rather than b y recombination, analogous to a case discussed b y R i t t n e r 16). It is also possible to find the appropriate solution if we apply the boundary conditions Ap = An = 0 at the end-electrode contacts. The result can be readily found employing a general analysis given recently b y V a n V l i e t and V a n d e r Z i e l 11). 3. Experimental results. Some efforts have been made to measure ambipolar fluctuations as predicted in the previous section. The effect has not been found before for several reasons. First, the condition *a < r is not easily satisfied. Only a small temperature region in the near-intrinsic range will be adequate; also the heat dissipation becomes excessive in order to meet this requirement for small lifetime samples. Moreover, the measurements should be done with a spectrum analyzer with a more or less continuously variable frequency band. This is clear from the calculated spectra in Figure 4. Therefore, measurements were made from 1--500 kc in steps of about 5 kc in the region where the effects were expected. For the available crystals the lifetime was, .after etching, of the order of 20 # sec. Larger samples might have had a larger T, b u t they have the disadvantage of a smaller resistance. At temperatures slightly above room temperature the noise was large enough to guarantee accurate measurements and currents for which Ta was equal or slightly less than r could be passed. The measured noise for a p-type crystal is shown in Figure 5, together with the calculated result, curve II of Figure 4. As is seen, both the magnitude of the noise as predicted b y equation (15) and the form of the spectrum agree very reasonably. The best fit is obtained for ra = 20/~ sec which is
718
J. E. HILL AND K. M. VAN VLIET
also within experimental error equal to the value calculated according to equation (2) from the measured resistivity of the crystal. The results were repeated for n-type germanium, see Figure 6.
p-Ge
]
~,
T=75°C Exp.results, result ~\ - - - -.-- Theoreficol Theo "~ ='~a=20~tse¢ ~b,~ for f°r''~
l/n°'se-'v"i
0.'~ - - ,hermo,-- ~ I
I0 Frequency
''
~-~
(Kc.)
K~
Fig. 5. Noise of a p-type germanium crystal with T= = 20/~ sec at T = 75°C (solid curve experimental, dotted curve calculated). The noise is represented by Ieq, defined by Sz = 2eleq. The crystal was in the near-intrinsic region (52°C; no ~ 30 × 1018/cm a P0 ~ 7 x 1013/cm3). The lifetime was ~ 25 # sec. The noise was measured in steps of 5 kc up to 400 kc for currents from 2 mA -- 17 mA. These values are such that for low currents ~= > ~, whereas for the highest currents ambipolar effects should become noticeable. This is very clearly demonstrated in Figure 6, which shows three minima for the highest currents. Tile frequency values are given in the Table I. The minima shift to higher frequencies for larger currents as is expected from the theory. It m a y also be noted, that the 1/[ noise content in Figure 5 increases as the current increases. This increase is more than expected from the difference in current dependence; it can be easily understood as follows. At low fields the noise, produced b y the electrodes does not show up, because of the probe method. [It was found that without probes, this noise rises b y a factor 10.] If the ambipolar drift time becomes sufficiently short, however, the probes loose their function, since the contact fluctuations are swept into the region between the probes. Hence, simultaneously with the occurrence of ambipolar effects, the 1//noise level rises. The fact tha~/the 1/[ noise does not extend beyond 10 kc but in fact, changes to 1/] 3 indicates that the shortest lifetime present in 1/[ noise (ill these crystals) is the lifetime of the minority carriers. This is to be expected from several of the l I~ models which have been
A M B I P O L A R T R A N S P O R T OF C A R R I E R D E N S I T Y F L U C T U A T I O N S
7 ]9
proposed. It indicates that 1/[ noise (caused b y "slow" surface states) modulates the generation and recombination rate of the carriers in the bulk or at the surface ("fast" surface states).
."- ooo
Id:
-,-
,o
N-Ge-I
\
%oo
4
~
,o
,~
Frequency (cycles) F i g . 6. N o i s e of a n n - t y p e g e r m a n i u m c r y s t a l i n t h e n e a r - i n t r i n s i c r e g i o n for various current strengths. TABLE I
I
T
17 ma lO
25/~ sec 25 25 25 25 25
8 6 4 2
Ta 12 ~ sec 20 25 33 5O I I00
calc. 1st rnim 85 kc 50 40 30 20 I0
I
calc. 2nd min.
obs. 1st rain.
obs. 2nd min.
170 kc I00 80 60 40
80 kc 55 45 ~35 ,~,15
170 kc 125 II0 80
2o
Finally, the noise was measured for the same crystal in the intrinsic region at fields sufficiently high, so that T was larger than the drift times for electrons and holes separately. The same number of frequencies was measured as before. The result is shown in Figure 7. Evidently the noise is of the form ~/(1 + ~o2~2) and no resonances occur, in accordance with the theory.
720
AMBIPOLAR TRANSPORT OF CARRIER DENSITY FLUCTUATIONS
I0
N-Ge-I
"~ '~t x
I=10 mo. R=1500~ T=80°C
o
E
t~a
t
I0
Io2 .Frequency (Kc.)
Fig. 7. Noise of t h e s a m e c r y s t a l as in F i g u r e 6, n o w a t a t e m p e r a t u r e of T = 80°C, a t w h i c h t e m p e r a t u r e t h e c r y s t a l is intrinsic.
Acknowledgements. We wish to thank Mr. H. C. M o n t g o m e r y of Bell Telephone Laboratories for providing us the bridge shaped crystals and for correspondence and interesting discussions about this subject. We are also indebted to Dr. A. V a n d e r Ziel for improvements in the manuscript. Received 24-4-58. REFERENCES 1) Hill, J. E. and Van Vliet, K. M., J. appl. Phys. 29 (1958) 177. 2) D a v y d o v , B. and G u r e v i e h , B., J. Phys. USSR 7 (1943) 138. 3) Rose, A., in "Photoconductivity Conference" Ed. Breckenridge, e~ al. J o h n Wiley, (1954), 34-44. 4) V a n Vliet, K. M. and Blok, J., Physica 22 (1956) 231. 5) Boer, K. W., et alii., Ann. Physik 17 (1956) 344. 6) M o n t g o m e r y , H. C., Private Communication. 7) S t 6 c k m a n n , F., Z. Physik 147 (1957) 544. 8) S t 6 c k m a n n , F., in "Photoconductivity Conference", Ed. B r e c k e n r i d g e , et al. J o h n W i l e y (1954). 9) Van R o o s b r o e c k , W., Phys. Rev. 91 (1953) 282. 10) K e i l s o n , J., J. appl. Phys. 24 (1953) 1198. 11) Van Vliet, K. M. and V a n der Ziel, A., Physica, 24 (1958) 415. 12) B u r g e s s , R. E., Brit. J. appl. Phys. 6 (1955) 185. 13) Gisolf, J. H., Physica 15 (19491 825. 14) Boer, K. W., Ann. Physik 14 (1954) 87; also 15 (1954) 55. 15) M o n t g o m e r y , H. C., Bell Syst. techn. J. 31 (1952) 950. 16) R i t t n e r , E. S., in "Photoconductivity Conference" Ed. B r e c k e n r i d g e et al. J o h n W i l e y (1954).
Physica XXIV 709-720
AMBIPOLAR TRANSPORT OF CARRIER DENSITY FLUCTUATIONS IN GERMANIUM*) Department of Electrical Engineering University of Minnesota, Minneapolis, Minnesota
by J. E. H I L L and K. M. VAN V L I E T Abstract
The spectrum of generation-recombination noise in semi-conductors may differ from the usual form at relatively high field strengths or relatively short drift times of the carriers. In contrast to previous opinions it is shown t h a t such a modification occurs only if the ambipolar drift velocity is sufficiently large. Accordingly, the sweeping of ~ r r i e r s through the sample by the electric field has no effect in material with a sino,~ type of carrier or in intrinsic material, but should show up in near-intrinsic ..... "s. Measurements, made to detect this effect, completely confirmed this behavior. ]~lum the "resonances" in the spectra the ambipolar drift time was determined and found to be in good agreement with the calculated values.
Introduction. The noise observed in high resistivity germanium crystals above ~-~ I00 c/s can entirely be attributed to statistical fluctuations in the rate of generation and recombination of holes and electrons, as has been reported before 1). For sufficiently low concentrations of impurity centers, which m a y occur either in the bulk or at the surface, the fluctuations in both carrier densities are almost completely correlated, i.e. An(t) ~. Ap(t). At sufficiently low electric fields the spectrum of the carrier fluctuations is the same as under thermal equilibrium. For larger field strengths an appreciable fraction of the carriers is carried off at the electrodes before recombination occurs. Under these circumstances the noise spectrum m a y be quite different from the low field spectrum. It is the purpose of this paper to discuss this effect and to present experimental results pertaining to this case. The effect of the carrier drift on the noise was first discussed by D a v y d o v and G u r e v i c h 2). Although not stated specifically, it seems that their analysis was concerned with a conductor, containing only a single type of carriers, e.g. electrons. According to their views, the correlation function, which normally has an exponential form, would be zero for times larger than the drift time ,g of the carriers from one electrode to the other electrode (,~ = = L//~nE). This idea is not correct as was shown before by several investigators 3)4)5). The general argument is that in order to maintain space-
*) Work supported by a U. S. Signal Corps Contract. -- 709 -Physica XXIV
710
J.E.
HILL AND K. M. VAN VLIET
charge neutrality, carriers start to move everywhere in the crystal simultaneously, when at some spot a fluctuation in the generation rate has occurred. If the contacts are ohmic, then electrons will move in at the negative electrode, while others are carried off at the positive electrode. This chain continues until most of the extra electrons have recombined after a time of the order of the lifetime T which m a y be much larger than *a. Consequently the correlation function is not affected b y the value of Ta. The "replenishment" of carriers itself also is a statistical process, since at the negative electrode a large reservoir of electrons is available. Therefore, it has been stated before by V a n V l i e t and B l o k 4) and independently by BOer et al. 5) that these injection-extraction fluctuations should give rise to noise of the order of 2eI per cycle in case that the l i f e t i m e , is much larger than the drift time *a of the carriers. This, however, is inconsistant with the picture just given s), since space-charge neutrality will be restored within a time of the order ~r, where ~r is the dielectric relaxation time. The injection-extraction fluctuations, therefore, will be strongly "sp~cecharge suppressed" and, in fact, will not exist in semiconductors. In insulators, like CdS an effect m a y occur at very low light levels. However, in that case the current flow is space-charge limited and a more drastic change of the noise picture is anticipated. We thus conclude that in a semiconductor with a single type of carriers (the minority carriers m a y be trapped and thus immobilized) the noise spectrum does not change in shape and no additional noise effects occur necessarily if we pass from the condition Ta >~ T to *a ~ *, up to such fields where the electrons become appreciably heated and w~ actually enter the pre-breakdown region. In semiconductors involving both electrons and holes the situation is different; this will be investigated in this paper.
2. Theory o~ /luctuations in mixed semiconductors. A very clear picture of arnbipolar motion of carriers in a semiconductor was recently published by S t ~ c k m a n n 7)s) which we will briefly review. In Figure I we will distinguish between three extreme cases. Let at t = 0 extra carriers be generated somewhere in the sample in a region/lx. If the carriers come from donor levels (Figure l a), the positive charge is fixed. Electrons moving out of the slice Ax to the positive electrode are replenished by others to preserve neutrality. Hence, the region of increased conductivity A~ remains at the same spot; the magnitude o f / t ~ decays, of course, with time. If holes and electrons are generated across the band gap in an n-type crystal, the picture changes (Figure l b). The electrons flowing to the right into the lower conductivity region will not produce an extra electron concentration there, since others flow out "simultaneously" (i.e. within a time of the order of the dielectric relaxation time), thus preserving spacecharge neutrality. The holes flowing to the left, however, are mainly neutra-
A M B I P O L A R T R A N S P O R T OF C A R R I E R D E N S I T Y F L U C T U A T I O N S
71 1
lized b y electrons going in the opposite direction. Thus, in the region to the left of the region which at t ---- 0 had an increased conductivity, the electron flow adjusts itself in such a w a y that at each moment An ~ Ap; the slice of increased conductivity, therefore, moves to the negative electrode with approximately the hole mobility. Finally, in an intrinsic semiconductor, (Figure l c) no shift of the concentration disturbance occurs, since the flow of carriers into the lower conductivity regions b y either carrier can be neutralized b y the same type of carrier. (I) Strongly N-type; transitions from donor levels Bn
Bn
t=0
t>0
(2) N-type; hole-electron pair transitions An~ &p Anj Ap
t=O
t>O
(3) Intrinsic ~n,~p
&n, Ap
HI"
tfO
t>O
F i g . 1. C a r r i e r c o n c e n t r a t i o n d i s t u r b a n c e i n : a) a n e x t r i n s i c s e m i c o n d u c t o r i n v o l v i n g d o n o r - c o n d u c t i o n b a n d t r a n s i t i o n s ; b) a n n - t y p e s e m i c o n d u c t o r n o t t o o f a x f r o m i n t r i n s i c s o t h a t h o l e e l e c t r o n p a i r s a r e g e n e r a t e d a c r o s s t h e b a n d g a p ; c) a n i n t r i n s i c c r y s t a l . ( c o m p a r e r e f e r e n c e 7).
This behavior is more rigorously described with the ambipolar transport equation for carriers as given b y v a n R o o s b r o e c k 9) which results from the continuity equations for holes and electrons separately under elimination of the term div E:
O(Ap)/Ot = -- qff= E . V(Ap) -- Ap/.r + D J 2 ( A p ) + g(r, t);
(1)
here . is the lifetime, E the electric field, /za and D= are the ambipolar mobility and diffusion coefficient respectively and g(r, t) is a stochastic source term. The quantities F= and D= are given as *)
#a =
(no -- Po)~n#~ ; #~no + #~Po
Da
=
(no + Po) D~Dn noDs + poD~
(2)
• ) Note t h a t /za and Da are n o t related b y the Einstein relation; other n a m e s for/~= are "group mobility" or "signal mobility". Expressions for/Za and Da in the ease t h a t the dielectric relaxation time is not small compared to "c have been given b y K e i 1 s o n 10) and by S t 8 c k m a n n s).
712
j.E.
H I L L AND K. M. VAN V L I E T
In strongly n-type material # a s s / ~ , for strongly p-type material/~a ~ ~ #n and in intrinsic material #a = 0 in accordance with Figure 1. The general solution of (1) is lengthy if we apply proper boundary conditions at the surface. Recently, a solution of (1) neglecting ambipolar effects was obtained 11). Here we shall neglect the diffusion telm since we mainly want to point out the effect of concentration shifts on the noise. We come back to this at the elid of this section. Assuming that the field is in the positive X-direction equation (1) reduces to 0(Ap)/at + q~a E 0(Ap)/0x = -- Ap/~ + g(x, t). (3) If recombination occurs mainly at the surface, t h e n , stands for the dominant surface transient time constant. We will be interested in the fluctuations in the total hole density given b y
AP(t) = foL Ap(x, t) dx.
(4)
R constant current source
C C.
Fig. 2. Geometry of crystals used in the measurements. The limits in the integral depend on the crystal geometry. Let us consider a bridge type sample as was actually used, Figure 2. The noise is measured between probes. Let us assume that the current is driven b y a perfect constant current generator. The voltage fluctuations between the probes are then given by A V 2 = I z AR 9., where R is the resistance between the probes. Hence, the integral in equation (4) extends only over the region (0 - - L ) between the probes. Although in the arrangement of Fig. 2 the noise is physically represented by a voltage generator, it is convenient to also introduce the equivalent short-circuited current generator for the noise between the probes,/lIss ~ = A V2/R 9". Then we find for the relation between /fiBs z and Ap2, or also for the relation between the spectral densities SI([) and S~([) of these respective quantities Sx(]) = Iz [(b + 1)/(bNo + P0)] 2 Sp(]),
(5)
where b =/.tn/#,~, and where No and Po are the total numbers of electrons and holes in the region between the probes.
A M B I P O L A R T R A N S P O R T OF C A R R I E R D E N S I T Y F L U C T U A T I O N S
713
T o find S~(]) we first consider the effect on the noise of fluctuations' generated at t = 0 in a slice between x' and x' + dx'. When we average over an ensemble for which Ap(x', O) = ,4po, the stochastic term drops out in equation (3). The remaining partial differential equation has the general solution p(x, 0 = e
-
(6)
where ~5 (y) is an arbitrary function. We have the condition that/tp(x, 0) = = 6(x -- x') Apodx', where $(x -- x') equals 1/dx' inside the interval (x', x' + dx') and zero outside; it represents a Dirac-delta function in the limit dx' ~ 0. This gives
lip(x, t) -~ Apo e -t/* 6(x --/~a, Et -- x')dx'.
(6a)
This is an obvious result, since the concentration disturbance progresses with the velocity #aE as pointed out in Figure 1. If we take into account diffusion, the concentration disturbance spreads out and the 6-function is replaced b y a more gaussian type of function, as can be found in the same way. Integrating (6a) with respect to x yields.
AP(t) = Apodx' e-t~* if 0 _< /~a,Et + x' < L AP(t) = 0
elsewhere
(7)
The inequalities in equation (7) limit the correlation time in an n-type sample (#a positive) to t <_ (L -- x')/l~E; in a p-type sample (/~a negative) the limit is given b y t ~ -- x'/#aE. These differences correspond to concentration shifts in the positive and negative X-direction respectively. In what follows we shall assume first that the crystal is n-type. Then we obtain for the noise due to the sources between x' and x' + dx', applying the WienerKhintchine theorem: dSp(/) = 4(Apodx')2foCL-~)l~"~ exp (-- u/~) cos cou du.
(8)
The fluctuations in any part dx' are caused b y generation and recombination and b y ambipolar diffusion (see end of this section). Both processes actually mean a transfer of conduction electrons to valence band electrons. Regardiess of which effect prevails, the total variance following from Fermi statistics is: (Ap0dx'F =
porto dx'/(no + Po).
(9)
We can now sum up the contributions from different slices dx'. The sources are statistically independent (at the same instant) if dx' is large compared to the scattering mean free path. As a good approximation it is, however, permissible to integrate (8), which yields 4pon0 f ~ f Ir'-x'll~'"~ Sp(/) = "no + Po dx' J0 exp (--u/-r) cos cou du
(10a)
714
J . E . HILL AND K. M. VAN VLIET
Changing the order of integration of this double integral yields: S p(/) 4 porto [ r4v,~ [ r,-v,.E~, = du dx' e-U~* cos oau du no +
Po Jo
= 4 ponoL no + Po where
fo
Jo
[e-ul* (1 -- U/ra)] cos cou du,
(lOb)
L L(#nno + I~po) -(11) ltZa[E Ino - - P01 t z n l ~ E is the ambipolar drift time. The absolute marks have been added to make the expression also valid for p-type semiconductors as is easily checked by repeating the above derivation for that case. The correlation function given by [] in (10b) is exactly of the form suggested before except for the fact that now the appropriate lifetime Ta enters into the expression. In semiconductors with a single type of carriers or in intrinsic semiconductors ra = oo and the correlation function is always exponential. The spectral density is obtained from (10a) or (10b). The result of the integration is found ill a paper by B u r g e s s lZ); in our case this yields: Sp(/) = (4ponoL'ca/(no + Po))W(cor, (.O'ra) (12) where 02/¢2 exp (--¢/0) W[O, ¢) ---- ~1 + 0 t ~ -------~ 1+ 0 + 1 + 09. [(1--02) cos¢--20sin¢]}. (12a) ~'a - -
-
t~
-
1--02
The low frequency value of W is given by W(O, O) -----('c/'ra)2 (e-'°1. -- 1) + "c/'ra
(13)
Introducing the normalized frequency response F(w'r, coTa) = W(coT, oa'Ca)/W(O, 0 ) w e find from (12), (13), (11)and (5):
SI(/) = 4qI
[~a .2 ] [ ~" + (e- * ' 1 . - 1) ~
b+2+ 1/b ]F(cor, COra)(14) [no/Po-- po/no]
For low field strengths or high values of ~a this expression reduces to a spectrum of the well-known form v/(1 -1- co2"r2). If Ta = r, the deviations are already noticeable; in that case
7
F" b+2W._l/b Sz(/) = (4/e) qI Llno/Po _ po/nolA
([ ~ ~o2~
Finally, in the case that *a ~ • we have F(O, ¢) m 2¢-2(1 -- cos ¢); hence S~r(/)=2qi I b + 2 + l / b ][sin½co-ra] 2 Ino/Po -- po/nol ½co'ra "
].(15)
(16)
Before proceeding we like to point out the connection with other methods employed before. D a v y d o v and G u r e v i c h 2) in their original derivation
AMBIPOLAR TRANSPORT OF CARRIER DENSITY FLUCTUATIONS
715
used the method of addition of elementary current pulses. This method in its simplest form (application of Campbell's or similar theorems) breaks down since the pulses are not independent. This affects the magnitude of the noise. The frequency dependence should be correct if the method is properly applied. According to Figure 1, one could consider as an elementary event the voltage pulse due to a hole-electron pair drifting with the ambipolar velocity #a~. On doing so, one has to take into account the statistical distribution of the lengths of the pulses and the statistical distribution of the origin of the pulses (x' in our notation), noting that the maximum drift length is L - x'. D a v y d o v and G u r e v i c h , although not.dealing with ambipolar transport, carried out this program correctly; accordingly, their frequency dependence is the same as in our case (equation 12a) if we replace their single carrier mobility # by #a. It is curious that in later versions of the "pulse addition method" different results were obtained since either the statistical distribution of the lengths of the pulses was ignored 13) or the difference in drift times for carriers generated at different spots x' was not taken into account 14). The above analysis, which is a collective carrier approach, is simpler because no information is needed about the behavior of the individual carriers. Moreover, it is better justified since the application of the charge neutrality implications to single carriers is highly speculative. I
s(f)
P 162
tO
I0
to~
Frequency (Kc.)
Fig. 3. Plot of equation (16).
We now return to the results given by equations (I 5) and (16). The spectrum of the noise, predicted by (16) is plotted in Figure 3. The striking fact is the appearance of "resonances" at high frequencies. A spectrum like this will never be measured as such, however, since the spectrum analyzer
716
j.E.
HILL AND K. M. VAN VLIET
averages o v e r a certain b a n d w i t h . I n Figure 4, c u r v e I, we p l o t t e d the result t h a t would be m e a s u r e d b y a s p e c t r u m a n a l y z e r with a b a n d w i d t h t h a t is 10% of the center frequency. I t is seen t h a t the higher order peaks average out due to the fact t h a t the b a n d w i d t h increases with increasing c e n t e r f r e q u e n c y of the band. [This is, of course, not necessary if f r e q u e n c y conversion involving n a r r o w low f r e q u e n c y filters is e m p l o y e d b u t usually s p e c t r u m analyzers below 500 kc do not operate this way.] Also p l o t t e d in Figure 4 is the s p e c t r u m o b t a i n e d f r o m e q u a t i o n (15) for r = T=. E v i d e n t l y , the peaks occur at n e a r l y the same frequencies, b u t are m u c h less pronounced. F o r o t h e r values of , / T a all varieties b e t w e e n the two curves in F i g u r e 4 are possible.
16 ~ o z
I: ~'=oo; integrated over 10% bandwidth
X: 1: =1:a
.N_ 0
E
o z
iff 2
~ff3
I
i
0.I
I
A
I I0
Fig. 4. Calculated noise according to equation (16) and equation (15). Curve I: T = co; spectrum of the noise integrated over a bandwidth B = 0.1 f. Curve II: v = va = 20/~ sec. W e shall now point out some limitations of the simple a p p r o a c h given above. Results different from the above ones m a y be o b t a i n e d if the c r y s t a l c u r r e n t is not driven b y a c o n s t a n t c u r r e n t generator. In t h a t case correlation effects occur between the various regions of the crystal as were observed b y M o n t g o m e r y 15). These effects can be c a l c u l a t e d in the same w a y as was done above for the noise. F i n a l l y we shall indicate w h a t modifications follow from inclusion of the diffusion term. L e t us first consider the case t h a t t h e noise is m e a s u r e d between probes which are not close to the e n d . contacts. T h u s end-effects can be ignored. T h e solution of e q u a t i o n (1) is now i n s t e a d of (6a) : Ap(x,
t)
A p ( x ' , O) d x ' (4~Dat)½ exp
r
t
L
r
(x - -
x ' - - #=Et)2 7 4Dat .] "
(17)
AMBIPOLAR TRANSPORT OF CARRIER DENSITY FLUCTUATIONS
717
The maximum spread of a disturbance, located at t = 0 at exaqtly x' is ,-~ (Da(L -- x')/po, E)~ < (DaL/l~o/E)t. In strongly extrinsic material we find from the Einstein relations that this spread is small compared to the distance between the probes if E L >> kT/q. This is always the case and consequently no influence of the diffusion is expected in extrinsic material. In intrinsic material the diffusion can have an appreciable effect. The noise resulting from (17) can be found in the same manner as before, or, in a more sophisticated approach, with the Smoluchowski method of probability-after effects. This results in: Sl(/)
x
=
412( no )(bpo+Po) Po no + Po bno + Po
foodu f; cLvf
dx'
2
cos wu L(4~Dau)~ exp
×
[
u r
4£)-au
a " (18)
For E sufficiently large, this expression should give the same result as equation (14). In an intrinsic semiconductor with/za = 0 and Dar >~ L 2 (18) reduces to a MacFarlane type spectrum. In such a semiconductor the response time is determined b y diffusion rather than b y recombination, analogous to a case discussed b y R i t t n e r 16). It is also possible to find the appropriate solution if we apply the boundary conditions Ap = An = 0 at the end-electrode contacts. The result can be readily found employing a general analysis given recently b y V a n V l i e t and V a n d e r Z i e l 11). 3. Experimental results. Some efforts have been made to measure ambipolar fluctuations as predicted in the previous section. The effect has not been found before for several reasons. First, the condition *a < r is not easily satisfied. Only a small temperature region in the near-intrinsic range will be adequate; also the heat dissipation becomes excessive in order to meet this requirement for small lifetime samples. Moreover, the measurements should be done with a spectrum analyzer with a more or less continuously variable frequency band. This is clear from the calculated spectra in Figure 4. Therefore, measurements were made from 1--500 kc in steps of about 5 kc in the region where the effects were expected. For the available crystals the lifetime was, .after etching, of the order of 20 # sec. Larger samples might have had a larger T, b u t they have the disadvantage of a smaller resistance. At temperatures slightly above room temperature the noise was large enough to guarantee accurate measurements and currents for which Ta was equal or slightly less than r could be passed. The measured noise for a p-type crystal is shown in Figure 5, together with the calculated result, curve II of Figure 4. As is seen, both the magnitude of the noise as predicted b y equation (15) and the form of the spectrum agree very reasonably. The best fit is obtained for ra = 20/~ sec which is
718
J. E. HILL AND K. M. VAN VLIET
also within experimental error equal to the value calculated according to equation (2) from the measured resistivity of the crystal. The results were repeated for n-type germanium, see Figure 6.
p-Ge
]
~,
T=75°C Exp.results, result ~\ - - - -.-- Theoreficol Theo "~ ='~a=20~tse¢ ~b,~ for f°r''~
l/n°'se-'v"i
0.'~ - - ,hermo,-- ~ I
I0 Frequency
''
~-~
(Kc.)
K~
Fig. 5. Noise of a p-type germanium crystal with T= = 20/~ sec at T = 75°C (solid curve experimental, dotted curve calculated). The noise is represented by Ieq, defined by Sz = 2eleq. The crystal was in the near-intrinsic region (52°C; no ~ 30 × 1018/cm a P0 ~ 7 x 1013/cm3). The lifetime was ~ 25 # sec. The noise was measured in steps of 5 kc up to 400 kc for currents from 2 mA -- 17 mA. These values are such that for low currents ~= > ~, whereas for the highest currents ambipolar effects should become noticeable. This is very clearly demonstrated in Figure 6, which shows three minima for the highest currents. Tile frequency values are given in the Table I. The minima shift to higher frequencies for larger currents as is expected from the theory. It m a y also be noted, that the 1/[ noise content in Figure 5 increases as the current increases. This increase is more than expected from the difference in current dependence; it can be easily understood as follows. At low fields the noise, produced b y the electrodes does not show up, because of the probe method. [It was found that without probes, this noise rises b y a factor 10.] If the ambipolar drift time becomes sufficiently short, however, the probes loose their function, since the contact fluctuations are swept into the region between the probes. Hence, simultaneously with the occurrence of ambipolar effects, the 1//noise level rises. The fact tha~/the 1/[ noise does not extend beyond 10 kc but in fact, changes to 1/] 3 indicates that the shortest lifetime present in 1/[ noise (ill these crystals) is the lifetime of the minority carriers. This is to be expected from several of the l I~ models which have been
A M B I P O L A R T R A N S P O R T OF C A R R I E R D E N S I T Y F L U C T U A T I O N S
7 ]9
proposed. It indicates that 1/[ noise (caused b y "slow" surface states) modulates the generation and recombination rate of the carriers in the bulk or at the surface ("fast" surface states).
."- ooo
Id:
-,-
,o
N-Ge-I
\
%oo
4
~
,o
,~
Frequency (cycles) F i g . 6. N o i s e of a n n - t y p e g e r m a n i u m c r y s t a l i n t h e n e a r - i n t r i n s i c r e g i o n for various current strengths. TABLE I
I
T
17 ma lO
25/~ sec 25 25 25 25 25
8 6 4 2
Ta 12 ~ sec 20 25 33 5O I I00
calc. 1st rnim 85 kc 50 40 30 20 I0
I
calc. 2nd min.
obs. 1st rain.
obs. 2nd min.
170 kc I00 80 60 40
80 kc 55 45 ~35 ,~,15
170 kc 125 II0 80
2o
Finally, the noise was measured for the same crystal in the intrinsic region at fields sufficiently high, so that T was larger than the drift times for electrons and holes separately. The same number of frequencies was measured as before. The result is shown in Figure 7. Evidently the noise is of the form ~/(1 + ~o2~2) and no resonances occur, in accordance with the theory.
720
AMBIPOLAR TRANSPORT OF CARRIER DENSITY FLUCTUATIONS
I0
N-Ge-I
"~ '~t x
I=10 mo. R=1500~ T=80°C
o
E
t~a
t
I0
Io2 .Frequency (Kc.)
Fig. 7. Noise of t h e s a m e c r y s t a l as in F i g u r e 6, n o w a t a t e m p e r a t u r e of T = 80°C, a t w h i c h t e m p e r a t u r e t h e c r y s t a l is intrinsic.
Acknowledgements. We wish to thank Mr. H. C. M o n t g o m e r y of Bell Telephone Laboratories for providing us the bridge shaped crystals and for correspondence and interesting discussions about this subject. We are also indebted to Dr. A. V a n d e r Ziel for improvements in the manuscript. Received 24-4-58. REFERENCES 1) Hill, J. E. and Van Vliet, K. M., J. appl. Phys. 29 (1958) 177. 2) D a v y d o v , B. and G u r e v i e h , B., J. Phys. USSR 7 (1943) 138. 3) Rose, A., in "Photoconductivity Conference" Ed. Breckenridge, e~ al. J o h n Wiley, (1954), 34-44. 4) V a n Vliet, K. M. and Blok, J., Physica 22 (1956) 231. 5) Boer, K. W., et alii., Ann. Physik 17 (1956) 344. 6) M o n t g o m e r y , H. C., Private Communication. 7) S t 6 c k m a n n , F., Z. Physik 147 (1957) 544. 8) S t 6 c k m a n n , F., in "Photoconductivity Conference", Ed. B r e c k e n r i d g e , et al. J o h n W i l e y (1954). 9) Van R o o s b r o e c k , W., Phys. Rev. 91 (1953) 282. 10) K e i l s o n , J., J. appl. Phys. 24 (1953) 1198. 11) Van Vliet, K. M. and V a n der Ziel, A., Physica, 24 (1958) 415. 12) B u r g e s s , R. E., Brit. J. appl. Phys. 6 (1955) 185. 13) Gisolf, J. H., Physica 15 (19491 825. 14) Boer, K. W., Ann. Physik 14 (1954) 87; also 15 (1954) 55. 15) M o n t g o m e r y , H. C., Bell Syst. techn. J. 31 (1952) 950. 16) R i t t n e r , E. S., in "Photoconductivity Conference" Ed. B r e c k e n r i d g e et al. J o h n W i l e y (1954).