Generation-recombination noise and its influence on the energy resolution of diffused silicon p-n junction radiation detectors

NUCLLAR

I N S I R U M L N T S A N I) MI T H O I ) S

46 (~967)

255-260;

~: ' , 4 O R I H - H O I . L A N D

PUBLISHING

CO.

GENERATION-RECOMBINATION NOISE AND ITS I N F L U E N C E ON "IHE ENERGY R E S O L U T I O N OF D I F F U S E D S I L I C O N p-n J U N C T I O N R A D I A T I O N D E T E C T O R S R Y DESHPANDE

Techmcal Phv~tc~ Dtvt~ton, Atomic Em'rgv E~tabh~hment Trombay, Bombay, Imha Recewed 10 Aprd 1966 In this paper a basic rnecham,,m hmJtmg the energy resolution of p-n junction nuclear radmt~on detectors ~s proposed ShockleyRead-Hall ,,taw, tics are apphed to determine the time con,,tant of the charge-fluctuation at the generation-recombination center,, Some factors governing th~s time con,,tant are d~scu,,sed A specific case of diffused ,,lhcon p-n junction detectors, for ~hJch

Information about the generation-recombination center~ is avadable, t,, then worked out The charge-fluctuation t~me constant 1~ used as a normahzlng parameter to evaluate the nol,,epower spectrum of the detector-amphfier system Finally, some desagn considerations of the system are presented

1. Introduction The energy resolution capabditJes of a semiconductor detector and amphfier system have been mvcstigated Jn detail by Gouldmg and Hansenl). They attribute the energy spread of a monoenerget~c hne to the various sources of electrical noise inherently present m the detector-amphfier system. In a well-designed practical system they find that the only important sources of noise contributing to the energy spread : e the input tube shot noise and the current no~se. The current noise includes noise due to the input tube gNd current, the current flowing m a reverse-bmsed p-n junction, and the surface current For a single d~fferentiatmg and integrating time constant r of the amphfier, the tube shot no~se decreases w~th increasing r whereas the current noise increases w~th increasing r. Therefore there exi~t~ an optimum amphficr time constant for which the best energy resolutions could bc obtained. In characterizing the noJse-bchawour of the detector, Goulding and Han~en have used the reverse current (leakage current) as the parameter determining the detector noise Their analys~s shows that the mean square noise w~ltage of the detector should decrease with the detector current. Hence a detector w~th low leakage current or operated such that the current is very small (for instance, on cooling) should show improved energy resolution. However, this assumes that the detector noise (when the leakage current is taken as the noise-criterion) is independent of the frequency band m which the noise voltage is measured. Obviously, this is a simphficatlon of the problem and, though a useful one, need not hold good always. In fact, on this ground it becomes difficult to understand why a diffused sdicon p-n junction detector prepared from h~gh-resistwlty sd~con fails to operate at low temperature', 2) where the leakage currents are extremely favourable.

Th~s immediately suggests that a leakage current fluctuation t~me constant should be incorporated into the theory. An amphfier operated at a certain cut-off frequency will therefore reject or partially accept the detector noise depending on the cut-off frequency and the fluctuation time constant The fluctuation time constant and its dependence on several factors will finally determine the energy resolution capabilities of the system. In thls paper we have undertaken an investigation into the nature and origin of the leakage current and, as we have suggested above, the fluctuation t~me constant to be as~ocmted with Jt But instead of dealing with the leakage current as the determining factor, we trace the origin of the detector no~se to the basic processes going on in a reverse-biased p-n junction. The reverse current itself depends upon the carrier generat~on and recombination through the agency of deep recombination centers in the forbidden band of the semiconductor3). The statistics of recombination processe~ in semiconductors have been worked out by Shockley and Read 4) and by Hall 5) and have been recently applied by Sah 6) and by Lauritzen 7) to the case of low-frequency noise in field-effect transistors We extend the theory of Sah to the case of no~se in detectors. The theory is compared with the experimentally observed energy resolutions for diffused silicon detectors8). The comparison becomes possible as the relevant data about the recombination centers is now available°). Finally, some design considerations are suggested for th~s specific case 2. Charge fluctuation time constant Consider a Shockley-Read-Hall model of generationrecombination centers in the forbidden zone of the semiconductor. These centers, as shown in fig. 1, are 255

256

n, Y. DESHPANDE (9

Ec

"fit

t )g. 1 Basic electron and hole capture and em~ssmn processes through deep recombmatmn centers. characterized by four capture and emission probabilities for electrons and holes, c,, e, and c o, %, respectwely. If N~ is the density of the recombination centers and f t h e probabfltty of a center being o e c u D e d by an electron, then the net capture rates, U~, and Uep of electrons and holes, respecttvely, are

densttms, q(6p-fn) This will be passed on as no>e, with t~me constant rf, in the external circmt. F r o m eq.(4) we calculate the fluctuation nine c o n s t a n t -rf f o r p-sflmon. Here we assume that the process treatment condittons introduce deep d o n o r levels which participate m the r e c o m b t m m o n mechanismg). The density of the d o n o r levels is 3 26 x 10~'~,; cm 3 at an energy level 0 3 2 e V measured from the valence band. The hole-capature cross-section, ap, for these levels is 1.16 × 10-16 cm 2 and the electroncapture cross-section, o,, ~s --. 10-) ~ cm 2 [tentative, ref.10)]. A s s u m m g t h e thermal velocmes of the earners to be 10 .7 cm/sec (at 300" K) the capture probabihtles are c. ~ 10 -8 cm3/sec and % = 1 16 x 10 -° cm3/sec. For p-silicon (Po > no) and the r e c o m b i n a u o n level m the lower half of the band gap (Pt > hi) as shown m fig. I, r t can be written as • , = [ ( P o + pt)cp]-'

(5)

where n and p are the electron and hole dens~tms. In thermal equlhbrium U¢,, =/./,~ = 0. Hence from eqs.(l) we have

Taking n , = l . 5 x 10t°/cm 3, E , - E ~ = 0 . 5 5 - 0 . 3 2 = 0.23 eV, we get p, = 1 1 × 10t4/cm 3. Po will depend upon the resistwny of the materml. In table 1 we have calculated rr for various values of the final resisuvtty of the p-type sthcon As is clear from this table, rt is essentially independent of the resistivity m the highresistivity region of interest in the field of radiatmn detectors. This also holds good for reverse-biased n-on-p sthcon juncttons. We can therefore further simplify eq. (5) and write,

en/c. = n,exp(E,-E,)/(hT) =- nt,

= (,,:,)-

U'~. = N , ( 1 - J ) n c . - - N , . f e n ,

U,. = Ndpcp- N,(I -t)ep,

= ,,,exp(E,-

-

(I)

p,.

(2)

In these expressions n, is the c a r e e r density and E, the Fermi level in the intrinsic semiconductor; E~ is the energy level of the r e c o m b m a t m n centers; n~ and Pt are the carrmr densit,es when the Fermt level coincides with the recombinatmn level. Under these conditmns, and considering only the long ttme constants, the rate of fluctuatmn of the charge in the recombination centers according to the theory o f Sah 6) is gwen by

Eqs. (6) and (2) can be used to determine the temperature-dependence of the fluctuation nine constant rr. In the temperature range such that p~ > Po, and assuming non-degenerate statistics to hold, we have ~r = ( c o P , ) - ' = [con, e x p ( E , - ED/I
TABLL 1 Fluctuatmn tmae constant (300" K).

This gives the charge fluctuatmn time constant ~t, (4)

where q is the electronic charge, and n o and Po the steady-state electron and hole densmes, respectwely. For the overall electrical neutrality o f the semiconductor, the charge fluctuation qan~ m the recombination centers, should be compensated by an equal charge fluctuatmn due to the fluctuation in the free carrier

(7)

where N~ is the effective density of states 111 the valence band. Taking ~) cp = 1.16 × 10 -~ cm3/~ec, N, = 6.36 × 1018[cm 3 and E r - E , = 0 3 2 e V , m fig. 2 we have

~(q6n~)/tt = -[c,(no+nl)+cp(Po + Pl)]qOn~. (3) z r = [c,(no + nl) + %(Po + P D ] - ' ,

(6)

}

PO

D

(cm-3)

(ohm" cm) 1o IOO 1 ooo ioooo >

I I I 1

3 3 3 3

× × x × <

l 0 Ls 1014 10is 1012

Tf-I

Tf

(see-t)

(see)

1 6 4 × 10('

6 10 × 10 .7

2 8 0 x 105

357 × 10 t,

I 4 3 × 105 1 29 x 105 1 27 x 105

699× 10 -0 7 7 5 x 10 (' 787 x 10-°

GENERATION-RFCOMBINATION NOISE It3

- - i

,;PC

JCO

2:>0 j

l

_ _

,

,

/ • Q.

/

~°

/

/

//

/

/

/

// !

1,',

1C

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I _ _ 4

257

frequency b a n d w i d t h Aco/2rr at a frequency o)/2n can be written as

( e2a) = [(q2/C2)~ Nrf(l - f ) A W] x x [(21rr)hA,,/(I +to2~t:)j,

where W is the d e p l e t i o n width o f the detector, C the c a p a c i t y and A its area. The three auxiliary e q u a t i o n s giving IV, C a n d J are as follows. F o r an a b r u p t n - o n - p j u n c t i o n detector, W = 3 . 2 x 1 0 - s (pV) tcm, where p is the resl~ttlvlty o f p-slhcon and 1/ the a p p l i e d reverse blasS). T r e a t i n g the j u n c t i o n as a parallel plate capacitor, C = 1.1 eA/(4nW)pF, where c ~s the dmlectrm c o n s t a n t (l 2 for s l h c o n ) . f l s given by'~)J = (1 +Po/Pl) - I Using eq. (8) we can calculate @ 2 ) m a p a r t i c u l a r frequency b a n d or else integrate it from 0o/27t = 0 to oo and find the total mean square noise voltage. But before c a r r y i n g out this c a l c u l a t m n we would like to express the ms noise in units m o r e useful in the tield o f nuclear spectroscopy, that is, express the noise in terms o f an equivalent energy o f the incident particle. F o l l o w i n g the p r o c e d u r e o f G o u l d m g and H a n s e n ' ) the ms noise in keV 2 is then given by ( E ~ ) = 1.05 x 10-4[,~N',J(1 - f ) A W] x

x [(2/a)hAvJ/(1 +o)2,~)]. (9)

Fig 2 Temperature dependence of the fluctuatton time con-

stant ft. p l o t t e d log r t as a function o f l / T ( t h e slope c o r r e s p o n d s to the actlvatlon energy o f the r e c o m b l n a t m n centers). It is clear that rr is a very steep function o f t e m p e r a t u r e and increases with decreasing t e m p e r a t u r e . Intuitively this means that, on cooling, the r e c o m b i n a t i o n centers b e c o m e m o r e effective as t r a p s and hold the charge for longer periods o f time. This will have the effect o f higher l f f n m s e at low temperatures. 3. Noise

power

spectrum

The fluctuatmn time c o n s t a n t rt, as d e t e r m i n e d in the last sectmn, gives some idea a b o u t the nature o f the noise f r o m the detector. However, to fully u n d e r s t a n d the n a t u r e o f the no~se ~t is necessary to k n o w the noise p o w e r s p e c t r u m . This is defined as the mean square noise voltage in a unit frequency b a n d w i d t h as a function o f frequency. The nmse s p e c t r u m associated wtth the fluctuauon u m e c o n s t a n t rf has been analysed by Sah 6) for the field-effect transistors. We assume that the t h e o r y developed for the depletion region o f the F E T ' s is a p p l i c a b l e for the detectors, even t h o u g h the fluctuatmn time c o n s t a n t s are much s h o r t e r in the present case. Hence following his analysis for the detectors, the mean square noise voltage ( e aa ) in a

(8)

Here we have taken 3.75 eV as the energy required to p r o d uce an electron-hole p a i r by the m o d e n t parttcle i ,). F r o m eq. (9) we can easily find the total ms d e t e c t o r noise, ( E 2 , ) k e V ~, on Integrating it over the entire frequency range oo/2n = 0 to 0"'. N o t e t h a t on rotegrating, the second b r a c k e t in eq. (9) becomes umty. A s s u m m g N, = 3.26 × 1014/cm3 a n d P l = 1.1 x 1014/ c m a, we have calculated (E~t)/A W, the ms noise m keV 2 per unit wflume, for several values o f the final resistivity o f p-sthcon; the results are shown m table 2, As a typical example, t a k m g p o = 1.3 x 10~3/cm a (for 1000 o h m . e r a p-S 0 14/= 10- 2 c m (for a bins o f 100 V) a n d A = 1 c m 2, w e get (]:,'2t) = 10 7 keV 2, o r ( E d t ) = 3.3 x 10 3 keV. Surprisingly this shows that the d e t e c t o r as a whole is extremely nmsy. However, the frequencyd e p e n d e n c e o f the noise comes as a saving grace and TABLE 2

Total mean square detector noise (300° K). P (ohm. cm) 10 100

I 000 10000

P0 (cm-3)

I f

l-f

10Is 7 8 0 x 458x I 3 x 1013 8 9 4 x , 1 3 x 1012 1989×

10-z 9 2 2 x 10-11 542 x I0-11 1.06x 10-I I 14x

I

f ' 1 3 x

I 3 X 1014

i

I

/(AW) (keV2/cm~) 10q 10 J 10 1 10-2

8 18x lOS 2 83 × 10~ 1 08 x 109 1 28 × 10 s

258

R.

Y.

DESHPANDF

m a k e s its use for particle s p e c t r o s c o p y possible only in c o n j u n c t i o n with p r o p e r filter n e t w o r k s and amplifier. To illustrate the point, we have calculated the ms noise per unit b a n d w i d t h as a function o f ~o for a few values o f rf. The results for ( E ~ ) = 10 v keY / are shown in fig. 3. As is clear f r o m the figure and as should be expected, the noise f r o m the d e t e c t o r is d o m i n a n t below the frequency c o r r e s p o n d i n g to co = 2n/rf. Also, below this frequency the ms noise increases with r r. On the other hand, b e y o n d co = 2n/rr, the ms noise decreases r a p i d l y as l / w 2. Thus, the occurrence of the d e t e c t o r noise p r e d o m i n a n t l y in the "'low" frequency region of the s p e c t r u m presents a prospect o f deslgmng suitable filter networks to suppress the noise as against the signal. W e e x a m i n e these aspects in the following section.

T h e time c o n s t a n t involved in the collection of charge p r o d u c e d by an incident lonlsing particle in the sensitive v o l u m e o f the d e t e c t o r is given by t~ = W2]F~V, where t¢ is the charge collection time (or the transit time), W the width o f the d e p l e u o n region, V the ,i

G = Gowr/(l

+C02~'2).

/

= < e~,> [(2/~)~,A,~1(I + ~ z ~ ) ] × ×1-o~'/(1 +o~:y].

In this expression ( E ~ ) is the total integrated ms noise f r o m the detector without the amplifier• Putting x = cozf and a = ~2/~f2, eq. (11) can be written as

(12)

\ \ .

.

.

.

(,1)

<{E~> = (e<~,)>[(21~)Axi(l +x2)] [ax'i(I +,~2)2].

\

.

(10)

Hence the equivalent ms noise at the o u t p u t terminal o f the amplifier is

4. Detector-amplifier system

"1

a p p l i e d reverse bias, and p the carrier mobility, tc will d e t e r m i n e the rise-time of the pulse from the detector. As a typical example, t a k i n g W = 10 -2 cm, V = 100 V and p = 5 0 0 c m 2 / V . s e c , t ¢ - - - 2 x 10 - ° s e e . As against this, the fluctuation time c o n s t a n t d e t e r m i n i n g the noise f r o m the detector is rf = 7 × 10 - " sec. Obviously, a p r o p e r selective network could be designed to filter out the signal and reject the noise for m a x i m u m signal to noise ratio. To analyse the p r o b l e m we therefore consider in c o n j u n c t i o n with the detector the frequency response o f an amplifier w~th time c o n s t a n t r and an a p p r o p r i a t e cut-off frequency ¢~J2n to be d e t e r m i n e d for o p t i m u m o p e r a t i o n The gain o f the amplifier varies with frequency and is given by

.

.

'~---x

F o r a = I, eq. (12) can be easily integrated over the entire frequency range, x = 0 to oo. In this case the equivalent noise is (Ea2~) = ~(EaZt). Thus the effect o f = rf is to reduce the noise just by a factor ~ 3 showing that the amplifier passes practically the whole o f the d e t e c t o r noise. As this case ~s not of any practical use, we have calculated the noise function

j(x) = axZ! {(1 + x Z ) ( l + ax2)2),

(13)

as a function o f x with a as the varmble parameter. The results are shown in fig. 4. As is clear f r o m the figure, well b e y o n d the m a x i m a o f f ( x ) the noise function has a slope o f - 4 on a log-log scale. Thus if we consider only a high-pass filter with a cut-off frequency lying on this slope, eq (13) can be simplified and written as

J(x) =

1!,,.,:.

(14)

Substituting eq (14) in (12) and integrating the noise f r o m the lower cut-off frequency defined by x = xc to oo we have

I,

(Lz~,) = (ls~,) (21'n)/(3ax~) o

I

,

Fig. 3 Total detector mean square no~se per umt bandwidth as a functmn of frequency ~ith rr as the varmble parameter

= (E~,) ~1i(12~4))~1(~,:,), where ~, =

(is)

I!J~=2nloo< The d e p e n d e n c e o f energy

GENERATION-RECOMBINATION

259

NOISE

resolution on various time constants is obvious from ~. ~° I" eq. (15) and need not be elaborated further. Taking a typical case of a = 1 0 -2 ( r f = 7 / l s e c , r = 0 7 / t s e c ) , x ¢ = 102 (r¢=0.45tlsec) and ( E a ] ) = 107 keV 2 ( p = 1000 ohm. cm, W = 1 0 - 2 c m , A = 1 cm 2, see table 2), the energy resolution is ( Ea, ,) = 15 keV. This seems to be a reasonable value agreemg with the experimentally observed best resolutionsS). It should, however, be realised that the resolution ts extremely sens~nve to the various time constants and therefore wdl also be sensitive to the several factors that determine these time constants. Here It should also be mentioned that there remains a large uncertainty, to the extent of a factor of 4, in the actual value of Tf itself. This arises because of , the uncertainty m the values of the physical constants of ,~ sdicon, hke n,, N , E,-E~, and the uncertainty in the energy level E, of the recombination centers. Thus the value of rf = 7,usec used in our calculations is actually on the lower side by th~s factor, or the resolution xs poorer by a factor of 2. 5. Other sources of noise In the above discussion we have been considering only the generation-recombination noise in the depeletion region of the detector and the effect of pulse shaping networks on the energy resolutions. But in addition to th~s we should also consider other sources of noise that can cause the broadening of the energy peak. As was mentioned m the Introducnon, the only other source of no~se that can be of importance m a practical system is the input tube shot noise. Following the analysis similar to the one gtven above for the detector and the procedure of Gouldmg and Hansen~), the tube shot noise ~s given by

10-4{51~TCZ/(~g,,qZ)}(l/rfaxc) (16) 1.05 x lO-4~51,TCZ/(rtgmqZ)}r¢/(2nrZ)keV 2.

~ E~,) = 1.05 x =

Here kT ~s the thermal energy, C the detector capactty, and gm the mutual conductance of the input tube. In this expression we are summing the shot noise only above the cut-off frequency defined by x = x¢. Agam, assummg a = 10- 2, x¢ = 102, rr = 7/tsec, gm 16 mA/V (for EC-1000), C = 1 0 2 P F (for W = 1 0 - 2 c m , A = I cm2), kT/q=25 mV, the input tube shot nmse is ( E ~ z ) = 1.5 keV. Although this noise Js much smaller m comparison wHh the detector noise at the rime constants used for calculations, it wdl be much higher at shorter nine constants. However, at very short rime constants the no~se theory developed for the detectors itself may not be strictly applicable. In such an eventuahty several other sources, hke collection time =

1~ ' l

1"~ /

l

' ,' x t

"q

1, ,"

1"

~Jtf,

Fig. 4 D e t e c t o r - a m p h f i e r n o i s e functzon .f(x) vs n o r m a h z e d frequency x=u~rr with normalized amplifier time constant a = r Z / r r 2 as the v a r i a b l e p a r a m e t e r .

fluctuation, fluctuation m carrier densities, will have to be considered in detail. Finally, we would hke to make a remark about the noise from the detector surface As with the reverse biased p-n junction, Goulding and Hansen t'8) assume that the noise from the surface wdl be determined by the surface current. Hence a well-compensated electrically neutral surface showing neghglble surface current12.13) wdl not contribute to the total noise. But in the light of the statistical theory developed for the bulk noise, this need not be true. A surface having a high denMty of surface traps will show neghglbly small leakage currents and still could be extremely nmsy 14.15). A similar statistical theory can be developed for the no~se from the surface m the presence of deep surface traps. 6. Summary L,m~tat~ons of the detector no~se theory based on the leakage current m a reverse-bmsed p-n junction were pointed out. It was suggested that, to determme the nature of the nmse from the detector, a fluctuat|on time

260

R. Y. DESHPANDE

c o n s t a n t should be associated w~th the leakage current. A statistical theory was then developed to determine the time c o n s t a n t of charge fluctuation at the generat~on-recombmat]on centers. The time c o n s t a n t was used to lind the noise power spectrum of the detector. Factors governing the time c o n s t a n t were investigated. A specific case of diffused ,silicon p-n j u n c t i o n detectors was then worked out. Experimental data a b o u t the r e c o m b i n a t i o n centers, introduced m sd~con during process treatment conditions, was used to e',tamate the noise. The noise was expressed m terms of an eqmvalent energy (in keV) of the m o d e n t particle. It was emphasized that the s e m i c o n d u c t o r detectors become useful only m c o n j u n c t i o n with proper selective networks. For the optma~zatlon of the energy resolution some design aspects of the detector-amphfier system were then analysed. Finally, sources other than the detector noise contrib u t i n g to the energy spread were briefly considered. The a u t h o r thanks Prof. D. Y. Phadke for his keen

interest in thc work and for providing him the necessary facdltles.

References 1) F. S. Gouldlng and W L Hansen, Nuc[ lnstr and Meth. 12 (1961) 249 2) W. L Hansen and M D. Roach, Lawrence Radiation Laboratory (1965) private comnlunlcatlon 3) C I Sah, R. N. Noyce and W. Shocklcy, Proc IRE 45 (1957) 1228. 4) W. Shockley and R. T. Read, l'hys R.ev 87 (1952) 835. 5) R. N Hall, Phys Rcv 87 0952) 387 t,) C. T Sah, Proc IEEE 52 (1964) 795. 7) p. O. Laurltzen, Sohd-Statc Electr 8 (1965) 41. 8) F.S Gouldmg, lEEETrans. Nucl Sol. NS-11 (1964) 177 ~) R Y. Deshpande, Solid-State Elcctr. 9 (1966) 265 10) R Y Dcshpande, unpubhshcd 11) 1- S Gouldmg, UCRL-16231 (1965) 27 12) W L. Hansen and F. S Gouldmg, Nucl. Instr. and Meth 29 (1964) 345 ~3) R P Lothrop, M D Roach and H. L SmLth, UCRL-11828 (1964) 225 14) R. Y Dc~hpande, Sohd-State Llectr 8 (1965) 313 l.~) R Y. Deshpandc, Sohd-Statc Electr. 8 0965) 619.