a physical model for the dependence of carrier lifetime on doping density in nondegerate silicon

8/16/2019 A physical model for the dependence of carrier lifetime on doping density in nondegerate silicon

1/7

Solid-State Electronics Vol. 25 No. 8 pp. 741-7 47 1982 0038-11011821080741-07503.0010

Printed in Great Britain. Pergamon Press Ltd.

P H Y S I C L M O D E L F O R T H E D E P E N D E N C E O F C R RIE R

L I FE T IM E O N D O P I N G D E N S I T Y I N

N O N D E G E N E R T E S IL I C O N t

J. G . FOSSUM an d D . S. L E E

Departm ent of E lectrical Engineering, University of Florida, GainesviUe ,FL 32611, U.S.A .

R e c e i v e d 12 Ju ly 1981; i n rev i s ed /or m 3 Decem ber 1981)

A lm rae t--A theoretical model that describes the depend ence of carrier lifetime on doping density, which is based

on the equilibrium solubility of a sing le defect in nondegene rately do pe d silicon, is developed. The m odel

predictions are consistent with the

longes t

measured hole and electron lifetimes reported for n-type and p-type

silicon, and hence imply a possibly fundam ental (unavoidable)defect in silicon. The de fect is accep tor-type and

is m ore soluble in n-type than in p-type silicon, which sugge sts a longer fundamental limit for electron lifetime than

for h ole lifetime at a given nondegenerate doping density. The prevalent, minimum density of the defect, which

defines these limits, occurs at the p rocessing temperature below which the defec t is virtually imm obile n the silicon

lattice. The analysis reveals that this temperature is in the range 300--400°C,and thus em phasizes, when related also

to com mon non-fundame ntal defects, the significanceof low -temperature processing in the fabrication of silicon

devices requiring long or well-controlledcarrier lifetimes.

Cn0

Cp-

D •

D s

D -

Eo

Ea

Ei

ga

h +

k

K t , K 2 , K 3

N A

N o

ND +

N a

N d~

N a -

N ~ o

Nsi

N

nl

P

T

i

U

PH

Tn

,rp

N O T A T I O N

electron capture param eter of (neutral) defect

hole capture param eter of (ionized) defect

unionized defect in external phase

unionized defect in sem iconductor (silicon) phase

ionized (acceptor-type) defect in semiconductor

(silicon) phase

activation energy for defect form ation in silicon

energy level (in energy gap) of defect state

intrinsic Fermi level

degeneracy factor of defec t energy state

hole in semicond uctor (silicon) phase

Boltzmann's constant

mass -action reaction (tempearture-depe ndent) con-

stants

acceptor dop ing density

donor do ping density

ionized donor density

defect density in semicond uctor (silicon)

(constant) defect density in external phase

unionized defect density in se micondu ctor (silicon)

ionized (acceptor-type) defect density in semicon-

ductor (silicon)

ionized (acceptor-type) defect density in undoped

sem iconductor (silicon)

atomic density of silicon

electron density

intrinsic carrier density

hole density

absolute temperature

effective temperature of defect formation in semi-

conductor (silicon)

net electron-hole recom bination rate

high-injection(P = N) carrier l ifetime

minority electron lifetime

minority hole lifetime

tThis work w as supported by Sand ia National Laboratories

under Con tract No. 23-11 61 and by U nited States-Spain

Cooperative Re search Grant No. T3773008.

INTRODU TION

T h e p e r f o r m a n c e o f v i r tu a l l y e v e r y s e m i c o n d u c t o r

d ev i ce o r i n t eg ra t ed c i r cu i t i s d ep en d en t o n t h e ca r r i e r

l i f e t imes i n t h e semico n d u c to r . F o r ex amp le , t h e p o wer -

co n v er s io n e f f i c i en cy o f so l a r ce l l s an d t h e q u an tu m

ef f ic i en cy o f d e t ec to r s d e p en d s t ro n g ly o n r eco m b in a t i o n

car r i e r l i f e t imes , an d t h e s i g n a l - t o -n o i se r a t i o o f d y n a mic

M O S m e m o r i e s in V L S I c i r c u i ts a n d t h e c h a r g e - t r a n sf e r

e f f i c i en cy o f CCDs d ep en d c r i t i ca l l y o n g en era t i o n ca r -

r i e r l i f e times . Becau se s i l ico n is t h e mo s t c o m mo n ly u sed

semico n d u c to r i n t h e e l ec t ro n i cs i n d u s t ry , an u n d er -

s t an d in g o f h o w ca r r i e r l i f e t imes a r e r e l a t ed t o t h e s i l i co n

g ro wth an d p ro ce ss in g h as p rac t i ca l s i g n i f i can ce .

T h e r e c o m b i n a t i o n a n d g e n e r a t i o n c a r d e r l if e t im e s a r e

d e f in e d b y t h e p r e d o m i n a n t c a r d e r r e c o m b i n a t io n -

g e n e r a t i o n m e c h a n i s m s [ l ] , m o s t c o m m o n l y t h e b a n d -

b a n d ( p h o n o n - a s s i s t e d ) A u g e r - i m p a c t p r o c e s s a n d t h e

cap tu re -em i ss io n (S h o ck l ey -Read -Hal l (S RH)[2 ] ) p ro cess

a t d e fec t s . Th e l a t t e r p ro cess i n v o lv es b o u n d s t a t es , o r

t r ap s , i n t h e en erg y g ap r esu l t i n g f ro m th e d e fec t s i n t h e

sem ico n d u c to r la t t ice . F o r n o n d e g en era t e ly d o p ed s i li -

co n , t h e S RH p ro cess i s g en era l l y d o m in an t , an d co n -

seq u en t l y t h e ca r r i e r l i f e t imes a r e co n t ro l l ed b y t h e

d en s i t i e s o f d e fec t s i n t h e s i l i co n . To ach i ev e o p t imal

d es ig n s o f s i l i co n d ev i ces an d c i r cu i t s, we m u s t k n o w th e

u p p er b o u n d s , o r fu n d am en ta l limi t s , f o r ca r r i e r

l i f e t imes , an d we mu s t u n d er s t an d h o w th e s i l i co n

p ro cess in g i n f l u en ces t h e so lu b i l i ty o f d e fec t s t h a t d e f i ne

th ese b o u n d s .

M a n y p rev io u s p u b l i ca t i o n s [3] h av e d i sc lo sed

measu red d ep en d en c i es o f ca r r i e r l i f e t ime o n d o p in g

d en s i t y i n s i l i co n . Ho wev er , b ecau se mo s t o f t h ese

o b s e r v e d d e p e n d e n c i e s r e s u lt e d f r o m n o n - f u n d a m e n -

t a l d e fec t s , e .g . imp u r i t i e s , an d b ecau se t h e d a t a were

o b t a in ed b y a v a r i e ty o f ex p er imen ta l t ech n iq u es , a l l o f

741


2/7

742 J. G . FOSSUM

which are subject to poss ibly s ignif icant er ror [3] , the

l i f e t imes a r e w ide ly sca t te r ed . Thus in f e r ences f r om

these m easurem ents are not gene ral ly s ignificant an d are

of ten mis lead ing .

To pr ovide a be t te r under s tanding of l i f e t ime- vs -

doping depend enc ies in s i l i con , w e deve lop a theo r e t ica l

mode l based on the pr esence of a s ing le de f ec t in non-

degener a te s i l i con . The mode l de r ives f r om the ana logy

to chem ica l r eac t ions of the dopan t - induced f or mat ion of

la t t i ce de f ec t s [4 , 5 ]. The de te r mina t ion o f the de f ec t

( so lu te ) dens i ty f o l low s f r om a char ac te r iza t ion , based

on the law of mass ac t ion , o f the doped semiconduc tor

( so lvent ) in the r mal equi l ib r ium a t the pr eva len t t em-

per a tur e o f de f ec t f or mat ion T~. D er ived expr es s ions f or

the dependence of the de f ec t dens i ty ( as sumed to be

homogeneous ) on the doping dens i ty in bo th n- type and

p- typ e s i l icon , i . e. Nd(No) and N d( N A ) , w hich depend

on Tf , enable the de te r mina t ion of the des i r ed depen-

denc ies of minor i ty ho le and e lec t r on l i f e t imes ,

:p(No)

and rn(NA).

To suppor t the mode l and to as say impor tan t de f ec t s ,

w e compar e theor e t ica l p r ed ic t ions w i th measur ements

of r ecombina t ion l i f e time- ver sus - doping dependenc ies .

By f ocus ing our a t t en t ion on those dependenc ies show -

ing the longest car r ier l i fe t im es ( -* 1 msec) repo r ted for

s i li con , w e de tec t a poss ib ly f undam enta l de f ec t in

s i li con . The d e f ec t , w hich i s acceptor - type , appear s to b e

f undamenta l ( unavoidable ) because ( a ) i t i s r ead i ly

ava i lab le in u n l imi ted quant i ty f or inco r por a t ion in to the

s i l icon la t t ice , and (b) i ts solubil i ty is fundam ental ly

r e la ted to the doping dens i ty and , a l though i t s dens i ty

can be in fluenced by the pr ocess ing , i t cannot be to ta l ly

e l imina ted f r om the la t t i ce . The de f ec t could the r e f or e be

a vacancy , bu t becau se s ingula r vacanc ies a r e uns tab le in

s i l i con a t r oom temper a tur e [ 6] , i t i s mor e l ike ly a

d ivacancy o r a vacancy complex [ 7] . O ur an a lys i s is no t

conc lus ive in th i s r egar d , bu t the r esu l t s a r e cur ious and

should induce addi t iona l w or k .

The ana lys i s sugges ts tha t the de f ec t i s mor e so luble in

n- type than in p- type s i l i con , and hence tha t the f un-

damen ta l l im i t f o r e lec t r on l i f e t ime i s longer than tha t f or

hole l i f e t ime a t a g iven doping dens i ty . The pr eva len t ,

min imum dens i ty of the de f ec t , w hich de f ines these

l imi t s, i s f r ozen- in a t the t emper a tur e TI be low w hich

the de f ec t i s v i r tua l ly immobi le in the s i l i con la t t i ce .

Compar i sons of our mode l p r ed ic t ions w i th u l t ima te

car r ier l i fe t ime data reveal that T1 is in the range 300-

400 C, and thereby emphasize the s ignif icance of low-

temper a tur e pr ocess ing in the f abr ica t ion of s i l i con

devices r equi r ing long or w e l l - cont r o l led ca r r ie r

l i fe t imes. For example, the ef f icacy of optimal annealing

schedules i s s t r e s sed , and f ur the r mor e a poss ib le phys i -

ca l mechanism under ly ing such annea l ing i s desc r ibed .

F ina l ly , the bas i s o f our ana lys i s , w hich i s concen-

t r a ted on a poss ib ly fundamenta l d e f ec t , i s used to

qua l i t a t ive ly d i scuss common ca r r ie r l i f e t imes in

s i l i con[ 3] tha t a r e de f ined by non- f undamenta l de f ec t s .

N LYSIS

Co nside r fi r s t n- typ e s i l icon. Kendall [8] has rep or ted

measur ements of rp(No) made on s i l icon w af e r s tha t had

and D . S. LEE

never been hea ted above 450° C. A l though i t w as no t

s tated whether or not this s i l icon was f loat-zone, we

sur mise tha t i t w as because the l i f e times a r e much longer

than those f ound in c r uc ib le - gr ow n s i l i con[ 3] . These

measur ements , w hich involved doping dens i t i e s r anging

from about 10 T cm - 3 to about 10T cm - 3 , y ie lded , to our

know ledge , the long es t r ecombina t ion hole l i f e t imes ever

r epor ted f or s i l i con . The da ta a r e desc r ibed w e l l by the

emp ir ical exp ress ion [9]

r p ( N o ) = t o p

N o

1 +

NOD

(1)

where ~'op= 4 .0 x 10-4 sec an d Noo = 7.1 x 10t~ cm-3; (1)

show s tha t r p va r ies inver se ly w ith N o w hen the r es i s -

t ivi ty is suf f ic iently low. Th e high-res is t iv i ty l i fe t ime

impl ied by ( 1) i s cons i s ten t w i th the long hole l i f e t imes

measur ed b y G r a f t and P ieper [ 10] .

The rp(ND) dependence in (1) is a lso consis tent with

r esu l t s o f d e te r min a t ions [ l 1 ], based on measur ed e lec -

t r i ca l cha r ac te r i s t i c s , o f r ecombina t ion hole l i f e t ime in

the base r eg ions of super ior

p+nn÷

f loat-zone s i l icon

sola r ce l l s f abr ica ted a t Sandia Labor a tor ies . The con-

s i s tency of these da ta and of those of G r a f t and

P ieper [10] , t aken f r om ca r e f u l ly pr ocessed d evices , w i th

those r epor ted by K enda l l [ 8] , t aken f r om unpr ocessed

mate r ia l , por tends the pr edominance of a f undam enta l

(unavoidable) defect whose prevalent , minimum solu-

b i l i ty in s i l i con r esu l t s r ead i ly f rom pr oper pr ocess ing .

The asympto t ic behavior of ( 1) i s cha r ac te r i s t i c o f a

s ingula rly ion ized , acceptor - type de f ec t in n- type s i li con .

W e now demo ns t r a te th is by de r iv ing a theor e t ica l mode l

f or the depen dence N d( N o) of the so lubi l i ty of the

def ec t on the doping dens i ty . W e then use the mod e l and

the ca r r ie r l i f e time da ta to es t ima te the phys ica l p r oper -

t i e s o f th i s poss ib ly f undamenta l de f ec t . This d e f ec t

char ac te r iza t ion a l so involves our de r iva t ion of the

s o l u i l i t y Na NA)of the same def ec t in p- type s i l i con ,

w hich leads to ~ 'n (N A ) pr ed ic t ions tha t conf or m c r ude ly

w i th measur ements [ 12 , 13] of u l t ima te r ecombina t ion

electron l i fe t imes in s i l icon.

W e emphas ize tha t th i s ana lys i s i s based on the

assumpt ion of

homogeneous

dis t r ibu t ions of de f ec t s in

the s i li con . Consequ ent ly i t i s appl icab le on ly to r eg ions

w hose vo lumes a r e much la r ge r than liNd such tha t

macroscopic ( vo lume- aver age) ca r r ie r l i f e times f a i th f u l ly

char ac te r ize the r ecombina t ion and gener a t ion of ho le -

e lec t r on pa i r s .

Cons ide r the de f ec t a s a so lu te tha t i s in ( chemica l )

equi l ib r ium w i th bo th a nondegener a te ly doped semi-

conduc tor ( the so lvent ) and an ex te r na l phase w i th con-

s tan t ac t iv i ty , w hich impl ies an un l imi ted sour ce f or the

( fundamental) defect[5] :

K

K

D e ~ D ~ D + h +.

(2)

In the revers ib le react ion s ind icated in (2) , D e and D s

r epr esen t the un ionized de f ec t in the ex te r na l and semi-

conduc tor phases , and D and h ÷ r epr esen t the ion ized


3/7

A physical model for the dependence of carrier lifetime on doping density in nondegenerate silicon

defect and a hole in the semiconductor. This approach,

which i s equivalent to tha t based on the Shockley-Last

theo ry [4], is used here beca use of the physical insight i t

provides and bec ause o f the simplifying assumptions i t

reveals; these a t tr ibutes w i l l becom e apparent as the

analysis is described.

The law of m ass action applied to (2) gives [5]

N d - P = K 2 N d ° = K 2 K , N d = - K 3

where N a - and N a° are the ionized and unionized defect

densit ies in the semiconductor,

N a

i s the constant

defect density in the external phase, and K~, K2, and K3

are mass -action (temperature-d ependen t) constants.

Combining (3) with the hole-electron mass-action law for

the nondegenerate semicond uctor ,

P N = n~

743

defec t . I t involves the degeneracy fac tor gd of the defec t

energy state, which is unknown, and i ts energy level Ed,

whose value relative to the intrinsic Fermi level E~

depends on Ts since both Ed and E~ vary with tem-

perature.

Because of the uncertainty in the energy-band struc-

ture a t T = TI, we can only estimate the relationship

between N 2 and Nd- , which we do as follows. For the

(3) particular cas e

N o < n i ( T s) ,

the hole and electron den-

sit ies at T = Ts are nearly equal, and for simplicity we

assume that approximately half of the defects are ion-

ized; this would be true, for example, if gd = 1 and

Ed - E~ at T = TI. Thu s fo r this particular case, Nd ---

N a - ( N o n O - - N ~ o ,

as implied b y (6 ). Since

N d ° and

N To are independent of N o, their relationship, subjec t to

the stated assumption, holds generally f o r a l l N t ~ Hence

using (6) , we can w rite the explicit dependence of

(4)

N d ( = N d - + N d°)

o n N o a s

where P and N are the hole and electron densit ies and n~

is the intrinsic carrier density, and with the condition for

electrical neutrali ty in the sem icond uctor dop ed with N o

donor impurit ies,

p - N + N o + - N a - = O ,

yields the following expression of the equilibrium solu-

bi li ty o f the ionized d efec t in the sem iconductor :

N D N D 2

where N ~o is the solubili ty of the ionized de fect in the

undoped semiconductor . In wri t ing (6) we have recog-

nized the c om mo n condi tions tha t a l l the do nor im-

purities are ionized, i.e. N o ~ - N o + , and tha t Nd - < ND in

extrinsic (n-typ e) si l icon.

No te that (6) is applicable at temperatures T high

enough that the defe ct is sufficiently mobile in the si l icon

to support the equilibrium condition (2) and to allow Nd-

and Nd to vary in accordance wi th (3) . During the

sil icon grow th or processing, wh en T finally drops below

some temperature Tf, the defe ct beco me s virtually im-

mobi le , and i ts densi ty i s hence f roz en in accordanc e

with (6) evaluated at T = TI; i.e. n~ = m ( T = T ~ ) and

N ~ o = N S o ( T = T s ) . The temperature TI, which we

estimate later, is thus the effective temperature of for-

mation of the defe ct and is fundamentally related to the

mobili ty o f the defe ct in si l icon. N ote th at if the si l icon is

cooled too quickly (quenched) dur ing the growth or

processing, the defects may freeze-in at a temperature

higher than Ts. The result ing d efect de nsity is then higher

than the fund am ental l imit corresponding to (6)

evaluated at T = Tg.

The total defect density Nd that is frozen-in at T = T~,

and hen ce affects ~-p at room temperature, is the sum o f

Na- given by (6) and Nd . We must therefore de termine

Nn at T = TI, which, as indicated b y (3), is independent

of Nt~ The general relationship betw een Nd and N~-,

i .e. the degree of ionization of the defect, is quite com-

plex, ev en i f we assume a sin gle energy level for the

N d( N o) = N d° { l + ~ - ~ + [ l + ( ~ n~ ) 2] '~ 2}

(7)

whe re N d and n~ are evaluated at T -- Tt. This m odel

implies carrier lifetimes that are self-consistent with the

(5) hole l ifetime data referred to earlier, as we will show,

and its underlying assumptions are no more uncertain

than the properties of the defec t at T = T~. N ot only may

these properties differ from tho se at room temperature,

but the nature of the d efect m ay be entirely d ifferent[14];

for example, si l icon vacancies, which may be stable at

(6) T -- T I [6] , might precipita te to form divacancies as T

drops to room temperature.

A simi lar t rea tment of the densi ty of the same defec t

in p-type sil icon yields

We have checked the genera l i ty of (7) and (8) by

examining the effects of assuming N f = f N a - where

f # 1 . For 10 2 < f -< l0 s , we h ave fou nd tha t the l i fet ime-

vs-doping dependencies implied by (7) and (8) do not

change significantly in the doping range where the SRH

proce ss predominates. Thus s ubject to the uncertainties

associated with the defect, (7) and (8) seem to be a

reasonable model.

In (7) and (8),

N a

which depends only on TI, is the

solubili ty of the neutral defect in si l icon at the effective

temperature of defect formation. M inimization of the

la tt ice f ree energy a t T = TI with respect to N 2

yields[15]

- E o

Na°(T i ) = N s , exp ( - -~ / )

(9)

where Ns~ is the atomic density of si l icon (=5×

1022 cm 3) and Ea is the activation ene rgy required to

form the defect in the si l icon latt ice.

The dependencies on doping dens ity given by (7) and

(8) are illustrated in Fig. 1. For low doping densities

[No , N a '~ ni(Ti)], both (7) and (8) yield N d

~- - 2Nd° T I ) .


4/7


5/7

A physical model for the dependence of carrier lifetime on doping density in nondegenerate silicon

At room temperature, the defe ct level appears to be

significantly above mid-gap, and the hole and electron

capture parameters seem to be comparable. U sing next

the high-No asymptot ic dependence of ¢p(No) given by

(1), we estimate n~ T1)= 5 x 1015cm 3, or

T¢=330°C. (17)

I f we a ssume tha t c o - - c , °~ 10-Scm3-sec -1 , wh ich i s

reasonable f or an accepto r-typ e trap located significantly

above mid-gap, then (14) and (15) imply tha t

Nd° Ti)-

10 cm 3. This solubili ty, with (9) and (17), corres pon ds

to

Eo

= 1.4 eV. (18)

Al though we cannot , based only on these crude est i -

mates o f the defe ct properties, unequivocally identify the

nature of the defect, i t is interesting to note that these

properties are characterist ic of a fundamental v aca ncy

com plex in si l icon. F or exam ple, (1 8) is approximately

the activation energ y of the si l icon diva canc y[7], and

(14)-(16) are reasonable estimates of properties of the

(dominant) acce ptor- type level associated w ith the

divacan cy [14] .

The significance of our estimate fo r Ts in (17) is not

manifested b y the absolute value of the temperature, but

by the fac t tha t i t i s

low

(300--400°C) relative to com m on

silicon proces sing temperatures. This result thus

emphasizes the importance o f low-temperature proce ss-

ing in the fabrication of si l icon devices requiring long or

well-controlled carrier lifetimes.

Th e fundam ental-limit carrier lifetime dependencies

zp(No) and r,(NA) derived from (7), (8), (10), (11), (12),

103

102

1

1

+io~++

745

(14), (15) and (1 6) are plotted in Fig. 2. Ultimate values of

mea sured carrier l ifetimes rep orted in [8], [11 ], and [12]

are included in the figure to indicate the general

theoretical-experimental con sistency . N ote that r ,(NA ) is

dramat ica l ly higher than rp(No=NA), and tha t the

difference increases, in accord anc e with Fig. 1, with

increasing doping densi ty as long as the SRH process

dominates the recombination. The z,(NA ) data[12] for

NA > 10 7 cm -3, which show a sharp decre ase in z, with

increasing NA, signify the onset of significant band-band

Auger-impact recombination[16]. Suc h an onset can not

be seen in the

zp No)

data[8], [11] in the nondegenerate

doping range because of the lower SRH zp(No) due to

the higher

Nd No).

We emphasize that the experimental data plotted in

Fig. 2 represent, in m ost cases , only the

ultimate

carrier

l ifetimes measured[8], [10], [11], [12], [13], and hen ce are

comparable to the predictions of our fundamental-l imit

l ifetime theo ry. It should be n oted tha t [13] also reports

measured values of

r

as high as 7000/zsec in 1 f l -cm,

p-type, float-zone sil icon. The discrepancy between

these l ifetimes and the predictions of our analysis is due

either to experimental error, which is likely in very-long-

lifetime measurements[17], [18], or to crud e assumptions

in our model . This discrepancy does not however dero-

gate our carrier l ifetime theory, which predicts the

observed t rends-- i .e , a s t rong

rp No)

dependence show-

ing hole l ifetimes that are considerably sho rter than the

electron l ifetimes shown by a relatively weak z,(NA)

depe nden ce--an d thus which encourages addit ional work.

DISCUSSION

The analysis described in the preceding section, which

is based on measurements that have yielded the

longest

\ A u g e r [ 1 6 1

• r p (ND )

\

I I l

1 0TM 10TM 1017 10TM

N D . N A ( C m - 3 )

Fig. 2. Fundam ental-limit min ority hole [¢p(N o)] and e lectron [ r n ( N A ) ] lifetime dependencies on

oping ensity

(continuous curves) in nondegenerate silicon as predicted by ou r mod el. The po ints represent the lon gest measured

hole (circles)J8], [11] and electron (squares)[12] recombination ifetimes reported. T he b roke n line characterizes the

band-ban d A uger-impact electron lifetime[16], wh ich underlies the sharp decreas e in the measured r,(Na ) for

N > 1 17

cm 3.


6/7

746

J. G . FossuM and D . S. LEE

recombination carrier l ifetimes ~ 1 msec) repo rted fo r

sil icon, has implied the significance of a possibly fun-

damental , acceptor-type defect, e.g. a si l icon divacancy.

Such a def ect is fundamental unavoidable), and hen ce

defines the fundamental l imits for carrier l ifetimes in

nondegenerate si l icon, becau se i t is readily available fo r

incorporation into the latt ice and because i ts solubili ty is

fundamentally determined b y the dopa nt typ e and den-

sity.

A theore tical model for the depen dence o f the defec t

solubili ty on doping density, for both n-type and p-type

nondegenerate si l icon, was derived. This model is base d

on the assumptions, in accordan ce with the l ifetime da ta,

that the defect can be only singularly ionized and that i t

produces only a s ingle important energy level . In con-

junction w ith the SRH recombination-generation theo ry,

our model yields theoretical dependencies of ultimate

carrier lifetimes on doping density in nond egenerate sili-

con in which Auger-impact processes are insignificant.

These results predict a strong, inverse dependence for

the fundamental l imit r p N o ) , but only a weak depen-

dence for r , NA). Conse quent ly in the fundamental

l imits, r , in p-typ e sil icon will excee d rp in n-ty pe sil icon

having the same doping density. This difference is most

pron oun ced in low-resist ivity, non degenerate si l icon.

Our model indicates the resultant defect density is

dependent on an effective temperature of defect for-

mation as defined by the si l icon growth or processing.

The minimum value TI of this temperature, wh ich results

in the minimum defec t densi ty and hence the fundamen-

tal-limit lifetimes, is the temperature below w hich the

defect is virtually immobile in the silicon lattice. Com-

parisons of our model predictions with the ult imate

lifetime data reveal that Tr is relatively low in the range

300--400°C. Th us the fi nal low-temperature proce ssing

steps in the fabrication of si l icon devices are cri t ical

when the fundamental-l imit l ifetimes are being ap-

proac hed . Th ese low-temperature steps mus t be defined

so as to ensure that the fundamental defect freezes-in at

T~, and does not assume a larger solubili ty defined by a

higher effective temperature of formation.

This conclusion suggests that annealing the si l icon at

Tf for a t ime long enough to allow the equilibrium,

detailed-balance conditions 2) and 3) to obtain will

minimize the density of the fundamental defect. It im-

plies then that, with respect to this defect, T~ is an

optimal annealing temp erature. Annealing at tem-

peratures abo ve Ts will result in a higher solubility o f the

defec t in accordance wi th our theory. Anneal ing a t tem-

peratures below Tt will be ineffective since the def ect is

immobile. This physical explanation is compatible with

experimental determinations[18], [19] of optimal anneal-

ing temperatures that yield near-ult imate e lectron

lifetimes in as-grown, p-type, float-zone sil icon. Maxi-

mum values of Tn generally resulted when the annealing

temperature wa s abou t 450°C, which is not inconsistent

wi th our

es t i ma t e

of TI.

Our analysis is based on the longest carrier l ifetimes

measured in nondegenerate si l icon, and hence pertains,

as we have suggested, to a possible fundamental defect.

Generally, carrier lifetimes in silicon devices are drama-

tically shorter than these ult imate values[3]. Shorter

carrier l ifetimes w ill result fro m non-optimal processing,

e.g. quenching at a high temperature, that induces higher

densit ies o f fundamental defe cts and/o r substantial den-

sit ies o f other, non-fundamental defec ts, e.g. impurit ies.

l ies and Soclof[20] inferred l ifetimes from measure-

men ts of minority-carrier diffusion lengths in n-typ e and

p-type si l icon, and found in both types monotonic

reductions in lifetime with increasing doping density. The

values of l i fe t ime they repor ted are much lower than

those plotted in Fig. 2, and thus imply the existence of

non-fundamenta l defec ts , probably hea vy meta ls , which

are com m on in si licon devices. Such de fects, e.g . gold

and copper, are generally amphoteric in si l icon and tend

to, be.cause of their ionization interaction, create traps o f

the type opposite to the doping impurity[21]. T his results

in strong inverse dependencies between carrier l ifetime

and doping density in both n-type and p-type sil icon l ike

those in [20].

Graft and Pieper[18] investigated the dependence of

carrier l ifetime in si l icon on heat treatment and found

varying results from one sil icon sample to the next

because of different densit ies of dislocations and

vacancy c lusters , both of which can be avoided and

hence are non-fundamenta l defec ts . They a lso found tha t

in float-zone silicon the carrier lifetime-vs-annealing

temperature characteristic has several maxima at

different temperatures in different samples. How ever ,

there i s a com mo n maximum at about 350°C. This opt i-

mal annealing temperature is cons istent with ou r estima-

tion of T~ and there fore m ay be associated with the

fundamenta l defec t . The other maxima are probably

associated w ith non-fundamental defects .

To achieve the fundamental-l imit l ifetimes that we

have discussed, the non-fundamental defe cts must be

removed or avoided by proper processing pr ior to the

final low-temperature steps that minimize the funda men -

tal defe ct density. Su ch processing a) requires float-zone

sil icon, which is relatively free of non-fundamental

defec ts associated with oxy gen and carbon ; b) excludes

high-temperature oxidation, w hich can create detrimental

stacking faults and other dislocations and defects; and

c) includes gettering, e .g . by p hos pho rus diffusion,

which can effectively remove certain impurit ies, e.g.

heavy meta ls .

Acknowledgments--The

authors thank H. J. Stein for a clarifying

discussion of defec ts in silicon and A. N eugroschel and F. A.

Lindholm for helpful comments.

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SSE Vol . 2 5 N o . 8 - - E

a physical model for the dependence of carrier lifetime on doping density in nondegerate silicon

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