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