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Microscopic theory of atomic diffusion mechanisms in silicon
]?hysica 127B (1984) 401--407 North-Holland, Amsterdam
IV~CROSCOk~IC T H E O R ' Y O F A T O m i C DEFFUSION M E C H A ~ S N I S
IN S I L I C O N
R. C A R * , P.J. KE/.,LY**, A. OSITI'YAMA'~ and S.T. PANT'ELIDES [BM Thomas J. Walson ~e,c;earch Center, Yorktowtz Heights, NYl0598, USA
Selt-interstitials in $i are known tO migrate atheroaalty at very tow temperatures (--4 K), In eontra~;t, at high temperatures (l t00-1.600 K). ~lf-diffvzlon has an activation energy of - 5 eV. We de.qcril~e re~:ttlts of self-consistent Green's-funetloa total energy eaieulations which, for the first time, provide detailed mJcroSeople und¢lrstandingof lhe mechanisms underlying these phenem~,e and reconcile the contrasting low- and high-temperature data.
A t o m i c diffusion is a fundamental solid-state process which plays a central role in the design of electronic devices. Yet, in semiconductors, espeeialiy Si, atomic*diffusion processes are poorly understood. The most fundamental of these processes is self-diffusion, namely the motion of Si atoms in the Si lattice, in this paper we focus on the question of self-diffusion in Si. We review brightly the key experimental inform~ttion and then present our ~ecent theoretical results [1] which allow us to resolve several existlng puzzles and reconcile seemiagly inconsistent data. A t low temperature,';, irradiation experiments have given valuable information about atomic motiort. High-ene~rgy electrons create vacancies which have been identified [2]. The vacancy migration barriers were obtained from thermal anneaging studies aad fotmd to b~ ~tnall and slightly d e p e n d e n t on the charge state (0.2-0.3 ,eV) [2]. Self-interstitials, on the other hand, have not been detected directly. Instead, after irradiation of Al-doped Si, int:erstitial AI was detected by etee~ron paramagnetic resonance (EPR) at roughly the same concentration as vacancies. Anatogous effects were seen in Ga- and B-doped Si [2]. The conclusion was that self-inter,;titials in p-type material are highly mobile, since they are * Present ~ddre.ss:$1SSA,Strada CC.ostieruII, Trieate. Italy. ~'* Present address.' Max-Planck Imtitut. Stuttg.u't, W. Germany. ~' Present address: Department of Physics, Uni,~ersity of T~kyo, Japan.
able to find a n d replace the substitutional impurities. D a t a suggest that this high mobility persists down to 4 K, so that migration is, for all practical purposes, athermal. At high temperatures ( t 2 0 0 - 1 6 5 0 K ) , selfdiffusion m e a s u r e m e n t s usiL~g radioactive tracer Si atoms found that the serf-diffusion coefficient D is well described by an Arrllenius relationship of the form D
=
Do exp(-O/kT),
(1)
with O ranging from 4.1 to 5,1 eV [3-5], Part of the uncertainty' i~n the value of O stems from the fact that different types of experiments are used in different temperature regimes. It appears that the high values, closer to 5 eV, are obtained from m e a s u r e m e n t s at relatively high temperatures ( 1 4 0 0 - I 6 5 0 K), whereas the smaller values, closer to 4 eV, are obtained from relatively lower temperatures ( 1 2 0 0 - 1 4 0 0 K ) [3-5]. A n o t h e r notable result of the tracer measurements is that the preexponential D O is larger than typical values in metals [3, 4]. Diffusion at high temperatures is generally believed to be mediated by thermally created intrinsic defects. Over the years, there have ~een advocates of vacancies, divacancies, .,~elfinterstitials, etc. [3--5]. No consensus has been achieved, however, because supporting m'gumeets in favor of different mechanisms derive from anaIysis of indirect experimental observations (e.g. of oxidation-induced stacking faults, gold diffusion, etc.) [3-5] which are also i~t well understood. Theory has not [:een particularly
0378-4363/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics PubIishing Division)
402
R. Cae et a L I Atomic diffttsi~n mechanisms in Si
helpful either, because calculations of the relevant quantities axe very difficult. If only o n e m e c h a n i s m is active, the diffusion coefficient is given by eq. (1), where
O = H,~+ H~.
(2)
Here, H~ and HM are the formation and migration enthalpies, respectively, of the defect mediating self-diffusion. All exis~.iug calculations for the vacancy find Q ' s that are too small (23 eV) [3-5]. For the interstitial, existing O ' s are either toe small (1-3 eV) [3-5] or too large [6]. Similarly, if only one mechanism is active, D O is proportional to exp(S~+$M), whc.~e S F and S M are the formation and migration entropies, respecti,~ely. Because D o is observed ~o be large, it has been inferred that the defe~:t o r defects mediating self-diffusion must r,~ave large iormation o r migration entropies [3-5]. Before we present our new theoretical results that bear on the issue of self-diffusion, it is useful to inquire what values of individual O ' s (i.e., H~+ H M) would be consistent with the experimenlal observations. It is possible that two or -14
i
i
-16
-22
- 2 4
--
6
7 ; oooo/'r
Fig,
I. Logarithmic plot
-
S of
-
9
[K'~
O~,a=D 1exp(-Ol,tkTl+
D~cxp.l'-O:lkT) with CRI=4cV and O_.=6¢V and D.:D, = 5:1 (solid linel, compared wi~'h z plot of D ~ De
exp(~O/kT} w~th O=SeV (dashed line) denaonstrallng ~ha! two terms may app..'ar as a sin~de exponcznt~al ~n a ~ivcn temperalure range.
more mechanisms are active simultaneou.sly and the s u m of two terms like eq, (1) still appears to have the £orm of a single exponential in the temperature range of the observations. ¥,le have explored this possibility and found that a n u m b e r of mechanisms with O ' s ranging between roughly 4 and 6 eV can give rise to a ou~rve that is consistent with observations. A s an illustration, in fig. 1 we show the result of two m e c h a n isms having Q = 4 and 6 eV, respectiv,:.qy, and preexponentiats in a ratio of 5 : 1. W e s~:,e that a slope of 5 eV results for most of the tempera~:ure range of interest, except for an upturn toward smaller slopes at low temperatures. A collection of the experimental data suggests that the selfdiffusion activation elzergy Q in Si behaves in just this m a n n e r [3],
2. 'I'heory Green's-function methods were shown in the last six years [7.8] to be very powerful in treating isolated point defects in infinite host crystals, without the limitations imposed by cluster or supeleell approximations. Until recently, it was possible to calculate only the charge densities, single-particle energy levels and wavefunctions. W e have now extended this developme~t to allo,~Y the calculation o~ total energies for various ator:aic arrangements [19]. W e use density functional theory and the Iocabdensiiy approximation with norm-conserving pseudopotentials [10]. W e reproduced the single-particle energy levels obtained by earli~::r Green's-functior. calculations [7, 8] and carried o~t a series of tests specifically designed to check the accuracy of the totalenergy results ~.l I]. T h e main result of technical significance is that, for accurate total-energy calculations, it is necessary to retain Oasis orbitals on considerably more shells ~ff atoms than is needed if one is onty interested in single-particle energy levels. For example, for the relaxed vacancy, it was necessary to retain orbitals on 46 atoms (five shells of a~oms around the central site), which corresponds to having orbital,'; on the
R, Car et al, I Atomic diffm'ion mecP.anisrr~ in Si "
T
i
i
i
Long-range lattice relaxation was included by the semiempirieal Keating model [12, 13]. The overall uncertainty in the cal~,,lations ranges from 0.5 to 3, eV and arises prim'drily from limitations imposed by computer capacity.
5
~4
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3
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I 20
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NUMBER OF ATOMS INCLUDED i
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T h e main results of our calculations are described in ref. 1. W e summ,'u-ize them here and discuss briefly how they relate to the experimen. tel data mentioned in the Introduction. O u r resuits for the formation enthalpy of the vacancy (V) are shown in fig. 3a. This figure reflects the well-known ne;gative-U properties of the vaca~.~cy, na~neiy that V + is not the equilibrium charge state for any Fermi-level position [13, 14], Fig. 3a looks somewhat different from the figures in ReL 12, because it incorporates the fact that the formation energy of a neutral vacancy does not depend on the Fermi energy, since no electron transfer to or from it occurs. For DuE purposes here, the most significant result is that
2
VACANCY ±
i
i
6
8
;
i 10
,_; ~ 12
.
p 14
16
M A X . D I S T A N C E O F A T O M S I N C L U D E D (a,u.)
~ s '~-~Fig. 2. Calculated tormati~n cmhalpy for a .,;elf-iater~titih] in Si at the tetrahedral sit,." as a Inaction of (ai the number (,f atoms and (b) the radial distance of atoms 0 a which ba~,is orbltals are retzined. In each ca~e, the two et~rvcs corre~pom~ to two different expressions for the total enel.gy, as discussed in ref. 11. T h e triangles m a r k complete shells of atoms.
9
~'~ . - ~
,
L. . . . . . . .
~.-~;'-
7
first and second neighbors of each displaced atom. For the self-irlterstifial at the teb'ahedralsite, without any lattice relaxation, the convergenoe rate is shown in fig. 2, where the upper curvt~ refers to the total-energy calculated using the eigenvalue expression and the lower curve to the: total-energy obtained with the kinetic-energy expression [11]. All the total-energy results described in this paper were obtained by allowing the ne,~,rest neighbors to relax to their optimum position.
,
1
T
SELF-INTERSTITIAL
÷+~
.........
.-~
. . . . . . . . . . . . . . . . . . . . . . .
0.0
0,2
0.4
0.6
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Er (oVt F{g..%. Calculated f n r m a d o n cnthMpi,:s of v~rious eh~r~c
stutes o" (a) O,e vacancy: and (b) the selr-intetstitiul a~. three ~ifferent s i t ~ (see fig. 4) in SL
41~
R. Car et a L t AtOmiCdiffusion mechanirms in Si
the formation enthalpy is large, of order 5 eV for all charge states [15], "Ibis is a rather r~ovel result and provides a natural connection with the lowtemperature annealing data according to which the vacancy migration energy is small, 0.2-0.3 eV. Using this migration energy (computer capacity did not allow us to obtain a reliable theoretical migration energy), the re:;ulting Q for self-diffusion is within the experim eLtal range, In contrast, earlier estimates of Q for the vacancy were small (of order 2 - 3 eV), leading to t h e need to postulate a much larger migration enthalpy at high temperatures. T h e large values of t h e vacancy formation enthalpies resulting from o u r calculations may raise concern about the resulting concentrations, if comparisons are to be made with estimates extracted from quenching experiments [16]. However, concentrations depend on both formation enthalpy and formation entzopy. | , a n n o o and A~lan [t7] recently carried out semiempirical calculations of the vacancy formation entropy atld fotmd that it can be quite large. W e should point out, however, that we are unable at present to calculate the formation enthaEpy of rebonded models of the vacancy in the spirit o1 rebonding known to occur on surfaces [I8]. Rebonded or "extended" vacancies may have smaller formation enthalpies and larger migration enthalpies with a r~et Q still of order 5 eV. If that were to occur, both simple and rebonded vacancies would mediate self-diffusion. O u r results fur the self-interstitial (I) reveal a very ric,h structure, In search of the equilibrium configura~.ion, we investigated several sites~ illustrated schematically in fig. 4. Our results for three sites are shown in fig. 3b, We see that the self-interstitial, just like the' vacancy, exhibits negatiue-U properties in the sens,~ that ~" is not Ihe equilibrium chark;e state for any Fermi-level position. Note, however, that the negative-U property is now accomplished by motion of the ~nterstitial to different siles as the charge state changes. Such charge-stale ifJstabil;ties underlie atl~erm;tl migration according tt~ the BourgoinCorbett mechanism [19]. In this particular case, in p-lype Si, the equilibrium site is the T site, where the imerstitial is in the 2" charge state and has an e m p t y loc:tlized stale of Tz symmetry at
°;:'o'o
.....
r.
/?
°°'"
oa,~/÷~ P""\.
Fig. 4. Schematic definitions o[ several configurationsof the self-interstitial in SI. T: Tetrahcdr'.d sitel H: Hcxag,onal site; B: Bond-cenlered site: S: (100"~split configuration; S': (] t0) split configuration. In the enleu[ations, lhe lattleu around the interstitial is allowed to relax. Severn] paths arc defined by the s~.tes~nvolved:TH is z path in the Jow-dcnsit'.¢channels, as is the HH path. The "r'BTH path eoIP.bincs three highsymmetry S~les and corresponds to motion along a (l] }) direction it~volvingcondnuoos exchange with atoms at lattice sites. The TS path corresponds to motion along a (10(3) direction, also involving continuous exchange with atoins a! laUice sites. The BS path is a simple path with the extra atom. winding through the bonds as proposed in tel 23, Other possible paths, not shown. ~e TS' and BS', both of which av," interstlti~lcy paths.
just about the conduction-band edge. Capture of one or two extra electrons m a k e s the center unstable against J a h n - T e l l e r distortions. Motion away from the ']" site result:; in a splitting of the T~ state, lowering of one c o m p o n e n t into the gap, and a linear gain in energy. O n the basis of this notion, we cons=:ructed a theory of carriercapture-enhanced and athermal migration which allows us to draw total-energy craves between high-symmetry sites and to determine when athermal migration is possible [2(I]. W c found that athermal migration is just barely possible along the T H path (fig, 4), which was proposed in ref. 21 and supported by ref. 22, A t h e n n a l migration is possible along the TB and T B T H paths (fig. -4), and likely along the TS path (fig. 4). The total-energy curves for the TH, TB and T B T H paths are shown in fig, 5, Athermal mi~ gration is not. however, possible along the BS path as proposed in ref, 23, We were unable to
405
R. Car et al. / Atomic ¢l~fu~ion raechania,z~" in Si
6
T
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~ [=r,rRtNSlC
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Fig. 5. Tcrl.al-energycurves for thrci: ¢I.iffererltcharge states of the interstitial along three dilf~t.'.ntpaths for three different Fermi levels, explore the TS' path because of the low symmetry of the S' configuration. According to fig. 5, atherrnal migra6oa is also possible in intrinsic and n-type material, but the final product of such migration has not been identified [2]. W e note, however, that, in e-type material,, a Si atom starting at an 14 site would most likely skip the high-energy T site and move in the channel along a relatively straight path (shown as H H in fig. 5), as envisioned in ref. 21. Finally, we turn to the high-temperature selfdiffusion regime where the Fermi level is, for most doping levels, in the midgap region. Fig. 5 reveals that just about all the charge states exist at several sites with roughly the same formation enthalpy and that migralion energies are very small. For self-diffusion, however, one needs paths that involve exchange with atoms at lattice sites, such as the TB, T B T H , TS, and TS' paths. We have n a m e d such paths interstitialcy paths since they contribute ~o sel~-diffusion via the interstitialcy mechanism. Other paths, such as the T H and BS paths, are referred to as simple paths. W h e n combined with interstitlalcy paths, simple paths can a u g m e n t self-diffusion, A s fig. 5 reveals, self-interStltials contribute to self-
diffusion with effective Q ' s of order 6-6.5 eV, which, in view of the theoretical uncertainty, overlaps the range of observed values. A s in the case of the vacancy, the large formation energy and small migration energy for the interstitial is quite r~ovel, It provides a natural explanation for the contrasting low- and higt,temperature data: At low T ( - 4 K ) , selfinterstitials are created by irradiation. The smaIT migration battlers are then required for athermal migration..At high temperatures, self-diffusion is limited primarily by the need to create selfinterstitials thermally. O n the other hand, as in the case of the vacancy, '.he large formation enthalpy may raise concerns about the net concentration of self-interstitials. In this case, tb,ere is no information ~ o m quenching measurements, but the large value observed for .Do suggests large formation and/or migration entropies. Large entropy can arise from the multitude of configurations and charge states of ~he selfinterStitial with roughly the same formation enthalpy. W e note that this result is remi~i~;cent of the proposal [3] that the self-interstitial exists in 'extended' forms, a notion that was introduced in order to account for the large preexponenfial in
406
R. Car ez aL / ~'! romie cliffaSian mechanism*" in Si
terms of configurational entropy. Our calculat i o n s p r o v i d e explicit d e s c r i p t i o n s of t h e v a r i o u s c o n f i g u r a t i o n s of the self-interstitial a n d also p r o v i d e d e t a i l e d m o t i o n a l m o d e l s , w h i c h are l a c k i n g in t h e discussions of ' e x t e n d e d ' f o r m s [3]. A g a i n , h o w e v e r , ,as in t h e case of t h e vacancy, o u r r e s u l t s d o not rule o u t the possibility of m o r e e x t e n d e d a n d r e b o n d e d f o r m s og the selfinterstitial, w h i c h m a y h a v e s m a t l e r f o r m a t i o n e n t h a l p i e s , l a r g e r m i g r a t i o n e n t b z l p i e s , a n d still c o n t r i b u t e t o self-diffusion with r o u g h l y the s a m e a c t i v a t i o n e n e r g y Q.
4. C o n c l u s i o n s In this p a p e r we d e s c r i b e d brief/y so~,~ne o f the m a i n results of e x t e n s i v e self-consistent G r e e n ' s f u n c t i o n t o t a l - e n e r g y calculations wh(~s,', o b j e c tive w a s t o e l u c i d a t e the microscopic m e c h a n i s m s of the self-intcrstitial's l o w - t e m p e r a t u r e u t h e r real m i g r a t i o n a n d of h i g h - t e m p e r a t u r e selfdiffusion in St. T h e s e results p r o v i d e a new set of ideas in t e r m s of which to u n d e r s t a n d t h e s e processes. T h e y have significant c o n s e q u e n c e s for the i n t e r p r e t a t i o n of several e x p e r i m e n t s , including i m p u r i t y d i f f u s i o n , - o x i d a t i o n - i n d u c e d s t a c k i n g fitults, etc. [3]. Such e x p e r i m e n t s h a v e been i n t e r p r e t e d o n the basis of q u a l i t a t i v e ass u m p t i o n s a b o u t f o r m a t i o n e n t h a l p i e s a n d entropies, i n c l u d i n g oversimplified a s s u m p t i o n s a b o u t the e n e r g y - l e v e l s t r u c t u r e of the selfinterstitial [ 3 . 2 4 ] , and various a s s u m p t i o n s a b o u t other, u n r e l a t e d p h e n o m e n a . C o n c l u s i o n s b a s e d on such analysis may, as a result, a p p e a r in conflict with o u r results. It is n e c e s s a r y at this point t o r e e x a m i n e the original d a t a a n d c o m pare t h e m d i t : c t l y with the n e w q u a n t i t a t i v e results, u n h a m g e r e d by e x t r a n e o u s a s s u m p t i o n s , Such a n a l y s i s Gf a v a i l a b l e e x p e r i m e n t a l d a t a will be r e p o r t e d e l s e w h e r e ,
Acknow:ledgement This w o r k was s u p p o r t e d in part by O N R Contract N00014-80-C-0679.
Re~erences [1] R, Car, P.J, Kelly, A. Oshiyama and S.T. Pantelides, Phys. Ray. Lett, 52 (1984L [2] For recent reviews and references to the original paper~,, see G.D. Watkins. In!a, Phys. Conf. Ser, 23 (I975) I, and G.D. Watkins. in; Deep Centers in Semiconductors. S.T. Pantelides. ed. (Gordon and Breach, New York, in press). [3] For recent reviews, see W. Frank, Festko~rperpmbleme 21 (t981); 221; W. Frank, U. Gosd.e, H Mehrer, and A, Seegec, in: Diffusion in Solids Ii. A,S,, Nowiek and G: I~ureh, ads. (Academic Press, New "~/~xk, in press), See also refs. ,6 and 5. [4] Atomic Diffusion in Semiconductors, D. Shaw, ed. (Plenum. New York, t973L [5] M. Lannoo and 3, Bourgoin, Point Defects in SemiConductors I (Springer, Berlin, 1981). [6] J,A, Van Vechten, Inst. Phys. C o a l Scr. 31 (1977) 441. [7] J. Bernhole. N.O. Lipari and S.T. Pantelides, Phys. Ray. Leu. 41 (1978) 895. G.A. P_,aratland M. Schluter. Phys, Ray, Lett. 41 (1978) 892, [8]. For a review of Green's function reztrlts see M. Saheffler, Festkorperprobleme 22 (1982) ~ 15. [9} We have implemented the formalism of A,R. Williams, P.J. Feibelman and N.D. Lang. Phys. Rev. B2fi (1982) 5433. which is particularly suitable when atoms are displaced from lattice sites, [101 For recent reviews of the theory and a0plieations to perfect ,n'ystals and surfaces, see the serle:~of articles in The Inhomogeneous Electron Gas, N,D:. March :rod S. Landqvist. ads, (Plenum, New York, 1984). [tl] R. Car. P.J. Kelly, A. Oshiyama and $.T. Pantelides, Phys, Ray. B, to be published. [t2] P.N. Keating, Phys. Rev, 145 (1966) 637. [13] O.A. Baraff. E.O. Kanu and M. Sf:hluter, P~ys. Ray. B21 (1950) 5662. [141 G.D, Watkins and .I.R. Troxell, Phys, Ray, Lett. 44 (1980) 593. [151 The formation energies o[ the vacancy are most likely upper bounds because more tlexible ba,;is sets, not currently feasible, m~y produce additiona'~ lattice relaxation. [16] See, e,g,, A. Chantre, M. Keehouane and D. BoIs. Physica 116B (L983) 547. [171 M, Lannoo and G. Allan, Phys. Ray. B 25 (1952) ~-089. [~8l K,C. Pand$y, Phy~. Ray. Lett. 49 (1~182) 223. [191 J. Bourgoie and ].W. Corbett. Pljys, Len. 38A (10721 135. [2Ol S.T. Pantelides, A. Oshiyama. R. Car and P.J, KcBy, Phys. Ray. B 30 (1984) 2260. f21] S.T. Pantdides, L leaner. M, Seheffier and J.P. Vignerott, Physica 116B (1983) 19; ,,;ee also S.T. Pantelides, in: Methods and Materials for Microelectror5¢ Technology. J. Bargon, ed. (Plenum, Nev~ York, 198,*). [22] Y, Bar-Yam and J.D. Joannopeulos. Phys. Ray. 1L~tl. 52 (1984) 1129: see ref, 20 for discussion of the eraSerion for athermal migration used ia this paper.
R. Car et a L / Atomic diffusion mcchanism,~ in Si ['23] G.D. Watkins, R.P. Messmer, C. Weigel, D. IPeak and J.W, Corbett, Phys. Rev. Lett. 27 (1971) ],573; C. Weigcl. D. P,z-~k,3".W. Corbett, R.P. Mcssmer and G.D. Watkins~ Phys. Rev. B 8 (1973) 2906. [24] For example, in much of the analysis based on selfinterstitials, an energy-level structure based on qua.lira-
407
tire arguments for an unhonded interstitial given by Blount in 1959 [E.I, Blount, J. Appl. Phys. 30 (19.59) 1218], Our results reveal that the energy-level structure of the sell-interstitial dependS sensitively on the site. N o n e of the sites we investigated so far supports a
negatively-charged state.
IV~CROSCOk~IC T H E O R ' Y O F A T O m i C DEFFUSION M E C H A ~ S N I S
IN S I L I C O N
R. C A R * , P.J. KE/.,LY**, A. OSITI'YAMA'~ and S.T. PANT'ELIDES [BM Thomas J. Walson ~e,c;earch Center, Yorktowtz Heights, NYl0598, USA
Selt-interstitials in $i are known tO migrate atheroaalty at very tow temperatures (--4 K), In eontra~;t, at high temperatures (l t00-1.600 K). ~lf-diffvzlon has an activation energy of - 5 eV. We de.qcril~e re~:ttlts of self-consistent Green's-funetloa total energy eaieulations which, for the first time, provide detailed mJcroSeople und¢lrstandingof lhe mechanisms underlying these phenem~,e and reconcile the contrasting low- and high-temperature data.
A t o m i c diffusion is a fundamental solid-state process which plays a central role in the design of electronic devices. Yet, in semiconductors, espeeialiy Si, atomic*diffusion processes are poorly understood. The most fundamental of these processes is self-diffusion, namely the motion of Si atoms in the Si lattice, in this paper we focus on the question of self-diffusion in Si. We review brightly the key experimental inform~ttion and then present our ~ecent theoretical results [1] which allow us to resolve several existlng puzzles and reconcile seemiagly inconsistent data. A t low temperature,';, irradiation experiments have given valuable information about atomic motiort. High-ene~rgy electrons create vacancies which have been identified [2]. The vacancy migration barriers were obtained from thermal anneaging studies aad fotmd to b~ ~tnall and slightly d e p e n d e n t on the charge state (0.2-0.3 ,eV) [2]. Self-interstitials, on the other hand, have not been detected directly. Instead, after irradiation of Al-doped Si, int:erstitial AI was detected by etee~ron paramagnetic resonance (EPR) at roughly the same concentration as vacancies. Anatogous effects were seen in Ga- and B-doped Si [2]. The conclusion was that self-inter,;titials in p-type material are highly mobile, since they are * Present ~ddre.ss:$1SSA,Strada CC.ostieruII, Trieate. Italy. ~'* Present address.' Max-Planck Imtitut. Stuttg.u't, W. Germany. ~' Present address: Department of Physics, Uni,~ersity of T~kyo, Japan.
able to find a n d replace the substitutional impurities. D a t a suggest that this high mobility persists down to 4 K, so that migration is, for all practical purposes, athermal. At high temperatures ( t 2 0 0 - 1 6 5 0 K ) , selfdiffusion m e a s u r e m e n t s usiL~g radioactive tracer Si atoms found that the serf-diffusion coefficient D is well described by an Arrllenius relationship of the form D
=
Do exp(-O/kT),
(1)
with O ranging from 4.1 to 5,1 eV [3-5], Part of the uncertainty' i~n the value of O stems from the fact that different types of experiments are used in different temperature regimes. It appears that the high values, closer to 5 eV, are obtained from m e a s u r e m e n t s at relatively high temperatures ( 1 4 0 0 - I 6 5 0 K), whereas the smaller values, closer to 4 eV, are obtained from relatively lower temperatures ( 1 2 0 0 - 1 4 0 0 K ) [3-5]. A n o t h e r notable result of the tracer measurements is that the preexponential D O is larger than typical values in metals [3, 4]. Diffusion at high temperatures is generally believed to be mediated by thermally created intrinsic defects. Over the years, there have ~een advocates of vacancies, divacancies, .,~elfinterstitials, etc. [3--5]. No consensus has been achieved, however, because supporting m'gumeets in favor of different mechanisms derive from anaIysis of indirect experimental observations (e.g. of oxidation-induced stacking faults, gold diffusion, etc.) [3-5] which are also i~t well understood. Theory has not [:een particularly
0378-4363/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics PubIishing Division)
402
R. Cae et a L I Atomic diffttsi~n mechanisms in Si
helpful either, because calculations of the relevant quantities axe very difficult. If only o n e m e c h a n i s m is active, the diffusion coefficient is given by eq. (1), where
O = H,~+ H~.
(2)
Here, H~ and HM are the formation and migration enthalpies, respectively, of the defect mediating self-diffusion. All exis~.iug calculations for the vacancy find Q ' s that are too small (23 eV) [3-5]. For the interstitial, existing O ' s are either toe small (1-3 eV) [3-5] or too large [6]. Similarly, if only one mechanism is active, D O is proportional to exp(S~+$M), whc.~e S F and S M are the formation and migration entropies, respecti,~ely. Because D o is observed ~o be large, it has been inferred that the defe~:t o r defects mediating self-diffusion must r,~ave large iormation o r migration entropies [3-5]. Before we present our new theoretical results that bear on the issue of self-diffusion, it is useful to inquire what values of individual O ' s (i.e., H~+ H M) would be consistent with the experimenlal observations. It is possible that two or -14
i
i
-16
-22
- 2 4
--
6
7 ; oooo/'r
Fig,
I. Logarithmic plot
-
S of
-
9
[K'~
O~,a=D 1exp(-Ol,tkTl+
D~cxp.l'-O:lkT) with CRI=4cV and O_.=6¢V and D.:D, = 5:1 (solid linel, compared wi~'h z plot of D ~ De
exp(~O/kT} w~th O=SeV (dashed line) denaonstrallng ~ha! two terms may app..'ar as a sin~de exponcznt~al ~n a ~ivcn temperalure range.
more mechanisms are active simultaneou.sly and the s u m of two terms like eq, (1) still appears to have the £orm of a single exponential in the temperature range of the observations. ¥,le have explored this possibility and found that a n u m b e r of mechanisms with O ' s ranging between roughly 4 and 6 eV can give rise to a ou~rve that is consistent with observations. A s an illustration, in fig. 1 we show the result of two m e c h a n isms having Q = 4 and 6 eV, respectiv,:.qy, and preexponentiats in a ratio of 5 : 1. W e s~:,e that a slope of 5 eV results for most of the tempera~:ure range of interest, except for an upturn toward smaller slopes at low temperatures. A collection of the experimental data suggests that the selfdiffusion activation elzergy Q in Si behaves in just this m a n n e r [3],
2. 'I'heory Green's-function methods were shown in the last six years [7.8] to be very powerful in treating isolated point defects in infinite host crystals, without the limitations imposed by cluster or supeleell approximations. Until recently, it was possible to calculate only the charge densities, single-particle energy levels and wavefunctions. W e have now extended this developme~t to allo,~Y the calculation o~ total energies for various ator:aic arrangements [19]. W e use density functional theory and the Iocabdensiiy approximation with norm-conserving pseudopotentials [10]. W e reproduced the single-particle energy levels obtained by earli~::r Green's-functior. calculations [7, 8] and carried o~t a series of tests specifically designed to check the accuracy of the totalenergy results ~.l I]. T h e main result of technical significance is that, for accurate total-energy calculations, it is necessary to retain Oasis orbitals on considerably more shells ~ff atoms than is needed if one is onty interested in single-particle energy levels. For example, for the relaxed vacancy, it was necessary to retain orbitals on 46 atoms (five shells of a~oms around the central site), which corresponds to having orbital,'; on the
R, Car et al, I Atomic diffm'ion mecP.anisrr~ in Si "
T
i
i
i
Long-range lattice relaxation was included by the semiempirieal Keating model [12, 13]. The overall uncertainty in the cal~,,lations ranges from 0.5 to 3, eV and arises prim'drily from limitations imposed by computer capacity.
5
~4
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3
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NUMBER OF ATOMS INCLUDED i
Ib)
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T h e main results of our calculations are described in ref. 1. W e summ,'u-ize them here and discuss briefly how they relate to the experimen. tel data mentioned in the Introduction. O u r resuits for the formation enthalpy of the vacancy (V) are shown in fig. 3a. This figure reflects the well-known ne;gative-U properties of the vaca~.~cy, na~neiy that V + is not the equilibrium charge state for any Fermi-level position [13, 14], Fig. 3a looks somewhat different from the figures in ReL 12, because it incorporates the fact that the formation energy of a neutral vacancy does not depend on the Fermi energy, since no electron transfer to or from it occurs. For DuE purposes here, the most significant result is that
2
VACANCY ±
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i
6
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i 10
,_; ~ 12
.
p 14
16
M A X . D I S T A N C E O F A T O M S I N C L U D E D (a,u.)
~ s '~-~Fig. 2. Calculated tormati~n cmhalpy for a .,;elf-iater~titih] in Si at the tetrahedral sit,." as a Inaction of (ai the number (,f atoms and (b) the radial distance of atoms 0 a which ba~,is orbltals are retzined. In each ca~e, the two et~rvcs corre~pom~ to two different expressions for the total enel.gy, as discussed in ref. 11. T h e triangles m a r k complete shells of atoms.
9
~'~ . - ~
,
L. . . . . . . .
~.-~;'-
7
first and second neighbors of each displaced atom. For the self-irlterstifial at the teb'ahedralsite, without any lattice relaxation, the convergenoe rate is shown in fig. 2, where the upper curvt~ refers to the total-energy calculated using the eigenvalue expression and the lower curve to the: total-energy obtained with the kinetic-energy expression [11]. All the total-energy results described in this paper were obtained by allowing the ne,~,rest neighbors to relax to their optimum position.
,
1
T
SELF-INTERSTITIAL
÷+~
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. . . . . . . . . . . . . . . . . . . . . . .
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stutes o" (a) O,e vacancy: and (b) the selr-intetstitiul a~. three ~ifferent s i t ~ (see fig. 4) in SL
41~
R. Car et a L t AtOmiCdiffusion mechanirms in Si
the formation enthalpy is large, of order 5 eV for all charge states [15], "Ibis is a rather r~ovel result and provides a natural connection with the lowtemperature annealing data according to which the vacancy migration energy is small, 0.2-0.3 eV. Using this migration energy (computer capacity did not allow us to obtain a reliable theoretical migration energy), the re:;ulting Q for self-diffusion is within the experim eLtal range, In contrast, earlier estimates of Q for the vacancy were small (of order 2 - 3 eV), leading to t h e need to postulate a much larger migration enthalpy at high temperatures. T h e large values of t h e vacancy formation enthalpies resulting from o u r calculations may raise concern about the resulting concentrations, if comparisons are to be made with estimates extracted from quenching experiments [16]. However, concentrations depend on both formation enthalpy and formation entzopy. | , a n n o o and A~lan [t7] recently carried out semiempirical calculations of the vacancy formation entropy atld fotmd that it can be quite large. W e should point out, however, that we are unable at present to calculate the formation enthaEpy of rebonded models of the vacancy in the spirit o1 rebonding known to occur on surfaces [I8]. Rebonded or "extended" vacancies may have smaller formation enthalpies and larger migration enthalpies with a r~et Q still of order 5 eV. If that were to occur, both simple and rebonded vacancies would mediate self-diffusion. O u r results fur the self-interstitial (I) reveal a very ric,h structure, In search of the equilibrium configura~.ion, we investigated several sites~ illustrated schematically in fig. 4. Our results for three sites are shown in fig. 3b, We see that the self-interstitial, just like the' vacancy, exhibits negatiue-U properties in the sens,~ that ~" is not Ihe equilibrium chark;e state for any Fermi-level position. Note, however, that the negative-U property is now accomplished by motion of the ~nterstitial to different siles as the charge state changes. Such charge-stale ifJstabil;ties underlie atl~erm;tl migration according tt~ the BourgoinCorbett mechanism [19]. In this particular case, in p-lype Si, the equilibrium site is the T site, where the imerstitial is in the 2" charge state and has an e m p t y loc:tlized stale of Tz symmetry at
°;:'o'o
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/?
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Fig. 4. Schematic definitions o[ several configurationsof the self-interstitial in SI. T: Tetrahcdr'.d sitel H: Hcxag,onal site; B: Bond-cenlered site: S: (100"~split configuration; S': (] t0) split configuration. In the enleu[ations, lhe lattleu around the interstitial is allowed to relax. Severn] paths arc defined by the s~.tes~nvolved:TH is z path in the Jow-dcnsit'.¢channels, as is the HH path. The "r'BTH path eoIP.bincs three highsymmetry S~les and corresponds to motion along a (l] }) direction it~volvingcondnuoos exchange with atoms at lattice sites. The TS path corresponds to motion along a (10(3) direction, also involving continuous exchange with atoins a! laUice sites. The BS path is a simple path with the extra atom. winding through the bonds as proposed in tel 23, Other possible paths, not shown. ~e TS' and BS', both of which av," interstlti~lcy paths.
just about the conduction-band edge. Capture of one or two extra electrons m a k e s the center unstable against J a h n - T e l l e r distortions. Motion away from the ']" site result:; in a splitting of the T~ state, lowering of one c o m p o n e n t into the gap, and a linear gain in energy. O n the basis of this notion, we cons=:ructed a theory of carriercapture-enhanced and athermal migration which allows us to draw total-energy craves between high-symmetry sites and to determine when athermal migration is possible [2(I]. W c found that athermal migration is just barely possible along the T H path (fig, 4), which was proposed in ref. 21 and supported by ref. 22, A t h e n n a l migration is possible along the TB and T B T H paths (fig. -4), and likely along the TS path (fig. 4). The total-energy curves for the TH, TB and T B T H paths are shown in fig, 5, Athermal mi~ gration is not. however, possible along the BS path as proposed in ref, 23, We were unable to
405
R. Car et al. / Atomic ¢l~fu~ion raechania,z~" in Si
6
T
H
r
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r
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.
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Fig. 5. Tcrl.al-energycurves for thrci: ¢I.iffererltcharge states of the interstitial along three dilf~t.'.ntpaths for three different Fermi levels, explore the TS' path because of the low symmetry of the S' configuration. According to fig. 5, atherrnal migra6oa is also possible in intrinsic and n-type material, but the final product of such migration has not been identified [2]. W e note, however, that, in e-type material,, a Si atom starting at an 14 site would most likely skip the high-energy T site and move in the channel along a relatively straight path (shown as H H in fig. 5), as envisioned in ref. 21. Finally, we turn to the high-temperature selfdiffusion regime where the Fermi level is, for most doping levels, in the midgap region. Fig. 5 reveals that just about all the charge states exist at several sites with roughly the same formation enthalpy and that migralion energies are very small. For self-diffusion, however, one needs paths that involve exchange with atoms at lattice sites, such as the TB, T B T H , TS, and TS' paths. We have n a m e d such paths interstitialcy paths since they contribute ~o sel~-diffusion via the interstitialcy mechanism. Other paths, such as the T H and BS paths, are referred to as simple paths. W h e n combined with interstitlalcy paths, simple paths can a u g m e n t self-diffusion, A s fig. 5 reveals, self-interStltials contribute to self-
diffusion with effective Q ' s of order 6-6.5 eV, which, in view of the theoretical uncertainty, overlaps the range of observed values. A s in the case of the vacancy, the large formation energy and small migration energy for the interstitial is quite r~ovel, It provides a natural explanation for the contrasting low- and higt,temperature data: At low T ( - 4 K ) , selfinterstitials are created by irradiation. The smaIT migration battlers are then required for athermal migration..At high temperatures, self-diffusion is limited primarily by the need to create selfinterstitials thermally. O n the other hand, as in the case of the vacancy, '.he large formation enthalpy may raise concerns about the net concentration of self-interstitials. In this case, tb,ere is no information ~ o m quenching measurements, but the large value observed for .Do suggests large formation and/or migration entropies. Large entropy can arise from the multitude of configurations and charge states of ~he selfinterStitial with roughly the same formation enthalpy. W e note that this result is remi~i~;cent of the proposal [3] that the self-interstitial exists in 'extended' forms, a notion that was introduced in order to account for the large preexponenfial in
406
R. Car ez aL / ~'! romie cliffaSian mechanism*" in Si
terms of configurational entropy. Our calculat i o n s p r o v i d e explicit d e s c r i p t i o n s of t h e v a r i o u s c o n f i g u r a t i o n s of the self-interstitial a n d also p r o v i d e d e t a i l e d m o t i o n a l m o d e l s , w h i c h are l a c k i n g in t h e discussions of ' e x t e n d e d ' f o r m s [3]. A g a i n , h o w e v e r , ,as in t h e case of t h e vacancy, o u r r e s u l t s d o not rule o u t the possibility of m o r e e x t e n d e d a n d r e b o n d e d f o r m s og the selfinterstitial, w h i c h m a y h a v e s m a t l e r f o r m a t i o n e n t h a l p i e s , l a r g e r m i g r a t i o n e n t b z l p i e s , a n d still c o n t r i b u t e t o self-diffusion with r o u g h l y the s a m e a c t i v a t i o n e n e r g y Q.
4. C o n c l u s i o n s In this p a p e r we d e s c r i b e d brief/y so~,~ne o f the m a i n results of e x t e n s i v e self-consistent G r e e n ' s f u n c t i o n t o t a l - e n e r g y calculations wh(~s,', o b j e c tive w a s t o e l u c i d a t e the microscopic m e c h a n i s m s of the self-intcrstitial's l o w - t e m p e r a t u r e u t h e r real m i g r a t i o n a n d of h i g h - t e m p e r a t u r e selfdiffusion in St. T h e s e results p r o v i d e a new set of ideas in t e r m s of which to u n d e r s t a n d t h e s e processes. T h e y have significant c o n s e q u e n c e s for the i n t e r p r e t a t i o n of several e x p e r i m e n t s , including i m p u r i t y d i f f u s i o n , - o x i d a t i o n - i n d u c e d s t a c k i n g fitults, etc. [3]. Such e x p e r i m e n t s h a v e been i n t e r p r e t e d o n the basis of q u a l i t a t i v e ass u m p t i o n s a b o u t f o r m a t i o n e n t h a l p i e s a n d entropies, i n c l u d i n g oversimplified a s s u m p t i o n s a b o u t the e n e r g y - l e v e l s t r u c t u r e of the selfinterstitial [ 3 . 2 4 ] , and various a s s u m p t i o n s a b o u t other, u n r e l a t e d p h e n o m e n a . C o n c l u s i o n s b a s e d on such analysis may, as a result, a p p e a r in conflict with o u r results. It is n e c e s s a r y at this point t o r e e x a m i n e the original d a t a a n d c o m pare t h e m d i t : c t l y with the n e w q u a n t i t a t i v e results, u n h a m g e r e d by e x t r a n e o u s a s s u m p t i o n s , Such a n a l y s i s Gf a v a i l a b l e e x p e r i m e n t a l d a t a will be r e p o r t e d e l s e w h e r e ,
Acknow:ledgement This w o r k was s u p p o r t e d in part by O N R Contract N00014-80-C-0679.
Re~erences [1] R, Car, P.J, Kelly, A. Oshiyama and S.T. Pantelides, Phys. Ray. Lett, 52 (1984L [2] For recent reviews and references to the original paper~,, see G.D. Watkins. In!a, Phys. Conf. Ser, 23 (I975) I, and G.D. Watkins. in; Deep Centers in Semiconductors. S.T. Pantelides. ed. (Gordon and Breach, New York, in press). [3] For recent reviews, see W. Frank, Festko~rperpmbleme 21 (t981); 221; W. Frank, U. Gosd.e, H Mehrer, and A, Seegec, in: Diffusion in Solids Ii. A,S,, Nowiek and G: I~ureh, ads. (Academic Press, New "~/~xk, in press), See also refs. ,6 and 5. [4] Atomic Diffusion in Semiconductors, D. Shaw, ed. (Plenum. New York, t973L [5] M. Lannoo and 3, Bourgoin, Point Defects in SemiConductors I (Springer, Berlin, 1981). [6] J,A, Van Vechten, Inst. Phys. C o a l Scr. 31 (1977) 441. [7] J. Bernhole. N.O. Lipari and S.T. Pantelides, Phys. Ray. Leu. 41 (1978) 895. G.A. P_,aratland M. Schluter. Phys, Ray, Lett. 41 (1978) 892, [8]. For a review of Green's function reztrlts see M. Saheffler, Festkorperprobleme 22 (1982) ~ 15. [9} We have implemented the formalism of A,R. Williams, P.J. Feibelman and N.D. Lang. Phys. Rev. B2fi (1982) 5433. which is particularly suitable when atoms are displaced from lattice sites, [101 For recent reviews of the theory and a0plieations to perfect ,n'ystals and surfaces, see the serle:~of articles in The Inhomogeneous Electron Gas, N,D:. March :rod S. Landqvist. ads, (Plenum, New York, 1984). [tl] R. Car. P.J. Kelly, A. Oshiyama and $.T. Pantelides, Phys, Ray. B, to be published. [t2] P.N. Keating, Phys. Rev, 145 (1966) 637. [13] O.A. Baraff. E.O. Kanu and M. Sf:hluter, P~ys. Ray. B21 (1950) 5662. [141 G.D, Watkins and .I.R. Troxell, Phys, Ray, Lett. 44 (1980) 593. [151 The formation energies o[ the vacancy are most likely upper bounds because more tlexible ba,;is sets, not currently feasible, m~y produce additiona'~ lattice relaxation. [16] See, e,g,, A. Chantre, M. Keehouane and D. BoIs. Physica 116B (L983) 547. [171 M, Lannoo and G. Allan, Phys. Ray. B 25 (1952) ~-089. [~8l K,C. Pand$y, Phy~. Ray. Lett. 49 (1~182) 223. [191 J. Bourgoie and ].W. Corbett. Pljys, Len. 38A (10721 135. [2Ol S.T. Pantelides, A. Oshiyama. R. Car and P.J, KcBy, Phys. Ray. B 30 (1984) 2260. f21] S.T. Pantdides, L leaner. M, Seheffier and J.P. Vignerott, Physica 116B (1983) 19; ,,;ee also S.T. Pantelides, in: Methods and Materials for Microelectror5¢ Technology. J. Bargon, ed. (Plenum, Nev~ York, 198,*). [22] Y, Bar-Yam and J.D. Joannopeulos. Phys. Ray. 1L~tl. 52 (1984) 1129: see ref, 20 for discussion of the eraSerion for athermal migration used ia this paper.
R. Car et a L / Atomic diffusion mcchanism,~ in Si ['23] G.D. Watkins, R.P. Messmer, C. Weigel, D. IPeak and J.W, Corbett, Phys. Rev. Lett. 27 (1971) ],573; C. Weigcl. D. P,z-~k,3".W. Corbett, R.P. Mcssmer and G.D. Watkins~ Phys. Rev. B 8 (1973) 2906. [24] For example, in much of the analysis based on selfinterstitials, an energy-level structure based on qua.lira-
407
tire arguments for an unhonded interstitial given by Blount in 1959 [E.I, Blount, J. Appl. Phys. 30 (19.59) 1218], Our results reveal that the energy-level structure of the sell-interstitial dependS sensitively on the site. N o n e of the sites we investigated so far supports a
negatively-charged state.