General Model of Diffusion of Interstitial Oxygen in Silicon and Germanium Crystals

Transcription

1 Generl Model of Diffusion of Interstitil Oxygen in Silicon nd Germnium Crystls Vsilii Guskov Institute of Solid Stte nd Semiconductor Physics, P. Brovk str. 17, Minsk, Belrus A theoreticl modeling of the oxygen diffusivity in silicon nd germnium crystls both t norml nd high hydrosttic pressure hs been crried out using moleculr mechnics, semiempiricl nd b initio methods. It ws estblished tht the diffusion process of n interstitil oxygen tom (O i ) is controlled by the optimum configurtion of three silicon (germnium) toms nerest to O i. The clculted vlues of the ctivtion energy E ( Si ) = 2.59 ev, E ( Ge) = 2.05 ev nd pre-exponentil fctor D 0 (Si) = 0.28 sm 2 s -1, D 0 (Ge) = 0.39 sm 2 s -1 re in good greement with experimentl ones nd for the first time describe perfectly n experimentl temperture dependence of the O i diffusion constnt in Si crystls (T= C). Hydrosttic pressure (P 80 kbr) results in liner decrese of the diffusion brrier ( P E ( P ) = ev kbr -1 for Si crystls). The clculted pressure dependence of O i diffusivity in silicon crystls grees well with the pressure enhnced initil growth of oxygen-relted therml donors. PACS numbers: Jt, Ct,81.40.Vw, 87.15Vv Development of theoreticl methods of determining the diffusivity of toms in crystls is of gret interest not only from fundmentl, but lso from prcticl point of view. The resoning is tht the tomic diffusion in crystls occurs very often under extreme conditions (very high tempertures, fields of stress etc) nd tht essentilly impedes, mkes expensive or even impossible n experimentl reserch. However till now there re mny obscure questions relting to the microscopic mechnism of diffusion in crystls, whenever migrtion of n impurity tom involves the brking nd forming of covlent bonds. It is common knowledge the diffusion of interstitil oxygen toms in silicon crystls is of crucil importnce in the processes of oxygen gglomertion (formtion of therml donors) nd in the gthering of metllic impurities in industril processing of silicon nd, s result, the experimentl mesurements of the diffusivity of oxygen in silicon hs received much ttention. As pointed out by Mikekelsen 1 most experimentl dt cn be consistently fit over wide temperture rnge ( K) by single expression of the form D = 0.13exp( 2.53 ev / kbt ) sm 2 s -1. The expression hs been obtined by fitting to dt from six independent experiments. This expression is generlly believed to be the intrinsic diffusion constnt involving oxygen jumping from bond-center to one of the six nerest bon-center cites. Severl theoreticl efforts hve ttempted to clculte the diffusion brrier, but different results were obtined. Thus the clculted vlues of the brrier re rnging from 1.2 ev 2, 2.0 ev 3 up to 2.3 ev 4, 2.5 ev 5. All these clcultions (except 6 ) ssume the sddle point configurtion for diffusion in (110) plne nd the midwy between the two bond-center sites. The remining degrees of freedom nd the position of the other Si toms were determined by totl-energy minimiztion. The resulting totl energy, mesured from the energy of the equilibrium configurtion, results the dibtic

2 ctivtion energy for diffusion. Using empiricl intertomic potentils Jing nd Brown 6 hve concluded tht the sddle point of O i migrtion is pst the midpoint, but their conclusion hs clled in question in 7. Moreover Rmmoorthy nd Pntelides 7 hve offered tht seemingly simple oxygen jump is ctully complex process cn be properly described in terms of coupled brriers by energy hypersurfce with n ctivtion energy is rnging from 2.2 ev up to 2.7 ev. In this connection it should be pointed out tht the clcultion of n ctivtion brrier is importnt, but not n ultimte point of the theoreticl description of diffusion constnt. The complete clcultion of the diffusion coefficient guesses clcultion nd the pre-exponentil fctor. Unfortuntely in the mjority of previously published works the preexponentil fctor ws not evluted t ll, nd the clculted vlue in 6 differs from the experimentl one more thn on n order of mgnitude. In this Letter, the simultion of diffusion of interstitil oxygen (Oi) in silicon nd germnium crystls under norml nd hydrosttic pressure (HP) is reported. The ctivtion brrier nd pre-exponentil fctor hve been clculted nd re in excellent greement with experimentl ones. To the best of my knowledge, no effects of HP on the Oi diffusivity hve been considered yet. Let us consider the physicl prmeters determining the diffusion process of n tom in crystl. Modeling by the method of csul wnderings results in the following generl expression for the diffusion constnt: 2 d N D = et Γ, (1) 2dim where d is diffusion jump distnce, N et is the number of the equivlent trjectories leving the strting point, dim is the dimension of spce, Γ is the verge frequency of jumps on the distnce d. In the cse of system consisting of N toms, using the rection-rte theory 8, the vlue of Г my be written in the following form N i = 1 ( o) i λ 1 i= 0 E Γ= exp, 2π N (2) kbt λ ( b) i where E is the dibtic potentil energy difference between the sddle point nd the stble one, λ i re the eigenvlues of the mtrix (with respect to mss-weighted internl coordintes) 2 Kij = Ueff / fi f j,ueff ( f1,..., f m ) denotes the potentil function s function of the internl degrees of freedom. The indices (b) nd (o) indicte tht the corresponding quntities re evluted t the sddle point nd locl minimum, respectively. Thus, the diffusion constnt D is determined by the following diffusion prmeters: the length of diffusion jumps (d), the diffusion brrier ( E ), the number of equivlent wys leving the strting point of diffusion jumps (N et ) nd the eigenvlues mtrix ( λ i ). The clcultion of diffusion prmeters ws performed in cluster pproximtion. For comprison with the previous clcultions different methods such s empiricl potentil (MM2), semiempiricl (AM1, PM3, PM5) nd b initio (RHF, LDA) hve been used for the clcultion of the cluster totl energy. Depending on the method of totl energy clcultion the cluster size ws vried 3 from 17 Si toms (b initio methods) up to 10 Si toms (semiempiricl nd empiricl potentil methods). Individul oxygen toms occupy interstitil bond-center (BC) position in silicon nd to diffuse by jumping between the neighboring BC sites. Hence the strting nd the finl points of the diffusion jump correspond to the equilibrium configurtion of n interstitil oxygen tom (O i ) in silicon. The clculted equilibrium configurtion of O i nd the locl vibrtion frequency of the symmetric stretching mode (B 1 ) tkes the following vlues: d Si-O =1.63 Å (6-31G ** ), Å

3 (MM2), 1.61 Å (PM5), Si-O-Si =161.6 o (6-31G ** ), o (MM2), 171 o (PM5), n O = 1214 cm -1 (6-31G ** ), 1091 cm -1 (AM1), 1078 cm -1 (LDA) nd re in good greement with experimentl 9, 10 nd recently clculted ones 11, 12. The clculted vlue of the potentil brrier for the rottion of O i round Si-Si xis equls DE j 20 mev (PM5). As DE j is much less thn k B T (t diffusion tempertures) n interstitil oxygen tom my jump on ny of six nerest Si-Si bonds nd, hence, in Formul (1) the prmeters N et = 6 nd d=1.9 Å. In the course of trnsition of O i tom from one equilibrium configurtion in nother the brking of old nd formtion of new covlent Si-O bonds tkes plce. The process of reconfigurtion of n electronic subsystem will occur in the cse when oxygen nd neighboring silicon toms owing to therml fluctutions get in the region of configurtion spce G (bounded by the criticl surfce S G ) where the electronic reconfigurtion leds to lowering of the crystl totl energy. It is cler tht for the given position of the oxygen tom there re mny configurtions of silicon toms for which the electronic reconfigurtion cn occur but ll of these configurtions re differ in the totl energy of crystl. Since the diffuse constnt exponentilly depends on the diffusion brrier (2) we should select the miniml vlue of E E = inf[ E ( S ) E ( O)], (3) cl G cl where Ecl ( S G ) nd Ecl ( O) re the totl cluster energy on the surfce S G nd in locl minimum O (equilibrium O i configurtion), respectively. In our simultion the vlue of E ws clculted s follows. The oxygen tom ws displced from the equilibrium O i configurtion long trjectory in the direction of the nerest Si-Si bond. Along the given trjectory the totl cluster energy hs been clculted nd mong the set of clculted trjectories the extreme trjectory stisfying condition (3) ws selected. It is significnt tht for the extreme trjectory (3) the sddle point of O migrtion is displced from the midpoint of the pth both for Si nd Ge crystls nd the displcement is fr more for Ge crystls. The simultion hs reveled n importnt fct for understnding of the diffusion process. Ecl ( SG ) nd hence E depends on the number of the nerest to O i silicon toms (n) involved in the minimiztion of the cluster totl energy. The diffusion brrier E ( n) decreses nd tends to 2 ev with increse of the number of Si toms involved in minimiztion (Fig 1.). Therefore, first of ll, we should determine how mny of the nerest to O i silicon toms re involved in the diffusion process. An O i tom cn overcome brrier t ny optimum configurtion of the nerest Si toms. However the reltive E (ev) 4,0 3,5 3,0 2,5 2, P(n)/P(3) N t N t Fig. 1. Diffusion brrier DE (n) s function of the number of Si toms involved in minimiztion of the totl cluster energy. In the inset the probbility P occ P dj of occurrence of n optimum configurtion out of n toms is presented (Dt(n)/t(n)=0.01).

4 number of O i toms diffused t the given optimum configurtion is proportionl to the product of probbilities of occurrence of the optimum configurtion (P occ ) nd probbility of diffusion jump ( P exp ( E ( n) / k T).The dj B probbility of occurrence of n optimum configurtion out of n toms hve been clculted on the bsis of geometricl definition of probbility ( problem of rndom collisions) nd in this cse the product PoccP dj my be written s τ ( n) Pn ( ) PoccPdj n τ ( n) n 1 E ( ) exp n, (4) kbt where n is the number of toms in the optimum configurtion, τ ( n) nd τ ( n) re the period of formtion nd lifetime of the given optimum configurtion, respectively, Ei ( n) is the diffusion brrier. Usully τ ( n)/ τ ( n) is much less thn one 8. The dependence P(n) s function of Si toms involved in the minimiztion is depicted in the inset of Fig. 1. Clcultions hve shown, tht P(n) hs shrp mximum t n=3 which more thn on the order of mgnitude exceeds P(n) for n=2, 4, 5 Hence, essentilly ll O i toms overcome the diffusion brrier when only three nerest Si toms re in the optimum configurtion, nd the diffusion prmeters should be clculted for the given configurtion. For this cse the following vlues of the diffusion brrier E = ev (AM1, PM3, PM5 method of clcultion) hve been obtined. Mtrix λ i necessry for the clcultion of the preexponentil fctor D0 ws evluted s follows. At the equilibrium configurtion of interstitil oxygen O i nd t the intersection point of the extreme trjectory of O i with surfce S G the squre-lw interpoltion of the potentil energy U ( f 1,..., f ) (f1 f m re coordintes of O i eff m nd nerest Si toms) hs been constructed nd λ i hs been obtined t once by digonliztion of Kij. In such mnner clculted vlue of the pre-exponentil fctor equls D 0 = sm 2 s -1. On Fig. 2 one cn see the excellent greement between the clculted nd experimentl temperture dependences of the diffusion coefficient in ll temperture rnge T= K. D, sm 2 s D(P)/D(0) P (GP) 0,6 0,9 1,2 1,5 1,8 1000/T, K -1 Fig. 2. Temperture dependence of diffusion constnt of interstitil oxygen tom in silicon. Points experiment [1], line theory. In the inset: solid line indictes the clculted pressure dependence of reltive coefficient of diffusion D(P)/D(0), T= 450 o C; points re the reltive concentrtion of oxygen therml donors s function of pressure (experiment 16 ). Being grounded on the procedure described bove, the diffusion coefficient of interstitil oxygen in germnium crystls hve been clculted lso. Clculted vlues of the ctivtion energy E (Ge) = 2.05 ev nd preexponentil fctor D 0 (Ge) = 0.39 sm 2 s -1 re in excellent greement with experimentl ones

5 DE exp (Ge) = ev, D exp (Ge) = 0.4 sm 2 s For better understnding of oxygen diffusion in silicon crystls the influence of hydrosttic pressure (P) on the diffusion coefficient hs been evluted. This is prticulrly interesting s the high hydrosttic pressure (HP) hs been found to enhnce strongly the oxygen gglomertion t elevted tempertures 14, 15, 16, 17. The origin of this unusul phenomenon hs been under debte nd remins open. To explin the HP effect in 15, 16 n enhncement nd in 17 n opposite effect of retrdtion of oxygen diffusion occurred t high tempertures under HP hs been suggested. In experimentl studies of gglomertion processes of oxygen in silicon hydrosttic pressure usully reches GP. At given pressures vrition of Si lttice constnt is reltively smll ( 0.1 Å) nd, hence, chnges of diffusion coefficient will be determined by vrition of E ( D exp ( E / kt) ) with pressure. To model the effect of pressure the cluster hs been conventionlly divided into internl (R < R 0 ) nd externl prts (R>R 0 ). The internl prt includes O i nd is selected in such mnner tht the increse in R 0 does not result in essentil chnge of the equilibrium structure of O i defect (Si-O bonds nd Si-O-Si ngle) t P=0 (usully R o equls 5-7 Å). The pressure hs been modeled by replcement of the equilibrium length of Si-Si bonds in the externl prt of the cluster with the length of Si-Si bonds tht re chrcteristic (clculted from experimentl vlue of Si compressibility modulus) for the given pressure. Upon minimiztion of the cluster totl energy, the lengths of Si-Si bonds t R>R 0 did not vry, nd the minimiztion ws crried out on the coordintes of oxygen nd silicon toms being in the internl prt of the cluster. The further evlution of E ( P) ws crried out similrly to the cse P=0. Clcultions hve reveled tht hydrosttic pressure leds to lowering the diffusion brrier E ( P) nd in the whole investigted intervl of pressures ( P 80 kbr ) is described well by the following expression: E ( P)/ E (0) = 1 γ P, (5) where γ = kbr -1, P is the hydrosttic pressure in kbr. The clculted pressure dependence of the O i diffusivity (without ny djustble prmeters) corresponds well to n enhnced growth of the oxygen-relted therml donors (TDs) observed experimentlly 15. Fig. 2 shows the clculted diffusion constnt of n interstitil oxygen tom in silicon nd the experimentlly observed dependence of reltive concentrtion of TD s function of pressure. One cn see tht the theoreticl curve is well consistent with shrp increse in the TD enhnced growth t P ~ 1GP. In summry, theoreticl modeling of the oxygen diffusivity in silicon nd germnium crystls t norml nd uniform pressure hs been presented. On the bsis of the results obtined it is possible to drw the following conclusions. Three nerest Si (Ge) toms re involved in n elementry oxygen jump from bond-center site to nother bond-center site long pth in the (110) plne. It is precisely their optimum position (corresponding to locl minimum of the crystl totl energy) determines the vlue of the diffusion potentil brrier of n interstitil oxygen tom in silicon nd germnium. The theoreticlly determined vlues of the diffusion potentil brrier nd preexponentil fctor re in excellent greement with experimentl ones nd describe very well the experimentl temperture dependence of diffusion constnt in Si crystls (T = C). Hydrosttic pressure ( P 80 kbr ) gives rise to the decrese of the diffusion potentil brrier in Si crystls nd ccordingly increses the diffusion coefficient. Such pressure dependence of O i diffusivity ppers most likely to be responsible for the HP enhncement in genertion of the oxygenrelted therml donors.

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