The W center in self-implanted silicon is the self-interstitial cluster I_3

a r X i v :c o n d -m a t /0202495v 1 [c o n d -m a t .m t r l -s c i ] 27 F e

b 2002

The W center in self-implanted silicon is the self-interstitial cluster I 3

Giorgia M.Lopez and Vincenzo Fiorentini

INFM and Dipartimento di Fisica,Universit`a di Cagliari,Cittadella Universitaria,I-09042Monserrato (CA),Italy

(February 1,2008)

We identify the W center in self-implanted crystalline Si with the three-membered self-interstitial cluster I 3on the basis of ?rst-principles density-functional-theory calculations matching all the known experimental signatures of the center (emission energy,extrinsic energy levels,activation energy and dissociation energy,local vibrational structure,and symmetry).PACS:61.72.Bb,61.72.Ji,71.55.Ht,85.40.Ry

Self-implanted Si is relevant to the pre-amorphization processes used at various stages of device fabrication,and as a model environment for the study of anomalous interstitial-driven impurity di?usion.A prominent de-fect in self-implanted Si is the so-called W center,which has been known and studied intensely for a long time [1,2],but not ?rmly identi?ed so far,although gener-ally accepted to originate from the clustering of excess interstitial Si atoms [3–5].By means of ?rst-principles density-functional calculations,we provide a conclusive identi?cation of the W center with the I 3three-membered self-interstitial cluster.Remarkably for a di?cult enter-prise such as the identi?cation of defects in crystals,our results match all the experimental signatures available for this center,and con?rm the relevance of ab initio calculations for the physics of defects and impurities in crystals.

Recent experimental studies [3,4],performed with a variety of combined techniques,including variable-dose implant,rapid thermal annealing (RTA),deep level tran-sient spectroscopy (DLTS),and photoluminescence (PL),have ?rmly identi?ed several experimental signatures of the W defect.By carefully cross-referencing a number of n -type samples with di?erent implant and RTA histories,Libertino et al.[3]identi?ed a low-dose implant regime in which the extended {311}defects –a common product of self-interstitial aggregation –do not form:small,nearly point-like entities are active instead.They suggested that the W photoluminescence line,a typical emission at 1.018eV in self-implanted Si,is indeed the main signature of these centers.A second signature identi?ed as strongly correlated with this emission,is a pair of extrinsic lev-els determined by DLTS at about 0.35and 0.6eV above the valence edge.The third signature brought forth by the analysis of the pertaining DLTS signals under ther-mal treatment is the activation energy of around 2.3eV needed to cause the essentially concurrent disappearance of these two levels upon RTA.In addition,the appear-ance of the W band is itself thermally activated,with a characteristic energy of about 0.85eV [4,6]–the fourth signature of W .A ?fth group of signatures stems from the vibronic structure of which the W emission is the zero-phonon line;this structure has been known for some

time [2]to include main peaks at 17meV and 40meV,and a local vibrational mode at 70meV [2].The symme-try of the center identi?ed via the stress response of the vibronic structure is at least C 3v .Finally,several indi-rect hints led many authors (see e.g.[3]for a summary)to postulate the most likely size of the cluster responsi-ble for these signatures to be n =3,with a high-symmetry compact structure,as opposed to the non-compact I 4and asymmetric I 2clusters found in recent calculations [7–9].In the following,we give evidence that the I 3Si tri-interstitial is to be identi?ed with the W center,as its properties explain all the mentioned experimental signa-tures.Our conclusion is based on a series of ab initio calculations on self-interstitial clusters in c -Si (full de-tails are reported elsewhere [7]),performed at the ?rst-principles level within density-functional theory in the generalized gradient approximation (GGA)[10],and the ultrasoft pseudopotential-plane wave repeated-supercell approach using the Vienna Ab-initio Simulation Pack-age code [11].We report data obtained with 32-and 64-atom supercells (the results are in fact quite insensitive to cell size),multiprojector ultrasoft pseudopotentials [11](with two s ,two p ,and one d projectors,r c =1.31?A ),a plane wave cuto?of 151eV,and a 444k-space summa-tion mesh (this yields 32k-points in the Brillouin zone for C 1,i.e.no,imposed symmetry;in occasional test we used 666and 888meshes,with 108and 256k-points re-spectively).Atomic geometries are relaxed until all force components are below 0.01eV/?A .The theoretical lattice

constant a Si =5.461?A is used throughout.The formation

energies and extrinsic levels are obtained in the usual way (see e.g.Refs.[12]and [13]),including multipole correc-tions for charged states [14].

We obtained the equilibrium structure of the I 3cluster by relaxation of a supercell containing an I 2di-interstitial cluster [7–9]and a single interstitial I 1in the adjacent tetrahedral site,both in the neutral state.The intersti-tial binds spontaneously to I 2,producing the I 3structure depicted in Fig.1.This structure agrees well with recent tight-binding [8]and ab initio results [15],although not with others [16](incidentally,the properties of the n =3cluster studied in [16]do not correlate well with those of the W center).The interatomic distances between the

four atoms (dark grey in Fig.

1)involved in the local structure of I 3are essentially identical,2.488±0.002?A or 1.054times the calculated Si-Si bond length of 2.365?A .All bonds are oriented along (110)-equivalent direc-tions,hence the four atoms form a perfect tetrahedron with edges aligned with the cubic (110)axes,and whose centerpoint is at a lattice site.The local symmetry of the cluster is therefore T d .This agrees with the response of the zero-phonon W line to applied stress and its depen-dence on the electric ?eld direction.The symmetry of the cluster,we point out,is obtained spontaneously in the simulated assembly of I 2and tetrahedral I 1without imposing any initial symmetry.

FIG.1.View of the structure of the I 3self-interstitial clus-ter in c -Si.Cluster atoms are dark grey,while their ?rst bulk-like neighbors,involved in the 70-meV local vibrational mode (see text),are light grey.Bulk atoms are white.

Studying the energies of di?erent charge states of the defect,we determined the extrinsic levels.I 3possesses two extrinsic thermal charging levels in the gap,namely ?(++/+)=0.35eV +E v and ?(+/0)=0.8eV +E v ,with E v the valence band top.These values are close to the experimental values of 0.35eV and 0.6eV.Since the sam-ples of interest here are n -type,these levels are ?lled and DLTS-detectable.Since a supersaturation of Si inter-stitials in c -Si produces moderate n -type conditions [7],this conclusion will hold even in as-implanted intrinsic samples.The DLTS signatures of the W center are thus reproduced by the I 3cluster.

To interpret the thermal evolution of these levels,we assume that their disappearance under thermal treat-ment is due to the evaporation of the cluster into single interstitials.For the n -type Fermi level of relevance here,we ?nd that all the interstitial complexes I 1,I 2,and I 3are in their neutral state.We then estimate the cost of splitting I 3into well-separated neutral I 2and (neutral dumbbell)I 1by direct comparison of calculated total en-ergies :this cost is 2.38eV,matching very closely the deactivation energies,2.28eV and 2.36eV (estimated experimental error ?15%),of the observed DLTS peaks,con?rming our previous attribution of those peaks to the I 3extrinsic levels.

After splitting an I 3into an I 2and an I 1,one may still have electrically active levels present,since both the lat-ter centers have extrinsic levels [7]in the 0.25-0.35eV and 0.6-0.8eV range.The observation of those levels is however preempted by two facts:a)I 2splits into two I 1’s with an energy cost of 1.5eV,hence the thermal treat-ment dissolving the I 3’s also gets rid of any I 2’s;b)an I 3-cracking thermal process should presumably cause the remaining I 1’s (also electrically active [7])to evaporate o?the sample,both because they are quite rapidly dif-fusing [17]neutral dumbbells,and because they have no e?ective capture centers,such as vacancies,available in su?cient concentration.

Of course the main signature of the center at issue is the W emission itself.The relevant emission energy is 1.02eV;since the material is n -type,the transition is probably bound-to-free.Since the gap of Si is about 1.1eV at the relevant temperature,the involved level lays ～80meV below the conduction edge.This does not im-ply,however,a delocalized state as in shallow impurities,the very existence of a sharp zero-phonon line in an in-direct gap material indicating a localized character.We ?nd the 0/–level of I 3at 1.10eV above the valence (k-points convergence was checked to better than ±10meV up to the (888)mesh;no change was observed between 32-,64-and 265-atom cells).This is clearly a very good candidate for the initial state of the emission:its en-ergy is quite close to the observed transition and,since emission is involved,the Franck-Condon principle dic-tates that the emission energy coincides with the charging level in this case.In addition,this state is empty in ther-mal equilibrium,and can thus receive the photoexcited electrons thermalizing down from the upper conduction states.We ?nd that the state is orbitally non-degenerate,in agreement with the analysis of the phonon replicas and with the absence of pseudo–Jahn-Teller distortions [2].To give a proper estimate of the position of the state with respect to the conduction band,we calculated the fundamental gap of Si using the expression,exact in the N →∞limit [18],

E gap =E tot (N +1)?2E tot (N )+E tot (N ?1),

(1)

using an undefected 64-atom supercell with N,N+1,and N–1electrons.The gap energy thus obtained is 1.13eV,which gives a 30meV binding energy.The satifactory agreement with experiment further con?rms our identi?-cation [19].

We now consider the activation barrier for the creation of the center,which was quanti?ed in 0.85eV in Refs.[4,6].The simplest interpretation is that the di?usive motion of the interstitials must be activated to achieve

clustering.It is natural to presume that at least two processes are to be activated to form I 3:the di?usion of I 1’s towards each other to form I 2,and the

di?usion of I 1towards I 2to form I 3.We estimated (see also [17])a minimum energy barrier for the dumbbell-to-dumbbell di?usion of 0.2eV via a tetrahedral site and 0.18eV via an hexagonal site.This is much lower that the experi-mental activation energy.We then hypothesized that I 1be subject to a local repulsion by its companion center (another I 1or the I 2),e?ectively increasing its di?usion barrier.To check this idea we compare the energy of a dumbbell self-interstitial and a tetrahedral I 1in the same cell in adjacent sites –i.e.the last saddle-point con?guration before I 2formation [7]–with the sum of the energies of an isolated dumbbell I 1plus an isolated tetrahedral I 1.The di?erence is an estimate of the local repulsion (if any)between single interstitials.We then calculate the same di?erence for I 2and tetrahedral I 1in the same cell,and isolated I 2plus isolated tetrahedral I 1.Again,the di?erence is an estimate of the local repulsion (if any)between I 1and I 2.In both cases we obtain an e?ective repulsion,of 0.64eV in the ?rst case,and 0.53eV in the second case (errors due to ?nite-size relaxation e?ects are at least one order of magnitude smaller [7]).Therefore,an e?ective repulsion acts locally between the precursors of I 3(I 1’s and I 2)The largest of these two re-pulsion energies,added to the normal di?usion barrier,provides an estimate of the maximum e?ective barrier.The result is 0.84eV,in close agreement with the ex-perimental estimate of 0.85eV.Thus,an e?ective local repulsion between the component centers of the I 3cluster explains quantitatively the activation-energy signature of the W -band center.

We now analyze the vibrational modes of the center.The zero-phonon W line splits under (110)-and (111)-oriented,but not under (100)-oriented stresses [2].This is quite compatible with the symmetry and orientation of the I 3complex,a tetrahedron with (111)-oriented axes and (110)-oriented bonds.The analysis of the phonon replicas of Ref.[2]suggests a dominance of couplings to the Si bulk phonon continuum.The main features in the replica spectrum [2]are broad structures associated with vibrational energies of 17meV and 40meV.Fur-ther,sharper but much weaker lines exist in the optical-mode energy region.An additional peak at 70meV below the zero-phonon line appears to involve [2]a local vibra-tional mode (the highest vibrational energy in Si bulk is ˉh ωSi TO (Γ)=64meV).

To identify possible local modes,we estimated the vi-brational frequency of selected normal modes of the clus-ter via the frozen-phonon method [20]in which a given displacement pattern is frozen into the lattice,and the force/displacement ratio for the atoms involved yields the mode’s harmonic force constant.The calculated Si LO-TO mode energy at Γof 62.4meV,–2.5%from experi-ment [21]sets a reliability reference.The internal vibra-tional modes of the cluster,i.e.those involving the four atoms of the cluster,are found to have energies in the range 35–45meV.The modes considered include most of the typical modes of a tetrahedron (breathing,twist,pinch,etc.)schematized e.g.in Fig. 2.5of Ref.[22].A detailed discussion will be presented elsewhere.The relatively low frequencies of the internal modes are un-surprising if we consider the weaker bonding within the cluster compared to the bulk:this is apparent from the (110)-plane slice of the charge density [23]through one of the internal cluster bonds of neutral I 3in Fig.2.The “translational”mode of I 3,in which the whole cluster moves with respect to the crystal,and which would have zero frequency for the cluster in free space,has a low en-ergy of 19meV.This is expected since the cluster is also quite weakly bound to the surrounding bulk (Fig.2).

FIG.2.Charge density contour plot in a (110)plane pass-ing through one of the internal bonds of the I 3tetrahedron.Two of the cluster atoms are visible (center top).The amount of bonding charge between the cluster atoms,and towards the bulk atoms,appears modest when compared to bulk-like bonds.

Their highest frequency being less than 45meV,none of the internal vibrations can explain the 70-meV local mode [2].Remarkably,indeed,it turns out that the local mode is not a proper cluster mode:after some search we identi?ed it with the vibration of each of the four cluster-adjacent atoms (light grey in Fig.1)in the (111)planes parallel to the corresponding triangular face of the tetrahedral cluster.The mode energy of 71.4meV matches well the observed 70meV shift of the local-mode replica.Its isotropic character (four equivalent atoms are involved,near each cluster face),and its strong ex-pected dependence on (110)-and (111)-oriented strains also agree with the observed properties of the local-mode replica [2].The mode clearly derives from a TO wave impinging on the I 3cluster from a (111)direction –pic-torially,a “breaker-on-the-shoal”e?ect.

As for the prominent 17meV and 40meV replica struc-tures,they can be interpreted as suggested [2]as the interaction signature of I 3with the bulk phonon contin-uum,with some contribution,we add,of the internal

modes of the cluster.The phonon dispersion and density of states of Si[24]suggest that the modes involved in the 5-meV–wide structure at about17meV are mainly the zone-border modes at the K and X points(e.g.theΣ3 TA mode at K,e.g.,has the correct18meV energy).The “translation”mode of the cluster mentioned earlier,with its energy of19meV,is also expected to be involved in this replica.For the40-meV feature,the phonon density-of-states weight comes mainly from the LA modes along theΛline,topping with the46-meV L2mode at L.As mentioned,the35–45meV internal modes of the clusters t are also most likely to be involved in the40-meV replica, superposed on the bulk continuum,and contributing to its substantial(10meV)linewidth.With these attribu-tions of vibrational signatures,in particular of the local mode,we conclude our identi?cation of the I3cluster with the W center.

In summary,ab initio calculations of emission,acti-vation,and dissociation energy,extrinsic levels,and vi-brational modes explain all the known experimental fea-tures associated with the W band in self-implanted Si (PL emission,DLTS signatures of electronic levels,RTA behavior,phonon replicas in the PL spectra)as being due to the tri-interstitial cluster I3.The W center remains thus unambiguously identi?ed with the I3self-interstitial cluster.

We acknowledge partial support from the Italian Min-istry of Research within the PRIN2000project“Non-equilibrium dopant di?usion in Si”,and from the Parallel Supercomputing Initiative of INFM.