Quantum Monte Carlo Study Silicon Defectsdownloads.hindawi.com/archive/2001/083797.pdf · QuantumMonteCarlo Study ofSilicon ... Kokubunji, Tokyo 185-8601 Wegive a brief description

VLSI DESIGN2001, Vol. 13, Nos. 1-4, pp. 229-235Reprints available directly from the publisherPhotocopying permitted by license only

(C) 2001 OPA (Overseas Publishers Association) N.V.Published by license under

the Gordon and Breach Science Publishers imprint,member of the Taylor & Francis Group.

Quantum Monte Carlo Study of SiliconSelf-interstitial Defects

W.-K. LEUNGa, R. J. NEEDSa’*, G. RAJAGOPALa, S. ITOHb and S. IHARAb

aTCM Group, Cavendish Laboratory, Madingley Road, Cambridge CB3 0HE, UK;bCentral Research Laboratory, Hitachi Ltd. (Japan), Kokubunji, Tokyo 185-8601

We give a brief description of the variational and diffusion quantum Monte Carlomethods and their application to the study of self-interstitial defects in silicon. Thediffusion quantum Monte Carlo calculations give formation energies for the most stabledefects of about 4.9 eV, which is considerably larger than the values obtained in densityfunctional theory methods. The quantum Monte Carlo results indicate a value for theformation+migration energy of the self-interstitial contribution to self-diffusion ofabout 5 eV, which is consistent with the experimental data.

Keywords: Self-interstitials; Silicon; Diffusion quantum Monte Carlo; Self-diffusion; Densityfunctional theory

1. INTRODUCTION

A sophisticated description of electron correlationis required to describe the structures and energeticsof solids. Quantum Monte Carlo methods in thevariational [1] (VMC) and diffusion [2, 3] (DMC)forms allow a direct assault on the many-bodySchr6dinger equation using statistical techniqueswhich are capable of yielding highly accurateresults. The great promise of these methods lies inthe fact that electron-electron correlations areincluded explicitly, essentially without approxima-tion, and the computational cost increases as thethird power of the number of electrons, which ismuch more favourable than other correlated wave

function techniques and allows applications to

large systems.In this paper we give a brief introduction to the

VMC and DMC methods and describe their

application to the study of self-interstitial defectsin silicon [4]. One of the important problems inthe manufacture of sub-micron devices is the dif-fusion of dopant impurity atoms during thermalprocessing, which limits how small they can bemade. To understand these effects requires aknowledge of diffusion on the microscopic scalein situations far from equilibrium. The diffusionof impurity atoms in silicon is controlled byintrinsic defects such as self-interstitials andvacancies, and it is therefore important to

* Corresponding author, e-mail: rnl @phy.cam.ac.uk

229

230 W.-K. LEUNG et al.

improve our understanding of the behaviour ofthese defects.

Unfortunately it has not been possible to detectthe self-interstitial directly, although their presencehas been inferred from various measurements [5].Measurements of the self-diffusion constant orself-diffusivity of silicon at high temperaturesusing radioactive isotopes of silicon as tracershave established an Arrhenius behaviour with anactivation energy in the range 4.1- 5.1 eV [6]. Theself-diffusivity, DsD, is usually written as the sumof contributions from independent diffusive me-chanisms. The contribution of a particular micro-scopic mechanism can be written as the product ofthe diffusivity, Di, and the concentration, Ci, of therelevant defect, i.e.,

DSD ’DiCi ZAiexp(-Ei/kBT), (1)

where the fi are correlation factors of order unity,A is a prefactor and Ei is the activation energy formechanism i. Various experimental studies haveindicated that in silicon two mechanisms of self-diffusion are important, vacancy diffusion andself-interstitial diffusion. The prefactors and acti-vation energies of the different mechanisms cannotbe measured directly but their values can bededuced by combining various experimental data.G6sele et al. [7] have estimated that EI 4.84eVfor interstitials and Ev--4.03eV for vacancies,while Bracht et al. [8] have given EI 4.95 eV andEv 4.14 eV. The prefactor is predicted to belarger for vacancy diffusion than for self-inter-stitial diffusion and therefore vacancy diffusiondominates at low temperatures while self-inter-stitial diffusion dominates at higher temperatures,with a crossover at about l100-1200K. Thesituation is highly controversial when it comes todetermining the individual values of D and Cx.Indeed, experimental data has been used tosupport values of the diffusivity of the siliconself-interstitial, D, which differ by ten orders ofmagnitude at the temperatures of around 800C atwhich silicon is processed [9].

Self-diffusion in silicon is both technologicallyimportant and imperfectly understood, and ouraim is to perform an accurate theoretical studywhich can help clarify the situation. In this paperwe describe quantum Monte Carlo calculations ofthe energies of self-interstitial defects in silicon,and compare our results with those of densityfunctional theory studies and with experiment.

2. THE VMC AND DMC METHODS

In the VMC method [1- 3] expectation values arecalculated as integrals over the configurationspace, which are evaluated using standard MonteCarlo techniques. The variational energy Ev isgiven by

f ,r (R)/-/C,r (R)dRff ’r(R)2 (’r(R)-//r(R))dR

f r(R)2dR(2)

where y is the trial wave function (assumed real),f/ is the Hamiltonian, and R denotes the 3N-dimensional vector of the electron positions. Inthe second line of Eq. (2) we have rewrittenthe numerator in terms of the local energyEL ff)T(R)-I/-/ff)T(R). This step is very importantfor the statistical evaluation of the energy.The energy can now be evaluated as an integralover a probability distribution, P(R)= ff)r(R)2/f ffr(R)2dR. Such an integral can be performedusing Monte Carlo techniques based on theMetropolis algorithm which generates a sequenceof points R; which are asymptotically distributedaccording to the required probability distribution.The variational energy is then given by

Ev f P(R)E(R)dR ZEL(Ri), (3)

where the electron configuration vectors Ri aredistributed according to P(R). The statistical error

SELF-INTERSTITIAL DEFECTS 231

in this estimate of Ev falls as the inverse of thesquare root of the number of sampling points.The accuracy of the VMC method is limited by

the quality of the trial wave function, and inpractice one has to use highly sophisticatedcorrelated wave-functions. Our trial wave func-tions are of the Slater-Jastrow type:

r(R) DDexp[J(R)], (4)

where and D and D are Slater determinates ofup- and down-spin single particle orbitals andexp[J(R)] is the Jastrow factor. In the calculationsdescribed here the single particle orbitals areobtained from local density approximation(LDA) calculations. We use Jastrow factors, whichcorrelate the motion of pairs of electrons, althoughhigher-body correlations can also be included. TheJastrow factor includes a number of parameterswhose values may be varied to optimize the wavefunction. For each of the defect systems our trialwave function contains 64 variational parameters,while for the perfect crystal calculation there were32 parameters. The optimal values were obtainedby minimizing the variance of the energy [11, 10],which has been found to be more numericallystable than minimizing the energy itself [12].The DMC method is a stochastic projector

method for solving the imaginary-time Schr6din-ger equation,

0(R, t)Ot ([/- Es)(R, t), (5)

where is a real variable measuring the progressin imaginary time, and Es is an energy offset.This equation has the property that an initialstarting state decays towards the ground statewave function (provided they have a non-zerooverlap). The time evolution of Eq. (5) may befollowed using a stochastic technique. The wavefunction (R, t) is represented by an ensemble of3N-dimensional electron configuration vectors,{Ri}, whose time evolution is governed byEq. (5). The Green function of Eq. (5) shows

that the rules of the evolution are randomdiffusive jumps of the configuration vectors fromthe kinetic term and removal/addition of config-uration vectors from the potential energy termwhich acts like a rate term.

Unfortunately this simple algorithm suffersfrom two serious problems. The first in that wehave implicitly assumed that t9 is a probabilitydistribution, although its fermionic nature meansthat it must have positive and negative parts. Todeal with the problem of the fermion antisym-merty we use the fixed-node approximation. Thenodal surface of a wave function is the surface onwhich it is zero and across which it changes sign. Ifwe force the time evolution of Eq. (5) to maintaina nodal surface consistent with the fermionicantisymmetry we can simulate the equationseparately in each nodal pocket. The fixed nodalsurface can be imposed by considering the evolu-tion of the mixed distribution f= T

, where T isthe known trial wave function discussed above.The nodal surface of is constrained to be thesame as that of (I) r and therefore f can beinterpreted as a probability distribution. Thefixed-node DMC method generates the distribu-tion f= Or, where is the best (lowest energy)wave function with the same nodes as Or. Thesecond problem is less fundamental but in practicevery severe. The required rate of removing/addingconfiguration vectors diverges when the potentialenergy diverges, which occurs whenever twoelectrons or an electron and a nucleus are

coincident, leading to poor statistical behaviour.Using the f distribution also introduces impor-tance sampling which greatly reduces the statisticalnoise. Normally we demand that (I)r has thecorrect cusp-like behaviour when two electronsor an electron and a nucleus are coincident [13],which removes the divergence in the removal/addition process.

Using the fixed-node approximation means thatwe solve independently in different nodal pockets,and at first sight it appears that we have to solvethe Schr6dinger equation in every nodal pocket,which would be impossible in large systems.


However, the tiling theorem for fermion ground silicon. There have, however, been manystates [14, 15] asserts that all nodal pockets are in theoretical studies of their structures. The mostfact equivalent and therefore one only needs to advanced of these have used the local densitysolve the Schr6dinger equation in one of them. approximation (LDA) to density-functional the-A simulation proceeds as follows. First we pick ory to calculate the defect formation energies

an ensemble of a few hundred configuration and energy barriers to diffusion [17-19]. Thevectors chosen from the variational distribution, consensus view is that the split-(ll0), hexagonalP(R). This ensemble is evolved in imaginary time and tetrahedral defects (see Fig. 1) are low inaccording to the rules of the importance sampled energy. Other defects, such as the split-(100)imaginary time Schr6dinger equation, which in- and bond centred interstitials, are calculated tovolves biased diffusion and addition/subtraction be much higher in energy and therefore we dosteps. The bias in the diffusion is caused by the not consider them here. We have also studiedimportance sampling which directs the sampling the saddle point of Pandey’s concerted exchangetowards parts of configuration space where r is mechanism [20] for self-diffusion, which involveslarge. After a period of equilibration the config- the exchange of neighbouring atoms in theuration vectors start to trace out the probability perfect lattice via a complicated 3-dimensionaldistribution f(R)/ff(R)dR. We can then start to path which allows the atoms to avoid largeaccumulate averages, in particular the DMC energy barriers.energy, Ez, which is given by As a preliminary to our main calculations we

Em= ffEdRffdR ZE(Ri), (6)

performed a thorough study at the LDA level [21]using a plane wave basis set and a norm-conserving pseudopotential to represent the Si4+

ions. We performed LDA calculations for thewhere the configuration vectors are distributed fully relaxed defect structures using supercellsaccording to f(R)/ff(R)dR. This energy expres- for which the perfect crystalline structure con-sion would be exact if the nodal surface ofr was tains 16, 54 and 128 atoms, respectively. Theexact, and the fixed-node error is second order in relaxed defect structures are almost independentthe error in the nodal surface of r. Fixed-node of the size of the supercell, and are illustratedDMC calculations have now been performed for in Figure 1. One of the intriguing features ofmany systems. The accuracy of the fixed node the various defect structures is the wide rangeapproximation can be tested on small systems and of interatomic bonding they exhibit. Thenormally leads to very satisfactory results [3]. bond length for the perfect crystal is 2.35.

This completes our brief introduction to VMC The nearest-neighbour defect bonds in theand DMC methods. We have left out many tetrahedral interstitial structure are significantlytechnical details of the calculations, and for more longer than the perfect crystal bonds at 2.44/,information on these aspects the reader is directed while at the saddle point of the concertedto Refs. [2,3, 16]. exchange the atoms are joined by a very short

bond of length 2.15]k. The variation in co-ordination number of atoms in the different

3. STRUCTURES OF THE SELF- defect structures varies from 3 (saddle pointINTERSTITIAL DEFECTS of concerted exchange) to 6 (hexagonal inter-

stitial). This wide range of interatomic bond-Experiments give no information about the ing poses a significant challenge to theoreticalstructure of single-atom self-interstitial defects in techniques.


Split-(l 10) interstitial Hexagonal. interstitial

Perfect crystal

Tetrahedral interstitial. Concerted exchange

FIGURE The split-(110), hexagonal, and tetrahedral interstitial defects, the saddle point of the concerted-exchange mechanism,and the perfect silicon crystal.

4. DEFECT FORMATION ENERGIES

The defect formation energies of the differentstructures calculated within the LDA are reportedin Table I. We also show values for the defectenergies calculated with the gradient-correctedPW91-GGA density functional [22]. From Table Iwe can see that the split-(ll0) and hexagonalinterstitials are the lowest energy defects with DFTformation energies of about 3.3eV (LDA) and3.8 eV (PW91-GGA). We have also calculated theenergy barriers to diffusive jumps between the lowenergy structures within DFT, finding a path forsplit-(ll0)-hexagonal diffusion with a barrier of

0.15 eV (LDA) and 0.20eV (PW91-GGA), and abarrier for hexagonal-hexagonal diffusive jumpsof 0.03 eV (LDA) and 0.18 eV (PW91-GGA). OurDFT results give the sum of the formation andmigration energies for self-interstitials as 3.5eV(LDA) and 4.0eV (PW91-GGA), which areconsiderably smaller than the activation energyfor self-interstitial diffusion deduced from experi-mental measurements of 4.84 eV [7] and 4.95 eV[8]. Although the LDA activation energy of4.45 eV and the PW91-GGA activation energy of4.80 eV for Pandey’s concerted exchange are with-in the experimental range for self-diffusion of 4.1-5.1 eV [6], our results indicate that the activation


TABLE LDA, PW91-GGA, and DMC formation energies in eV of the self-interstitial defects and the saddle point of the concerted-exchange mechanism

Defect LDA GGA DMC

Split- 110 3.31 3.84 4.96(28)Hexagonal 3.31 3.80 4.82(28)Tetrahedral 3.43 4.07 5.40(28Concerted Exchange 4.45 4.80 5.78(27)

energies for interstitial mediated self-diffusion arelower.We conclude that the DFT results for self-

interstitial diffusion in silicon do not afford asatisfactory explanation of the experimental tem-perature dependence of the self-diffusivity. Giventhe doubts about the accuracy of the DFTfunctionals and the technological importanceof the problem, we have recalculated the energiesof the defect structures using the DMC methodusing the 16 and 54 atom supercell-structuresobtained from our LDA study. First we calculatedthe cohesive energy of silicon within the VMC andDMC methods by performing calculations for theatomic ground state and for the perfect solid usingthe 54-atom simulation cell. The resulting VMCand DMC cohesive energies of 4.48(1) eV and4.63(2)eV per atom, respectively, are in excel-lent agreement with the experimental value of4.62(8) eV per atom.The results of our DMC calculations for the

defect formation energies are shown in Table I.The clearest conclusion is that the DMC forma-tion energies are roughly eV larger than thePW91-GGA values and 1.5eV larger than theLDA values. Within DMC the hexagonal inter-stitial has the lowest formation energy, and thesplit-(110) interstitial is slightly higher in energy,while the tetrahedral interstitial has a significantlyhigher energy. The saddle point of the concertedexchange has an energy which is too high toexplain self-diffusion in silicon.

Since mapping the energy barriers to diffusion iscomputationally prohibitive within DMC weestimate the migration energy as follows. Thetetrahedral interstitial is a saddle point of apossible diffusion path between neighbouring

hexagonal sites. The DMC formation energy of5.4 eV for the tetrahedral interstitial is therefore anupper bound to the formation +migration energyof the hexagonal interstitial. The true forma-tion +migration energy is expected to be less thanthis, and we can use the following argument toobtain a crude estimate of it. Within the LDA wehave found a diffusion path for the hexagonalinterstitial with a barrier of only 25% of thetetrahedral-hexagonal energy difference. Applyingthe same percentage reduction to the DMC barriergives an estimate of the formation+migrationenergy of the hexagonal interstitial of 5eV. Thisestimate is in good agreement with experimentalvalues of the activation energy for self-interstitialdiffusion of 4.84eV [7] and 4.95 eV [8].

5. CONCLUSIONS

In conclusion we have given a brief introduction tothe VMC and DMC methods and described theirapplication to the problem of the energetics of self-interstitial defects in silicon. Our calculationsdemonstrate the importance of a proper treatmentof electron correlation when calculating defectformation energies in silicon. The LDA andPW91-GGA functionals do not provide a satisfac-tory explanation of the experimental temperaturedependence of the self-interstitial contribution tothe self-diffusivity because they predict forma-tion +migration energies which are too small. Thelarger defect formation energies found in ourDMC calculations indicate a possible resolution ofthis problem. This may be an important step inimproving our understanding of self-diffusion insilicon.


Acknowledgments

We thank Paul Kent and Randy Hood for helpwith the calculations. Financial support wasprovided by the Engineering and Physical SciencesResearch Council (UK), Hitachi Europe Ltd.and Hitachi Japan Ltd. The DMC calculationswere performed on the Hitachi SR2201 at theUniversity of Cambridge High Performance Com-puting Facility and at Hitachi’s Central ResearchLaboratory.

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[21] We used the LDA of Perdew, J. P. and Zunger, A. (1981).Phys. Rev. B, 23, 5048 who parametrized the DMC data ofCeperley, D. M. and Alder, B. J. (1980). Phys. Rev. Lett.,45, 566.

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Quantum Monte Carlo Study Silicon Defectsdownloads.hindawi.com/archive/2001/083797.pdf · QuantumMonteCarlo Study ofSilicon ... Kokubunji, Tokyo 185-8601 Wegive a brief description