distorted silicon hydrides— a comparative study with various density functionals

Distorted Silicon Hydrides—A Comparative Study with VariousDensity Functionals

THOMAS KRÜGER, ALEXANDER F. SAXInstitut für Chemie, Karl-Franzens-Universität Graz, Strassoldogasse 10, A-8010 Graz, Austria

Received 31 March 2000; accepted 14 July 2000

ABSTRACT: We aim at an understanding of the so far unknown nature of thelocalized reaction centers in amorphous, hydrogenated silicon (a-Si:H), which areresponsible for the Staebler–Wronski effect. For this reason we have examined thesuitability of various density functionals to supply reliable information onstrained and defective silicon hydrides up to Si16H37. Five combinations ofexchange and correlation functionals have been tested that represent the threepossible ways to improve the local spin density approximation. In cases wherehigh-end quantum-chemical methods can be employed, most results obtained bythe density functionals are in at least satisfactory agreement with the referencevalues. The description of larger systems is reasonable. From this we concludethat the use of density functionals in embedding procedures to describe verylarge silicon frameworks is promising. c© 2000 John Wiley & Sons, Inc.J Comput Chem 22: 151–161, 2001

Keywords: a-Si:H; Staebler–Wronski effect; silicon hydrides; density functionals

Introduction

F or application in optoelectronic devices as, forexample, solar cells, amorphous hydrogenated

silicon (a-Si:H) could serve as a cheap and tech-nically more versatile replacement for crystallinesilicon. The most important characteristic featuresof the material are shown in the following. Note that

Correspondence to: T. Krüger; e-mail: [email protected]

Contract/grant sponsor: Austrian FWF; contract/grant num-ber: S57910-CHE

differences in the manufacturing process are quiteoften responsible for the observed range of varia-tion.

1. Hydrogen contents between 1 and about 15atom % have been detected. The distribu-tion seems to be inhomogeneous. Hydrogen ismostly bound as Si—H, but variable shares ofSiH2 bonding have been found as well. Underextreme circumstances even molecular H2 hasbeen observed.1 – 4

2. The material contains up to 5×1019 microvoidsper cm3. It has been shown that, by keepingthe hydrogen content below 4 atom %, also

Journal of Computational Chemistry, Vol. 22, No. 2, 151–161 (2001)c© 2000 John Wiley & Sons, Inc.

KRÜGER AND SAX

nearly void-free samples can be produced.Most device-quality a-Si:H, however, is inter-spersed with cavities. Assuming a sphericalstructure the mean void radius amounts to3.3–4.3 Å, which corresponds to 16–26 miss-ing atoms. The mean hydrogen content pervoid is estimated to 5–9 bonded H atoms,i.e., the inner void surfaces are only slightlyhydrogenated.5 – 7

3. a-Si:H contains 1018–1020 strained Si—Si bondsand about 1015–1016 paramagnetic centers percm3. It is proven that these defects are localizedradical centers of subvalent (threefold coor-dinated) Si called native dangling bonds. Theirdistribution depends on the manufacturingconditions, but it seems to be rather homoge-neous with a distance ≥10 Å. Some of thesenative dangling bonds may be located in thevicinity of a hydrogen that could be responsi-ble for the observed shift in the ESR hyperfinesplitting.3, 8 – 14

Because of the above-mentioned microvoids andthe vast amount of strained (weakened) Si—Sibonds, and because it is interspersed with hydro-gen, it is generally believed that a-Si:H is a materialwith substantial intrinsic mechanical stress.

The practical use of a-Si:H, however, is limitedby the so-called Staebler–Wronski effect (SWE).15, 16

Phenomenologically, the SWE is a photoelectronictwo-step process: during several hours of exposureto sunlight both the photoconductivity and the darkconductivity of a-Si:H decreases significantly. Thisconductivity loss, however, can be fully reversed bythermal annealing in the dark (T ≥ 150◦C). Thewell-established facts are as follows:

1. During 1–100 h of light exposure a decreasein both photoconductivity (about one orderof magnitude) and conductivity in the dark(at least four orders of magnitude) has beenobserved.15 – 18

2. The spin density (the number of paramagneticcenters) increases by about one to two ordersof magnitude. These additional paramagneticcenters are called light-induced or metastabledangling bonds. The saturation value of thespin density is independent of (i) the temper-ature during exposure, (ii) the initial density,and (iii) the carrier generation rate. It is, how-ever, approximately proportional to the bandgap and the hydrogen content. The increasein spin density is qualitatively independent ofthe photon energy (if E ≥ 1.2 eV) and the light

intensity (at least in the range between 0.1 and3.0 W/cm2). It is obviously coupled to the con-ductivity loss, but no one-to-one correlationhas been established.11, 12, 17, 19 – 25

3. By thermal annealing in the dark at a tem-perature above 150◦C the original state of thematerial can be fully restored.15 – 17, 19, 20

Despite a vast amount of experimental informa-tion on both device-quality a-Si:H and the SWE,1 – 30

its mechanism is still in question and discussed con-troversially.31 – 36 However, there is general agree-ment that a-Si:H possesses a lot of strained, weak-ened, and unsaturated Si—Si bonds, and it is,therefore, supposed that these localized defects arethe precursors (A) of the light-induced danglingbonds (see, e.g., ref. 26). Taking together both theexperimental evidence and the different mechanis-tic conceptions, one explanation of the SWE is asfollows:

1. Exposure of A to light excites an electron froma bonding to an antibonding orbital. In this ex-cited state A may trap an energetically nearbyelectron from the conduction band tail. Byrelaxation of this intermediate, presumably ac-companied by radiationless deactivation, themetastable dangling bond B is formed. It is sit-uated at about 0.5 eV above the A level.

2. The thermal reformation of A (the annealingprocess) may be considered the reverse of thephotoelectronic formation of B. The thresholdfor thermal reformation of A amounts to about1.1 eV.

Recall that both A and B are geometrically local-ized defects embedded in the amorphous bulk, butneither the nature of A and B nor the reaction pathsand the role of hydrogen migration are known. Theonly idea that already can be excluded is that thenative dangling bonds act directly as electron traps,because in this case negatively charged closed-shellsilicon centers would be generated, and accordinglya decrease in spin density should be observed.

Strategy and Methods

The aim of our efforts is to contribute to theunderstanding of the SWE. In a first step we willconcentrate on the nature of the localized reactioncenters A and B. Our strategy is as follows:

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DISTORTED SILICON HYDRIDES

1. A and B are represented by suitably modeledsilicon clusters that are treated by means ofhigh-level quantum-chemical (QC) methods.

2. The influence of the surrounding a-Si networkis described using an embedding procedureof the above-mentioned small clusters into amuch larger Si frame that is treated semiem-pirically or by molecular mechanics.

The QC method has to provide reliable geome-tries and relative total energies of strained and de-fective Si clusters with or without hydrogen atomspresent. Also, hydrogen migrations should be de-scribed correctly. Moreover, it should allow for rea-sonable electron affinities for both closed-shell andopen-shell systems. Besides this, it would be bene-ficial if the HOMO–LUMO gaps would yield goodestimates of excitation energies. In the present arti-cle it will be examined whether customary densityfunctionals meet with this requirements. We do notaim at the presentation of tests of the embeddingprocedure itself. This will be the topic of a subse-quent article.

We have tested five combinations of exchangeand correlation functionals that represent the threepossible ways to improve the local spin densityapproximation (LSDA): inhomogeneity correction,modeling of exchange and correlation hole, and hy-brid exchange (see Table I). The following function-als have been employed: Becke’s 198837 and Gill’s1996 exchange functional38 as well as Perdew’s1986 correlation functional39 present inhomogeneitycorrections to the LSDA. The hole modeling con-cept has been realized in the Perdew–Wang 1991functionals40, 41 for both exchange and correlation,and the idea of hybrid exchange became popularwith Becke’s B3LYP42 – 44 including correlation ac-cording to Lee, Yang, and Perdew.

These DFT results have been compared tocoupled-cluster calculations perturbatively includ-ing triple excitations from the Hartree–Fock deter-minant (CCSD(T)).68 In open-shell cases the corre-sponding unrestricted Hartree–Fock determinant has

TABLE I.

Functional Type of Exchange Type of Correlation

B3LYP hybrid inhom. correctionBP86 inhom. correction inhom. correction(PW91)2 hole modeling hole modelingBPW91 inhom. correction hole modelingG96PW91 inhom. correction hole modeling

been used. For open-shell DFT calculations, with theexception of Si13H23 (see the related discussion in alater section), the unrestricted Kohn–Sham scheme(UKS) has been employed. All calculations havebeen performed with Gaussian 9845 using the stan-dard 6-31G∗∗ basis set, unless otherwise stated. Theabsolute energies of all the species calculated areavailable on request.

Results

SMALL SILICON HYDRIDES

The Structures of Disilyne, Disilene,and Disilane

Despite its small size, Si2H2 provides an interest-ing example of multiple Si—Si bonds in connectionwith an unusual structure. Similar atom arrange-ments could be present at the inner surface of mi-crovoids. It has been shown that in the ground statethe Si—Si bond is doubly H-bridged so that C2v

symmetry is obtained. Other structures as, for ex-ample, a planar-trans orientation yielding C2h sym-metry, are energetically unfavorable. For a selectionof experimental and theoretical papers see refs. 46 –56. The results of our geometry optimization aregiven in Table II. Surprisingly, all examined den-sity functionals show the same behavior: The Si—Hdistances and the H—Si—H angles are slightly over-estimated. The same is true for the Si—Si distanceof the doubly bridged structure, whereas the cor-responding distance of the planar structure is tooshort. Also the dihedral angle of the doubly bridgedstructure is underestimated throughout. To sum up,it can be said that (i) the overall deviations fromthe reference structures are quite small, and that(ii) the various density functionals yield very sim-ilar results. Also, the energy difference between thedoubly bridged and the planar isomer is reproducedquite well.

The effect of a basis set enlargement on the mini-mum geometry is presented in Table III. It is obviousthat there is no significant difference between thedensity functionals and the reference method.

Si2H4 is the most simple example of a com-pound formally containing a Si—Si double bond.Its equilibrium structure is flexed, and possessesC2h symmetry.47, 48, 51 This structure is rendered cor-rectly by all density functionals. BP86 yields a Si—Sidistance of 2.192 Å, in excellent agreement with thereference value of 2.198 Å. As in the case of disi-lyne, the Si—H distances are slightly overestimated

JOURNAL OF COMPUTATIONAL CHEMISTRY 153

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TABLE II.Si2H2, C2v (Doubly Bridged) vs. C2h (Planar, Trans) (R in Å, E in kcal/mol).

CCSD(T) B3LYP BP86 (PW91)2 BPW91 G96PW91

C2vR(Si—Si) 2.218 +0.004 +0.021 +0.016 +0.017 +0.013R(Si—H) 1.664 +0.022 +0.032 +0.028 +0.030 +0.0296 (H—Si—H) 48.2 +0.6 +0.5 +0.5 +0.5 +0.66 (H—Si—Si—H) 105.1 −2.0 −2.3 −2.4 −2.5 −2.7

C2hR(Si—Si) 2.129 −0.018 −0.008 −0.012 −0.010 −0.014R(Si—H) 1.486 +0.010 +0.024 +0.020 +0.022 +0.0216 (H—Si—H) 123.4 +1.1 +0.9 +1.0 +1.0 +1.2

E(C2h)− E(C2v) +14.2 +18.1 +17.0 +17.3 +17.0 +17.0

[1.498 vs. 1.478 Å, obtained by CCSD(T)]. The H—Si—Si angles are too large by 0.3◦, and the dihedralangle is underestimated by 1.1◦. The reference val-ues are 117.4◦ and 43.2◦, respectively. Similar valuesare obtained by use of the other functionals. B3LYP,however, shows some larger but uncritical devia-tions from the reference values: R(Si—Si)= 2.178 Å,R(Si—H) = 1.485 Å, 6 (H—Si—Si) = 118.5◦, and6 (H—Si—Si—H) = 39.5◦.

Also, the disilane minimum (D3d)57 is very wellreproduced by BP86. The H—Si—Si angle devi-ates by +0.2◦ from the reference value of 110.4◦.Again, the Si—H distance is slightly overestimated(1.500 vs. 1.481 Å). The length of the Si—Si sin-gle bond (2.353 Å) is in good agreement withthe result of the coupled-cluster calculation, whichamounts to 2.339 Å. The other functionals yield es-sentially the same results, with the exception thatthe B3LYP Si—H bond length overestimation is low-est (+0.007 Å).

Singlet Excitation Energies

The description of excited-state properties withinDFT is notoriously difficult. Nevertheless, onemight be tempted to interpret the orbital energy

differences ωjk := εj − εk as excitation energies, be-cause in most density functional calculations thelowest virtual orbitals attain a negative energy, andthereby are considered more reasonable than thecorresponding Hartree–Fock orbitals (at least theymay be applied in a qualitative manner in MOarguments58). Despite its lack of a rigorous basis,this interpretation could prove useful for certain ap-plications due to its extreme simplicity. It is wellknown that the ωjk deviate by 10–50% from the trueexcitation energies1E := Ej − Ek (ref. 59), but if, fora certain class of molecular systems, this deviationwould be constant, it could be scaled out. So the ex-penditure to compute true excitation energies couldbe avoided.

The triple-zeta standard basis set cc-pVTZ60 – 65

has been used throughout to determine the respec-tive reference value, which is a true excitation en-ergy in the above sense.

First of all, we have examined the vertical π →π∗ excitation energy (au → bg) of C2h-disilyne. Theresults are shown in Table IV. Most density func-tional orbital gaps overestimate 1E by about 1 eV.The value obtained with B3LYP, however, is morethan twice as large as the true energy. This had to

TABLE III.C2h-Si2H2, cc-pVTZ vs. 6-31G∗∗ Basis Set (R in Å, E in kcal/mol).

CCSD(T) B3LYP BP86 (PW91)2 BPW91 G96PW91

C2hR(Si—Si) −0.007 −0.007 −0.007 −0.007 −0.006 −0.006R(Si—H) +0.006 −0.002 −0.002 0.0 −0.001 −0.0016 (H—Si—H) +1.5 +0.5 +0.6 +0.5 +0.6 +0.5E(cc-pVTZ)− E(6-31G∗∗) −69.0 −34.6 −34.2 −34.3 −33.8 −33.5

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TABLE IV.Excitation Energies (E in eV).

Excitation Basis Set CCSD(T) B3LYP BP86 (PW91)2 BPW91 G96PW91

C2h-Si2H2, 6-31G∗∗ 3.98 2.77 2.78 2.78 2.81π → π∗ cc-pVTZ 1.84 3.99 2.76 2.77 2.78 2.80

D2h-Si2H4, 6-31G∗∗ 1.93 1.00 1.00 1.00π → π∗ cc-pVTZ 0.36

D3d-Si2H6, 6-31G∗∗ 4.34 3.12 3.12 3.14 3.17σ → σ ∗ cc-pVTZ 2.39

be expected, because B3LYP contains Hartree–Fock-type exchange and, therefore, must yield worsevirtual orbital energies. The true B3LYP excitationenergy amounts to 1.06 eV, which is nearly thesame as the orbital gap values obtained by the otherfunctionals, i.e., twice the expenditure results in thesame quality. Note that there is no effect at all if thesmaller 6-31G∗∗ basis set is used instead of cc-pVTZ.

According to the standard model of the SWE,the localized electronic excitation of strained re-gions plays a decisive role in the formation of themetastable defects. We have investigated two cor-responding model systems, namely (i) the verticalπ → π∗ excitation (b3u → b2g) of planar (D2h-) disi-lene, Si2H4, at a Si—Si distance 40% larger than theequilibrium distance, and (ii) the vertical σ → σ ∗excitation (a1g → a2u) of disilane in its D3d ground-state configuration where again the Si—Si distancehas been prolonged by 40%. Recall that this is avalue typically discussed in connection with theSWE (see, e.g., ref. 32). The orbital gaps have beendetermined by use of the 6-31G∗∗ basis set. The re-sults (see Table IV) are qualitatively the same as inthe case of disilyne discussed above. B3LYP yieldsenergies that are unreasonably high with respect tothe reference values. It is true that also the otherdensity functionals overestimate the excitation en-ergies, but the absolute difference is much smaller(about 0.6 eV for both Si2H4 and Si2H6) and essen-tially the same throughout the series.

Singlet–Triplet Transitions

Normally Si tends to favor low-spin electron con-figurations. If, however, Si—Si bonds are strained,or in the case of formal multiple bonds, also tripletstates could play a role in certain processes. On thebasis of the results of the preceding subsection theperformance of density functionals has been testedusing two representatives only, B3LYP and BP86.

The vertical transition energy from the 1A1 equi-librium ground state of the doubly H-bridged dis-ilyne to the lowest triplet (3A2) yields a value of2.10 eV (B3LYP) and 1.98 eV (BP86), respectively.These results are in good agreement with the valueof 2.24 eV obtained by the coupled-cluster referencecalculation.

In planar disilene the HOMO (b3u) is the bondingπ-orbital, whereas the LUMO (b2g) is of π∗-char-acter. The corresponding triplet state is 3B1u, and wehave examined the transition from the 1Ag groundstate in the case of a significantly strained Si—Sibond. At an increased Si—Si distance of 3.020 Å(140% of the equilibrium distance of 2.157 Å) thetriplet lies 0.45 eV above the singlet. Stretching thebond even further the triplet energy rises faster thanthe energy of the singlet so that the energy gapincreases. At 5.0 Å it amounts to 1.24 eV. Usingdensity functionals instead of CCSD(T), the situa-tion changes completely. None of the two densityfunctionals is able to reproduce the correct behav-ior even qualitatively! At 140% of the equilibriumdistance BP86 yields an energy difference of 0.17 eV,whereas the B3LYP value (0.01 eV) is even worse.At larger distances 1Ag and 3B1u cross, and the 5.0 Åenergy gap amounts to −1.11 eV and −0.73 eV forB3LYP and BP86, respectively. Note that this willcause a wrong dissociation behavior.

In contrast to the situation with planar Si2H4, theelectronic structure of disilane is quite simple so thata reasonable description by means of DFT should beexpected. This is the case indeed. B3LYP and BP86yield singlet–triplet gaps of 2.30 eV and 2.34 eV,respectively, whereas the coupled-cluster referencevalue amounts to 2.26 eV.

The Dissociation of Disilane

The reaction Si2H6 → 2SiH3 is reliably describedby DFT as far as the energy difference is concerned:


KRÜGER AND SAX

the five functionals yield dissociation energies be-tween 71.5 and 75.9 kcal/mol, which agrees verywell with the reference value of 73.2 kcal/mol. Thedissociation behavior (i.e., the curve shape), how-ever, is as ill-represented as is known from the HFlevel of theory. With the four nonhybrid function-als, and at a Si—Si distance of 20 Å, the total energyof the electronic state corresponding to the equilib-rium geometry is about 0.035 Hartree higher thantwice the total energy of the ground state of thedissociation product SiH·3. With B3LYP, this gap iseven pronounced (0.056 Hartree). Note that similarresults are obtained with the correlation-consistentcc-pVDZ basis set. Obviously, the amount of corre-lation energy provided by the density functionalsis not sufficient to reproduce the correct behaviorwhich, for example, is obtained by use of MC-SCF.

Electron Affinities

Removing one hydrogen atom from disilane, theradical H3Si—Si·H2 is formed. By addition of oneelectron the closed-shell anion Si2H−5 is obtained.To determine the electron affinity, the geometriesof both the radical and the anion have been opti-mized on the coupled-cluster level of theory. Theradical has been described by an unrestricted wavefunction. Improved total energies have been cal-culated at these stationary points by replacing thesmaller basis set (6-31G∗∗) by a triple-zeta basis (cc-pVTZ) endowed with additional smooth functionsat the radical center (AUG-cc-pVTZ). The electronaffinities are 1.85 eV (BP86) and 1.73 eV (B3LYP), re-spectively. These values are in good agreement withthe reference value of 1.66 eV.

A more critical case is the addition of an electronto disilene in its nonplanar C2h ground state, be-cause thereby an open-shell anion is produced. Pro-ceeding essentially in the same way as before with,of course, AUG-cc-pVTZ on both silicon atoms, theelectron affinity turns out to be 1.06 eV (compare to1.03 eV in ref. 66). BP86 yields 1.36 eV, whereas us-ing B3LYP we obtain 1.27 eV. The agreement is notso good as for the above-mentioned closed-shell an-ion but can, nevertheless, be accepted. Note that inboth cases the density functionals overestimate theaffinity.

The Isomerization of Si2H4

The performance of density functionals regard-ing hydrogen migration has been examined bymeans of the isomerization reaction of disilene tosilylsilylene:67 H2Si—SiH2 → HSi—SiH3. With all

TABLE V.Isomerization of H2Si—SiH2 (E) to H∗Si—SiH3 (P) viaT (R in Å, Relative Energy in kcal/mol).

CCSD(T) B3LYP BP86

E (C2h)R(Si—Si) 2.198 −0.020 −0.006R(Si—H) 1.478 +0.007 +0.0206 (H—Si—Si) 117.3 +1.2 +0.66 (H—Si—Si—H) 43.2 −2.7 −1.0Erel. 0.0 0.0 0.0

T (C1)mean |1R| 0.015 0.026mean |16 | 0.7 1.0Erel. +19.2 +14.8 +13.1

P (Cs)R(Si—Si) 2.389 +0.017 +0.018R(H∗—Si) 1.518 +0.013 +0.0286 (H∗—Si—Si) 89.4 −0.5 −1.7Erel. +8.0 +6.0 +9.0

methods of interest (CCSD(T) and the two densityfunctionals BP86 and B3LYP) the geometries of theeduct, the product, and the transition state havebeen optimized independently. The results can befound in Table V. Note that for the sake of simplicityonly mean deviations from the CCSD(T) structureof the transition state have been given. The struc-tural parameters are well reproduced by the DFTmethods. Although the density functionals under-estimate the height of the barrier E → T → P, theenergetical order is correct.

Tetrasilyl, the Most Simple Monovoid

By removal of the central silicon atom from thetetrahedral tetrasilylsilane, Si(SiH3)4, the most sim-ple example of a monovoid is created. The resultingsystem contains four unpaired electrons. Because itis well known that silicon has a strong tendencyto favor low-spin states, only singlets have beenconsidered. We have optimized the geometry oftetrasilyl in three different symmetry variants (Td,D2d, and C2v) keeping fixed the absolute positionsof the 12 hydrogen atoms to simulate the restrictiveaction (the stiffness) of a large silicon bulk. Table VIpresents the results. The coupled-cluster calcula-tions show that a D2d arrangement of the Si atomsis preferred. However, the curvature of the potentialenergy surface is very flat. This corresponds to thefact that in none of the three structures stable Si—Sisingle bonds are formed. Obviously the Si—H force

156 VOL. 22, NO. 2


TABLE VI.(SiH3)4, Structural Comparison (R in Å, RelativeEnergy in kcal/mol).

CCSD(T) B3LYP BP86

TdR(Si–center) 2.319 2.273 2.246Erel. +0.2 +5.4 +4.5

D2dR(Si–center) 2.315 2.257 2.2296 (Si–center–Si) 108.9 107.5 107.5Erel. 0.0 0.0 0.0

C2vR1(Si–center) 2.320 2.257 2.224R2(Si–center) 2.387 2.282 2.253Erel. +2.2 +19.7 +18.9

constant is so strong that a real relaxation of the fourSi atoms cannot take place. In D2d, the shortest Si—Si distance is still 3.752 Å. Note that this value isonly 0.1 Å lower than the distance of next but oneneighbours in c-Si.

Faced with the enormous complexity of tetrasi-lyl, the density functionals stand their ground.Although the energy differences between the D2dminimum and the Td and C2v configurations, re-spectively, are overemphasized, the ranking is cor-rect. Both BP86 and B3LYP tend to distinct shorterSi—Si distances. Recall that in all silicon hydridesinvestigated in this article the Si—H distances areprolonged with respect to the CCSD(T) reference.The weaker Si—H bonds produced by the densityfunctionals allow for more relaxation of the Si sub-system.

It should be mentioned that Si4H12 dissociatesinto two disilane molecules, if all restrictions areremoved and a total geometry optimization in C1symmetry is performed.

Conclusions

The analysis has shown that the nonhybrid den-sity functionals meet with the requirements de-scribed earlier. For our further investigations wehave selected BP86 out of this group of function-als because of its conceptual simplicity and commonuse. B3LYP has been ruled out because of the unsat-isfactory excitation energy estimates, which are dueto the inclusion of Hartree–Fock exchange. More-over, said B3LYP estimates change dramatically if,instead of using the orbital gap, a true 1E is calcu-lated. Also, this is due to the Hartree–Fock characterof the corresponding exchange functional. Conse-

quently, the other density functionals do not showthis behavior. All functionals, however, fail to de-scribe both strained Si—Si double bonds and thedissociation behavior of Si—Si single bonds.

MODELING LARGER DEFECT-CONTAININGCLUSTERS

All calculations reported in this section have beenperformed using BP86 and the 6-31G∗∗ basis set un-less otherwise stated. Geometry optimizations havebeen performed so that the absolute positions of theatoms of the outer sphere are kept fixed to simulatethe effect of a large and stiff silicon bulk. Accordingto the already mentioned low-spin preference of sil-icon, we have confined our search to singlet statesonly.

Si13H22

By removal of one SiH2 group (labeled r) fromSi14H24 (Fig. 1) a cluster is created that contains twoonly threefold coordinated silicon atoms (labeled aand b, respectively) with one lonely electron each.Derived from the tetrahedral Si14H24 the symme-try is lowered to C2v. This cluster may serve as anexample for a hollow defect at an inner or outer sur-face. Two fourfold coordinated Si atoms (labeled cand d, respectively) are located perpendicularly tothe a–b axis. These four atoms, a, b, c, and d, de-fine a one-sided open cavity in the formerly closedSi14H24 cluster. The positions of a, b, c, and d havebeen optimized. The ground state is 1A1. Althoughatoms a and b tend to form a new bond that isclearly indicated by the decrease of R(a–b) from3.840 Å in Si14H24 to 3.606 Å, the strain exerted by

FIGURE 1. Si14H24.


KRÜGER AND SAX

the neighboring Si atoms is sufficient to hold themin a distance significantly larger than any usual Si—Si single bond.

The well localized a–b bonding HOMO is of a1symmetry. The LUMO (b2) is its antibonding ana-logue. The gap amounts to 1.34 eV which, also ifdownscaled by at least 0.6 eV (see earlier), fits intothe mechanistic concept of the SWE. Note that thecorresponding value of the unperturbed Si14H24 sys-tem is as large as 4.58 eV. This clearly indicates thatthe exictation necessary in the first step of the SWEcan take place at structural defects only.

The vertical electron attachment energy has beendetermined employing restricted open-shell DFTand the AUG-cc-pVTZ basis on atoms a to d. It turnsout to be 2.56 eV for the 2B2 state. The additionalcharge is equally distributed over the four inneratoms. The spin density on a and b amounts to 0.36each, whereas it is negligibly small for the saturatedatoms c and d.

Si13H23

Inserting one hydrogen atom into the cavity ofSi13H22 yields the radical Si13H23. Restricted open-shell DFT has been used, and we allowed for com-plete geometry relaxation of the inner 5 atoms. Theresulting minimum conformation is of Cs symmetry.The additional hydrogen is bound to atom a, andpoints in the direction of the removed SiH2 unit. Itdoes not provide a bridge between a and b. Detailsare given in Table VII. It is to be expected that theremaining unpaired electron should be localized atthe opposite atom (b). In fact, the spin density at bamounts to 0.79. It is worth noting that the addi-tional hydrogen is very strongly bound. The energyof the dissociation into Si13H22 and H is as large as

TABLE VII.Properties of Si13H23 (R in Å, Energies in eV).

R(a–b) 3.918R(a–H) 1.511R(b–H) 2.807R(c,d–H) 4.116Spin density at b 0.79Ediss 3.32HOMO type localized at bLUMO type delocalizedHOMO–LUMO gap 1.39Eva

a 2.12

a Vertical electron attachment energy, calculated with theAUG-cc-pVTZ basis set on atoms a to d and the in-cavity hy-drogen.

FIGURE 2. Si14H20.

76.6 kcal/mol so that at least in this defect modelno hydrogen migration could occur. If, however, aninterstitial hydrogen atom moves through the six-membered rings of a silicon lattice, a barrier of only1.0 kcal/mol has to be overcome. We have shownthis by analyzing the potential energy surface of Henclosed in Si14H20 (Fig. 2).

Si16H36

If we remove the central Si atom from the tetra-hedral cluster Si17H36 (Fig. 3), again a model of amonovoid defect in a silicon lattice is obtained. Aswe have seen in the case of tetrasilyl, tetrahedralsymmetry will not be maintained (see earlier) if thefour atoms surrounding the removed center of theoriginal cluster are allowed to relax. Accordingly,the geometry of the void has been optimized in D2d

symmetry. The results can be found in Table VIII.As for Si13H22, the ground state is 1A1. This singlet,however, is composed of four effectively unpairedelectrons. Again, the strain of the outer sphere pre-vents the formation of normal Si—Si bonds. This is

FIGURE 3. Si17H36.

158 VOL. 22, NO. 2


TABLE VIII.Properties of Si16H36 (D2d) (R in Å, Energies in eV).

R(a,b,c,d–center) 2.262R(a–b) = R(c–d) 3.5906 (a,b–center–bottom) 52.56 (c,d–center–bottom) 127.5HOMO type b2; a–b, c–d bondingLUMO type e; a–b (c–d) antibondinggap 0.22Eva

a 2.38Anion: charge on a,b −0.36Anion: charge on c,d +0.03Anion: spin density on a,b 0.35Anion: spin density on c,d 0.02

a Vertical electron attachment energy, calculated with theAUG-cc-pVTZ basis set on atoms a to d.

also the reason for the extremely small gap, whichsurely would rise significantly if said bonds couldbe realized. From this we learn that, to create a re-liable model of voids in a-Si:H, more silicon shellshave to be wrapped around the void and relax-ation of the total cluster has to be admitted. Thisis what we aim at by using the already mentionedembedding procedure, which will be the topic of asubsequent article.

Si16H37

This cluster is obtained from Si16H36 by plac-ing one hydrogen atom inside the void. In the caseof the hollow defect example Si13H22, only twounpaired electrons are present. Due to the strainexerted by the outer cluster sphere, they cannot pro-duce a normal Si—Si bond, but nevertheless, theyoccupy one common orbital, which has bondingcharacter. The true void in tetrahedral Si16H36, onthe other hand, is surrounded by four equivalentatoms and dangling bonds, respectively, so that fourequivalent unpaired electrons have to be placed insuitable orbitals. However, by distortion towardsD2d this problem can be reduced to a 2 × 2 case ef-fectively similar to Si13H22. But if one hydrogen isbond to a silicon belonging to the inner void sur-face, three lonely electrons are left over. We haveto expect quasi-degeneration of several double- andsingle-occupied and corresponding unoccupied re-stricted open-shell orbitals, and because the overallsymmetry is reduced to Cs at least, orbital mixingbecomes highly probable. It is evident that a systemlike this cannot be described by one single deter-minant. Instead, it is a typical multiconfigurationsituation. Recall, however, that dealing with degen-

erate ground states is not possible within the scopeof DFT. Nevertheless, a unique single-particle den-sity could be found even in this case if the restrictionto pure spin states would be released. If α- and β-electrons occupy different orbitals, greater flexibilityof the wave function is achieved. In consequence,we might hope for removal of the degeneracy. Butit remains open to question whether the densityobtained is reasonable indeed. At least the expecta-tion value of S2 should become significantly largerthan 0.75.

This is exactly what we have observed. Introduc-tion of the innervoid hydrogen reduces the overallsymmetry from D2d to Cs. Employing unrestrictedopen-shell SCF, a minimum on the potential sur-face is reached if the Si(a)—H bond length amountsto 1.496 Å. The Si—Si distance between a and theopposite silicon b is as large as 3.906 Å, whereas cand d are even more farther apart (4.098 Å), whichmeans that the void has expanded. Recall that thedistance between next but one neighbors in c-Siamounts to 3.840 Å. Because 〈S2〉 = 1.27, the cor-responding Kohn–Sham state is considerably spincontaminated.

Summary

We aim at an understanding of the so far un-known nature of the localized reaction centers ina-Si:H that are responsible for the Staebler–Wronskieffect. For this reason we have examined the suit-ability of various density functionals to supplyreliable information on strained and defective hy-drogenated silicon clusters up to Si16H36. The resultsare as follows:

1. Equilibrium geometries, energy gaps betweendifferent local minima on the same hypersur-face, isomerization reactions, dissociation en-ergies, and electron affinities are reproducedwell by all of the functionals in question.

2. In contrast, it is scarcely possible to approx-imate electronic excitation energies by thecorresponding orbital gaps. The true excita-tion energies are overestimated by 0.6–1.0 eVthroughout, so that at best some qualitativestatements can be made. The hybrid functionalB3LYP turned out to be inappropriate (whichhas already been expected).

3. None of the examined density functionals isable to describe strained Si—Si double bondsproperly. Also, the dissociation behavior ofsingle bonds is reproduced incorrectly.


KRÜGER AND SAX

4. Both hollow defects and monovoids can bemodeled, and the electronic states obtainedare reasonable. If, however, hydrogen atomsare inserted, the electronic environment be-comes quite unusual, and a multiconfigura-tion treatment would be more suitable thanany density functional approach. Neverthe-less, one may assume that larger voids willallow for comparatively normal bonding rela-tions as, for example, strained double bonds,so that also in these cases DFT might bepromising.

Acknowledgment

Support for this work was within the scope of thejoint German–Austrian silicon research focus (Siliz-iumschwerpunkt).

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distorted silicon hydrides— a comparative study with various density functionals

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