Si isotopic structure of the infrared absorption of the fully hydrogenatedvacancy in siliconB. Clerjaud and B. Pajot Citation: J. Chem. Phys. 138, 184505 (2013); doi: 10.1063/1.4803547 View online: http://dx.doi.org/10.1063/1.4803547 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v138/i18 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

Downloaded 27 May 2013 to 80.250.180.203. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

THE JOURNAL OF CHEMICAL PHYSICS 138, 184505 (2013)

Si isotopic structure of the infrared absorption of the fully hydrogenatedvacancy in silicon

B. Clerjaud1,2,a) and B. Pajot1,2

1UPMC Université Paris 06, UMR 7588, INSP, F-75005, Paris, France2CNRS, UMR 7588, INSP, F-75005, Paris, France

(Received 4 March 2013; accepted 18 April 2013; published online 13 May 2013)

An infrared absorption spectrum consisting in three lines observed around 2223 cm−1 at liquid he-lium temperature characterizes a defect common in silicon crystals containing hydrogen. Severalinvestigations of this spectrum have converged towards its assignment to a fully hydrogenated latticevacancy defect V(Si–H)4. However, the fact that the ratios of the intensities of the three lines havebeen reported to be proportional to the natural abundances of the three silicon isotopes suggests thatonly one Si atom is involved in the defect, apparently contradicting the above assignment. In thispaper, the spectroscopic investigation of this defect is revisited and the Si-related isotopic structuresof V(Si–H)4 and V(Si–D)4 defects are modeled. It is shown that the near proportionalities observedbetween the intensities of these lines and the abundances of the Si isotopes are fortuitous. Our anal-ysis of the isotope dependence of the 2223 cm−1 line finds the V(Si–H)4 model to be correct andthat the model of a single interstitial Si atom complexed with H can be rejected. The investigationis extended to the analysis of tetra-hydrogenated vacancy trapped by a carbon atom. © 2013 AIPPublishing LLC. [http://dx.doi.org/10.1063/1.4803547]

I. INTRODUCTION

Atomic hydrogen can be introduced in crystalline sili-con by growing the crystal in a hydrogen-containing atmo-sphere or by irradiating samples with high-energy protons ordeuterons. The growth in a hydrogen atmosphere producesa somewhat homogeneous hydrogen distribution while underirradiation hydrogen remains located within the penetrationdepth of the ions.

The presence of hydrogen is attested from the observa-tion in the samples of vibrational absorption bands at fre-quencies characteristic of the stretching of chemical bondsinvolving a hydrogen atom and an atom of the crystal. Thevibrational modes giving these absorptions are termed as“local” as they do not propagate in the crystal because of theirrelatively high frequencies.

Room temperature (RT) absorption measurements ofsilicon samples irradiated with 400 keV-protons have re-vealed several H related vibrational bands with wavenum-bers between 1835 and 2210 cm−1 characteristic of Si–Hstretch modes, together with the electronic absorption near5500 cm−1 of the silicon divacancies (V2) produced by theirradiation.1

The local vibrational mode (LVM) reported at 2210 cm−1

at RT1 was also observed in float-zone (FZ) silicon grown ina H2 atmosphere.2 This LVM was found to shift to a higherfrequency (2223 cm−1) and to sharpen considerably whenmeasured at liquid helium temperature (LHeT), allowing toobserve weak low-frequency satellites attributed to 29Si–Hand 30Si–H modes, as the intensities of the three peaks werefound to be roughly proportional to the natural abundances

a)Author to whom correspondence should be addressed. Electronic mail:bernard.clerjaud@upmc.fr

of the three Si isotopes2 (28Si: 0.9223, 29Si: 0.0467, and 30Si:0.0310). The presence of hydrogen was confirmed by the ob-servation in FZ silicon grown in a D2 atmosphere of the ab-sorption at LHeT of a LVM at 1617 cm−1, as the 2223/1617ratio (1.375) is comparable to the square root of the ratio(1.390) of the reduced masses of the Si–H/Si–D oscillators.2

In samples grown in a mixed (H2 + D2) atmosphere, besidesthe LVMs at 2223 and 1617 cm−1, four new LVMs were ob-served in the immediate vicinity of the former LVMs. Thiswas reasonably interpreted by the presence of several hydro-gen atoms in the center giving the 2223 and 1617 cm−1 LVMs,and by a partial hydrogen isotopic substitution during a mixedhydrogenation. The observation of only one Si–H or Si–DLVM associated to the above center under H2 or D2 growthseemed to imply a tetrahedral (Td) symmetry of this center, asa lower symmetry would have produced more than one LVM.It was therefore suggested that the center responsible of the2223 cm−1 LVM was either an interstitial silane molecule (i-SiH4) or a lattice vacancy V whose four dangling bonds havetrapped a H atom to give V(Si–H)4, noted VH4 for simplicity,with four equivalents Si–H bonds,2 shown in Fig. 1.

The Td symmetry of the center associated with the2223 cm−1 LVM was confirmed by piezospectroscopic mea-surements, and this mode attributed to the T2 stretch mode ofthe Si–H bonds of VH4 (T2 denotes a triply degenerate irre-ducible representation (IR) of Td).3

It has been stated4 that, in silicon, stretching modesof H-related defects with wavenumbers below 2000 cm−1

are connected with H interstitial complexes while thosewith wavenumbers above 2000 cm−1 are connected with H-vacancy related defects. Therefore, Mukashev et al.4 con-cluded that the 2223 cm−1 LVM was due to VH4. This attribu-tion is substantiated by ab initio calculations of the frequency

0021-9606/2013/138(18)/184505/11/$30.00 © 2013 AIP Publishing LLC138, 184505-1

Downloaded 27 May 2013 to 80.250.180.203. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

184505-2 B. Clerjaud and B. Pajot J. Chem. Phys. 138, 184505 (2013)

FIG. 1. Ball and stick model of the VH4 centers in silicon or germanium.The Si–H bonds point towards the missing atom along trigonal axes.

of the T2 mode of VH4 in silicon, which give ≈ 2300 cm−1

(Ref. 5) or 2344 cm−1 (Ref. 6), compared to 1802 cm−1 fori-SiH4 trapped in silicon.7

In addition to the above-discussed triply degenerate T2

mode, a center like VH4 or i–SiH4 also displays a singly de-generate breathing stretch mode with A1 symmetry, where thefour Si–H bonds of the center vibrate in phase. Such a modeis infrared (IR)-inactive, but Raman-active. In silicon samplesimplanted with protons at room temperature, a Raman scat-tering line was indeed measured at 2257 cm−1 at LHeT, andattributed to the A1 mode of VH4.8 In condensed solid SiH4,below 50 K, wavenumbers of ≈ 2170 and 2176 cm−1 havebeen reported for the A1 and T2 stretch modes, respectively.9

It has to be noted that not only LVMs of fully hydro-genated vacancies have been reported, but also those of par-tially hydrogenated vacancies VH, VH2, or VH3 in silicon.8

A H-related absorption mode at 2178 cm−1 at RT hasalso been reported in proton implanted samples10 and in FZsilicon grown in hydrogen atmosphere.11 This mode was ob-served at LHeT in C-rich silicon12 at 2183.32 (2183) cm−1

with a full width at half maximum (FWHM) of 0.12 cm−1.The observation and intensity of this LVM were correlated tothe ones of a LVM at 2826.86 (2827) cm−1, attributed to aC–H or N–H stretch mode.12 The observation of the 2183 and2827 cm−1 LVMs has recently been reported after appropri-ate annealing in multicrystalline (mc) silicon samples with acarbon concentration of 8.8 × 1017 cm−3 initially annealed at1250 ◦C in a H2 atmosphere and rapidly quenched at RT.13

These two LVMs, together with a weaker one at 2214.6 cm−1

(2214.5 cm−1 in Ref. 12), have been convincingly attributedto a VH3 HC center13 (the presence of C was notably attestedfrom the shift of the 2827 cm−1 LVM to 2819 cm−1 in a sili-con sample containing 13C). This center can be seen as a VH4

entity trapped by a C atom.13 In silicon samples grown in a D2

atmosphere, the Si–D and C–D LVMs corresponding to the2183 and 2827 cm−1 LVMs have been reported at LHeT at1587.8 and 2105.1 cm−1, with FWHMs of 0.2 and 0.3 cm−1,respectively.12

It has been shown14 that in C-rich silicon, VH4 begins tomigrate above 550 ◦C and then is trapped at substitutional Cthus forming VH3 HC centers.

Even though many arguments converge towards the attri-bution of the 2223 cm−1 mode to the T2 mode of VH4, it isnot obvious to reconcile its Si-related isotopic structure withthe VH4 attribution. The intensities of the three componentsnear 2223 cm−1 have been considered until now to be propor-tional to the abundances of the three Si isotopes. This wouldmatch perfectly the interstitial silane molecule interpretationas only one silicon atom is involved in this molecule. In theframe of the VH4 interpretation, it would imply that the lineat 2223 cm−1 involves four 28Si atoms bonded to H while thetwo lower wavenumbers satellites would involve three 28Siand one 29Si or 30Si atoms.15 There are however problemswith such a description. If the center with four 28Si has Td

symmetry and only the IR-active T2 mode of vibration, thesymmetry of the center lowers to C3v when one of the 28Siatoms is replaced by a 29Si or 30Si atom, and the triply degen-erate mode splits into a doubly and a singly degenerate modes.One expects therefore five IR-active modes of vibration: onedue to V(28Si–H)4, two to V(28Si–H)3(29Si–H), and two toV(28Si–H)3(30Si–H). Another problem comes from the factthat if V(28Si–H)4 has only one configuration, there are fourpossible orientations in the lattice for V(28Si–H)3(29Si–H) andV(28Si–H)3(30Si–H). This should clearly impede a simple di-rect proportionality between the intensities of the modes andthe abundances of the silicon isotopes.

In order to clarify the situation, in this paper, the Si-related isotopic structures of the stretch modes of VH4, VD4,and VH3 HC are revisited and modeled.

II. EXPERIMENTAL RESULTS

Experiments have been performed with FZ silicon sam-ples cut from crystals grown in H2 or D2 atmospheres andcontaining, for some of them, large concentrations of carbon.These samples were cooled near LHeT in a continuous flowcryostat, and their absorption measured at a spectral resolu-tion of 0.04 cm−1 using a BOMEM DA8 Fourier transformspectrometer. For this spectral resolution, considering that in-strumental broadening can be neglected when the true FWHMof a line is about three times the spectral resolution, there isa small instrumental broadening of the (Si–H) lines shownhere, which depends on their true width. In order to evaluatethe integrated intensities of the absorption lines, an empiricalbaseline has been drawn for each line and the area betweenthis baseline and the line has been measured.

The absorptions of the stretch modes of VH4 around2223 cm−1 and of VD4 around 1617 cm−1 are shown inFigs. 2(a) and 2(b), respectively. In Fig. 2(a), the FWHM ofline 1 is 0.11 cm−1. In additions to the lines noted 1, 2, and3 in Fig. 2(a), some humps can also be observed, with onenoted by an asterisk. These weak features are discussed inAppendix B.

In Fig. 2(b), line 1 at 1617.53 cm−1 is due to VD4.The observed FWHM is ≈0.06 cm−1 and the true FWHMshould be ≈0.05 cm−1. In this spectral region, despite theevacuation of the spectrometer, the residual absorption ofvery intense vibrational lines of water vapor can still be ob-served. Line a in Fig. 2(b) is such a line (its exact position is1616.714 cm−1)16 and it has been wrongly attributed to a

Downloaded 27 May 2013 to 80.250.180.203. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

184505-3 B. Clerjaud and B. Pajot J. Chem. Phys. 138, 184505 (2013)

(a)

(b)

FIG. 2. Absorption of the stretching modes of (a) VH4 in a sample grown inH2 atmosphere, (b) VD4 in a sample grown in D2 atmosphere. In both spectra,the spectral resolution is 0.04 cm−1. The peak wavenumbers and the relativeintensities of the components are indicated in the figures and the spectralspans are the same in (a) and (b).

Si–D mode of VD3H in Ref. 12. Line b at 1616.51 cm−1

can indeed be attributed to a Si–D mode of VD3H instead ofSi–D of VD2H2. The signal-to-noise (S/N) ratio of line 2 at1616.3 cm−1 is not large but this feature can nonetheless beattributed to a Si-isotope-related mode of VD4. In this paper,the wavenumber difference between the main line and oneof its associated satellites will be denoted, may be abusively,isotope shift or isotope splitting (IS). The ISs are ≈0.8 and≈1.6 cm−1 for VH4 and ≈1.2 cm−1 for VD4; the uncertaintyis too large to firmly attribute the weak feature near 1615.0–1615.2 cm−1 to a silicon isotope related structure of VD4. Forcomparison, the 28Si–29Si and 28Si—30Si ISs of the T2 modeof gaseous SiH4 are 1.53 and 2.97 cm−1, respectively.17

The line at 2183 cm−1, due to the VH3 HC center, alsoshows a Si-isotope-related structure, which is displayed inFig. 3.

The wavenumbers of the main lines are indicated inFigs. 2(a) and 2(b), and 3 together with the ratios of their in-tegrated intensities with the ones of lines 1.

The anharmonicity of the T2 modes of VH4 and VD4 insilicon is attested by the observation of overtones of thesemodes at 4388.54 and 3208.07 cm−1, with relative intensitiesof ≈0.8% with respect to the fundamentals.12 The FWHM ofthe overtone of T2(VH4) is 0.6 cm−1, but the S/N ratio pre-

FIG. 3. Absorption of the 2183 cm−1 LVM of VH3 HC in a sample from acrystal with a large C concentration grown in a H2 atmosphere. The spectralresolution is 0.04 cm−1.

cludes the observation of a Si isotope structure of this line(see Fig. 2 of Ref. 12). The overtone of the 2183 cm−1 LVMis observed at 4312.85 cm−1.12

III. MODELING OF VH4

A model similar, in spirit, to the one proposed by BechNielsen et al.18 for VH4 in silicon is used here. This model,where bending modes are not considered, has also been usedfor VH4 trapped by a carbon impurity (VH3 HC).13 In the caseof V(Si–H)4, it restricts the center to four convergent Si–Hbonds along trigonal axes of the silicon lattice coupled witheach other. Each Si–H oscillator, constrained to vibrate alonga trigonal axis, can be reduced to a one-particle system havingthe reduced mass of the oscillator and the modeled center hasonly four degrees of freedom, and thus four modes of vibra-tion. The effect of the rest of the surrounding crystal is takeninto account by defining the reduced mass μ of each oscillatorby

1

μ= 1

mH+ 1

χmSi. (1)

In this expression, first proposed by Thomas and Hopfield,19

mH is the mass of the H atom, mSi the mass of the Si iso-tope bonded to H, and χ is a factor, usually larger than unity,representing the interaction of the Si atom bonded to H withthe surrounding crystal. The model is investigated in the har-monic approximation; as the anharmonicity of the Si–H bondsis independent of the silicon isotope involved,20 anharmonic-ity will not affect noticeably the Si related ISs. The coordi-nate of virtual particle i (i = 1, 2, 3, or 4) is defined by itsdeparture ri from its equilibrium location; ri is taken as pos-itive when the distance to the center of the vacancy is largerthan the equilibrium distance as shown in Fig. 4. The ri’s usedin this paper have therefore signs opposite to those used inRef. 18.

In this section, a model describing the centers with fourand three 28Si atoms is presented. In Appendix B, the modelis extended to all possible isotopic configurations of VH4

and VD4.

Downloaded 27 May 2013 to 80.250.180.203. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

184505-4 B. Clerjaud and B. Pajot J. Chem. Phys. 138, 184505 (2013)

4

3

4444444444444444444444444444444444

33333333333333333333333333333333333333333333333

FIG. 4. Displacement coordinates of the particles for VH4. The origin of eachcoordinate, marked by a dot, corresponds to the equilibrium of the system.The broken line arrows indicate the directions of the electric dipoles inducedby the IR-active normal modes of the center.

A. Isotope shifts and splittings

In the model proposed here, the interaction between os-cillators i and j is described by an elastic force between par-ticles i and j. In principle, the equilibrium bond length de-pends upon the isotope mass,20 but it has been quoted thatthis effect is negligible for the Si isotopes.20 Therefore on theone hand, the equilibrium locations of the four particles are atthe same distance from the center of the vacancy, and on theother hand, the Si–H force constants K and the inter-oscillatorforce constants k are considered as independent of the siliconisotope mass. As the ri’s are considered to be very small as

compared to the distance between the center of the vacancyand the equilibrium location of the particles, to the first order,the change of the orientations of the inter-oscillator forces canbe neglected. In the context of this approximation, one sees inFig. 5(a) that the increase in inter-bond length is (ri + rj)sin(θ /2) where, for VH4, θ = 2 arcsin

√2/3 is the angle between

two Si–H bonds. Therefore, in the harmonic approximation,the interaction between oscillators i and j is described byforces whose norms are

‖Fij‖ = k

√2

3(ri + rj ), (2)

and by a potential energy,21

Epij = 1

3k(ri + rj )2. (3)

Particle i is submitted to three inter-oscillator interactions.The projection on the Si–H axis of oscillator i of the totalforce acting on particle i is therefore

Fi = −Kri − 2

3k

∑j �=i

(ri + rj ). (4)

If the reduced mass of the Si–H oscillator i is μi, the equationof motion of particle i is

μi

d2ri

dt2= −(K + 2k)ri − 2

3k

∑j �=i

rj . (5)

The V(28Si–H)4 center is considered first and the reducedmass of the 28Si–H oscillator is noted μ. In the context ofthis model, the angular frequencies ω of the eigenmodes ofthe center are solutions of

∣∣∣∣∣∣∣∣K + 2k − μω2 2k/3 2k/3 2k/3

2k/3 K + 2k − μω2 2k/3 2k/32k/3 2k/3 K + 2k − μω2 2k/32k/3 2k/3 2k/3 K + 2k − μω2

∣∣∣∣∣∣∣∣= 0. (6)

This system has a triply degenerate solution corresponding toa mode with T2 symmetry at angular frequency

ωT2 =√

K + 4k/3

μ. (7)

It also has a non-degenerate solution, corresponding to an IR-inactive mode belonging to the A1 representation of the Td

group

ωA1 =√

K + 4k

μ. (8)

Values of the force constants K and k can be derived fromthe experimental values of the T2 mode (2223 cm−1) andof the A1 mode of V(28Si–H)4 (2257 cm−1) at LHeT.8 Forχ values in the range 1–3, they yield K = 280 Nm−1 andk = 3.3 Nm−1.

The V(28Si–H)3(29Si–H) and V(28Si–H)3(30Si–H) combi-nations mentioned before involve a center with three 28Si–H

bonds and one ASi–H bond where A = 29 or 30. Let us, forinstance, assume that the ASi–H bond is bond 1 of Fig. 4,with reduced mass μ′. This center has C3v symmetry wherethe C3 axis is the axis of bond 1. The angular eigenfrequen-cies are now given by

∣∣∣∣∣∣∣∣K + 2k − μ′ω2 2k/3 2k/3 2k/3

2k/3 K + 2k − μω2 2k/3 2k/32k/3 2k/3 K + 2k − μω2 2k/32k/3 2k/3 2k/3 K + 2k − μω2

∣∣∣∣∣∣∣∣= 0. (9)

Downloaded 27 May 2013 to 80.250.180.203. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

184505-5 B. Clerjaud and B. Pajot J. Chem. Phys. 138, 184505 (2013)

θ θθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθθ

α αααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααααα

β

(a)

(b)

FIG. 5. Definition of the angles introduced in Secs. III and IV. (a) corre-sponds to VH4 and VD4 and (b) corresponds to VH3 HC. The heavy brokenlines indicate the inter-particles equilibrium distances, whereas the full linesindicate the out of equilibrium inter-particles distances.

This system has a doubly degenerate solution correspondingto a mode transforming like the E representation of the C3v

group, with angular frequency

ωE =√

K + 4k/3

μ, (10)

and two nondegenerate A1 symmetry modes with angular fre-quencies

ω2A±

1= 1

2

(K + 10k/3

μ+ K + 2k

μ′

±√(

K + 10k/3

μ− K + 2k

μ′

)2

+ 16k2

3μμ′

⎞⎠ . (11)

The + sign corresponds to a very weakly IR-active mode, notreported up to now, and the − sign to a fully IR-active mode.

One notes that the angular frequency of the E mode isexactly the same as the one of the T2 mode of the purelytetrahedral V(28Si–H)4 center. Thus, line 1 in Fig. 2(a) at2222.97 cm−1 is the superposition of the T2 mode of theV(28Si–H)4 center and of the E modes of the V(28Si–H)3(29Si–H) and V(28Si–H)3(30Si–H) centers while lines 2 and 3 in thisfigure correspond to the A−

1 modes of V(28Si–H)3(29Si–H)and V(28Si–H)3(30Si–H) centers, respectively. Similarly, line1 of Fig. 2(b) is the superposition of the T2 mode of V(28Si–D)4 and of the E modes of V(28Si–D)3(29Si–D) and V(28Si–D)3(30Si–D) and line 2 in the same figure is ascribed to theA−

1 mode of V(28Si–D)3(29Si–D).

One could indeed use the values of the force constantsdetermined previously to obtain directly the angular frequen-cies of the A1 modes. However, as μ and μ′ are quite close toeach other, the expressions for these modes can take simplerforms by introducing a dimensionless parameter ε such as

1

μ′ = 1

μ(1 − ε). (12)

In the cases considered in this paper, ε is positive and of theorder of 10−3. It is therefore legitimate to perform an expan-sion of ωA−

1limited to the terms linear with respect to ε. One

obtains

ω2A−

1= K + 4k/3

μ

(1 − 3

4ε

)= ω2

T2

(1 − 3

4ε

). (13)

To first order in ε,

ωA−1

= ωT2

(1 − 3ε

8

), (14)

and the IS is thus

SωT2 = ωT2 − ωA−1

= 3

8εωT2 , (15)

where the dimensionless factor S is denoted IS factor.Similarly, the IS of the A+

1 mode with respect to A1 is

ωA1 − ωA+1

= 1

8εωA1 . (16)

ωT2 and ωA1 are known from experiments and the onlyevaluation of ε is necessary to compare Eqs. (15) and (16)with the experimental results. For χ = 1, i.e., using the sim-ple diatomic mass, one obtains, for the infrared lines of the29Si and 30Si components of VH4, IS of 0.99 and 1.92 cm−1,respectively, and for those of VD4, IS of 1.39 and 2.70 cm−1,respectively. These values are slightly larger than the observedISs in Figs. 2(a) and 2(b) (0.82 and 1.61 cm−1 for VH4 and1.2 cm−1 for V(28Si–D)3(29Si–D)). Using χ = 1.2, the valuethat has been used in Ref. 5 for the investigation of VH4 insilicon, ε29(H) = 0.9966 × 10−3, ε30(H) = 1.927 × 10−3,ε29(D) = 1.937 × 10−3, and ε30(D) = 3.745 × 10−3 yielding28Si–29Si and 28Si–30Si ISs of 0.83 and 1.61 cm−1 for VH4

and 1.17 and 2.27 cm−1, respectively, for VD4. These valuesmatch rather well the observed ISs in Figs. 2(a) and 2(b) andthey confirm the validity of the model used. From Eq. (14), thewavenumber predicted for the A−

1 mode of V(28Si–D)3(30Si–D) is 1615.26 cm−1.

It must be noted that, as shown in Appendix A, the iso-topic shifts of a hypothetical i–SiH4 molecule should be abouttwice larger than those measured and modeled for VH4.

The ISs of mode A+1 that is practically only Raman ac-

tive should be almost one third of the values of the ISs of theIR spectrum. As a spectral resolution of 7 cm−1 was used inRef. 8, the ISs of the Raman mode were impossible to detectin these experiments.

B. Intensities of the absorption lines

The vibration-induced dipoles depend only very weaklyon the silicon atom masses. Therefore, for calculating the in-tensities of the absorption lines, one neglects the difference

Downloaded 27 May 2013 to 80.250.180.203. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

184505-6 B. Clerjaud and B. Pajot J. Chem. Phys. 138, 184505 (2013)

between the μ and μ′ reduced masses. In the context of thisapproximation, vibrations of the A+

1 mode do not induce anyelectric dipole. If one uses the bond numbering of Fig. 4 andassume that the eventual non-28Si is in location 1, the com-ponents of the electric dipoles induced by the vibrations aregiven by22

⎛⎝px1

py1

pz1

⎞⎠ = −a

√μ

2√

3

⎛⎝

√2 (2r4 − r2 − r3)√

6 (r3 − r)3r1 − (r2 + r3 + r4)

⎞⎠ , (17)

where the x1, y1, and z1 axes correspond to the 〈112〉, 〈110〉,and 〈111〉 axes, respectively, and a is a constant describing thestrength of the coupling of the radiation to the local mode.22

For the centers with four 28Si atoms, px1 , py1 , and pz1 are thecomponents of electric dipole induced by T2 mode; for thecenters with three 28Si atoms, pz1 is induced by the A−

1 modeand px1 and py1 are the components of electric dipole inducedby the E mode.

Normalized to unity, the natural abundances of the Si iso-topes are: P28 = 0.9223, P29 = 0.0467, and P30 = 0.0310.The probability of occurrence of a V(28Si–H)4 center is�28 = P 4

28. The probability of occurrence of a V(28Si–H)3(29Si–H) center with a 29Si atom in position i and three28Si atoms in locations j �= i is π29 = P29P

328, but there are

four possible locations for the 29Si atom. Therefore, the globalprobability for finding a V(28Si–H)3(29Si–H) center is �29

= 4P29P328 and in the same way, the probability for finding a

V(28Si–H)3(30Si–H) center is �30 = 4P30P328. When N is the

total number of V(Si–H)4 centers per unit volume, the inten-sity of the integrated absorption of the T2 mode of VH4 in-volving four 28Si–H bonds is22 IT2 = πNP 4

28a2/2ncε0 where

n is the silicon refractive index, c the speed of light in vac-uum, and ε0 the permittivity of vacuum. The intensity of theabsorption of the E mode of the V(28Si–H)3(29Si–H) center isgiven by I 29

E = 4πNP29P328a

2/3ncε0 and for the A−1 mode of

the same center I 29A−

1= 4πNP29P

328a

2/6ncε0,22 with equiva-

lent expressions for the V(28Si–H)3(30Si–H) centers. Lines 1in Figs. 2(a) and 2(b) are the superposition of IT2 , I 29

E , andI 30

E , therefore their intensities are I1 = IT2 + I 29E + I 30

E . Lines2 in Figs. 2(a) and 2(b) correspond to the A−

1 modes of thecenters involving three 28Si and one 29Si, therefore I2 = I 29

A−1;

and for line 3 in Fig. 2(a) I3 = I 30A−

1. Therefore,

I2

I1= 4P29

3P28 + 8P29 + 8P30= 0.055, (18)

I3

I1= 4P30

3P28 + 8P29 + 8P30= 0.037. (19)

In Table I, a comparison between the experimental spec-troscopic characteristics and those obtained from the abovemodel is given for V(Si–H)4 and V(Si–D)4. The values of therelative intensities given by Eqs. (18) and (19) are those inthe column “Model 1.” One notices a satisfactory agreementbetween the experimental and modeled values, even thoughthe experimental values of the intensities seem slightly largerthan those deduced from the model.

TABLE I. Experimental and modeled characteristics of the main lines of theVH4 and VD4 spectra. The experimental values are at 6 K. χ = 1.2 and naturalabundances of the Si isotopes have been used. Model 1 considers only cen-ters involving three or four 28Si whereas model 2, described in Appendix B,considers all possible combinations of the first silicon neighbors. The ratio ofthe natural abundances of the Si-isotopes are P29/P28 = 0.051 and P30/P28

= 0.034. The asterisk refers to the line marked by an asterisk in Fig. 2(a); thisline is explained in the context of model 2.

Wavenumber (cm−1) Relative intensity

Line Experiment Model Experiment Model 1 Model 2

1 2222.97 ± 0.01 1 1 1VH4 2 2222.14 ± 0.04 2222.14 0.063 ± 0.004 0.055 0.055

3 2221.35 ± 0.06 2221.36 0.043 ± 0.004 0.037 0.036* 2221.9 ± 0.1 2221.9 0.010 ± 0.007 0.007

1 1617.53 ± 0.03 1 1 1VD4 2 1616.3 ± 0.1 1616.36 0.06 ± 0.02 0.055 0.055

3 1615.26 0.037 0.036

IV. MODELING OF VH4 TRAPPED BYSUBSTITUTIONAL CARBON (VH3 HC)

A. Isotope shifts and splittings

Let us consider the VH3 HC center, noted also here V(Si–H)3 HC involving first three 28Si–H bonds. Its point groupsymmetry is C3v , with the trigonal axis along the C–H bond.The same notations as in Sec. III are used. At a differencewith VH4, the value of the angle θ between the Si–H bondsis not precisely known. For small displacements, the increasein inter-Si–H bond length is (ri + rj)sin (θ /2) to the first or-der; therefore, in the harmonic approximation, the interactionbetween Si–H oscillators i and j is described by forces whosenorms are

‖Fij‖ = k(ri + rj ) sin(θ/2), (20)

and by a potential energy

Epij = 1

2k(ri + rj )2 sin2(θ/2). (21)

The C–H bond involved in VH3 HC is numbered 1. Asthe C atom involved in the center relaxes away more thanthe Si atoms do, two angles are necessary for describing thecoupling between the C–H and Si–H bonds; the α and β an-gles shown in Fig. 5(b) are chosen. Out of equilibrium, in thesmall displacements approximation, the increase of the Si–H/C–H inter-bond distance is r1cos α + ricos β. Denoting kC

the force constant associated with the coupling of C–H andSi–H bonds, the interaction between these bonds is described,in the context of the harmonic approximation, by the potentialenergy

Ep1i = 1

2kC(r1 cos α + ri cos β)2, (22)

and by forces whose norms are

‖F1i‖ = kC(r1 cos α + ri cos β). (23)

Denoting KC the force constant associated with the C–H bondand μC the reduced mass of the associated oscillator, the equa-tion of motion of particle 1 for a purely stretching motion in

Downloaded 27 May 2013 to 80.250.180.203. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

184505-7 B. Clerjaud and B. Pajot J. Chem. Phys. 138, 184505 (2013)

the harmonic approximation is

μCd2r1

dt2= −(KC + 3kC cos2 α)r1 − kC cos α cos β

∑i �=1

ri,

(24)and those of particles i �= 1 are

μi

d2ri

dt2= −[K + 2k sin2(θ/2) + kC cos2 β]ri

− kC cos α cos β r1 − k sin2(θ/2)∑j �=1,i

rj . (25)

If one defines

G = KC + 3kC cos2 α − μCω2, (26)

H = kC cos α cos β, (27)

I = K + 2k sin2(θ/2) + kC cos2 β − μω2, (28)

J = k sin2(θ/2), (29)

the V(28Si–H)3 HC center has stretching modes whose angularfrequencies are roots of the equation∣∣∣∣∣∣∣∣

G H H H

H I J J

H J I J

H J J I

∣∣∣∣∣∣∣∣= 0. (30)

One sees that if I = J, the three last lines of the determinantare identical; this means that I = J corresponds to a doubly de-generate mode, i.e., the E mode of vibration of the center (Edenoting irreducible representation of the C3v group). The an-gular frequency of the E mode of vibration is therefore givenby

K + 2k sin2(θ/2) + kC cos2 β − μω2E = k sin2(θ/2), (31)

or

ωE =√

K + k sin2(θ/2) + kC cos2 β

μ. (32)

Let us now consider the VH3 HC center with C1h symme-try involving two 28Si atoms and one 29Si or 30Si atom. Withatom 4 taken as the 29Si or 30Si atom, the stretching motion ofthis atom is described by

μ′ d2r4

dt2= −[K + 2k sin2(θ/2) + kC cos2 β]r4

− kC cos α cos β r1 − k sin2(θ/2)∑j=2,3

rj . (33)

As ε 1, to the first order, μ′ = μ(1 + ε). Therefore, thestretching angular frequencies of the center are given by∣∣∣∣∣∣∣∣

G H H H

H I J J

H J I J

H J J I − εμω2

∣∣∣∣∣∣∣∣= 0. (34)

One notes that for I = J, the second and third lines of the de-terminant are identical. This means that I = J corresponds toa non-degenerate solution of Eq. (34), i.e., in this case, the A′′

mode of vibration (A′′ denotes an irreducible representationof the C1h group). Therefore, the angular frequency of the Emode of the center involving three 28Si and the one of the A′′

modes of the center involving two 28Si atoms and one 29Si or30Si are identical. The line observed at 2183.32 cm−1 is thusthe superposition of the absorption of the E (representationof the C3v group) mode of the V(28Si–H)3 HC center and ofthe A′′ (representation of the C1h group) modes of the V(28Si–H)2(29Si–H) HC and V(28Si–H)2(30Si–H) HC centers.

The angular frequencies of the three other modes areroots of the equation

(I − J )[G(I + 2J ) − 3H 2] − εμω2[G(I + J ) − 2H 2] = 0.

(35)The A′ mode involved in the structure around 2183cm−1 hasan angular frequency very close to ωA′′ . Therefore, one focuson the ωA′ solution of this equation such as I = J + δμω2

A′ , δbeing a very small dimensionless quantity. The previous equa-tion thus can be written

δ[G(3J + δμω2A′ ) − 3H 2] − ε[G(2J + δμω2

A′ ) − 2H 2] = 0.

(36)Neglecting the second order terms, i.e., the terms proportionalto δ2 and δ ε, it becomes

δ(3GJ − 3H 2) − ε(2GJ − 2H 2) = 0, (37)

or

δ = 2

3ε. (38)

Therefore,

K + 2k sin2(θ/2) + kC cos2 β − μω2A′

= k sin2(θ/2) + 2

3εμω2

A′ (39)

or

μω2A′

(1 + 2

3ε

)= K + k sin2 (θ/2) + kC cos2 β = μω2

E.

(40)One thus obtains, to first order of ε,

ωA′ = ωE

(1 − ε

3

). (41)

Taking for factor χ in the reduced masses μ and μ′

the same value 1.2 as for VH4, one obtains for the 29Si and30Si isotopic components of ωA′ ISs of 0.73 and 1.40 cm−1,respectively, agreeing well with the experimental values ofFig. 3 (0.76 and 1.4 cm−1).

B. Intensities of the absorption lines

In this subsection, the relative intensities of the threecomponents of the structure around 2183 cm−1 are calculated.As in Sec. III B, the differences between the μ and μ′ reducedmasses are neglected. With the C atom in location 1, the com-ponents of the electric dipole induced by the E mode of the

Downloaded 27 May 2013 to 80.250.180.203. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

184505-8 B. Clerjaud and B. Pajot J. Chem. Phys. 138, 184505 (2013)

center involving three 28Si are given by(px1

py1

)= −b

√μ

6

(2r4 − r2 − r3√

3 (r3 − r)

), (42)

where constant b describes the strength of the coupling of theradiation to the local mode.22

The probability for a V(28Si–H)3 HC center with theC atom in location 1 is π28 = P 3

28, therefore, the overall prob-ability for finding V(28Si–H)3 HC is �28 = 4P 3

28, as there arefour possible locations for the C atom.

For a V(28Si–H)2(ASi–H) HC center where A = 29 or 30,if the C atom is in location 1 and the ASi atom in location4, px1 is induced by the A′ mode and py1 by the A′′ mode.The probability to have a configuration with the C atom in lo-cation 1, a 29Si atom in location 4 and two 28Si in locations2 and 3 is π ′

29 = P29P228. As there are three possible config-

urations with a C atom in location 1, one 29Si and two 28Siatoms in the center, the probability for such a configuration isπ29 = 3P 2

28P29. As there are four possible locations for the Catom, the overall probability to have a configuration with aC atom, a 29Si and two 28Si atoms is �29 = 12P 2

28P29. In thesame way, the probability to have a center with a C atom, a30Si atom and two 28Si atoms is �30 = 12P 2

28P30.With N denoting the total number of VH3 CH cen-

ters per unit volume, the integrated intensity of the ab-sorption of the E mode of the V(28Si–H)3 HC center isIE = 4πNP 3

28b2/3ncε0.22 The intensities of the A′ and A′′

modes of the V(28Si–H)2(29Si–H) HC center are the same:I 29

A′ = I 29A′′ = 12πNP 2

28P29b2/6ncε0, with equivalent expres-

sions for the A′ and A′′ modes of the V(28Si–H)2(30Si–H) H Ccenter.

For comparison with the experimental spectrum in Fig. 3,line 1 at 2183.32 cm−1, is the superposition of the E modeof V(28Si–H)3 HC and of the A′′ modes of V(28Si–H)2(29Si–H) HC and V(28Si–H)2(30Si–H) HC; its intensity is there-fore I2183.32 = IE + I 29

A′′ + I 30A′′ . Lines 2 and 3 in Fig. 3, at

2182.56 and 2181.9 cm−1, are due to the A′ modes of V(28Si–H)2(29Si–H) HC and V(28Si–H)2(30Si–H) HC, respectively.Thus, I2182.56 = I 29

A′ and I2181.9 = I 30A′ . Therefore,

I2182.56

I2183.32= 3P29

2P28 + 3P29 + 3P30= 0.067, (43)

I2181.9

I2183.32= 3P30

2P28 + 3P29 + 3P30= 0.045. (44)

Table II gives a comparison of the experimental results withthose from the model developed in Sec. IV, and it is seen thatthe ratios in Eqs. (43) and (44) are within the error bars of theexperimental results.

A striking result is that neither the relative angular fre-quencies nor the relative intensities of the lines depend onthe strength of the coupling between C–H and Si–H oscilla-tors. This means that the results obtained are also valid if thiscoupling is zero corresponding in fact to VH3. Therefore, thesilicon related isotopic structure of the VH3 absorption line at2155 cm−1 (Ref. 8) should be very similar to the one of the2183 cm−1 line of VH3 CH. Unfortunately, to our knowledge,a detailed spectrum of the 2155 cm−1 line of VH3 is not avail-

TABLE II. Experimental and modeled characteristics of the three main linesin the spectrum of VH4 trapped by carbon. The experimental values are at6 K. χ = 1.2 has been used in column 3 and natural abundances of the Siisotopes have been used in column 5.

Wavenumber (cm−1) Relative intensity

Line Experiment Model Experiment Model

1 2183.32 ± 0.02 1 12 2182.56 ± 0.06 2182.59 0.07 ± 0.02 0.0673 2181.9 ± 0.1 2181.92 0.05 ± 0.02 0.045

able in the literature,23 impeding comparisons with the modeldeveloped in this section.

V. CONCLUSION

In this paper, it has been shown that the Si-related iso-topic structures of the IR absorption lines around 2223/1617cm−1 observed in silicon samples grown under H2/D2 atmo-spheres or implanted with protons/deuterons are wholly com-patible with fully hydrogenated monovacancies (VH4/VD4). Itturns out that if the intensities of the three main lines of thespectra look proportional to the abundances of the silicon iso-topes, the physical modeling does not substantiate further thissimilarity. The most intense absorption line is the superposi-tion of the absorptions due to the triply degenerate mode ofthe V(28Si–H)4 center and of modes of VH4 centers involving29Si or/and 30Si nearest neighbors. The experimental obser-vation of several small features in addition to the three mainlines around 2223 cm−1 in hydrogenated samples providesin fact an additional argument for assigning the center re-sponsible of this absorption to V(Si–H)4 as interstitial silanemolecules would give rise to only three components whoseintensities should strictly scale with the abundances of the sil-icon isotopes.

The model developed in this paper should in princi-ple also apply to tetra-hydrogenated vacancies in other four-fold coordinated materials such as germanium or III-V com-pounds. In germanium samples implanted with protons at30 K, a LVM at 2061.5 cm−1 is observed at LHeT after an-nealing at room temperature, but it is no longer observed af-ter annealing at about 635 K.24 The splitting under stress ofthis LVM has led to its attribution to the T2 stretch mode ofthe VH4 center.24 In deuteron-implanted samples, the stretchmode of VD4 is located at 1488.5 cm−1.24 There is no indica-tion of a Ge isotope effect of these LVMs, which should be,in any case, very difficult to observe because of the numberof isotopic components involved, as there are five Ge isotopesall of which have natural abundances above 7%. Moreover,for Ge, the ε factors defined in Eq. (12), and therefore theISs of the lines are smaller than in silicon as the average Gemass is about 72 u. A sharp LVM at 2315 cm−1 is observed atLHeT in indium phosphide samples from LEC crystals grownwith water-containing encapsulant.25, 26 It has been ascribedto the T2 stretch mode of an In vacancy “decorated” by fourH atoms (VIn(P–H)4),25–28 but P has only one natural isotopeand no comparison is possible. To our knowledge, LVMs of

Downloaded 27 May 2013 to 80.250.180.203. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

184505-9 B. Clerjaud and B. Pajot J. Chem. Phys. 138, 184505 (2013)

VH4 like centers have not been reported in other III-V or inII-VI materials.

The isotopic structure of the line around 2183 cm−1 dueto VH4 trapped by a C atom has also been investigated. Thecomponent at 2183.32 cm−1 is the superposition of the doublydegenerate mode of the center involving three 28Si neighborsand of non-degenerate modes of centers involving on the onehand two 28Si and a 29Si neighbors and on the other hand two28Si and a 30Si neighbors; the two other components observedare due to non-degenerate modes of centers involving, respec-tively, on the one hand two 28Si and a 29Si neighbors and onthe other hand two 28Si and a 30Si neighbors. The model pre-dicts that VH4 trapped by other impurities than carbon, or iso-lated VH3, should show Si-related isotopic structures almostidentical to the one of the VH3 HC center; it would be of in-terest to test this prediction.

APPENDIX A: ISOTOPE SHIFTS OF A HYPOTHETICALi-SiH4 MOLECULE

For estimating the isotope shifts of a hypothetical i-SiH4

molecule, a valence force approximation of the molecule isused. Moreover, as the radial force constant is more than oneorder of magnitude larger than the transverse force constant,the latter one is neglected. In the context of this crude model,the angular frequency of the T2 mode of vibration is givenby20

ω =√

K

(1

mH+ 4

3mSi

)=

√K

μT, (A1)

where K is the Si–H force constant and μT the reduced massof the T2 mode. In the same spirit as in Secs. III and IV, de-noting μT(A) the reduced mass of the T2 mode correspondingto the Si-isotope with A nucleons, one can define a dimension-less parameter γ A such as

1

μT (A)= 1

μT (28)(1 − γA) . (A2)

Numerically, γ 29(H) = 1.567 × 10−3 and γ 30(H) = 3.030× 10−3. Denoting ωA the angular frequency of the T2 modeof the ASiH4 molecule, one obtains

ωA = ω28

√(1 − γA) ≈ ω28

(1 − γA

2

)(A3)

and thus

ω28 − ωA = γA

2ω28. (A4)

One can test the validity of Eq. (A4) on the free SiH4

molecule. Using for this molecule the experimental valueω28 = 2189.16 cm−1 quoted in Ref. 17, one obtains ISs of1.72 and 3.32 cm−1 for 29SiH4 and 30SiH4, respectively. Tak-ing into account the crudeness of the model used in this ap-pendix, these ISs compare reasonably with the experimentalvalues17 of 1.53 and 2.97 cm−1.

A hypothetical i-SiH4 molecule in silicon having ω28

= 2222.97 cm−1 would have ISs around 1.7 and 3.4 cm−1

for i-29SiH4 and i-30SiH4, respectively. These values are al-most twice larger than the one measured experimentally in

Fig. 2(a). This discrepancy provides a supplementary argu-ment for rejecting the attribution of the spectrum shown inFig. 2(a) to i-SiH4.

APPENDIX B: EXTENSION OF THE MODEL FOR V H4

In this appendix, VH4 centers with less than three 28Sinearest neighbors of the vacancy are considered. All the pos-sible Si isotopic configurations of the VH4 center are thus ex-plored.

The techniques used in Secs. III and IV for the modelingof VH4 and VH3 HC can be extended for modeling VH4 cen-ters involving less than three 28Si. The results for all possiblesilicon related configurations are given in Table III. The S andε’s coefficients appearing in this table are defined in Sec. IIIand the new parameters ξ±, η±, and ζ± are defined below as

ξ± = 3 (ε29 + ε30) ±√

9 (ε29 + ε30)2 − 32ε29ε30

8, (B1)

η± = 3 (ε30 − 2ε29) ±√

9 (ε30 − 2ε29)2 + 32ε29 (ε30 − ε29)

8,

(B2)

ζ± = 3 (ε29 − 2ε30) ±√

9 (ε29 − 2ε30)2 − 32ε29 (ε30 − ε29)

8.

(B3)The wavenumbers given in Table III have been calculated us-ing χ = 1.2.

When taking into account all possible Si isotopic config-urations, the line of VH4 at 2222.97 cm−1 is shown to be thesuperposition of six modes compared to the three ones con-sidered in Sec. III. The intensities of the three componentsomitted in Sec. III are quite weak in silicon having naturalisotopic composition and therefore the intensity ratios calcu-lated in Sec. III are only very weakly affected. The column“Model 2” of Table I gives the relative intensities of VH4 is-sued from the model taking into account the six componentsof the 2222.97 cm−1 line.

Six modes also coincide at 2221.86 cm−1 and a mode at2221.90 cm−1 almost coincides with these six modes. Theseseven modes correspond to the line marked by an asterisk inFig. 2(a) observed experimentally at 2221.9 cm−1. The rela-tive intensity of this line issued from the model is 0.007 thatcompares relatively well with the experiment (0.010) takinginto account the large uncertainty on the experimental value.

Several other modes in Table III have noticeable in-tensities. The most intense ones, corresponding to V(28Si–H)2(29Si–H)(30Si–H), are at 2222.30 and 2221.21 cm−1; thesevalues match well humps appearing on the high energy tail ofline 2 and on the low energy tail of line 3 in Fig. 2(a). How-ever, these humps are too weak for allowing to state defini-tively that they correspond to real lines. It has to be notedthat these two modes have not been considered in the mea-surements of the intensities of lines 2 and 3 of Fig. 2(a);therefore, the measured intensities of these two lines might beoverestimated.

Downloaded 27 May 2013 to 80.250.180.203. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

184505-10 B. Clerjaud and B. Pajot J. Chem. Phys. 138, 184505 (2013)

TABLE III. Modeled wavenumbers (cm−1) and relative intensities of the Si isotopic combinations of the Si–H stretch modes of VH4 in silicon. S is the isotopeshift factor and the wavenumber in column 5 is (1 − S) × 2222.97 cm−1. The wavenumbers in column 5 are calculated for χ = 1.2. The intensities in column6 are expressed in terms of the abundances P28, P29, and P30 of the Si isotopes and those in column 7 are for the natural Si abundances.

Isotopic configuration Symmetry group IR S Wavenumber (cm−1) Intensity proportional to Intensity (a.u.)

(28Si–H)4 Td T2 0 2222.97 3P284 2.1708

(28Si–H)3(29Si–H) C3v E 0 2222.97 8P283 P29 0.2931

A1 3ε29/8 2222.14 4P283 P29 0.1466

(28Si–H)3(30Si–H) C3v E 0 2222.97 8P283 P30 0.1946

A1 3ε30/8 2221.36 4P283 P30 0.0973

(28Si–H)2(29Si–H)(30Si–H) C1h A′′ 0 2222.97 12P282 P29P30 0.0148

A′ ξ−/2 2222.30 12P282 P29P30 0.0148

A′ ξ+/2 2221.21 12P282 P29P30 0.0148

(28Si–H)2(29Si–H)2 C2v B1 0 2222.97 6P282P29

2 0.0111

A1 ε29/4 2222.42 6P282P29

2 0.0111

B2 ε29/2 2221.86 6P282P29

2 0.0111

(28Si–H)2(30Si–H)2 C2v B1 0 2222.97 6P282P30

2 0.0049

A1 ε30/4 2221.90 6P282P30

2 0.0049

B2 ε30/2 2220.83 6P282 P30

2 0.0049

(28Si–H)(29Si–H)3 C3v A1 ε29/8 2222.69 4P28 P293 0.0004

E ε29/2 2221.86 8P28 P293 0.0008

(28Si–H)(30Si–H)3 C3v A1 ε30/8 2222.43 4P28 P303 0.0001

E ε30/2 2220.83 8P28 P303 0.0002

(28Si–H)(29Si–H)2(30Si–H) C1h A′ (η− + ε29)/2 2222.65 12P28 P292P30 0.0007

A′ (η+ + ε29)/2 2221.13 12P28 P292P30 0.0007

A′′ ε29/2 2221.86 12P28 P292P30 0.0007

(28Si–H)(29Si–H)(30Si–H)2 C1h A′ (ζ− + ε30)/2 2222.58 12P28 P29P302 0.0005

A′ (ζ+ + ε30)/2 2221.46 12P28 P29P302 0.0005

A′′ ε30/2 2220.83 12P28 P29P302 0.0005

(29Si–H)3(30Si–H) C3v E ε29/2 2221.86 8P293 P30

A1 (ε29 + 3ε30)/8 2221.09 4P293 P30

(29Si–H)2(30Si–H)2 C2v B1 ε29/2 2221.86 6P292 P30

2

A1 (ε29 + ε30)/4 2221.35 6P292 P30

2

B2 ε30/2 2220.83 6P292 P 2

30

(29Si–H)(30Si–H)3 C3v A1 (3ε29 + ε30)/8 2221.60 4P29 P303

E ε30/2 2220.83 8P29 P303

(29Si–H)4 Td T2 ε29/2 2221.86 3P294 1.43 × 10−5

(30Si–H)4 Td T2 ε30/2 2220.83 3P304 2.8 × 10−6

It would be indeed of interest to measure the VH4 andVD4 spectra in isotopically enriched silicon crystals in orderto compare them with the results predicted for lines that arebelow the detection limit in silicon with natural abundances ofthe silicon isotopes. In quasi-monoisotopic 28Si, 29Si, and 30Sisamples implanted with protons, Si-isotope dependent LVMsof the partially hydrogenated VH2 defect have indeed beenmeasured.29

At last, it must be noted that the feature marked by anasterisk in Fig. 2(a) could not be interpreted in the frame ofan interstitial silane molecule. Therefore, it could provide anadditional argument for assigning the center responsible ofthe IR structure discussed in this paper to VH4.

1H. J. Stein, J. Electron Mater. 4, 159 (1975).2G. R. Bai, M. W. Qi, L. M. Xie, and T. S. Shi, Solid State Commun. 56,277 (1985).

3B. Bech Nielsen, J. Olajos, and H. G. Grimmeiss, Phys. Rev. B 39, 3330(1989).

4B. N. Mukashev, M. F. Tamendarov, and S. Z. Tokmoldin, Mater. Sci. Fo-rum 38–41, 1039 (1989).

5B. Bech Nielsen, L. Hoffmann, M. Budde, R. Jones, J. Goss, and S. Öberg,Mater. Sci. Forum 196–201, 933 (1995).

6Y. K. Park, S. K. Estreicher, C. W. Myles, and P. A. Fedders, Phys. Rev. B52, 1718 (1995).

7B. Hourahine, R. Jones, S. Öberg, and P. R. Briddon, Phys. Rev. B 59,15729 (1999).

8E. V. Lavrov, J. Weber, L. Huang, and B. Bech Nielsen, Phys. Rev. B 64,035204 (2001).

9A. Nucara, P. Calvani, S. Lupi, and P. Roy, J. Chem. Phys. 107, 6562(1997) and references therein.

10N. N. Gerasimenko, M. Rollé, L.-J. Cheng, Y. H. Lee, J. C. Corelli, andJ. W. Corbett, Phys. Status Solidi B 90, 689 (1978). In this reference, thewavenumbers are overestimated by 7-8 cm−1; the value quoted for the 2178cm−1 line is 2185 cm−1.

11S.-F. Cui, Z.-H. Mai, and L.-Z. Qian, Sci. Sin., Ser. A 27, 213(1984).

12B. Pajot, B. Clerjaud, and Z.-J. Xu, Phys. Rev. B 59, 7500 (1999). The2183.23 cm−1 value given in this reference results from an accidental in-version of the last two digits.

13C. Peng, H. Zhang, M. Stavola, W. B. Fowler, B. Esham, S. K. Estre-icher, A. Docaj, L. Carnel, and M. Seacrist, Phys. Rev. B 84, 195205(2011).

14S. K. Estreicher, A. Docaj, M. B. Bebek, D. J. Backlund, and M. Stavola,Phys. Status Solidi A 209, 1872 (2012).

Downloaded 27 May 2013 to 80.250.180.203. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

184505-11 B. Clerjaud and B. Pajot J. Chem. Phys. 138, 184505 (2013)

15B. Pajot and B. Clerjaud, Optical Absorption of Impurities and Defectsin Semiconducting Crystals II. Electronic Absorption of Deep Centresand Vibrational Spectra, Springer Series in Solid-State Sciences Vol. 169(Springer, Heidelberg, 2013).

16J. M. Flaud, C. Camy-Peyret, and R. A. Toth, Selected Constants of Wa-ter Vapour Lines Parameters, International Tables of Selected ConstantsVol. 19 (Pergamon Press, Oxford, 1981).

17G. R. Wilkinson and M. K. Wilson, J. Chem. Phys. 44, 3867(1966).

18B. Bech Nielsen, L. Hoffmann, and M. Budde, Mater. Sci. Eng., B 36, 259(1996).

19D. G. Thomas and J. J. Hopfield, Phys. Rev. 150, 680 (1966).20S. W. Biernacki and B. Clerjaud, Phys. Rev. B 63, 075201 (2001).21In Refs. 13 and 18, Epij = fijrirj is used. Therefore in these references,

part of Epij is included in the potential energy describing, in principle, the

interaction between the H atom and the Si atom to which it is bonded. Con-sequently, the physical meaning of the parameters quoted in these works isdifferent from the choice made here.

22B. Clerjaud and D. Côte, J. Phys.: Condens. Matter 4, 9919 (1992).23This line appears in Fig. 1(a) of Ref. 18, but the temperature and the spectral

resolution used did not allow revealing its Si-related isotopic structure.24M. Budde, Ph.D. dissertation, University of Aarhus, Denmark, 1998.25B. Clerjaud, D. Côte, and C. Naud, J. Cryst. Growth 83, 190 (1987).26B. Clerjaud, D. Côte, and C. Naud, Phys. Rev. Lett. 58, 1755 (1987).27R. Darwich, B. Pajot, B. Rose, D. Robein, B. Theys, R. Rahbi, C. Porte,

and F. Gendron, Phys. Rev. B 48, 17776 (1993).28W. Ulrici, A. Kwasniewski, M. Czupalla, and M. Neubert, Phys. Status

Solidi B 242, 873 (2005).29T. Ohya, K. M. Itoh, R. N. Pereira, and B. Bech Nielsen, Jpn. J. Appl. Phys.

44, 7309 (2005).

Downloaded 27 May 2013 to 80.250.180.203. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions