The rovibrational-translational states of a hydrogen molecule moving in a cage site in Si, when subjected to an electrical field arising from its surroundings, are investigated. The wave functions are expressed in terms of basis functions consisting of the eigenfunctions of the molecule confined to move in the cavity and rovibrational states of the free molecule. The energy levels, intensities of infrared and Raman transitions, effects of uniaxial stress, and a neighboring oxygen defect are found and compared with existing experimental data

A. Mainwood

A.M. Stoneham

B. Clerjaud

B. Hourahine

C.G. Van de Walle

E.E. Chen

E.U. Condon

E.V. Lavrov

I. Vasiliev

J.D. Poll

J.K. Nørskov

J.W. Corbett

N. Bras

R. Jones

R.E. Pritchard

S.K. Estreicher

V.P. Markevich

W. Beall Fowler

English

Hourahine, B.

Jones, R.

University of Strathclyde Institutional Repository

Strathprints Institutional RepositoryHourahine, B. and Jones, R. (2003) Infrared activity of hydrogen molecules trapped in Si. PhysicalReview B: Condensed Matter and Materials Physics, 67 (12). pp. 121205-1. ISSN 1098-0121Strathprints is designed to allow users to access the research output of the University of Strathclyde.Copyright c© and Moral Rights for the papers on this site are retained by the individual authorsand/or other copyright owners. You may not engage in further distribution of the material for anyprofitmaking activities or any commercial gain. You may freely distribute both the url (http://strathprints.strath.ac.uk/) and the content of this paper for research or study, educational, ornot-for-profit purposes without prior permission or charge.Any correspondence concerning this service should be sent to Strathprints administrator:mailto:strathprints@strath.ac.ukhttp://strathprints.strath.ac.uk/Infrared activity of hydrogen molecules trapped in SiB. Hourahine and R. JonesSchool of Physics, The University of Exeter, Exeter EX4 4QL, United Kingdom~Received 27 January 2003; published 19 March 2003!The rovibrational-translational states of a hydrogen molecule moving in a cage site in Si, when subjected toan electrical field arising from its surroundings, are investigated. The wave functions are expressed in terms ofbasis functions consisting of the eigenfunctions of the molecule confined to move in the cavity and rovibra-tional states of the free molecule. The energy levels, intensities of infrared and Raman transitions, effects ofuniaxial stress, and a neighboring oxygen defect are found and compared with existing experimental data.DOI: 10.1103/PhysRevB.67.121205 PACS number~s!: 61.72.Bb, 71.55.2i, 85.40.RyHydrogen molecules have long been expected to be low-energy defects in Si,1,2 but the observation by infrared ~IR!absorption of what appeared to be isolated hydrogen mol-ecules trapped at cage sites was quite unexpected.3,4 Whilethere is no difficulty in understanding that molecules trappedat low-symmetry sites in a range of materials6,7 would be IRactive, the same is not true with molecules at tetrahedral cagesites in Si. The usual argument to explain their activity is thatthe lattice electric field Fi induces a dipole moment a i jF j onthe molecule in direction i.5 Here, a i j is the molecular po-larizability. However, the electric field at the Td cage site isactually zero, and a zero-point movement of the center ofmass of the molecule has been invoked to provide aninteraction.8,9 Now, if the molecule can freely rotate, Condonnoted10 that this dipole moment couples with the electricfield of the IR radiation and leads to the same selection rulesas for Raman scattering: namely, D j50 or 2. In addition,transitions involving both ortho-H2 with j51, and para-H2with j50, should occur and hence four IR transitions11 areexpected at low temperatures. However, experiments4 indi-cate that only one mode at 3618.4 cm21 is detected. This hasled to suggestions that the molecule is static but recentstudies9,12 demonstrate that this is not correct and the modearises from a D j50 transition from a degenerate j51 state.Thus it appears that the para-H2 species is not IR active –possibly because of symmetry reasons.12 A further difficultythen emerges when considering the transitions for HD. Twotransitions are detected9 corresponding to D j51 and D j50. The first disagrees with Condon’s analysis and impliesthat something new is needed.We show here, using a simple model, that the infraredintensity does not arise from the polarization induced in themolecule undergoing zero-point motion, but from a polariza-tion of the Si lattice by the electric field arising from thequadrupole moment of a rotating molecule. Further, we showthat the ground state of the para-H2 species transforms as A1and the only dipole allowed transition involves a so far un-detected D j52 transition. Transitions between T2 states ac-counts for the observed D j50 transition. The effect of anonuniform electric field due to the lattice on the molecule iscrucially important in understanding the activity of HD. Weshow that two transitions with D j50 and 1 then arise. More-over, the model accounts for the effect of stress and a nearbyoxygen impurity on the molecular spectrum.The Hamiltonian of the molecule moving in the cage sur-rounding a tetrahedral site contains kinetic and potentialterms dependent on the center-of-mass coordinates, definedby spherical polar coordinates (R ,Q ,F), as well as internalcoordinates defined by the molecular length and orientationin space (r ,u ,f) with the axes assumed parallel to the cubeedges. For a spherical cavity, the translational motion leadsto a center-of-mass wave function described byf L(R)Y LM(Q ,F), and the internal motion to rovibrationalwave functions xn(r2re)Y jm(u ,f). For illustrative pur-poses, we take f L(R) to be a d function so that the center ofmass is at a fixed distance, R;0.13 Å, from the Td site: avalue consistent with the zero-point displacement found bymolecular dynamics.13 Our results are, however, insensitiveto R. The electric field Fi arising from the lattice is modeledby placing charges of 4e on each Si atom and 22e at eachbond center and, where relevant, an additional 2e/9 at abond centered oxygen interstitial. Such a field is at the upperlimit of what is to be expected. We have assumed that thereis no charge density at the Td site. In fact, this leads to areduction in the vibration frequency of the molecule whichwe can take into account by reducing the molecular stretchforce constant.14 The potential energy of the molecule de-pends on the electric field Fi and its gradient Fi , j at thecenter of the molecule and is given by6,72 12 a i jFiF j213 Q i jFi , j ,where the polarizability and quadrupole moment are ex-pressed in terms of the molecule alignment r bya i j5a~r !d i j1g~r !3r2 ~3rir j2r2d i j!, ~1!Q i j512r2 Q~r !~3rir j2r2d i j!.This energy depends on both the position of the center ofmass and the molecular alignment. Figure 1 shows this mo-lecular potential energy, which is dominated by the quadru-pole term, of the H2 molecule when the center of mass islocated at (R ,Q ,F), with R50.13 Å, while aligned alongu5Q ,f5F with r equal to the equilibrium bond length re .We note that variations in the potential are ;6200 cm21and directions where the molecule is aligned away from thefour Si neighbors are favored. The weak potential is consis-tent with ab initio calculations15 and would not prevent mo-lecular rotation.16 This justifies our use of basis states madefrom those of a freely rotating molecule. The HamiltonianRAPID COMMUNICATIONSPHYSICAL REVIEW B 67, 121205~R! ~2003!0163-1829/2003/67~12!/121205~4!/$20.00 ©2003 The American Physical Society67 121205-1matrix elements for each vibrational state n , in the absenceof Fi , are \v0(n11/2)1Bn j( j11)/m12LD/M . Here,v0 ,Bn ,m , and M are the fundamental stretch frequency,rovibrational constant, reduced mass, and total molecularmass. 2D/M is the energy separation between L50 and 1cavity states that we take for H2 to be ;1000 cm21 whichagain can be related to the energy profile of the moleculearound the Td site.13 A Taylor series expansion of a(r),g(r),and Q(r) about re with the molecular constants taken fromRefs. 7 and 17 enabled the potential-energy matrix elementsto be found numerically using Lebedev’s method. Matrix el-ements of this Hamiltonian between different rotational, j,6, and cavity states, L,2, were found for each n and theenergy levels and wave functions determined.The intensity of an infrared transition between states un50,n& and un51,m& is related to the transition dipole mo-ment Pi(n ,m) in direction i. This has two contributions: thefirst comes from the induced dipole, as given by Condon,and the second comes from the dipole induced in the Silattice by the molecule. The second term is found from thefield Fi(a) arising from the quadrupole moment of the mol-ecule at the site of an atom a with atomic polarizabilityaa :23Pi~n ,m !5K n50,nUa i jF j1(aaaFi~a !Un51,mL;A \2mv0K nUH a i j8 F j1(a aaFi8~a !J UmL ,~2!where the prime denotes the derivative with r. The effectivecharge for the oscillator is A$2mv0(n ,m8 uP(n ,m)u2/(3\)%,18where the sum is over degenerate n and m states. The firstterm in Pi is zero when R50 as then Fi vanishes and is oforder 1023e for R;0.13 Å. The second term dominates andFig. 2 shows that its z component has a magnitude compat-ible with the experimental effective charge of ;0.08e .4 Thestrength of a Raman transition is proportional to(n ,m ,i , j8 (^nua i j8 um&)2. Condon’s result is easily derived fromEqs. ~1! and ~2! when Fi is a fixed uniform field.Rotating H2 by 180° about its center of mass leaves theHamiltonian and dipole moment unchanged. This symmetryallows us to classify each state as ‘‘even,’’ 1 , or ‘‘odd,’’2 , corresponding to para- and ortho-hydrogen, respectively.For H2, the ground state has A11 symmetry and is derivedfrom orbitals with even j, while the first excited state has T22symmetry and arises from orbitals with odd j. Both stateshave contributions from the L50 and L51 cavity orbitals.Figure 3 shows the ground-state wave function that is peakedin the troughs of the potential, i.e., in the direction awayfrom the Si atoms.FIG. 1. Potential energy ~per centimeter! of the H2 molecule at(R ,Q ,F) with R50.13 Å and aligned along (Q ,F). The direc-tions to the four Si neighbors correspond to Q554.5°,F5135°and 315°, Q5125.3°,F545° and 225°, and to peaks in the po-tential.FIG. 2. z component of effective charge vector a i j8 (u ,f)F j1(aaaFi8(u ,f ,a), in units of e, of a H2 molecule at the Td site vsu ,f describing its alignment. Note that the effective charge variesby 60.15e .FIG. 3. A11 wave function for center of mass of H2 molecule at(R ,Q ,F), with R50.13 Å, and aligned along (Q ,F). Note thatthe peaks correspond with directions away from the four Si neigh-bors. The wave function is normalized over the surface of a sphere.RAPID COMMUNICATIONSB. HOURAHINE AND R. JONES PHYSICAL REVIEW B 67, 121205~R! ~2003!121205-2The dipole moment transforms as T21 and hence an orbit-ally degenerate state must comprise the initial or final IRtransition and parity must be conserved. Thus, as noted inRef. 9, transitions between A11 states are IR inactive, in con-trast with transitions between T22 states of ortho-hydrogen.Figure 4 shows the lowest-energy states and their symmetriesfor n50 and 1 along with the intensities of the IR andRaman-active transitions between them. An IR-active transi-tion @not shown in Fig 4~a!# involving para-H2 between A11and T21, derived primarily from a j52 orbital, occurs;340 cm21 above the ortho-transition with an effectivecharge of ;0.1e , but this has not been detected. The onlydetected transition, labeled a in Fig 4~a!, has a strength ingood agreement with the calculation. Very recently, Ramanscattering has detected the transition between A11 states inboth H2 and D2.19For HD on the other hand, rotating the molecule by 180°about its center of mass displaces the center of the moleculeto a new location. Its interaction with the surrounding latticeis then changed, as the field Fi is nonuniform, and theHamiltonian and dipole moment are now different. Parity isthen not conserved and transitions between A1 and T2 statesare now possible as shown in Fig. 4~b!. At low temperatures,only the A1→T2 b transition is seen but the stronger transi-tion between T2 states, labeled a, is detected at *23 K aftermolecules have been thermally excited from A1 to T2. Theexperimental ratio of the effective charges for the a and btransitions is 2.6:1 compared with 2:1 found here. Theweaker T2→A1 transition c has not been reported to date.The calculated n50,A1-T2 separation of 74 cm21 is in verygood agreement with the experimental value of 71 cm21.Application of a general stress results in a displacement ofthe lattice and a change to the field Fi . The symmetry is nowlowered leading to a splitting of the T22 levels. We take theatomic displacements to be given by elasticity theory andFig. 5 shows the calculated splittings in the T22 manifolds forH2. We find the piezospectroscopic parameters Bg , Cg , Be ,and Ce ~Fig. 5!, respectively, to be 4.3, 5.4, 4.6, and5.5 cm21 GPa21. These are in excellent agreement with ex-perimental values of 4.5, 5.4, 4.5, and 5.4 cm21 GPa21found by assuming identical shifts for n50 and n51. ForD2, we get 4.1, 5.3, 4.2, and 5.9 cm21 GPa21 demonstratingthat the splittings are almost independent of isotopic mass inagreement with earlier theoretical and experimentalfindings.12 As has been noted previously, this result arisesfrom the mass independence of the interaction givenin Eq. ~1!.The presence of oxygen at a neighboring bond center siteleads to an additional electric field and a change in symmetryto C1h , or to C‘h if the lattice field is negligible. This alsoresults in a splitting of the T2 states and to a greater numberof IR transitions. We assume that the perturbation by oxygencan be modeled by a single point charge, and neglect thestrain due to the impurity, and show in Fig. 6 the spectra forH2, HD, and D2. Experimentally for H2, transitions a, b, andc are detected20 in agreement with the calculations with theenergy separations c-b and a-b being 6.1 and 57.9 cm21,respectively, compared with 10 and 68 cm21 found here.For HD, four transitions, h, b, a, and d are observed—twoweak ones from the ground and two from the first excitedFIG. 4. Calculated rovibrational levels of ~a! H2, ~b! HD, and~c! D2 at the Td site in silicon. For H2 and D2, only transitions a areIR active with effective charges of 0.1e , while both a and b areRaman active with relative strengths of 3:1. For HD, transitions a,b, and c are all infrared active with effective charges of 0.12e ,0.06e and 0.06e . Only transitions a and d are Raman active withrelative strengths of 3:1. The nuclear wave functions and degen-eracy have not been included in the symmetry classification of thestates or the transition strengths. FIG. 5. The calculated splitting of the T22→T22 transitions of H2at the Td site for @100# and @111# applied stress. The effectivecharges of the split components are shown along with polarizationselection rules.RAPID COMMUNICATIONSINFRARED ACTIVITY OF HYDROGEN MOLECULES . . . PHYSICAL REVIEW B 67, 121205~R! ~2003!121205-3state. We find that transitions a –h are all active with transi-tions b –e all being of similar activity. The energy separa-tions and transition strengths found here for all the defectsare not especially sensitive to the parameters R and D , but inthe case of oxygen, they are sensitive to the assumed chargeon oxygen and clearly the model will only obtain the correcttransitions if this is modeled correctly. If the lattice polariza-tion is reduced with respect to the charge on oxygen, thenagreement is found with experiment in that only two domi-nant transitions h, b from the ground, and a, d from the firstexcited state, occur. The separation of b and h according toour theory is 40 cm21 and about twice the observed value of19 cm21 but the separation between a and d is 48 cm21 infair agreement with experiment at 58.6 cm21. Much betteragreement with the experimental ortho-para splitting hasbeen found by a similar method but using a quite differentperturbing potential.21In conclusion, we have found that the IR activity of themolecule does not require a movement of the molecule awayfrom the tetrahedral site, as has been thought previously, buta polarization of the surrounding lattice. The effectivecharges relating to the strengths of the IR transition are thenthe same for H2 as D2 and in agreement with the data, unlikea model where the activity arises from a displacement of themolecule from the lattice site.21 The splitting of the linesobserved under stress, or by the presence of a neighboringoxygen impurity, can be modeled in a simple way and theresults are in reasonable agreement with experiment. Theanalysis shows that there are other as yet undetected IR tran-sitions at low temperatures albeit with low intensities. Themethod we have introduced is clearly applicable to mol-ecules in a wide range of materials, e.g., GaN.Finally, we note an interesting consequence of the split-ting of the T22 state of H2 in the presence of oxygen. Theenergy separation between the energetically lowest ortho-A82state and para-A81 states is only 82 cm21 for H2 near O,compared with 118 cm21 for H2 at a Td site. This impliesthat ortho molecules would favor the former sites. It isknown that H2 is weakly bound to oxygen with a bindingenergy about 0.28 eV.22 Upon warming to around 30 °C,molecules move reversibly between the two sites. Thusortho-H2 near oxygen sites is in equilibrium with the samenuclear species at Td sites. This is also true for the paraspecies. However, the binding energy of ortho-H2 at O sitesis lower by 118282 or 36 cm21 than for para-H2. Thus, theequilibrium fraction of ortho- to para-H2 species at oxygensites should be about 20% larger than at Td sites. Such aneffect has not yet been reported.We thank Ron Newman for many discussions during thecourse of this work.1 A. Mainwood and A.M. Stoneham, Physica B & C 116, 101~1983!.2 J.W. Corbett et al., Phys. Lett. A 93, 303 ~1983!.3 R.E. Pritchard et al., Phys. Rev. B 56, 13 118 ~1997!.4 R.E. Pritchard et al., Phys. Rev. B 57, 15 048 ~1998!.5 We use Einstein summation convention for repeated indexes6 J.D. Poll and J.L. Hunt, Can. J. Phys. 63, 84 ~1985!.7 N. Bras, J. Chem. Phys. 110, 5943 ~1999!.8 A.M. Stoneham, Phys. Rev. Lett. 84, 4777 ~2000!.9 E.E. Chen et al., Phys. Rev. Lett. 88, 105507 ~2002!.10 E.U. Condon, Phys. Rev. 41, 759 ~1932!.11 This ignores splitting of the j52 state into T21E .12 E.E. Chen et al., Phys. Rev. Lett. 88, 245503 ~2002!.13 S.K. Estreicher et al., J. Phys.: Condens. Matter 13, 6271 ~2001!.14 J.K. Norskov, Phys. Rev. B 20, 446 ~1979!.15 C.G. Van de Walle, Phys. Rev. Lett. 80, 2177 ~1998!.16 B. Hourahine et al., Phys. Rev. B 57, 12 666 ~1998!.17 G. Herzberg, Spectra of Diatomic Molecules, 2nd ed. ~Van Nos-trand Reinhold, New York, 1950!.18 B. Clerjaud and D. Coˆte, J. Phys.: Condens. Matter 4, 9919~1992!.19 E.V. Lavrov and J. Weber, Phys. Rev. Lett. 89, 215501 ~2002!.20 E.E. Chen, M. Stavola, and W. Beall Fowler, Phys. Rev. B 65,245208 ~2002!.21 W. Beall Fowler, P. Walters, and M. Stavola, Phys. Rev. B 66,075216 ~2002!.22 V.P. Markevich and M. Suezawa, J. Appl. Phys. 83, 2988 ~1998!.23 We take a to be 20 a.u. and related to the bulk Si dielectricconstant by the Clausius-Mossotti relation a5(3/e0)3(er21)/(er12) ~see Ref. 24!.24 I. Vasiliev, S. O¨ g˘u¨t, and J.R. Chelikowsky, Phys. Rev. Lett. 78,4805 ~1997!.FIG. 6. Calculated rovibrational levels of H2, HD, and D2 nearOBC . Only transitions arising from ground and first excited state aregiven. For H2 and D2 transitions a –c are infrared active with ef-fective charges of 0.10e , 0.06e , 0.02e and 0.10e , 0.07e , 0.03e ,respectively. Transitions b and c are also strongly Raman activewith relative strengths 1:1. For HD, transitions a –h are all infraredactive with effective charges of 0.08e , 0.06e , 0.06e , 0.06e , 0.06e ,0.05e , 0.04e , and 0.03e . Only transitions b and d are stronglyRaman active with equal strengths. The E state of HD is split byabout 9 cm21.RAPID COMMUNICATIONSB. HOURAHINE AND R. JONES PHYSICAL REVIEW B 67, 121205~R! ~2003!121205-4

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Infrared activity of hydrogen molecules trapped in Si

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