15
Vibrational spectra of vitreous germania from first-principles Luigi Giacomazzi, P. Umari, and Alfredo Pasquarello Ecole Polytechnique Fédérale de Lausanne (EPFL), Institute of Theoretical Physics, CH-1015 Lausanne, Switzerland and Institut Romand de Recherche Numérique en Physique des Matériaux (IRRMA), CH-1015 Lausanne, Switzerland Received 14 July 2006; revised manuscript received 2 September 2006; published 27 October 2006 We report on a first-principles investigation of the structural and vibrational properties of vitreous germania v-GeO 2 . Our work focuses on a periodic model structure of 168 atoms, but three smaller models are also studied for comparison. We first carry out a detailed structural analysis both in real and reciprocal spaces. Our study comprises the partial pair correlation functions, the angular distributions, the total neutron correlation function, the neutron and x-ray total structure factors, and the Faber-Ziman and Bhatia-Thornthon partial structure factors. We find overall good agreement with available experimental data. We then obtain the vibra- tional frequencies and eigenmodes. We analyze the vibrational density of states in terms of Ge and O motions, and further in terms of rocking, bending, and stretching contributions. The inelastic neutron spectrum is found to differ only marginally from the vibrational density of states. Using a methodology based on the application of finite electric fields, we derive dynamical Born charge tensors and Raman coupling tensors. For the infrared spectra, we calculate the real and imaginary parts of the dielectric function, including the high-frequency and static dielectric constants. The Raman spectra are shown to be sensitive to the medium-range structure and support an average Ge-O-Ge angle of 135°. We identify the shoulder X 2 as a signature of breathing O vibrations in three-membered rings. Four-membered rings are found to contribute to the main Raman peak. We advance an interpretation for the shoulder X 1 in terms of delocalized bond-bending modes. We derive bond polarizability parameters from the calculated Raman coupling tensors and demonstrate their level of reliability in reproducing the spectra. The calculated vibrational spectra all show good agreement with the respective experimental spectra. DOI: 10.1103/PhysRevB.74.155208 PACS numbers: 61.43.Fs, 78.30.j, 63.20.e, 71.15.Mb I. INTRODUCTION Disordered oxides, such as vitreous silica v-SiO 2 and vitreous germania v-GeO 2 , are currently key materials in many technological applications, ranging from optical fibers 1 to Si-based microelectronic devices. 2 Furthermore, these ox- ides represent archetypical materials for the study of the vit- reous state. Hence both technological and fundamental inter- ests justify the substantial efforts which have been devoted to the detailed characterization of their atomic structure. Experimental probes, such as neutron or x-ray diffraction, provide direct information on the structural arrangement in these oxides. The short-range order is characterized by regu- lar bonding configurations in which the cation Si or Ge is located at the center of corner-sharing tetrahedra having O atoms at their corners. The disorder sets in at medium-range distances through a distribution of bond angles on O atoms 3,4 and through a statistics of different ring sizes. 5 While the short-range order is well-described by diffrac- tion probes, structural arrangements on medium range dis- tances are more difficult to access because of the variety of interfering correlations. Interestingly, the networks of v -SiO 2 and v-GeO 2 present noticeable differences on these length scales. The average Ge-O-Ge bond angle in v-GeO 2 is 133° Ref. 4, significantly lower than the corresponding Si- O-Si bond angle in v-SiO 2 151°, Ref. 6. Consequently, v -GeO 2 is characterized by a higher packing density of tetra- hedra than v-SiO 2 . 7 These differences also suggest that the ring statistics of the two materials differ, favoring higher concentrations of small rings in v-GeO 2 . More generally, un- derstanding the structure of v-GeO 2 might provide informa- tion on structural reorganizations occurring in v-SiO 2 when subject to external pressure. 8 Structural information can in principle also be accessed in an indirect way through vibrational spectroscopies. For these oxide glasses, vibrational spectra are currently routinely measured through inelastic neutron scattering INS, infrared spectroscopy, and Raman scattering. The INS spectrum of these oxide glasses is expected to closely resemble the vibra- tional density of states. 9,10 Indeed, the interaction between the neutron and the lattice vibrations is mediated by atomic neutron scattering lengths, which are well-known experimen- tally and do not depend on the electronic structure in the oxide glass. At variance, the infrared and Raman spectra show important frequency-dependent modulations with re- spect to the vibrational density of states. 1113 In these mea- surements, the electronic structure plays an important role in determining the coupling factors. Consequently, infrared and Raman spectra may differ considerably from the vibrational density of states. For instance, in the Raman spectrum of v -SiO 2 two sharp lines appear 14,15 which do not have counter- parts in the other vibrational spectra. It has recently been shown 12,13 that Raman spectra in dis- ordered oxides are highly sensitive to the oxygen bond angle distribution, thus offering an indirect structural probe for the connections between tetrahedra. This sensitivity is particu- larly valuable since it specifically highlights medium-range arrangements, which are more difficult to access through dif- fraction probes. However, the interpretation of vibrational spectra in terms of structural correlations is not trivial and can occur only through accurate theoretical modeling. Mod- eling approaches do not only face the difficulty of estimating coupling factors, but also require viable model structures and PHYSICAL REVIEW B 74, 155208 2006 1098-0121/2006/7415/15520815 ©2006 The American Physical Society 155208-1

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Page 1: Vibrational spectra of vitreous germania from first-principles

Vibrational spectra of vitreous germania from first-principles

Luigi Giacomazzi, P. Umari, and Alfredo PasquarelloEcole Polytechnique Fédérale de Lausanne (EPFL), Institute of Theoretical Physics, CH-1015 Lausanne, Switzerlandand Institut Romand de Recherche Numérique en Physique des Matériaux (IRRMA), CH-1015 Lausanne, Switzerland

�Received 14 July 2006; revised manuscript received 2 September 2006; published 27 October 2006�

We report on a first-principles investigation of the structural and vibrational properties of vitreous germania�v-GeO2�. Our work focuses on a periodic model structure of 168 atoms, but three smaller models are alsostudied for comparison. We first carry out a detailed structural analysis both in real and reciprocal spaces. Ourstudy comprises the partial pair correlation functions, the angular distributions, the total neutron correlationfunction, the neutron and x-ray total structure factors, and the Faber-Ziman and Bhatia-Thornthon partialstructure factors. We find overall good agreement with available experimental data. We then obtain the vibra-tional frequencies and eigenmodes. We analyze the vibrational density of states in terms of Ge and O motions,and further in terms of rocking, bending, and stretching contributions. The inelastic neutron spectrum is foundto differ only marginally from the vibrational density of states. Using a methodology based on the applicationof finite electric fields, we derive dynamical Born charge tensors and Raman coupling tensors. For the infraredspectra, we calculate the real and imaginary parts of the dielectric function, including the high-frequency andstatic dielectric constants. The Raman spectra are shown to be sensitive to the medium-range structure andsupport an average Ge-O-Ge angle of 135°. We identify the shoulder X2 as a signature of breathing Ovibrations in three-membered rings. Four-membered rings are found to contribute to the main Raman peak. Weadvance an interpretation for the shoulder X1 in terms of delocalized bond-bending modes. We derive bondpolarizability parameters from the calculated Raman coupling tensors and demonstrate their level of reliabilityin reproducing the spectra. The calculated vibrational spectra all show good agreement with the respectiveexperimental spectra.

DOI: 10.1103/PhysRevB.74.155208 PACS number�s�: 61.43.Fs, 78.30.�j, 63.20.�e, 71.15.Mb

I. INTRODUCTION

Disordered oxides, such as vitreous silica �v-SiO2� andvitreous germania �v-GeO2�, are currently key materials inmany technological applications, ranging from optical fibers1

to Si-based microelectronic devices.2 Furthermore, these ox-ides represent archetypical materials for the study of the vit-reous state. Hence both technological and fundamental inter-ests justify the substantial efforts which have been devoted tothe detailed characterization of their atomic structure.

Experimental probes, such as neutron or x-ray diffraction,provide direct information on the structural arrangement inthese oxides. The short-range order is characterized by regu-lar bonding configurations in which the cation �Si or Ge� islocated at the center of corner-sharing tetrahedra having Oatoms at their corners. The disorder sets in at medium-rangedistances through a distribution of bond angles on O atoms3,4

and through a statistics of different ring sizes.5

While the short-range order is well-described by diffrac-tion probes, structural arrangements on medium range dis-tances are more difficult to access because of the variety ofinterfering correlations. Interestingly, the networks of v-SiO2 and v-GeO2 present noticeable differences on theselength scales. The average Ge-O-Ge bond angle in v-GeO2 is133° �Ref. 4�, significantly lower than the corresponding Si-O-Si bond angle in v-SiO2 �151°, Ref. 6�. Consequently, v-GeO2 is characterized by a higher packing density of tetra-hedra than v-SiO2.7 These differences also suggest that thering statistics of the two materials differ, favoring higherconcentrations of small rings in v-GeO2. More generally, un-derstanding the structure of v-GeO2 might provide informa-

tion on structural reorganizations occurring in v-SiO2 whensubject to external pressure.8

Structural information can in principle also be accessed inan indirect way through vibrational spectroscopies. For theseoxide glasses, vibrational spectra are currently routinelymeasured through inelastic neutron scattering �INS�, infraredspectroscopy, and Raman scattering. The INS spectrum ofthese oxide glasses is expected to closely resemble the vibra-tional density of states.9,10 Indeed, the interaction betweenthe neutron and the lattice vibrations is mediated by atomicneutron scattering lengths, which are well-known experimen-tally and do not depend on the electronic structure in theoxide glass. At variance, the infrared and Raman spectrashow important frequency-dependent modulations with re-spect to the vibrational density of states.11–13 In these mea-surements, the electronic structure plays an important role indetermining the coupling factors. Consequently, infrared andRaman spectra may differ considerably from the vibrationaldensity of states. For instance, in the Raman spectrum of v-SiO2 two sharp lines appear14,15 which do not have counter-parts in the other vibrational spectra.

It has recently been shown12,13 that Raman spectra in dis-ordered oxides are highly sensitive to the oxygen bond angledistribution, thus offering an indirect structural probe for theconnections between tetrahedra. This sensitivity is particu-larly valuable since it specifically highlights medium-rangearrangements, which are more difficult to access through dif-fraction probes. However, the interpretation of vibrationalspectra in terms of structural correlations is not trivial andcan occur only through accurate theoretical modeling. Mod-eling approaches do not only face the difficulty of estimatingcoupling factors, but also require viable model structures and

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accurate vibrational properties. The aim of an optimal mod-eling approach then consists in determining a structuralmodel through a virtuous circle in which intermediate mod-els progressively show improved comparisons with experi-mental spectra.

An appropriate theoretical approach should meet severalrequirements. Structural models for disordered materials canbe generated through simulation approaches of varying lev-els of complexity. For instance, for v-SiO2 and v-GeO2, clas-sical molecular dynamics simulations yield model structureswhich generally compare well with diffraction data.16,17 Inprinciple, further structural refinement could be achievedwith ab initio methodologies. However, as far as the vibra-tional properties are concerned, classical modeling ap-proaches are generally not sufficiently accurate.9,18 Further-more, the modeling of infrared and Raman coupling factorsrequires an explicit treatment of the electronic structure. Anadditional constraint results from the necessity of treatingmodel systems of relatively large size in order to describe thedisordered nature of the oxide in a statistically appropriateway.

Density functional methodologies appear at present mostsuitable to address these issues. Vibrational properties ofboth crystalline19 and disordered materials9,13,20 are de-scribed with great accuracy. The description of the electronicstructure gives access to infrared11,21 and Ramanintensities.22–24 Algorithmic improvements and the availabil-ity of powerful computer resources now allow one to treatsystems containing up to several hundred atoms. In particu-lar, a recent development which consists of applying a finiteelectric field in conjunction with periodic boundary condi-tions is instrumental to the calculation of coupling factors forsystems of relatively large size.25,26

In this work, we address the vibrational spectra of vitre-ous germania within a density functional framework. Wehere provide a comprehensive analysis that complements andextends the concise account reported previously.13 We startour investigation by analyzing the structural properties of themodel structure generated in the latter work. In Ref. 13, wecompared the calculated neutron structure factor to corre-sponding experimental data. We here extend the comparisonto experiment by considering the total neutron pair correla-tion function, the total x-ray structure factor, and partialstructure factors both in the Faber-Ziman and Bhatia-Thornthon formulations. Next, we address the vibrationalproperties. The vibrational eigenmodes are not only analyzedin terms of germanium and oxygen weights but also in termsof stretching, bending, and rocking motions.27 The vibra-tional density of states is compared to the inelastic neutronspectrum finding the same kind of correspondence as forvitreous silica.9,10 We then turn to the dielectric properties. InRef. 13, we presented the imaginary part of the dielectricfunction and the energy loss function in comparison withexperimental results. Here, we give the real part of the di-electric function, for which we could not find any experi-mental counterpart. In particular, we also calculate the staticdielectric constant. Furthermore, we address the dynamicalBorn charges, the relevant coupling factors in the infraredspectra, and show a clear relation between the oxygen dy-namical Born charges and the local structural environment.

Our study devotes particular attention to the Raman spectrabecause of their sensitivity to medium-range structural prop-erties. We extend our analysis by generating three additionalmodel structures showing various Ge-O-Ge bond-angle dis-tributions and ring statistics. In this way, our analysis clearlyillustrates the dependence of the Raman spectra on thesemedium-range structural properties. In particular, ourassignment13 of the X2 shoulder to three-membered rings ishere further supported through projections on ring vibrations.We also investigate the origin of the X1 shoulder on the op-posite side of the main Raman peak. We find that the under-lying eigenmodes correspond to bond-bending motions in theGe-O-Ge bridge, leading us to suggest that this feature re-sults from diffuse network motions rather than from vibra-tions localized within specific structural subunits. In the finalpart of our study, we focus on the Raman coupling factorsand derive therefrom optimal bond-polarizability parameters.

This paper is organized as follows. In Sec. II, we outlinethe generation procedures that gave rise to the model struc-tures of v-GeO2 used in this work. In Sec. III, we address thestructural properties of v-GeO2 focusing on the largest ofthese model structures. Section IV is devoted to the analysisof the vibrational properties, including the vibrational den-sity of states and the inelastic neutron spectrum. Section Vcontains the infrared properties, including the dielectric func-tion, the static dielectric constant, and the dynamical Borncharges. Section VI addresses the Raman spectra. In particu-lar, we focus on the relation between the Raman spectra andthe intertetrahedral angle distribution, on the origin of thefeatures X1 and X2, and on the derivation of optimal bondpolarizability parameters. The conclusions are given in Sec.VII.

II. MODELS GENERATION

All the models of v-GeO2 generated in this work are sub-ject to periodic boundary conditions, have cubic simulationcells, and have the experimental density ��3.65 g/cm−3�.7The analysis is mainly based on a structural model of 168atoms, hereafter referred to as model I. For comparison, wealso generated three smaller models, containing either 72�models II and III� or 36 atoms �model IV�. These modelsshow different structural properties as reflected by their in-tertetrahedral angular distributions and ring statistics.5

Model I was generated according to the following proce-dure. First, we carried out classical molecular dynamics28 ofSiO2 at 2.7 g/cm3 corresponding to the experimental pack-ing density of tetrahedra in v-GeO2. The system was firstheated up to a temperature of 3500 K and thermalized for aperiod of 50 ps. Then, we quenched the temperature to300 K at a rate of −13 K/ps by scaling the velocities. After asubsequent thermalization of 25 ps, we obtained a disorderedstructure consisting of a chemically ordered network ofcorner-sharing tetrahedra. We then transformed the obtainedstructure to a model of v-GeO2 by rescaling the simulationcell by the Ge-O/Si-O bond length ratio and by further op-timizing the atomic positions through damped first-principlesmolecular dynamics.29,30

We generated model II according to a similar procedure.However, in this case, the classical molecular dynamics of

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SiO2 was carried out at the experimental density of silica.28

The model was then transformed by rescaling the simulationcell to the experimental density of v-GeO2. The structuralproperties were finally adjusted through a final first-principles relaxation.

Model III was generated by replacing Si with Ge in amodel structure of v-SiO2, generated previously by first-principles molecular dynamics.31 We obtained a model of v-GeO2 by rescaling the simulation cell and by carrying out afirst-principles relaxation as for model II. Model IV was gen-erated in the same way as model II, but with a smaller num-ber of atoms.

In the present work, we performed electronic and struc-tural relaxations using first-principles methodologies,29,30 asprovided in the QUANTUM-ESPRESSO package.32 The ex-change and correlation energy was accounted for through ageneralized gradient approximation �GGA�33 to density func-tional theory. Plane wave basis sets with energy cutoffs of 25and 250 Ry were used to expand the electron wave functionsand the electron density, respectively. Core valence interac-tions were accounted for by a norm-conserving pseudopoten-tial for Ge �Ref. 34� and an ultrasoft one for O.35 The wavefunctions were expanded at the sole � point of the Brillouinzone, as justified by the large size and the large band gap ofour systems.

The electronic density of states of model I is presented inFig. 1. The origin of the bands in terms of atomic orbitals issimilar to the case of SiO2:36 the lowest band arises fromO 2s states, the low-energy side of the central band resultsfrom the bonds between Ge sp3 and O 2p orbitals. The high-energy side of this central band consists of O 2p lone pairs,which define the top of the valence band. The low-energypart of the conduction band mainly consists of Ge orbitals.

The calculated band gap �2.6 eV� significantly underesti-mates the experimental value �5.6 eV, Ref. 37�, as usual indensity functional schemes. The present electronic density ofstates is consistent with previous calculations for �-quartzGeO2 and v-GeO2.38

For all our models, we obtained the normalized vibra-tional modes � n and frequencies �n through the diagonaliza-tion of the dynamical matrix. The index n labeling the vibra-tional modes runs from 1 to 3N, where N is the total numberof atoms in the model. The dynamical matrices were calcu-lated numerically by taking finite differences of the atomicforces.9 Atomic displacements of ±0.1 bohr were used inthese calculations.

III. STRUCTURAL PROPERTIES

A. Real-space properties

1. Short-range order

We analyzed the short-range order in model I by focusingon the structural properties of the basic structural unit corre-sponding to the GeO4 tetrahedron. The Ge-O bond-lengthand the O-Ge-O bond-angle distributions are shown in Fig.2. The mean values of these distributions are reported inTable I, together with the corresponding values of the otherthree models. The bond length in these models ��1.78 Å� isslightly longer than the experimental value �1.73 Å, Ref. 4�,an effect that should be attributed to our setup for the de-scription of the electronic structure.13 The O-Ge-O angle dis-tribution of model I is centered around the tetrahedral anglewith a standard deviation of �6°. The O-Ge-O angle distri-

FIG. 1. Electronic density of states �DOS� for model I of v-GeO2. The highest occupied states are aligned at 0 eV. A Gaussianbroadening of 0.25 eV is used.

FIG. 2. Distribution of the O-X-O angle and of bond length dXO

�X=Ge,Si� for model I of v-GeO2 �solid� and for a model of v-SiO2 generated according to a similar procedure �dashed� �Ref.39�. The bond lengths are rescaled with respect to their averagevalue. Gaussian broadenings of 2.5° and 0.005 Å are used.

TABLE I. Structural properties of our models of vitreous GeO2: the number of atoms �N�, the averageGe-O-Ge angle, the average O-Ge-O angle, and the average bond length �dGeO�. The respective standarddeviations are given between parentheses. Experimental estimates for the angles and the bond lengths aretaken from Refs. 4 and 57, respectively.

N � Ge-O-Ge � O-Ge-O dGeO ��

Model I 168 135.0° �10.6°� 109.4° �5.8°� 1.780 �0.014�Model II 72 127.0° �12.3°� 109.4° �5.7°� 1.770 �0.015�Model III 72 121.3° �9.9°� 109.5° �8.9°� 1.775 �0.020�Model IV 36 130.2° �10.9°� 109.4° �6.4°� 1.771 �0.011�Expt. 133° 109.4° 1.73

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bution of the other models show the same mean value. Thestandard deviations are also similar, except for model IIIwhich evidently suffers from the absence of a thermalizationstep in the generation procedure.

To address the degree of distortion in the GeO4 tetrahedra,we compared the bond-length and bond-angle distributionsin Fig. 2 with those of a model structure of SiO2 generatedaccording to a similar procedure.39 This comparison showsthat the relative bond-length distribution in v-GeO2 and v-SiO2 is similar. However, the bond-angle distribution of v-GeO2 is noticeably broader than that of v-SiO2, indicatingthat the tetrahedra in the former case show a larger departurefrom the ideal geometry. This property is conveniently illus-trated by introducing the distortion parameter �V. This pa-rameter is defined for each tetrahedron as the ratio betweenits volume and the volume of the ideal tetrahedron with thesame average bond length. In our models of v-GeO2, theratio �V is always found to be smaller than 1.0. We foundthat the distribution of �V for v-GeO2 is significantly broaderthan the corresponding distribution for v-SiO2 �Fig. 3�.

2. Medium-range order

In this section, we analyze structural properties on lengthscales beyond that of the single tetrahedron. We describe theconnectivity between tetrahedra by focusing on the Ge-O-Gebond-angle distribution and the ring statistics.5

As shown in Fig. 4, the Ge-O-Ge angle distribution differsmarkedly from one model to the other. In particular, the dis-tribution of model I shows an average value of 135° and astandard deviation of 10.6°, in good agreement with param-eters extracted from diffraction data �133°, 8.3°�.4 The other

models show distributions with lower mean values, rangingbetween 120° and 130°. The main parameters of these distri-butions are summarized in Table I.

Figure 5 shows the ring statistics for our model structures.The four model structures agree in indicating the six-membered ring as the dominant ring size. However, signifi-cant differences between the models occur for smaller ringsizes. It is particularly interesting to focus on three-membered rings, the smallest ring size found in our models.Model I contains four three-membered rings. Models II andIII have two and five of such rings, respectively. No three-membered ring is found in model IV. The three-memberedrings are quasiplanar, as can be inferred from the sum � overall bond angles in the ring. In model I, � averages to 696°,only slightly lower than the ideal value of 720°. In thesmaller models II and III, � averages to 687.2° and 691.7°,respectively. The planarity in three-membered rings is drivenby the fact that the average Ge-O-Ge angles in such rings�123° in model I� are considerably lower than the averageGe-O-Ge in v-GeO2 �133°, Ref. 4�. In the planar configura-tion, these angles are maximized.40

It is also worthy of note that models with similar Ge-O-Ge angle distributions do not necessarily show similarring statistics. Models II and III show very similar Ge-O-Geangle distributions, with a large weight in the range between110° and 130°. However, their ring statistics differ markedlyfor three- and four-membered rings �Fig. 5�. On the otherhand, models I and II share similar ring statistics, but theirGe-O-Ge angle distributions show considerably different pa-rameters �Table I�.

3. Pair correlation functions

In a binary system, the partial pair correlation functiong���r� gives the ratio between the density of atoms of species� at a distance r from an atom of species � and the averagedensity �� of atoms of species �:41

g���r� =1

N����

I��,J��

��r − �RJ − RI�� , �1�

where N� corresponds to the number of atoms of species �.We calculate partial pair correlation functions in the har-

monic approximation using the vibrational eigenmodes andfrequencies.42 At 300 K, this has been found to be a good

FIG. 3. The distribution of the distortion parameter �V is givenfor the same models considered in Fig. 2. We found a spread of0.011 and 0.020 for v-SiO2 and v-GeO2, respectively.

FIG. 4. Distribution of the Ge-O-Ge intertetrahedral angle forour four models of v-GeO2: model I �solid�, model II �dot-dashed�,model III �dotted�, and model IV �dashed�. A Gaussian broadeningof 2.5° is used.

FIG. 5. Ring statistics of our four models of v-GeO2 accordingto the minimal path analysis �Ref. 5�.

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approximation for oxide glasses.42,43 With respect to the al-ternative approach based on averaging over molecular dy-namics trajectories in which the ionic motion is treated clas-sically, this formulation offers the advantage of including thezero-point motion in the description.42,43

In our calculation, we replace the � functions in Eq. �1�with Gaussian functions with a variance 2 given by42

2 = ��d · �uI − uJ��2 , �2�

where uI is the displacement of the Ith atom with respect tothe equilibrium position RI, and d is a unitary vector alongthe direction of RJ−RI. The brackets �¯ indicate a thermalaverage obtained as follows:42

�uIiuJj = �n

�n

�Iin

MI

�Jjn

MJ�nB��n� +

1

2� , �3�

where the indices i and j label the Cartesian directions, MIcorresponds to the mass of the Ith atom, and the temperaturedependence enters through the Boson occupation numbernB�E�= �exp E /kBT−1�−1, where kB is the Boltzmann factor.

In Fig. 6, we give the partial pair-correlation functions,gGeO�r�, gGeGe�r�, and gOO�r�, corresponding to model I. Thefirst peak of gGeO�r� is located at 1.77 Å and agrees closelywith the Ge-O bond length �Table I�. From a spherical inte-gration of gGeO�r� up to its first minimum at 2.25 Å,41 wederive an average coordination number of 4.01, consistentwith the absence of coordination defects in our model. Thefirst peaks of gGeGe�r� and gOO�r� correspond to Ge-Ge andO-O distances of 3.25 and 2.88 Å, respectively.

The ratio between the positions of the first peaks ingGeO�r� and gOO�r� gives another measure of the degree ofdistortion. For model I, this ratio is 0.6154, in good agree-

ment with the experimental value of 0.6141 estimated fromneutron diffraction.44 For comparison, this ratio is equal to3/8 0.6124 for the ideal tetrahedron.

Neutron diffraction data are often presented in the form ofthe neutron total correlation function TN�r�, which can beexpressed in terms of the pair correlation functions:7

TN�r� = 4����

c�c�b�b�g���r� , �4�

where c�=N� /N and b� are the concentration and the neu-tron scattering length of species �, and � is the number den-sity of the sample. For comparison with experiments, it isnecessary to take into account the experimental resolution inreal space, expressed by the reduced neutron peak functionP�r�:7

TN� �r� = Nuc�0

TN�x��P�r − x� − P�r + x��dx , �5�

where Nuc is the number of atoms in the unit of composition�Nuc=3 for GeO2�. Figure 6�b� gives TN�r� calculated formodel I. The first peak of TN�r� corresponds to the first peakof gGeO�r� and is located at 1.77 Å. The second and thirdpeaks of TN�r� correspond to the first peaks in gOO�r� andgGeGe�r� and are found at 2.92 and 3.27 Å, respectively,slightly shifted with respect to the partial pair correlationfunctions because of the multiplicative factor r in Eq. �4�. Asshown in Fig. 6�b�, TN�r� and TN� �r� are essentially indistin-guishable beyond the first narrow peak located at 1.77 Å.The experimental broadening introduced by the peak func-tion causes the first peak to decrease to half its height, whichis essential to find good agreement with the experimentalcorrelation function.45

B. Reciprocal space properties

1. Total structure factor

To avoid Fourier transformations, direct comparisons be-tween diffraction data and model structures are preferablyperformed in reciprocal space. Following Waseda,41 we ex-press the total structure factor as a function of the exchangedmomentum Q:

S�Q� = ��,�

�c�c��1/2 f��Q�f��Q��f2�Q�

S���Q� , �6�

�f2�Q� = ��

c�f�2�Q� , �7�

where f� are atomic scattering factors and S���Q� are thepartial structure factors in the Ashcroft-Langreth formula-tion:

S���Q� =1

N�N����

J=1

N�

�K=1

N�

e−iQ·�R�J−R�K�� − �Q,0� ,

�8�

where the brackets �¯ indicate a thermal average. For iso-tropic materials, the structure factors only depend on the

FIG. 6. �a� Pair-correlation functions �PCF� gGeO�r� �solid�,gGeGe�r� �dashed�, and gOO�r� �dot-dashed�, calculated in the har-monic approximation for v-GeO2 �model I�. �b� Theoretical neutrontotal correlation function TN�r� �dot-dashed�. Its counterpart TN� �r��solid� accounts for the experimental resolution and permits com-parison with experiment �dotted� �Ref. 45�.

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modulus of Q and are obtained by a spherical average.46

Adopting the harmonic approximation, the thermal aver-age can be calculated by means of Eq. �3�. The partial struc-ture factors then simplify to

S���Q� =1

N�N���

JK

e−WJK�Q�eiQ·�R�J−R�K� − �Q,0� , �9�

where the exponents in the Debye-Waller factors are givenby

WJK�Q� =1

2��Q · �uJ − uK��2 . �10�

In neutron diffraction, the interactions between the incom-ing neutrons and the nuclei are described by the neutronscattering lengths b�: f��Q�=b�. In our calculations, we tookthe neutron scattering lengths bGe=8.1929 fm and bO=5.805 fm.47 The calculation of S�Q� corresponds to the av-erage over a discrete set of Q vectors of modulus Q. Only Qvectors compatible with the periodicity of the simulation cellare retained. Therefore the result corresponds to the infinitelyrepeated model.48 In Fig. 1 of Ref. 13, we compared thecalculated neutron static structure factor S�Q� of model Iwith experiment,49 finding excellent agreement. In particular,model I describes well the first sharp diffraction peak at�1.6 Å−1 that is generally taken as a signature of intermedi-ate range order.50,51 Beyond 7 Å−1, the oscillations of thetheoretical S�Q� are slightly out of phase with respect to theexperimental one. As discussed previously,13 this effectarises because of the slight overestimation ��2% � of theGe-O bond length in the simulation �cf. Table I�.

Our structural model can also be compared with x-raydiffraction data.49 Following Waseda,41 the x-ray structurefactor is also given by Eq. �6�, where the f��Q� now stand forthe atomic x-ray scattering factors.52 We note that the use ofEq. �6� for x-ray diffraction implies the approximation thatthe electron density results from a superposition of electrondensities of isolated atoms. In Fig. 7, we compare the x-raystructure calculated for model I with its experimentalcounterpart.49 As for the neutron structure factor, theoreticaland experimental peak positions are in good agreement. Inparticular, the first sharp diffraction peak and the peak at�2.6 Å−1 show a stronger intensity than in the neutron struc-

ture factor �cf. Fig. 1 of Ref. 13�, in accord with the experi-mental trend.49 This behavior results from the important con-tribution of Ge-Ge correlations, which carry a larger weightin the x-ray structure factor.

2. Partial structure factors

Vitreous GeO2 is a material for which all the partial struc-ture factors have been obtained experimentally. In Fig. 8, wecarry out the comparison between our theoretical results andthe experimental data of Ref. 53 in the Faber-Ziman formu-lation, which is simply related to the Ashcroft-Langreth onegiven in Eq. �8�.41 The agreement between theory and experi-ment is very impressive. In particular, the first sharp diffrac-tion peak is well-reproduced in each of the partial structurefactors.

In fundamental glass science, it is currently of interest54 tounderstand the decomposition of the FSDP in partial struc-ture factors according to the Bhatia-Thornton formulation�BT�.55 The BT structure factors distinguish between fluctua-tions of number density and of concentration, resulting in thefollowing partial structure factors: SNN�Q� �number-number�,SNC�Q� �number-concentration�, and SCC�Q� �concentration-concentration�. In this way, separate information is providedon topological and chemical order, through SNN�Q� andSCC�Q�, respectively. In Fig. 9, the calculated BT structurefactors are compared with corresponding experimentaldata.53 The agreement between theory and experiment isagain excellent for all three BT structure factors. Further-more, in relation with an on-going debate in theliterature,54,56 it is interesting to note that both the theoretical

FIG. 7. Theoretical total X-ray structure factor �solid� for modelI of v-GeO2, compared with its experimental counterpart �circles��Ref. 49�. The structure factor is given as a function of an adimen-sional scattering vector, scaled by the average Ge-O bond distancedGeO �dGeO

expt =1.73 Å, dGeOtheo =1.78 Å�.

FIG. 8. Theoretical Faber-Ziman partial structure factors �solid�for model I of v-GeO2, compared with experimental data �circles�of Ref. 53: �a� SGeO�Q�, �b� SOO�Q�, and �c� SGeGe�Q�. The scatter-ing vector is scaled as described in Fig. 7.

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and experimental results show the appearance of a smallFSDP in the SCC. The origin of this peak is not yet clarified.However, we remark that it is not related to the occurrence ofcoordination defects, since these are absent in our model.

We also compared our theoretical results with the experi-mental data in Ref. 57 �not shown�. This comparison shows aless impressive agreement, particularly in the region of thefirst sharp diffraction peak.

IV. VIBRATIONAL PROPERTIES

We focus in this section on the analysis of the vibrationaldensity of states �v-DOS�, which underlies all the vibrationalspectra. The v-DOS, Z���, is expressed as

Z��� =1

3N�

n

��� − �n� , �11�

in terms of the vibrational frequencies �n.In Fig. 10, the v-DOS is decomposed according to the

weights of the two species: Z���=��Z����. The partial den-sity of states Z���� is defined by

Z���� =1

3N�

I

N�

�n

��In�2��� − �n� , �12�

� n being the eigenvector corresponding to the frequency �n.Figure 10�a� shows that the ratio between the O and Geweights is close to the concentration ratio in the range150–700 cm−1. Below 150 cm−1 this ratio approaches 1:1,while it increases to 5:1 above 700 cm−1. This implies that

the motions of the two species are strongly correlated up to700 cm−1, particularly below 150 cm−1. The correlationabove 700 cm−1 is significantly weaker, since the contribu-tion of the heavier Ge atoms becomes less important in thehigh-frequency range. At variance with the case of v-SiO2which shows a peak with dominant cation weight at�800 cm−1, such a peak does not occur in the v-DOS ofv-GeO2. This should be related to the higher Ge mass whichcauses a shift of this band to lower frequencies thereby fa-voring the interaction with O modes.

We further decomposed the oxygen contribution to thev-DOS according to three orthogonal directions defining thelocal environment of the O atoms �Fig. 10�b��.27 The princi-pal directions associated to the Ge-O-Ge bridge define therocking, bending, and stretching directions, respectively.9

The lowest vibrational band ��380 cm−1� prevalently fea-tures O rocking motions, while the band extending from380 to 700 cm−1 is mainly composed of O bending motions.The decomposition in Fig. 10�b� shows that the vibrationalmodes above �700 cm−1 are mostly given by O stretchingmotions. The stretching contribution features a double peak,very similar to vitreous silica.58 To address this splitting, it isconvenient to organize the stretching modes according to theirreducible representations of the tetrahedron A1 and T2.9 TheA1 mode corresponds to an in-phase motion of the four Oatoms towards the central Ge atom. In the T2 modes, two Oatoms move closer to the central Ge atom, while the othertwo move away. By projecting on A1 and T2 representations,we show that also for v-GeO2 this doublet stems from dis-tinct vibrational modes of the tetrahedra �Fig. 10, inset�.Therefore the origin of this splitting should not be assignedto a longitudinal-optic/transverse-optic �LO-TO� effect.9

Experimental access to the v-DOS is nontrivial. The ex-perimental spectrum which most closely resembles thev-DOS is the inelastic neutron spectrum.58,59 In Fig. 2 of our

FIG. 9. Theoretical Bhatia-Thornton partial structure factors�solid� for model I of v-GeO2, compared with experimental data�circles� of Ref. 53: �a� SNN, �b� SCC, and �c� SNC. The scatteringvector is scaled as described in Fig. 7.

FIG. 10. Vibrational density of states Z��� �solid� and its de-compositions. In �a�, Z��� is first decomposed into O and Geweights. In �b�, the O weight is further decomposed according tothe O motion along the rocking �R, dashed�, bending �B, dot-dashed�, and stretching �S, dotted� directions. Inset: Projection onsymmetry-adapted modes of the GeO4 tetrahedra: T2 �dashed� andA1 �dotted�. A Gaussian broadening of 19 cm−1 is used.

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previous work,13 we compared the theoretical and experi-mental inelastic neutron spectra,60,15 finding good agreement.Here, we compare the theoretical inelastic neutron spectrumto the actual v-DOS �Fig. 11�. The comparison shows thatthe two spectra differ only slightly. The main differencesconcern the intensities in the low and high frequency bands,below 300 cm−1 and above 700 cm−1, respectively.

The vibrational frequencies and modes also give access tothe vibrational amplitudes.42 For our model of v-GeO2 andfor a temperature of 300 K, we obtained mean square dis-placements for O and Ge atoms of 0.26 and 0.16 Å, respec-tively. These values are slightly higher than found forv-SiO2 �0.21 Å for O and 0.14 Å for Si, cf. Ref. 42� andreflect the higher density of soft phonon modes in v-GeO2.For the fluctuations of Ge-O bond lengths at 300 K, wefound a thermally averaged value of 0.045 Å. This is veryclose to an experimental estimate obtained from neutron dif-fraction �0.042 Å, Ref. 44�, indicating that structural disorderonly marginally affects the fluctuations of the Ge-O bondlength.

V. INFRARED SPECTRA

A. Born charge tensors

The coupling between the atomic displacements and theelectric field is described by the Born effective charge ten-sors Z*.61 We calculated these tensors by taking finite differ-ences of atomic forces FI with respect to the electricfield:25,26

ZI,jk* =

�FIj

�Ek, �13�

where I labels the atom and where the lower case indicescorrespond to Cartesian directions.

The charge tensors Z* were calculated for all O and Geatoms in model I. Because of the local tetrahedral symmetryaround the Ge atoms, the average Z* tensors for the Ge atomsare essentially isotropic. For these atoms, we calculated anaverage isotropic charge of 3.44 with a standard deviation of0.14. We represent the O tensor within a local reference setbased on the orientation of the Ge-O-Ge bond �cf. Sec. IV�.We take the x direction along the bisector of the Ge-O-Ge

angle, the y direction normal to the plane of the Ge-O-Gebridge, and the z direction orthogonal to the previous two.The average Z* for O atoms reads

ZO* = �− 1.12 0.00 − 0.02

0.00 − 1.05 − 0.03

0.00 0.01 − 2.99� . �14�

This average tensor is almost diagonal, with diagonal valuestypical for O bridge structures.62,11 In particular, for the dis-placements along the stretching direction �z direction� thecoupling is noticeably stronger. This is related to the fact thatsuch displacements involve a much larger chargetransfer.63,62 The distribution of the isotropic O Born chargehas an average value of −1.72 and a width of 0.09. In Fig.12, we correlate the isotropic O Born charge with the corre-sponding Ge-O-Ge angle. The Born charge is found to de-crease with increasing bond angle. A similar trend was alsoobserved for v-SiO2.11

The properties of the isotropic Born charge distributionsof our models are summarized through their averages andwidths in Table II. The differences between the models areconsistent with their Ge-O-Ge bond-angle distributions �Fig.4 and Table I�. Indeed, the isotropic O Born charge consis-tently follows the same dependence on the Ge-O-Ge angle asfound for model I �Fig. 12�. The isotropic Ge Born chargesalso depend on the Ge-O-Ge bond angles through a localdynamical charge neutrality property, which is found to berespected in v-GeO2 to a similar extent as in v-SiO2.11

In the following, we describe the coupling to individualvibrational modes. It is therefore convenient to introduce theoscillator strengths F n:

FIG. 11. Calculated neutron vibrational density of states �v-DOS� of v-GeO2 �solid� at room temperature compared to theactual v-DOS �dashed�. In the calculation of the neutron spectrum�Ref. 9� we used transferred momenta �Ref. 60� in the range0.5–4.5 Å.

FIG. 12. Isotropic oxygen Born charge vs Ge-O-Ge angle forour four models of v-GeO2: model I �open squares�, model II �opendisks�, model III �closed disks�, and model IV �closed squares�.

TABLE II. High frequency �� � and static dielectric constants��0� for models I–IV of vitreous GeO2. Average isotropic Borncharge for O �ZO

* � and Ge �ZGe* �, together with the standard devia-

tions of their respective distributions �in parentheses�.

� �0 ZO* ZGe

*

Model I 2.79 6.28 −1.72 �0.09� 3.44 �0.14�Model II 2.53 6.65 −1.65 �0.09� 3.29 �0.12�Model III 2.50 6.56 −1.61 �0.08� 3.21 �0.11�Model IV 2.57 6.11 −1.68 �0.08� 3.36 �0.07�

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F jn = �

Ik

ZI,jk* �Ik

n

MI

. �15�

B. Dielectric tensor

We calculated the high-frequency dielectric tensors � forour models of v-GeO2 through second derivatives of the en-ergy with respect to the electric fields.25,26 The dielectric ten-sor calculated for model I is almost isotropic, as expected foran amorphous system:64

� = � 2.81 − 0.03 0.00

− 0.03 2.78 − 0.04

0.00 − 0.04 2.77� . �16�

From the isotropic part of the dielectric tensor �

=Tr�� � /3, we obtained an average dielectric constant of2.79, only slightly larger than the experimental value of 2.58�Ref. 65�.

We evaluated the static dielectric constant using the cal-culated oscillator strengths and vibrational frequencies:11

�0 = � +4�

3V�

n

�F n�2

�n2 , �17�

where V is the volume of the periodic simulation cell. Formodel I, we obtained �0=6.28. For comparison, we report inTable II also the dielectric constants calculated for modelsII–IV. The high-frequency dielectric constants of these othermodels is generally smaller than for model I because offinite-size effects associated to their smaller size.66

C. Dielectric function

The real and imaginary parts of the dielectric responsefunction, �1��� and �2���, are given by67,11

�1��� = � −4�

3V�

n

�F n�2

�2 − �n2 , �18�

�2��� =4�2

3V�

n

�F n�2

2�n��� − �n� . �19�

The dielectric function described above gives access to allthe dielectric properties. In particular, the energy loss func-tion is obtained as −Im�1/�����. However, it was foundconvenient to access the latter function by a directcalculation:67,26

− Im� 1

����� =4�2

V�� �2�n

�q · F n�2

2�n��� − �n� . �20�

For isotropic systems, the energy loss function in Eq. �20�can be averaged over all directions of q. Here, we used thethree Cartesian directions for this average.

In Ref. 13, we compared the calculated dielectric function�2 and the energy loss function −Im�1/�� of model I withavailable experimental spectra,15 finding overall good agree-

ment. In particular, the relative intensities of the three mainpeaks were well-reproduced and their LO-TO splittingsshowed fair agreement with experiment.13

In Fig. 13, we show the real and imaginary parts of thedielectric function calculated for models I–IV. The four mod-els give spectra with a similar overall shape. This similaritystems from the common short-range order which dominatesthe infrared spectra.12 However, in the central band rangingbetween 400 and 700 cm−1, we observe minor differencesbetween the models. In both �1 and �2, model I gives featureswhich are shifted towards lower frequencies by �50 cm−1

with respect to those of models II–IV. These features corre-spond to O bending modes and their frequencies decreasewith increasing Ge-O-Ge bond angle.68 Hence informationconcerning the bond-angle distribution can be gained fromthe location of the peak at 550 cm−1 in the experimental �2.The frequency of this peak in model I is found to slightlyunderestimate the experimental one, while all other modelsgive higher frequencies. In consideration of the overall lowerfrequencies in our simulation,13 this result further indicatesthat the bond-angle distribution of model I is more reliablethan those of the other models.

VI. RAMAN SCATTERING

In a first-order Stokes process of Raman scattering, anincoming photon of frequency �L and polarization eL givesan outgoing photon of frequency �S and polarization eS anda vibrational excitation of frequency �n=�L−�S. The re-duced Raman cross section is given by69

d2

d�dE� �

n

�eS · Rn · eL�2��E − �n� , �21�

where E is the exchanged energy, and Rn the Raman sus-ceptibility associated to the normal mode n:

FIG. 13. �a� Real and �b� imaginary parts of the dielectric func-tion for our four models of v-GeO2: model I �solid�, model II �dot-ted�, model III �dot-dashed�, and model IV �dashed�. A Lorentzianbroadening of 25.7 cm−1 is used for the real part, a Gaussian broad-ening of 19 cm−1 is used for the imaginary part.

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Rijn = V�

Ik

��ij

�RIk

�Ikn

MI

, �22�

where the dielectric polarizability tensor �ij is expressed as

��ij

�RIk=

1

V

�2FIk

�Ei�E j. �23�

We here calculated Raman spectra through the applicationof finite electric fields to our periodic model structure.25 Weevaluated the tensor �ij by numerically calculating secondderivatives of the atomic forces with respect to the field. Weused electric fields with intensities of E=0, ±0.01 a.u. A de-tailed description of our methodology is given elsewhere.26

Raman spectra were calculated for incoming and outgoingphotons with parallel �HH� and perpendicular polarizations�HV� for our four models. We here focused on the reducedRaman spectra, which do not depend on the Bosonic occu-pation number and thus better highlight the dependence onthe coupling tensors. In Fig. 14, we compare the reduced HHand HV Raman spectra of model I with the respective ex-perimental ones.15 The agreement is excellent and of compa-rable quality with respect to that recorded for the nonreducedRaman spectra.13,70 Apart from an overall shift to lowerfrequencies,13 the theoretical spectra reproduce well the prin-cipal peak and the high frequency doublet. Furthermore, wenote that the HH and HV Raman spectra in Fig. 14 are scaledby the same factor. Therefore the comparison also shows thatthe ratio between the integrated HH and HV intensities isclosely reproduced in our simulation.

A. Dependence of HH Raman spectrum on Ge-O-Ge angle

We argued in Ref. 13 that the HH Raman spectrum ofv-GeO2 shows a strong dependence on the intermediaterange order through the Ge-O-Ge bond angle distribution, asobserved previously for v-SiO2.12,26 We found that the HHRaman spectrum is dominated by the coupling to O bendingmotions. A further analysis revealed that the tensors �� /�R

associated to these motions are almost isotropic. Identifyingthe bisector direction of the Ge-O-Ge bond angle by eb, weexpress the coupling factor for each O atom in terms of avolume-independent scalar:

f I =1

3V Tr��

k

��

�RIkeIk

b � . �24�

Figure 15 shows that the HH Raman spectrum below700 cm−1 is reproduced up to 90% when only the isotropicpart of the coupling to O bending motions is accounted for.

The coupling factors f I associated to the O atoms inmodel I show a clear correlation with the corresponding Ge-O-Ge bond angle �I �Fig. 16�. The observed dependence isconsistent with the relation12

f I = ��/3�cos��I/2� , �25�

which holds for the bond-polarizability model71 for a systemof regular tetrahedral units. A one-parameter least-squared fitgives a value of 101 bohr2 for the parameter �.

In Fig. 17, we illustrate the dependence of the dominantcoupling factor on the Ge-O-Ge angle by comparing the HHRaman spectra obtained for the different model structures. Itis convenient to discuss these spectra on the basis of the

FIG. 14. Theoretical reduced HH and HV Raman spectra �solid�for model I of v-GeO2, compared to corresponding experimentalspectra from Ref. 15 �dotted�. The theoretical HH spectra is scaledto match the integrated intensity of the experimental spectrum. Thesame scaling factor is then applied to the HV spectrum. A Gaussianbroadening of 19 cm−1 is used in the theoretical spectra.

FIG. 15. Reduced HH and HV Raman spectra �solid� comparedto spectra obtained for specific approximations of the coupling ten-sors, for model I of v-GeO2. The dotted curve in the HH spectrumcorresponds to retaining only the isotropic component of these ten-sors associated to oxygen bending motions. The dot-dashed curvescorrespond to the result obtained within the bond-polarizabilitymodel with optimal parameters.

FIG. 16. Coupling factor f I vs Ge-O-Ge angle for all O atoms inmodel I of v-GeO2 �disks�. The curve corresponds to the best fitwith Eq. �25�, obtained for �=101 bohr2.

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bond-angle distributions given in Fig. 4. The spectrum ofmodel I shows the best agreement with experiment. Thisprovides support for its Ge-O-Ge bond-angle distributionwhich has a mean value at 135°. Experimental support forthis mean value also comes from diffraction4 and NMRexperiments.72 The other model structures show bond-angledistributions shifted to lower Ge-O-Ge angles, particularlyfor models II and III. Correspondingly, the HH Raman inten-sity is enhanced in the frequency region between 500 and700 cm−1, at higher frequencies than the main peak in modelI, consistent with the relation between bond angle and bend-ing frequency.68 We note the peculiar behavior in the fre-quency region around 500 cm−1, where the intensity variesconsiderably according to the considered model. We deferthe discussion of this behavior to Sec. VI B.

B. Raman shoulders and small rings

The Raman spectrum of v-SiO2 shows two particularlysharp lines, D1 and D2, which appear in addition to broaderbands due to the structural disorder.14 A first-principlesinvestigation73 has definitely assigned the origin of theselines to localized vibrations in small rings, as proposed byGaleener.40 From the intensities of these lines, an estimate ofthe concentration of three- and four-membered rings in v-SiO2 could be derived.12 A recent analysis applied to v-B2O3 has analogously succeeded in giving an estimate ofthe fraction of B atoms in boroxol rings.20

Despite the structural similarity between v-SiO2 and v-GeO2, the HH Raman spectrum of the latter glass does notshow any sharp feature. This contrasts with the higher pack-ing density of tetrahedral units and with the lower averagebond angle on the O atoms which both suggest a higherconcentration of small rings in v-GeO2.13 However, the Ra-man spectrum is characterized by two shoulders known as X1and X2 occurring on either side of the main Raman peak. Inanalogy with the D1 and D2 lines, the X2 shoulder has tenta-tively been assigned to three- or four-membered rings on thebasis of its behavior under isotopic substitution and neutronbombardment.74,75 This assignment received support fromcalculations on a Bethe-lattice cluster.76 However, the natureof X1 has remained more elusive. Isotopic substitution ex-periments indicate that the vibrations contributing to X1

mainly involve Ge motion with a modest admixture of Omotion.74

In order to identify the contribution of three-memberedrings to the HH Raman spectrum, we projected the vibra-tional eigenmodes onto O breathing motions in these rings,prior to the calculation of the Raman intensities. For model I,the Raman projection obtained in this way gives a broadpeak in the range from 500 to 550 cm−1 �Fig. 18�a��. Similarprojections carried out for the other model structures also fallin this frequency range �Fig. 18�b��. The three spectra in Fig.18�b� show a shoulder or second peak close to 600 cm−1.This effect is associated to the overestimated spread of theGe-O-Ge angles in the three-membered rings �typical stan-dard deviations between 5° and 7°� and arises from the finitesize of our models. Indeed, considering the most planar ringsin each of our models �Fig. 18�c��, we found that the widthof such projections narrows considerably,77 reducing in par-ticular the weight close to 600 cm−1. For our models, thissuggests that the average peak position for a planar ring oc-curs at �500 cm−1, in fair agreement with the position of theX2 shoulder.13 Furthermore, upon 16O→ 18O substitution, theprojection peak for three-membered rings is found to shift byabout −26 cm−1, in excellent accord with the experimentalshift of −25 cm−1.74,13 Hence our analysis supports the as-signment of the shoulder X2 to oxygen breathing vibrationsin three-membered rings.

The intensity of the X2 shoulder conveys informationconcerning the concentration of three-membered rings in

FIG. 17. HH Raman spectra for our four models of v-GeO2:model I �solid�, model II �dot-dashed�, model III �dotted�, andmodel IV �dashed�. A Gaussian broadening of 19 cm−1 is used.

FIG. 18. �a� HH reduced Raman spectrum �solid� for model I ofv-GeO2, together with HH Raman spectra obtained for vibrationaleigenmodes projected on breathing O vibrations in three- �dotted�and four-membered rings �dashed�. �b� Projections as in �a� forthree-membered rings belonging to model I �dotted�, model II�solid�, and model III �dashed�. In model IV these rings are absent.�c� Projections as in �b� but only for the most planar ring in eachmodel. A Gaussian broadening of 19 cm−1 is used.

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v-GeO2. To derive an estimate of this concentration, it isinstructive to compare the Raman intensities of our modelsin the frequency region at �500 cm−1 �Fig. 17�. Model IIIand to a lesser extent model II show strong peaks at thesefrequencies, which stand out with respect to the spectrum ofmodel I. This relates to the relatively higher concentrationsof three-membered rings in these models. Conversely, modelIV, in which such rings are absent, shows a clear minimum inthis frequency range of the spectrum. The ring statistics inFig. 5 imply that 31% �13%� of the O atoms in model III �II�belong to such rings, compared to a ratio of 11% in model I.Comparison with experiment leads to the suggestion that theamount of O atoms belonging to three-membered rings in v-GeO2 situates slightly above 10%. This value is consider-ably larger than the estimate advanced for v-SiO2 ��0.22%,Ref. 12� and indicates that the connectivities of tetrahedra inv-SiO2 and v-GeO2 differ substantially.

To elucidate the origin of the feature X1 appearing at347 cm−1 in the Raman spectrum of v-GeO2,74,76 we firstenquired the role of four-membered rings by inspecting thecorresponding Raman projections. However, as shown inFig. 18�a�, we found that these projections for model I givetheir largest contribution in correspondence of the principalRaman peak, without any specific relation to X1.

To get insight into the kind of vibrational motions in-volved in the frequency region of X1, we considered themodes associated to the Ge-O-Ge bridge including the Gemotions. We focused on O bending motions which dominatethe Raman spectrum in this frequency range �Fig. 15�.Searching for associated Ge motions which give maximalRaman projections below 400 cm−1, we found symmetric Gemodes approximately oriented along the Ge-Ge direction asdepicted in Fig. 19 �B1�. The Raman projection averagedover all Ge-O-Ge bridges in model I is shown in Fig. 19.This projection clearly yields an enhanced contribution forfrequencies below 350 cm−1 in the range of the shoulder X1.Upon 16O→ 18O, the main peak of the B1 mode is found toshift by −7 cm−1, in good agreement with the experimentalshift74 of X1 �−5 cm−1�. We therefore suggest that the X1

shoulder originates from B1 modes. Since the Ge-O-Gebridges form the basic structure of the network, this analysisfurther suggests that X1 does not result from vibrations local-ized on a structural subunit but rather corresponds to diffusemotions throughout the network.

We also searched for coupled O and Ge motions givingmaximal projection on the high-frequency side of the bend-ing band. This search yielded symmetric Ge motions ori-ented along the Ge-O as shown in Fig. 19 �B2�. The averageRaman projection on B2 modes gives a dominant contribu-tion between 350 and 700 cm−1 with a peak at �530 cm−1.We note that the breathing motions at the origin of the X2shoulder can be pictured as a superposition of B2 modeslocalized within three-membered rings.

For comparison, we also show in Fig. 19 the average Ra-man projection on O bending motions irrespective of theassociated Ge vibrations. This comparison clearly shows thatthe bending contribution results from two separate subbands,well represented by B1 and B2 vibrations. We also note thatthe average Raman projection on O bending motions doesnot differ significantly from the bending contribution to the

v-DOS �Fig. 10�, however, focusing on model I, we note thatthe HH Raman spectrum differs considerably from thev-DOS, especially below 700 cm−1.68,78 In particular, themain Raman peak at 410 cm−1 does not occur in correspon-dence of a peak in the v-DOS. This peak falls very close tothe dip at 350 cm−1 in the v-DOS, in overall good agreementwith the experimental situation.78 As remarked previously,78

the occurrence of a peak at the edge of the B2 band is a resultof correlations which considerably enhance the Raman inten-sity. Since the modes in the B1 band contribute to the Ramanspectrum through the same coupling mechanism, we expectthat the Raman intensities at the upper edge of this bandcould also undergo similar enhancements. The spectrum re-sulting from the combination of these contributions wouldthen give rise to a peculiar X1 shoulder at frequencies slightlylower than the dip in the v-DOS.

C. Parameters for the bond polarizability model

The bond polarizability model71 has successfully been ap-plied to the calculation of Raman intensities in a large varietyof systems.69 In particular, the bond polarizability model hasbeen shown to give a reliable description of the Raman in-tensities in SiO2.24,79 For future reference, we derive in thissection optimal bond-polarizability parameters from ourfirst-principles calculation of the tensors �� /�R.

In the bond polarizability model, the polarizability is de-scribed in terms of bond contributions:

�ij =1

3�2�p + �l��ij + ��l − �p��RiRj

R2 −1

3�ij� , �26�

where R=RJ−RI is a vector which defines the direction andthe distance of a pair of nearest neighbor atoms at sites RI

FIG. 19. Average Raman intensity �solid� obtained for vibra-tional eigenmodes projected on specific motions of the Ge-O-Geunit. The average is carried out over all units in model I of v-GeO2. The illustrated motions correspond to the choices whichmaximize the intensity in either the �a� low- �below �350 cm−1� or�b� high-frequency band �between 350 and 700 cm−1�, correspond-ing to the B1 and B2 motions discussed in the text. The relativeweight of Ge and O motions in the B1 and B2 motions has beentaken to correspond to the relative contribution of the two species inthe respective frequency range of the v-DOS �Fig. 10�. The dottedcurve in both panels corresponds to a projection on O bendingmotions irrespective of the Ge motions.

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and RJ. The parameters �l and �p correspond to the longitu-dinal and perpendicular bond polarizability, respectively. Thebond polarizability model further assumes that the bond po-larizabilities �l and �p only depend on the length of thebond. Thus the derivative of the bond polarizability withrespect to the displacement of the atom J reads:

��ij

�RJk=

1

3�2�p� + �l���ijRk + ��l� − �p���RiRj −

1

3�ij�Rk

+��l − �p�

R��ikRj + � jkRi − 2RiRjRk� , �27�

where R is a unit vector along R, and �l� and �p� are thederivatives of the bond polarizabilities with respect to thebond length. Therefore when only one type of bond occurs,the bond polarizability model is completely defined by threeparameters:

� = 2�p� + �l�, � = �l� − �p�, � = ��l − �p�/R . �28�

We then obtain the tensor �� /�R which appears in Eq. �23�by dividing the tensor �� /�R by the volume V.

We determined optimal parameters as follows. First, wefixed the parameter � which is responsible for the principalRaman peak, as described in Sec. VI A. Focusing on modelI, the other parameters were then obtained by minimizing thesum of squared differences between the components of thetensor �� /�R calculated within the bond polarizabilitymodel and within our first-principles scheme. This results inthe following parameters for v-GeO2 �expressed in bohr2�:

� = 101.0, � = 25.0, � = 1.4. �29�

It is interesting to compare these values with those obtainedfor �-quartz following a similar optimization scheme:24

� = 45.5, � = 11.6, � = 3.3, �30�

expressed in the same units as the values for v-GeO2. Thecomparison between the absolute values of these parametersclearly indicates that the derivatives of the polarizability forthe Ge-O bond are roughly twice as large as those for theSi-O bond. The ratio � /� is similar for the two oxides,whereas there is a more significant difference for � /�. Thelatter is, however, less reliable because of the minor contri-bution of � to the tensors �� /�R.

In Fig. 15, we compare the Raman spectra for model Icalculated within our first principles scheme and within thebond polarizability model with optimized parameters. Over-all, the bond polarizability model reproduces quite faithfullythe Raman spectra. The residual differences reflect the limi-tations of the bond polarizability model. In particular, wefound deviations of �8% for the main Raman peak. Thislevel of accord is consistent with the average deviations of�15% recorded for the bond polarizability model when ap-plied to �-quartz.24

VII. CONCLUSIONS

In this work, we aimed at gaining structural informationon a disordered network-forming material such as v-GeO2through the analysis of its vibrational spectra. We performeda comprehensive investigation involving inelastic neutron,infrared, and Raman spectra. Our approach goes beyond thesimple vibrational density of states to model the specific cou-pling tensors involved in each of the analyzed spectra. Toachieve a reliable degree of accuracy, our analysis is carriedout within a fully consistent density functional scheme, in-volving the calculation of vibrational frequencies and eigen-modes, dynamical Born charge tensors, dielectric constants,and Raman coupling tensors.

The present application to v-GeO2 shows that the infor-mation contained in the individual vibrational spectra mightdiffer considerably. The inelastic neutron spectrum is shownto reproduce quite accurately the vibrational density ofstates. The infrared spectra are mainly dependent on the ba-sic structural unit showing a weak dependence on their con-nectivity in the network. Indeed, various models with similarshort-range properties give infrared spectra with only mar-ginal differences. At variance, the Raman spectra is shown tosensitively depend on the intertetrahedral angle distribution,thereby conveying information on the connectivity of tetra-hedra in the network. Our study shows that a Ge-O-Ge an-gular distribution with a mean value of 135° is consistentwith the Raman spectrum, in accord with diffraction andNMR experiments. Furthermore, we assign the shoulder X2appearing in the Raman spectrum to three-membered ringsand provide an estimate of their concentration. The shoulderX1 is instead attributed to diffuse bending vibrations. Hencethis analysis clearly demonstrates that the combined consid-eration of various vibrational spectra provides an invaluabletool to reveal the properties of the underlying network struc-ture.

Our approach starts with the generation of viable struc-tural models. Through comparisons between experiment andtheory it becomes possible to establish relations betweenspecific features in the spectra and the underlying structures.Understanding the origin of specific features then leads toinformation on how the model structures should be amelio-rated. Application of this methodology in an iterative mannercould clearly establish a virtuous cycle in which model struc-tures become progressively more realistic.

ACKNOWLEDGMENTS

We thank P. S. Salmon for providing us with the experi-mental data of Ref. 53 in electronic format. Support is ac-knowledged from the Swiss National Science Foundation un-der Grants No. 200021-103562/1 and No. 200020-111747/1.The calculations were performed on the cluster of CSEA-EPFL, on the cluster PLEIADES of EPFL, and at the SwissCenter for Scientific Computing �CSCS�.

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