Abstract.: We present first-principles calculations for the optical properties of germanium dioxide in the rutile structure. The electronic band structure has been calculated self-consistently within the local density approximation using the full-potential linearized augmented plane wave method. The electronic band structure shows that the fundamental energy gap is direct at the center of the Brillouin zone. The determinant role of a band structure computation with respect to the analysis of the optical properties is discusse

Baltache, H.

Daul, C.

Khenata, R.

Sahnoun, M.

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Eur. Phys. J. B 45, 455–458 (2005)DOI: 10.1140/epjb/e2005-00219-y THE EUROPEANPHYSICAL JOURNAL BOptical properties of germanium dioxide in the rutile structureM. Sahnoun1,a, C. Daul1, R. Khenata2, and H. Baltache21 De´partement de Chimie, Universite´ de Fribourg, Chemin du Muse´e 09, 1700 Fribourg, Switzerland2 Departement de Technologie, Universite´ de Mascara, Route de Mamounia, 22000 Mascara, AlgeriaReceived 22 February 2005Published online 13 July 2005 – c© EDP Sciences, Societa` Italiana di Fisica, Springer-Verlag 2005Abstract. We present first-principles calculations for the optical properties of germanium dioxide in therutile structure. The electronic band structure has been calculated self-consistently within the local den-sity approximation using the full-potential linearized augmented plane wave method. The electronic bandstructure shows that the fundamental energy gap is direct at the center of the Brillouin zone. The de-terminant role of a band structure computation with respect to the analysis of the optical properties isdiscussed.PACS. 71.15.Ap Basis sets (LCAO, plane-wave, APW, etc.) and related methodology (scattering methods,ASA, linearized methods, etc.) – 74.25.Jb Electronic structure – 74.25.Gz Optical properties1 IntroductionGermanium dioxide is particulary interesting to study be-cause it is similar to silicon dioxide in many respects.GeO2 occurs in three stable forms at ambient tempera-ture [1]. The ﬁrst form has the rutile tetragonal struc-ture (P42/mnm) with octahedral coordination [2]. Thesecond form has the trigonal structure of α-quartz [3], withtetrahedral coordination. The third form is glassy, hav-ing the same short-range correlation [4]. The tetragonaland trigonal crystalline forms diﬀer substantially in theirchemical properties, especially solubility, in their densitiesand indices of refraction. Over the past three decades var-ious physical and thermodynamical properties of rutile-type GeO2 have been investigated: elastic properties [5],compressibility [6], entropy and enthalpy [7], thermal ex-pansion [8], structure [9], melting phenomena [10], elec-tronic properties [11–16] and high-pressure behavior [18].The response function for the optical properties of asolid, describing the absorption or emission of electrons orphotons, is the dielectric function. One may calculate thedielectric function from accurate band structure calcula-tions, thereby allowing one to assign the spectral featuresto speciﬁc excitations within the Brillouin zone, as wellas obtaining interband topologies. However, to our knowl-edge, up to now, there has been no calculation of the op-tical properties of the rutile-type GeO2 that includes thefull transition matrix. The determination of optical prop-erties requires, apart from the Kohn-Sham eigenvalues,the explicit use of the wave functions which should be cal-culated as accurately as possible. The aim of this paper isa e-mail: mohammed.sahnoun@unifr.chto present the results of a theoretical investigation of theelectronic structure and optical spectra of GeO2 whichcrystallizes in a rutile-type lattice.The organization of the paper is as follows: we explainthe computational method in Section 2. The results arepresented and discussed in Section 3 for the electronicstructure and the optical properties. A brief conclusionis drawn in Section 4.2 Computational methodGeO2 crystallizes with the bulk rutile structure(space group P42/mnm) [2,19,20]. In Figure 1 aschematic representation of the lattice is depicted. Theatoms coordinates are Ge: (0,0,0), (12 ,12 ,12 ) and O:±(x, x, 0), ± (12 + x, 12 − x, 12 ). The structure consists ofchains of GeO6 octahedra and each pair shares oppo-site edges. Each Ge atom is surrounded octahedrally bysix oxygen atoms, whereas each oxygen is surrounded bythree Ge atoms arranged as corners of an equilateral trian-gle. The structure has a 6:3 coordination. Each octahedronis not regular, showing a slight orthorhombic distortionand is in contact with ten neighbor octahedra, two shar-ing edge oxygen pairs and eight sharing corner oxygenions [21].The geometry of rutile-type structure of GeO2 hasbeen optimized by varying the positional parameterand cell length, and calculating the total energy, un-til a minimum has been found, using the ﬁrst-principlesFP-LAPW method in a recent implementation [22–25].This code uses a basis set obtained by dividing the unitcell into non-overlapping atomic spheres (centered at the456 The European Physical Journal BFig. 1. Rutile structure, blue spheres represent germaniumatoms and red spheres the oxygen atoms.atomic sites) and an interstitial region. In the atomicsphere, a linear combination of radial functions timesspherical harmonics is used and in the interstitial re-gion, a plane wave expansion is augmented by an atomic-like function in every atomic sphere. This approach hasbeen extensively tested and is among the most accu-rate methods for performing electronic structure calcu-lations for crystals. The exchange and correlation po-tential within the Local Density Approximation (LDA)is calculated using the scheme of Hedin-Lundqvist [26].The sphere radii of Ge and O have been chosen as 1.8and 1.6 atomic units (a.u.), respectively. Within thesespheres, the charge density and potential are expandedin terms of crystal harmonics up to an angular momen-tum l = 10. For the calculation of the optical properties, adense mesh of uniformly distributed k-points is required.Hence, the Brillouin zone integration was performed us-ing the tetrahedron method with 105 k-points in the ir-reducible part of the Brillouin zone without broadening.Well converged solutions are obtained for RMT .Kmax = 7,RMT labelling the atomic sphere radii, and Kmax the in-terstitial plane wave cut-oﬀ. The optical properties havebeen obtained through the frequency-dependent dielectricfunction, ε(ω) = ε1(ω) + iε2(ω), using the formalism ofEhrenreich and Cohen [27]. The imaginary part of the di-electric function is given asε2(ω) =e2πm2ω2∑v,c∫BZ|Mcv(k)|2δ[ωcv(k) − ω]d3k (1)where the integral is over the ﬁrst Brillouin zone,Mcv(k) = 〈uck|ê · ∇|uvk〉 are the dipole matrix elementsfor direct transitions between valence- and conduction-band states, ωcv(k) = Eck − Evk is the excitation en-ergy, ê is the polarization vector of the electric ﬁeld, anduck(r) is the periodic part of the Bloch wave functionfor a conduction-band state with wave vector k. Excita-tion energies between the valence-band and conduction-band states up to approximately 19 eV from the valence-band edge were considered. Brillouin-zone integration hasbeen performed within the linear analytic tetrahedronscheme [28].Table 1. Calculated structural parameters of GeO2. Latticeconstants are in A˚.Lattice Present Experiments Calculationsconstants [19] [20] [29] [30]a 4.3897 4.4066 4.4066 4.3838 4.3515c 2.8740 2.8619 2.8619 2.8637 2.8628x 0.3070 0.3059 0.3060 0.3059 0.3060Table 2. Critical point energies (eV) for polarization perpen-dicular (E⊥) and parallel (E‖) to the c axis. The notation ofthe optical peaks is arbitrary.E⊥ Energy E‖ EnergyA1 6.38 X 7.37A2 8.12B1 10.42 Y1 8.62B2 11.23 Y2 9.60B3 12.52 Y3 10.29Y4 11.04C1 14.48 Z1 13.62C2 15.42 Z2 14.36Z3 15.423 Results and discussionsThe calculated structural parameters of the rutile GeO2are given in Table 1. They have been obtained as a result ofthe minimization of the lattice ground-state energies. Theobtained results are compared with available experimen-tal data and other calculations. A good agreement withina 1% interval is found. In Figure 2, we show the bandstructure for GeO2. The tightly bound O 2s is the lowestbands and it has 1.8 eV of bandwidth. The upper valencebands with a width of 10.49 eV, are composed mainlyof O 2p states. A 1.80 eV direct-forbidden gap occursat Γ and is expected to be lower than the experimentalvalue. The fundamental absorption edge has been studiedby Stapelbroek and Evans [17]. The fundamental gap wasconcluded to be direct and optically-forbidden of 4.68 eVmagnitude. Christie and Chelikowsky obtained a valueof 2.4 eV by using a pseudopotential density functionalmethod [12]. The lower conduction-band states, with afull width of 11.10 eV, are composed primarily of O 2pand Ge s-p states. The imaginary part of the dielectricfunction for both E⊥ and E‖ polarizations are shown inFigure 3. The obtained interband transition energies areindicated with arrows and are listed in Table 2. LDA un-derestimates the excitation energies as evident by the shiftin the fundamental absorption edge to lower energies.The fundamental absorption edge for E⊥ occurs at∼1.80 eV, resulting from transitions between the top-most valence band and the bottom of the conductionband along the Γ point. The two prominent features,designated as A1 and A2 in Figure 3, are attributed tosplitting in the O 2px,y orbitals. The ﬁrst major peak,which we attribute to A1, occurs at 6.38 eV and resultsM. Sahnoun et al.: Optical properties of germanium dioxide in the rutile structure 457Fig. 2. Band structure of GeO2 along the high-symmetry di-rections of the irreductible Brillouin zone. The valence-bandmaximum is taken as the zero of energy.from transitions between valence band 15 to the ﬁrstconduction band along the R–Γ direction (we label thevalence bands from 1–16 and the conduction bands from1 upwards). The second peak, which we assign to A2, oc-curs at 8.12 eV and results from transitions between va-lence bands 11 and 12 to the two ﬁrst conduction bandsalong the R point. The next three features, which weattribute to B1, B2 and B3 occur at 10.42, 11.23 and12.52 eV, respectively; the ﬁrst one results from transi-tions between valence band 14 to the third conductionband along the X–M direction, the second one results fromtransitions between valence band 13 to the third conduc-tion band along Λ direction. And the third peak resultsfrom transitions between valence bands 14 and 15 to theconduction bands 5 and 6 along M–Γ and ∆ directions, re-spectively. The next high-energy structures are associatedwith C1 and C2 which occur at 14.48 and 15.42 eV, andresult from transitions between valence bands 14 and 13to the conduction bands 8 and 9 in the neighborhood of Rand along M–Γ , respectively.The absorption edge for E‖ polarization occurs at∼1.80 eV resulting from transition between the top ofthe valence band and the bottom of the conductionband with the majority of the structure occurring on the∆–X–M–Σ plane. As seen in Figure 3, one prominent fea-ture at 10.29 eV, labelled B3, is obtained for E‖ and isattributed to transitions from the O 2pz orbitals. Theﬁrst major peak for E‖ polarization occurs at 7.37 eVand results from transitions between valence band 12 toFig. 3. The imaginary part of the dielectric function for po-larization vectors perpendicular E ⊥ and parallel E ‖ to thec axis. The labels are chosen arbitrary at critical points. Theonsets are included to indicate the fundamental absorptionedge.the bottom of the conduction band along X–M direction.Near the prominent feature Y3, located at 10.29 eV, weﬁnd three additional features. The ﬁrst one appears at8.62 eV, attributed to Y1 and resulting from transitionsbetween valence band 11 to the second conduction bandalong the M–Γ direction. The second feature, labelled Y2,occurs at 9.60 eV and results from transitions betweenvalence bands 13–14 to the conduction bands 4–5 in theneighborhood of the X–M direction. The third feature, la-belled Y4, occurs at 11.04 eV, and results from transitionbetween valence band 13 to the third conduction band.Three other features on the high-energy side appear asshoulders at 13.62, 14.36 and 15.42 eV. The ﬁrst transitionbetween valence band 15 and conduction band 8, locatedin the neighborhood of Γ–X direction, is associated withpeak Z1. The next major peak is a result of transitionsbetween valence bands 13–14 and conduction bands 7 lo-cation in the neighborhood of the R–Γ direction and isassociated with peak Z2. The last peak labelled Z3 resultsfrom transition between valence bands 13 to conductionband 9 in the neighborhood of the R–Γ direction.4 ConclusionWe have used the new FP-LAPW method, within theLDA for exchange-correlation potential, to investigate the458 The European Physical Journal Bband structure and the optical properties of GeO2 inthe rutile structure. The geometry of rutile-type struc-ture of GeO2 has been optimized and good agreement isobtained with experiment. The fundamental band gap ofthe rutile GeO2 is direct at the center of the Brillouinzone. From the imaginary part of the dielectric function,we have studied the diﬀerent optical transitions, in whichwe have presented the diﬀerent peak structures associatedwith their assignment of the direct interband transitions.Furthermore, we are able to account for diﬀerences in elec-tronic polarizations for the dielectric function calculation.This work was supported by the Swiss National Science Foun-dation and the Swiss State Secretariat for Education andScience.References1. J. Muller, H.R. Blank, J. Am. Chem. Soc. 46, 2338 (1924)2. A.W. Laubengauer, D.S. Morton, J. Am. Chem. Soc. 54,2303 (1932)3. W.H. Zachariasen, Z. Kirist. 67, 226 (1928)4. T.G. Kujumzelis, Z. Phys. 100, 221 (1935)5. R.C. Liebermann, Phys. Earth Planet. Inter. 7, 461 (1973)6. R. Hazen, L.M. Finger, J. Phys. Chem. Solids 42, 143(1981)7. A. Navrotsky, J. Inorg. Nucl. Chem. 33, 1119 (1971)8. K.V.K. Rao, S.V.N. Naidu, L. Iyengar, J. Am. Ceram.Soc. 51, 467 (1968)9. W.H. Baur, A.A. Khan, Acta Cryst. B 27, 2133 (1971)10. 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Optical properties of germanium dioxide in the rutile structure

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Optical properties of germanium dioxide in the rutile structure

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