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Ab initio structural___ electronic__ and optical properties of orthorhombic CaGeO3

Journal of Solid State Chemistry 180(2007)974–980

Ab initio structural,electronic and optical properties of

orthorhombic CaGeO 3

J.M.Henriques a ,E.W.S.Caetano b,?,V.N.Freire c ,J.A.P.da Costa c ,E.L.Albuquerque a

a

Departamento de F?

′sica Teo ′rica e Experimental,Universidade Federal do Rio Grande do Norte,59072-900Natal,Rio Grande do Norte,Brazil b

Centro Federal de Educac -a ?o Tecnolo

′gica do Ceara ′,Avenida 13de Maio,2081,Ben?ca,60040-531Fortaleza,Ceara ′,Brazil c

Departamento de F?

′sica,Universidade Federal do Ceara ′,Centro de Cie ?ncias,Caixa Postal 6030,Campus do Pici,60455-760Fortaleza,Ceara ′,Brazil Received 23September 2006;received in revised form 13December 2006;accepted 26December 2006

Available online 10January 2007

Abstract

Orthorhombic CaGeO 3is studied using density-functional theory (DFT)considering both the local density and generalized gradient approximations,LDA and GGA,respectively.The electronic band structure,density of states,dielectric function and optical absorption are calculated.Two very close indirect (S !G )and direct (G !G )band gap energies of 1.68eV (2.31eV)and 1.75eV (2.41eV)were obtained within the GGA (LDA)approximation,as well as the effective masses for electrons and https://www.sodocs.net/doc/0917465999.html,paring with orthorhombic CaCO 3(aragonite),the substitution of carbon by germanium changes the localization of the valence band maximum of the indirect transition,and decreases by almost 2.0eV the Kohn–Sham band gap energies.r 2007Elsevier Inc.All rights reserved.

PACS:61.43.Bn;71.15.Mb;72.80.Sk;78.20.Ci

Keywords:Orthorhombic CaGeO 3;Structural properties;Band structure;Effective masses;Optical absorption;Dielectric function

1.Introduction

Calcium germanate (CaGeO 3)belongs to the pyroxenoid group,with each germanium (coordination number IV)atom connected to six oxygen atoms,forming a tridimen-sional array of GeO 6tilted octahedra (see Fig.1).The calcium atoms are inserted between them,resembling the structure of CaSiO 3in the perovskite phase.Under normal conditions of pressure and temperature,CaSiO 3assumes a triclinic structure,with the unit cell containing two chains of tilted SiO 4tetrahedral groups along the b -axis.At ambient conditions,CaGeO 3crystals are orthorhombic (Pbnm symmetry),while under pressure it changes ?rst to a rhodonite-like structure at 6GPa,which is further trans-formed into a perovskite-form at about 15GPa [1].Since

CaGeO 3can be quenched to ambient condition,together with CaSiO 3,MgSiO 3and others ABO 3structures,it has being mainly studied for the understanding of the mechanisms that lead to the stabilization of the various perovskite structures,a long standing problem in material science [2].

The major calcium germanate analogs are the silicate CaSiO 3and the carbonate CaCO 3.The former has geophysical importance as one of the main materials considered in modelling the dynamics and evolution of the Earth’s lower mantle;the latter is one of the most abundant minerals in the Earth surface,with industrial (plastic,rubbers,papers,paints,etc.)and implant applica-tions.There are many theoretical ab initio studies on CaSiO 3in the perovskite phase [2–7],but only one on CaSiO 3triclinic [8].In this case,the band structure,density of states,Kohn–Sham band gap,effective masses,and optical absorption were obtained by ?rst-principles quan-tum mechanical calculations after optimization of the unit cell parameters and atomic coordinates.On the other hand,

https://www.sodocs.net/doc/0917465999.html,/locate/jssc

0022-4596/$-see front matter r 2007Elsevier Inc.All rights reserved.doi:10.1016/j.jssc.2006.12.029

?Corresponding author.Departamento de F?sica,Universidade Federal

do Ceara ,Centro de Cie ncias,Caixa Postal 6030,Campus do Pici,60455-760Fortaleza,Ceara

,Brazil.Fax:+558540089450.E-mail address:ewcaetano@https://www.sodocs.net/doc/0917465999.html, (E.W.S.Caetano).

calcium carbonate(CaCO3)has three polymorphs:calcite (rhombohedral),aragonite(orthorhombic),and vaterite (orthorhombic).A theoretical study on the structure and bonding of calcite was performed by Skinner et al.[9].They have obtained an indirect energy gap E GeD!ZT?4:4?0:35eV for calcite,which is well below the experimental value of6:0?0:35eV[10].Recently,Medeiros et al.[11] have investigated,by quantum mechanical?rst-principles calculations,the structural,electronic,and optical proper-ties of CaCO3aragonite(orthorhombic).The band gap energy was predicted to be$1eV smaller than that of calcite(rhombohedral),and two absorption regimes were shown to exist:the?rst one with weak absorption in the 4.4–6.3eV range associated to transitions involving mainly p-like states of dominating oxygen character;the other with stronger absorption,ruled by p!d transitions.

For the best of our knowledge,there is no scienti?c report so far on ab initio calculations of orthorhombic CaGeO3properties.The focus of this work is to simulate structural and optoelectronic properties of calcium germa-nate by applying quantum chemical?rst-principles calcula-tions using the density functional theory(DFT)within the local density and generalized gradient approximations framework,LDA and GGA,respectively.A comparison of the results with those of CaSiO3triclinic and CaCO3 aragonite is performed,revealing the effect of the C;Si! Ge replacement on the electronic structure by looking to the changing pattern of the atomic contributions to the density of states in the simulated crystals,and to the energy band structure as well as optical properties.2.Calculation methodology

Quantum mechanical?rst-principles calculations for orthorhombic CaGeO3were performed using the CASTEP code[12]within the DFT formalism[13,14].To improve over the total local-density approximation(LDA)[14,15] to the exchange-correlation(XC)energy[16],the general-ized gradient approximation(GGA)was also considered [17,18].For geometry optimization,ultrasoft Vanderbilt-type pseudopotentials[19]were used considering the following electronic con?gurations:Ca-3s23p64s2,Ge-4s24p2,and O-2s22p4.Ultrasoft pseudopotentials help to reduce the computational cost of?rst-principles electronic structure calculations by decreasing the energy cutoff of the plane-wave basis set.The XC functional for the LDA calculation is standard[20,21],while for the GGA the Perdew–Burke–Ernzerhof(PBE)functional[22]was cho-sen.Results for bulk materials obtained with the PBE functional are,in general,very similar to the commonly used PW91functional[23].A Monkhorst–Pack6?6?4 sampling was used to evaluate integrals in the reciprocal space[24].This Monkhorst–Pack grid is enough to give a well-converged electronic structure due to the?atness of the CaGeO3valence bands.

The lattice parameters and the atomic positions were optimized by seeking a total energy minimum for the CaGeO3unit cell,which is depicted in Fig.1.The unit cell dimensions and internal atomic coordinates of orthorhom-bic CaGeO3measured by X-ray diffraction[25]were used as input.In order to perform the geometry optimization,

Fig.1.Crystal structure of orthorhombic CaGeO3.Each germanium(IV)atom is connected to six oxygen atoms,forming a tridimensional array of GeO6 tilted octahedra with calcium atoms inserted between them.Top:different views of the unit cell and atomic nomenclature.Bottom:tilted GeO6octahedra build up the orthorhombic phase.

J.M.Henriques et al./Journal of Solid State Chemistry180(2007)974–980975

the following convergence thresholds were considered for two successive self-consistent steps:(i)total energy change smaller than0:5?10à5eV=atom;(ii)maximum force over each atom below0.01eV/A;(iii)pressure smaller than 0.02GPa;(iv)maximum atomic displacement not exceed-ing0:5?10à3A.The Pfrommer et al.minimizer[26]was employed to carry out the unit cell optimization.For each self-consistent?eld step,the electronic minimization parameters were:(i)total energy/atom convergence toler-ance of0:5?10à6eV;(ii)eigenenergy threshold of 0:1190?10à6eV at most;(iii)a convergence window of three cycles.

The basis set cutoff energy for our calculations was 500eV and its quality was kept?xed while taking into account changes of the unit cell volume throughout the search for the optimal geometry.In order to secure the accuracy of our results,we performed geometry optimiza-tions using a larger energy cutoff of600eV.The comparison of both outputs revealed that an increase of 100eV in the quality of the basis set decreases the unit cell total energy by only0.004%at most,and the?rst three decimal digits of the lattice parameters do not change at all in both LDA and GGA computations.

After obtaining the unit cell and atomic positions,the electronic band structure and the density of states(total and partial,and the relative contribution of each atom) were evaluated,as well as the dielectric function and optical absorption.We have used the same XC functionals of the energy minimization,although replacing the ultra-soft pseudopotentials by norm-conserved ones[27]with an energy cutoff of700eV.Such replacement is necessary due to limitations of the CASTEP code to include some contributions to the optical properties related to the use of ultrasoft pseudopotentials.Increasing the energy cutoff of the basis set to800eV changed the electronic eigenenergies by1meV at most.Effective masses at the extrema of the valence and conduction bands were estimated by quadratic interpolation of the corresponding band curves following the scheme of Henriques et al.[8]. The complex dielectric function and the optical absorption aeoTof orthorhombic CaGeO3were calculated following the same scheme of previous works[8,11].This scheme works as follows:?rst,optical absorption was obtained from the imaginary part of the complex dielectric function, given by

e2eoT?2e2p

Oe0

X

k;v;c

jh c c

k

j^uár j c v

k

ij2deE c kàE v kà_oT,(1)

where o is the light frequency,c c

k and c v

k

are,respectively,

the conduction and valence band wavefunctions at k calculated within the DFT approach.The vector points along the polarization of the incident electric?eld.The real part e1of the dielectric function is obtained from the imaginary part e2using the Kramers–Kronig relationship [28,29].The imaginary and real parts of the refraction index(k and n,respectively)are related to e1and e2through:

e1?n2àk2e2?2nk.(2) Finally,the optical absorption is given by

IeoT?

2k o

c

.(3) Although other physical parameters(re?ection coef?cient, loss function and optical conductivity)were obtained by using our numerical approaches,they are not presented here.

3.Results and discussions

3.1.Volume and geometry optimization

The structural parameters of orthorhombic CaGeO3 calculated after geometry optimization through the DFT-LDA and the DFT-GGA approximations are shown in Table1,together with the experimental data of Sasaki et al.

[25].In both cases,a good agreement with the experimental data is observed.The lattice parameters calculated within GGA are consistently bigger than those found through LDA calculation and the experimental data.They are coherent with the results found by comparison to other compounds[17,18]and the well-known underbinding effect for this kind of functional.The GGA overestimates the actual orthorhombic CaGeO3lattice parameters a,b,and c by 1.14%, 2.01%,and 1.34%,respectively.The GGA-calculated unit cell volume is 4.56%bigger than the experimental one,while the LDA-calculated is 5.23% smaller,since LDA overestimates the strength of intera-tomic interactions.After convergence using the GGA functional,the?nal average pressure wasà0:0076GPa, and the symmetrized stress tensor components were xx?0:002046GPa, yy?0:010546GPa,and zz?0:010233GPa.The stress is hydrostatic.

A comparison of the bond lengths and angles between our theoretical results for orthorhombic CaGeO3and orthorhombic CaSiO3,the latter with lattice parameters a?5:0517A,b?5:0517A,c?7:1442A,and space group Pbnm[30],is shown in Table2.We did not present a comparison with orthorhombic CaCO3(vaterite)due to the different connectivity exhibited by carbon and oxygen atoms(in this CO3compound,planar groups,not

Table1

Comparison between experimental[25]and calculated structural para-meters for the CaGeO3orthorhombic unit cell

Exp.(A)LDA(A)LDA-Exp.(%)GGA(A)GGA-Exp.(%)

a 5.2607 5.1624à1.87 5.3205t1:14

b 5.2688 5.1832à1.62 5.3749t2:01

c7.44527.3086à1.837.5452t1:34 Exp.e?A3TLDAe?A3TGGAe?A3T

V206.36195.56à5.23215.77t4:56

J.M.Henriques et al./Journal of Solid State Chemistry180(2007)974–980 976

octahedra,are formed).We see that,as expected,calculated Ge–O bond lengths are smaller and larger than X-ray measurements,for LDA and GGA approximations,respectively.In comparison to CaSiO 3,the Ge–O bond is

larger than the Si–O bond by 0.1A

approximately.The shortest distance between Ge and Ca atoms is also presented,and is smaller than the experimental value

(3.105A

)for both LDA (3.021A )and GGA (3.101A )calculations,the GGA value being very close to the

experimental data (only 0.004A

smaller).The O–Ca closest neighbors are separated by 2.284A

(LDA), 2.331A (GGA),and 2.346A

(experimental).On the other hand,Ca–Ca closest neighbors are 3.685A

apart from experi-mental data,but are closer within the LDA (3.607A )approximation,and farther within the GGA (3.710A

)one.Ge–O–Ge angles are considered along the z -axis and within the xy plane.The LDA values are smaller than but closer to experimental data in comparison with GGA values.O–Ge–O angles within GeO 6are also presented in crescent order,a o b o g .LDA values are larger than experimental ones for both b and g ,but smaller for a .For the GGA angles,comparing with the X-ray measurements we have smaller g (by only à0:04%),while a and b are larger.In perovskite CaSiO 3all O–Si–O (Si–O–Si)angles are 90 (180 )as expected due to the absence of tilting in SiO 6octahedra.3.2.Band structure,density of states and effective masses Fig.2shows the band structure near the main Kohn–Sham band gap of CaGeO 3according to the GGA (solid)and LDA (dotted)calculations.LDA eigenenergies are larger for the conduction bands and smaller for the valence bands,resulting in larger LDA band gaps in comparison to GGA.This difference is almost constant,practically the same amount of 1eV if we consider the direct gaps at the S ,X ,Y ,and U !T valence band extrema.At the G point,however,this discrepancy decreases to 0.6eV,indicating that a rigid shift of the LDA eigenenergies is not good enough to fairly reproduce the GGA results.

Considering the full GGA band structure (not shown here),there are four energy bands close to à38eV,originating from Ca-3s levels (see Fig.3),12bands near à20eV with strong contribution from Ca-3p states,and 12bands between à19and à15eV which are mainly O-2s in character.The uppermost valence bands count is 36,resulting mostly from O-2p orbital.Twenty conduction bands were calculated,mixing contributions from calcium,oxygen,Ge-4s and Ge-4p states,as indicated in https://www.sodocs.net/doc/0917465999.html,paring the density of states predicted for CaGeO 3with the calculated for CaSiO 3[8],we note the absence of Ca-3d contributions to the lowermost conduction bands.It seems that the replacement of Si by Ge in some way prevents the formation of Ca 3d -related excited electronic levels,at least for the ?rst 20conduction bands of orthorhombic CaGeO 3.This result,however,must be considered with some care,because it is well known that simple GGA calculations tend to poorly describe empty states and d states (both ?lled and empty).

Table 2

Comparison between experimental [25]and calculated interatomic distances and angles for the CaGeO 3orthorhombic crystal

Smallest interatomic distance (A )Angle (degree)Ge–O

Ge–Ca

O–Ca

Ca–Ca

Ge–O–Ge O–Ge–O z

xy a b g LDA

1.897 3.021

2.284

3.607158.222157.37290.00690.27790.833GGA 1.941 3.101 2.331 3.710155.33615

4.74490.41790.62290.752Exp

1.897 3.105

2.346

3.685160.354158.940

90.26390.269

90.792

Si–O

Si–Ca O–Ca Ca–Ca Si–O–Si O–Si–O CaSiO 3

1.786

3.094

2.526

3.572

180

18090

90

90

Data of orthorhombic CaSiO 3is also shown [30].Ge–O–Ge angles are taken for bonds along z -axis and parallel to the xy plane,while O–Ge–O angles are presented in crescent order (a ,b and g ).The remaining O–Ge–O angles belonging to the GeO 6octahedra are obtaining by subtracting these values from 180

.

1086420.0-0.4-0.8-1.2-1.6

E n e r g y (e V )

S

ΓZ

T Y

S

X U

R

Γ

U

T

Γ

Y

Fig. 2.Orthorhombic CaGeO 3band structure calculated using LDA (dotted)and GGA (solid)functionals.The top valence bands have been aligned at the G point (zero energy).

J.M.Henriques et al./Journal of Solid State Chemistry 180(2007)974–980

977

The smallest Kohn–Sham band gap calculated is indirect,between the valence band maximum at S eà0:5;0:5;0Tand the conduction band minimum at G (0,0,0).For the LDA calculation,the S !G energy gap is 2.31eV,while for the GGA we obtain 1.68eV.We note that due to the approximate nature of DFT functionals,the theoretically calculated energy gaps are inaccurate and somewhat smaller than the experimental measurements.Therefore,the smallest actual band gap of CaGeO 3must be larger than 2.31eV.The G !G direct transition involves an energy change of 2.41eV in the LDA,and 1.75eV in the GGA.In contrast,according to the LDA calculations of Henriques et al.[8],CaSiO 3presents a wide indirect Kohn–Sham band gap of 5.43eV between the Q and G points,and a direct band gap of 5.52eV for G !G transitions.

The CaCO 3aragonite polymorph has also an orthor-hombic unit cell.Recently performed LDA and GGA calculations [11]previewed very close indirect [E G ;LDA eX !G T?4:00eV and E G ;GGA eX !G T?4:29eV]and direct [E G ;LDA eG !G T?4:01eV and E G ;GGA eG !G T?4:27eV]energy gaps.Consequently,the substitution of carbon by germanium,which modi?es the contribution to the valence electronic con?guration from C-2s 22p 4to Ge-4s 24p 2,changes the localization of the valence band maximum of the indirect transition from the point of symmetry X to S ,and decreases by almost 2.0eV the Kohn–Sham band gap energies.

In Table 3we summarize the effective masses obtained for CaGeO 3.It can be seen that LDA masses are always smaller than GGA masses for electrons and holes.Hole effective masses are large (in particular,valence bands along lines to the R ,U ,T and Y points are too ?at to allow a meaningful estimate)and very anisotropic,varying between 1and 4:3m 0at the S point (m 0is the free space electron mass).Along the S !G direction we have two degenerate bands with distinct curvatures and,therefore,two hole masses:a light hole (lh)close to 1m 0,and a heavy hole (hh)almost 4times larger.Electron masses,on the other hand,are smaller and almost isotropic,varying between 0.35and 0:44m 0for the LDA calculation,and between 0.39and 0:50m 0for the GGA calculation.A similar behavior for the hole and electron effective masses (large and anisotropic hole masses,light and almost isotropic electron masses)was observed for CaSiO 3wollastonite [8].3.3.Optical properties

Migas et al.[31]reported ?rst-principles results for the ground-state properties,band structures,density of states and dielectric functions of Ca 2X compounds,where X ?Si,Ge,Sn,and Pb,considering cubic and orthor-hombic phases.Their calculations were carried out using ultrasoft pseudopotentials and the full potential linearized augmented plane wave (FPLAPW)method in both LDA and GGA frameworks.Their ultrasoft pseudopotential approach is essentially the same we are employing here for structural and electronic calculations,without taking into account quasiparticle corrections.More recently,

Lebe

gue et al.[32]recently computed the quasiparticle

160120

80400

86420

864

2086420

ENERGY (eV)

P O D S (E L E C T R O N S /e V )

Fig.3.Partial density of states for electrons in orthorhombic CaGeO 3including contributions from all atoms and contributions from each atom according to the angular momentum.Solid lines:s contribution;dotted lines:p contribution.

Table 3

Carriers effective masses of CaGeO 3orthorhombic along some symmetry directions Valence band LDA GGA Conduction band LDA GGA m lh eS !G T 1.00 1.23m e eG !S T0.410.46m hh eS !G T 4.19 4.26m e eG !Z T0.350.39m h eS !Y T 1.89 2.39m e eG !R T0.420.47m h eS !X T

1.25

1.43

m e eG !U T0.380.41m e eG !T T0.440.50m e eG !Y T

0.42

0.49

J.M.Henriques et al./Journal of Solid State Chemistry 180(2007)974–980

978

properties of Ca2Si using the all electron GW approxima-tion based on the projector-augmented-wave method (PAQ),exploring both the orthorhombic and the cubic https://www.sodocs.net/doc/0917465999.html,paring the resulting band structures with those obtained employing the LDA,there is,as expected,a very pronounced difference of energies between the conduction and valence states using the GW approximation,as compared with the DFT results.In particular,the minimum band gap for interband transitions at the G point is1.02eV within the GW approximation,but only 0.30eV using only DFT.

The dielectric function is affected by the quasiparticle self-energy correction and the local-?eld effects,due to the explicit dependence of the dielectric function er;r0;oTon r and r0and not only on j ràr0j[32].For the static dielectric function quasiparticle correction is unnecessary,because it is a ground state feature.On the other hand,the dielectric constant calculated by taking into account quasiparticle corrections is useful only if excitonic effects are considered. This occurs because excitonic effects tend to shift oscillator strength toward lower energies,cancelling in part or completely the quasiparticle correction to the static di-electric function.Electron–hole interaction effects on the dielectric constant,however,are beyond the reach of computational methods due to convergence problems[33]. Beyond that,dielectric properties calculated for some calcium compounds are impaired when the pseudopoten-tial for Ca does not take into account additional semicore electrons[34].In view of these considerations,the results presented here for orthorhombic CaGeO3must be con-sidered with some caution.Due to the limited computa-tional power available to our research team,we have not considered the quasiparticle corrections and Ca semicore electrons,so the dielectric function peaks we present here must be energy shifted by an unknown amount in order to be compared with experimental results,and their intensities must be regarded only as broad indications.Keeping these facts in mind we note,however,that the peaks for the dielectric functions calculated by Lebe gue et al.[32]are in general resized,but not very energy shifted,when local-?eld effects are included in the dielectric function calcula-tions.Indeed,their calculated dielectric functions consider only the self-energy correction at the G point,adopting this value to rigidly shift the energies of all unoccupied DFT bands in order to produce an average GW correction to the band structure,arguing at the same time that the choice of the correct quasiparticle energies,instead of the DFT shifted energies,will lead only to minor corrections in the dielectric function,hardly noticeable when one looks at the calculated optical spectra.So we presume that the results we show here,despite the absence of many particle re?nements and local-?eld effects,are useful for further experimental investigations of CaGeO3optical properties. We would like also to point out that a work similar to ours was carried out for perovskite CaTiO3,and the calculated dielectric functions were found to be in good agreement with the experimental data[35].

Fig.4depicts the calculated real and imaginary parts of the dielectric function and optical absorption for CaGeO3 with incident light polarized along different crystal direc-tions(100and111)and polarized light incident on a polycrystalline sample.In both LDA and GGA ap-proaches,the complex dielectric function of orthorhombic CaGeO3is very similar for light polarized along the crystalline directions100,010,and001,but for the111 direction the optical response was more intense as a consequence of the alignment of Ca–O dipoles with the111 polarized electric?eld.The GGA calculation of the real 1eoTand imaginary 2eoTparts of the complex dielectric function are presented at the top and bottom of Fig.4, respectively.The calculated static dielectric constant for polycrystalline CaGeO3is 0?5:1.The imaginary part 2eoT,depicted at the bottom part of Fig.4,is closely related to the optical absorption,which is shown as an inset at the bottom of Fig.4.We see some small peaks closer to the main Kohn–Sham band gap occurring at2.3,2.7–2.9, 3.2–3.4,and3.7–3.9eV,respectively.For energies between 2.3and7eV,approximately,optical absorption increases slowly.Starting from7eV to larger energies,however,we note that the optical absorption increases more quickly.

15

10

5

-5

ε

1

15

10

5

ε

2

010203040

Energy (eV)

010203040

Energy (eV)

100

111

POLYCRYSTALLINE

A

B

S

O

R

P

T

I

O

N

(

a

.

u

.

)

246810

ENERGY (eV)

Fig.4.Real(top)and imaginary(bottom)components of the orthor-hombic CaGeO3dielectric function for polarized incident light and light incident on a polycrystalline sample.The inset depicts optical absorption aeoT.

J.M.Henriques et al./Journal of Solid State Chemistry180(2007)974–980979

We suppose that this change of regime is due to the appearance of transitions involving mainly O-2p valence states and mainly Ca-4s conduction states for energies larger than7eV.The slow increase observed for energies smaller than7eV involves transitions between O-2p valence states and mainly Ge-4s conduction states(see Fig.3).

4.Conclusions

In this work,we have obtained the structural,electronic and optical properties of orthorhombic CaGeO3using quantum chemical?rst-principles calculations.The struc-tural parameters of orthorhombic CaGeO3,calculated after geometry optimization,show a good agreement with the experimental data.The GGA approximation over-estimates the actual orthorhombic CaGeO3lattice para-meters by2%at most,while the LDA lattice parameters are underestimated by practically the same amount.The smallest energy gap between valence and band conductions is indirect between the S and G points,corresponding for the LDA and GGA calculation to 2.31and 1.68eV, respectively.The G!G direct transition is2.41eV(LDA) and1.75eV(GGA).In comparison,according to the LDA calculations[8],CaSiO3presents a wide indirect band gap of5.43eV between the Q and G points,and a direct band gap of 5.52eV for G!G transitions,while CaCO3 orthorhombic(aragonite)exhibit very close indirect X!

G and direct band gaps[11].LDA effective masses are always smaller than GGA masses for electrons and holes, with hole effective masses larger and very anisotropic. Electron masses are smaller and almost isotropic.In both LDA and GGA approaches,the dielectric function of orthorhombic CaGeO3is very similar for incident light polarized along the crystalline directions100,010,and001, but for the111direction the optical response was more intense as a consequence of the partial alignment of Ca–O electric dipoles with the111polarized electric?eld.In the energy range between2.3and7eV,approximately,optical absorption increases slowly,but from7eV to larger energies,optical absorption increases more quickly due to the appearance of transitions involving mainly O-2p valence states and mainly Ca-2s conduction states(for energies larger than7eV).

Acknowledgments

VNF,JAPC,and ELA are senior researchers from the Brazilian National Research Council CNPq,and would like to acknowledge the?nancial support received during the development of this work from the Grants CNPq-CTENERG504801/2004-0and CNPq-Rede NanoBioes-truturas555183/2005-0.J.M.H.was sponsored by a graduate fellowship from the Brazilian National Research Council(CNPq)at the Physics Department of the Universidade Federal do Rio Grande do Norte.We also thank the referees for the valuable suggestions and corrections proposed to improve this paper.

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