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Physica B 405 (2010) 3427–3432

Contents lists available at ScienceDirect

Physica B

0921-45

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/physb

First-principles study of the electronic and the optical properties of In6Se7

compound

T. Ben Nasr n, H. Ben Abdallah, R. Bennaceur

Laboratory of Physics of Condensed Matter, Department of Physics, Faculty of Sciences of Tunis, University Campus, 2092 El Manar, Tunisia

a r t i c l e i n f o

Article history:

Received 11 November 2009

Received in revised form

10 April 2010

Accepted 8 May 2010

Keywords:

DFT

GGA

FP-LAPW

Semiconductors

Electronic structure

Dielectric function

Optical properties

26/$ - see front matter & 2010 Elsevier B.V. A

016/j.physb.2010.05.017

esponding author. Tel.: +21625212720; fax:

ail address: tarek.ben-nasr@laposte.net (T. Be

a b s t r a c t

The electronic and the optical properties of hexaindium heptaselenide In6Se7 have been reported using

the full potential linearized augmented plane wave (FP-LAPW) method as implemented in the Wien2k.

In this approach, the alternative form of the generalized gradient approximation (GGA) proposed by

Engel and Vosko (EV-GGA) was used for the exchange correlation potential. The calculated band

structure shows a direct band gap (Z–Z). The contribution of different bands was analyzed from the

total and the partial density of states curves. Moreover, the optical properties including the dielectric

function, absorption spectrum, refractive index, extinction coefficient, reflectivity and energy-loss

spectrum are all obtained and analyzed in details within the energy range up to 20 eV.

& 2010 Elsevier B.V. All rights reserved.

1. Introduction

The study on In–Se based materials is motivated by thepotential application of their physical properties (e.g. optical,electrical and phase change ability) in photovoltaic cells [1–3],solid-state batteries [4,5], phase change memories [6,7] and in themanufacture of detectors of ionizing radiation [8]. Research on theIn–Se phase diagram has been reported by many authors [9–11]but still remains incompletely studied [12,13]. The most stablephases existing in the In–Se system are InSe, In2Se3, In4Se3 andln6Se7 as reported by [14,15]. These compounds are studied by theX-ray absorption spectroscopy (EXAFS) [14] for crystal chemicalcharacterization of mixed valent indium chalcogenides. In6Se7

belongs to the III–VI compound semiconductors group and washardly investigated due to the difficulty of growing large singlecrystals. In the literature, some authors have studied experimen-tally this compound. Only very few works report on the crystalstructure and some physical properties [16–18], particularly,thermoelectric power and conduction mechanism [19] and theoptical constants of the evaporated In6Se7 [20]. To the best of ourknowledge, there are no earlier theoretical studies on the In6Se7

properties, despite the importance of these studies to give usinformation on this compound. The aim of the present work is tothrow more light on the electronic and the optical properties of

ll rights reserved.

+21671885073.

n Nasr).

In6Se7 compound using the first-principles calculations based onthe density functional theory (DFT) [21] and using the fullpotential linearized augmented plane waves (FP-LAPW) method[22,23] as implemented in Wien2k code [24]. To describe theexchange and correlation potential, we use the Engel Voskogeneralized gradient approximation (EV-GGA) [25].

2. Computational details

In the present calculation, we employ the FP-LAPW method asembodied in the Wien2K code in a scalar relativistic versionwithout spin orbit coupling. In6Se7 crystallizes in the monoclinicstructure, and belongs to the space group P21/m (C2

2h). There aretwo formula units per cell and all In and Se atoms are on thespecial position 2e (x, ¼, z) [17]. The atomic positions and thelattice parameters are given by Walther et al. [17]. In Fig. 1, werepresent a crystal structure of In6Se7. In this work, we haveconsidered the electronic configurations of In6Se7 as In:[Ni]4s24p64d104d105s25p1 where [Ni] represents the core states,(4s24p6) the semi-core states and (4d105s25p1) the valence statesof In atoms and Se: [Ar]3d104p44s2 where [Ar] represents the corestates and (3d104p44s2) the valence states of Se atoms. Themuffin-tin (MT) spheres radii were chosen to be 2.5 and 2 a.u.for In and Se, respectively. The potential and charge densityrepresentations inside the MT spheres are expanded withlmax¼10. For expansion of the basis function, we set

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Fig. 1. The crystal structure of In6Se7.

3500-209351,0

-209350,5

-209350,0

-209349,5

-209349,0

Tota

l ene

rgy

(Ry)

Volume (a.u.)

In6Se7

75007000650060005500500045004000

Fig. 2. Total energy versus unit cell volume of In6Se7 using GGA calculation.

Table 1Calculated equilibrium lattice constants, bulk modulus (B0) and pressure

derivative of bulk modulus (B0

0) of In6Se7 compound.

a (A) b (A) c (A) B0(GPa) B0

0

GGA 10.887 4.690 20.386 20.751 3.898

exp. [17] 9.433 4.064 17.663 � �

T. Ben Nasr et al. / Physica B 405 (2010) 3427–34323428

RMTKmax¼7, which controlled the size of the basis set, where Kmax

is the plane wave cut-off and RMT is the smallest of all atomicsphere radii. We have obtained 5165 LAPWs and 192 local orbitalschosen for In-s, In-p and In-d states and Se-s states. We found 300k-points in the irreducible wedge of the Brillouin zone (BZ), whichcorresponds to 1500 k-points in the whole (BZ) for self-consistentcalculations. The self-consistent calculations are considered to beconverged when the total energy of the system is stable within10-4 Ry. A total of 14 iterations were necessary to achieveself-consistency.

3. Results and discussion

3.1. Structural optimization

Fig. 2 presents the total energy variation as a function of theprimitive unit cell volume for In6Se7 obtained from the GGAcalculation. The curve of Fig. 2 was fitted to the Murnaghanequation of state [26] so as to determine the equilibriumstructural parameters. The calculated equilibrium latticeconstants, bulk modulus B0 and pressure derivative B00 are

collected in Table 1, together with available experimentalinformation. It is clearly seen that the GGA overestimates thelattice parameter; this finding is consistent with the general trendof this approximation.

3.2. Electronic properties

3.2.1. Density of states and band structure

The total and partial densities of states of In6Se7 arerespectively illustrated in Figs. 3 and 4. Low-lying bandsresulting mainly from Se-s orbitals were localized in the energyrange of about �11.4 to �10.8 eV below EF. Then, there is anempty region between �10.8 and �4.6 eV below the Fermi level,where the bottom of the valence band begins. The highestoccupied crystal orbitals (HOCO) are constructed by Se-4p as adominant component and In-5p. The lowest unoccupied crystalorbitals (LUCO) are mostly the antibonding orbitals of In-5p andSe-4p. The HOCO comes from sp–p interaction between Se-4p andIn-5p in the energy range of �1.97 eV to the Fermi level (0 eV),and the corresponding (sp�p)n antibonding orbitals locate above2.59 eV. The bands dispersing from �5 to �3.01 eV areconstructed by selenium 4p and indium 5s. The LUCO in therange of 0.38–1.48 eV is constituted by the antibonding orbitalsSe-4p and In-5s. Since the maximum of highest occupied valencebands (HOVB) and the minimum of lowest unoccupiedconduction bands (LUCB) both occur at the Z-point, In6Se7 is adirect semiconductor.

As described above, In6Se7 is a direct band gap semiconductor(Fig. 5) and the band gap value is somewhat determined by theenergy of the conduction band. The In-s band is important indetermining the band gap of In6Se7. In fact, the band gap dependson the energy of the Se-4p–In-s antibonding level.

The comparison between the calculated and the experimentalband gaps are shown in Table 2. The forbidden energy gapwas found to be 0.38 eV. However, the band gap of this materialis underestimated by the DFT, when compared with theexperimental data. It is well known that the self-consistent DFT

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Fig. 3. Calculated total density of states (DOS) of In6Se7.

Fig. 4. The angular momentum decomposition of the atom-projected densities of

states in In6Se7.

Fig. 5. The band structure of In6Se7 calculated by the FP-LAPW method.

Table 2Calculated and experimental energy gaps of In6Se7.

LAPW Exp.

Eg (eV) 0.38a 0.54b

0.71c

1.87d

a Reference. Our result.b Ref. [35]c Ref. [36]d Ref. [37]

Fig. 6. Charge density contour of the total valence charge of In6Se7.

T. Ben Nasr et al. / Physica B 405 (2010) 3427–3432 3429

usually underestimate the energy gap. On the other hand, it isthought that the technology used to grow this material mayinfluence its physical properties; this may explain why we have adiscrepancy between the experimentally band gap values.

3.2.2. Charge density

To visualize the nature of the band character, we calculate thetotal charge density depicted in Fig. 6. The total charge valencedensity for In6Se7 is plotted in a plane containing the In–Se bonds.

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Table 3Calculated ionicity factor fi of In6Se7.

EAS (eV) EVB (eV) fi

LAPW 5.92 11.48 0.51

Table 4The calculated static dielectric constant e1(0), n(0) and the positions of Plasmon

peaks appearing in L(o) for the three polarization directions.

E:x E:y E:z

e1 (0) 16.07 13.01 14.35

n (0) 4.00 3.60 3.78

Plasmon peaks in L(w) (eV) 14.68 15.95 15.39

Fig. 7. The real part e1(o) and the imaginary parts e2(o) of the dielectric function

of In6Se7.

T. Ben Nasr et al. / Physica B 405 (2010) 3427–34323430

The electron distribution indicates that the In–Se bond is acovalent bond. The case of ionicity is interesting since it can berelated to properties of the charge density on the whole andfeatures in the band structures. The iconicity, which is directlyassociated with the character of the chemical bond, provides us ameans for explaining and classifying the properties of this III–VIcompound. It is well known that the ionicity character is highlydependent on the total valence charge density by calculating thecharge distribution. To obtain an estimated value of the ionicityfactor for In6Se7 compound, we have used an empirical formula[27]. In this approach the ionicity parameter is defined as

fi ¼1

21�cos p EAS

EVB

� �� �ð1Þ

where EAS is the antisymetrie gap between the two lowest valencebands, and the EVB is the total valence band width. The calculatedionicity factor from the adopted method is given in Table 3. Theobtained results show that the In6Se7 is more ionic than GaAs andInAs materials. In fact, the ionicity factors of these III–Vcompounds are estimated to be respectively equal to 0.31 and0.35 [28].

3.3. Optical properties

The optical properties of matter can be described by thecomplex dielectric function e(o), which represents the linearresponse of the system to an external electromagnetic field with asmall wave vector. It can be expressed as

eðoÞ ¼ e1ðoÞþ ie2ðoÞ ð2Þ

Calculations ignore excitonic effects but include the local fieldeffect. There are two contributions to e(o), namely, intraband andinterband transitions. The contribution from intraband transitionsis important only for metals. The interband transitions can furtherbe split into direct and indirect transitions. We neglect theindirect interband transitions, which involve scattering of pho-nons and are expected to give only a small contribution to e(o).The momentum matrix elements in the dielectric function havebeen calculated on a grid of 3000 special k-points for In6Se7. Tocalculate the direct interband contribution to the imaginary partof the dielectric function e2(o), one must sum all possibletransitions from occupied to unoccupied states. The imaginarypart e2(o) can be calculated by the following formula

e2ðoÞ ¼4pe2

m2o

Zd3k

Xn,n0

/kn9P9kn0S�� ��2fknð1�fkn0 ÞdðEkn�Ekn0�_oÞ

ð3Þ

where e is the electron charge, m is the mass, O the volume, e0kn

the Fermi distribution, :o the energy of the incident phonon, P

the momentum operator and 9kn04 the crystal wave function.The evaluation of matrix element in Eq. (2) is done over the MTand interstitial regions separately. Further details about theevaluation of matrix elements are given elsewhere [29]. Thesummation over the BZ in Eq. (2) is performed using a linearinterpolation on a mesh of uniformly distributed points, i.e. thetetrahedron method [30]. e2(o) in Eq. (2) is calculated in theirreducible part of the Brillouin zone (IBZ). The dielectric functionis a tensor, and the calculation of all components of this tensor is acomplicated one. So we must introduce the correct symmetry of

the crystal to reduce the number of components. For monoclinicstructures, we need to calculate five components of the totaldielectric function [29]. For In6Se7 structure, [a¼g¼901,b¼100.921], the dielectric tensor is then

Imexx 0 Imexz

0 Imeyy 0

Imexz 0 Imezz

0B@

1CA

We can restrict our considerations to the diagonal matrixelements if we consider that the amplitude of exz is negligible incomparison to the amplitude of the diagonal. This is simply due tothe fact that the angle between the z and x axes, b, is 100.921.Thus, we obtain dielectric functions resolved into three cartesiancomponents. Furthermore, this decomposition stays valid for all

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T. Ben Nasr et al. / Physica B 405 (2010) 3427–3432 3431

optical constants given later. The real part e1(o) of the dielectricfunction e(o) is derived from e1(o) by the Kramers–Kronigtransformation.

e1ðoÞ ¼ 1þ2

pM

Z1

0

o0e2ðo0Þo02�o2

do0 ð4Þ

where M implies the principle value of the integral.The optical constants such as refractive index n(o) and the

extinction coefficient k(o), are calculated in terms of the real andthe imaginary parts of the complex dielectric function as

Fig. 8. Calculated optical constants of In6Se7. (a) Absorption spectrum (104/cm), (b) refr

follows [31]

n¼e1þ e2

1þe22

� �1=2� �1=2

ffiffiffi2p ð5Þ

k¼�e1þ e2

1þe22

� �1=2� �1=2

ffiffiffi2p ð6Þ

Furthermore, the other optical parameters, such as reflectivityR(o), absorption coefficient I(o) and energy-loss function L(o),are derived immediately from e1(o) and e2(o) [32]. For the optical

active index, (c) extinction coefficient, (d) energy-loss spectrum and (e) reflectivity.

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T. Ben Nasr et al. / Physica B 405 (2010) 3427–34323432

properties calculations of In6Se7, we have corrected our band gapvalue with 0.16 eV by using the correction scissors and we choosea fine k-mesh (more k-points than chosen for SCF cycle). In thiswork, we have found good results for 690 k-points in the IBZ,which originated from 3000 k-points in BZ.

The calculated real and imaginary parts of the dielectric functione(o) for In6Se7 are shown for the three polarization directions inFig. 7 as a function of the photon energy :o. The imaginary parte2(o) of the dielectric function shown in Fig. 7(b) indicates thatIn6Se7 is anisotropic and its maximum values are around 1.54, 1.68and 1.47 eV for E:x, E:y and E:z, respectively. These peaks thatappeared in Fig. 7(b) belong to an energy transition between someorbital corresponding to certain energy because e2(o) is related tothe DOS. In comparison with Fig. 4, it can be recognized that thepeaks are mainly due to transitions from Se-4p valence bands toIn-5s conduction bands. Above the main peaks, the next structuresare peaks from 3.23 to 7.72 eV, which arise mainly from directtransitions between Se-4p and In-5p orbitals. In the curve e2(o), thelowest peaks appear at 8.43, 8.54 and 8.64 eV respectively for E:x,E:y and E:z, and are related to direct transitions between Se-4sand In-5s orbitals. At still higher energy, the spectrum is withoutstructures and decays very rapidly with photon energy.Unfortunately, there are no experimental data concerning theorigin and the positions of these peaks to make a comparison.The results for the dispersive part of the dielectric function, e1(o) forthe compound under investigation, are given in Fig. 7(a). At highfrequencies, the zero crossing of e1(o), which corresponds to thelocation of the screened plasma frequency is located at 14.48 eV. Thestatic dielectric constant e1(0) is given by the low energy limit ofe1(o). It is necessary to emphasize that we do not include phononcontributions to the dielectric screening, and e1(0) corresponds tothe static optical dielectric constant eN. The obtained opticaldielectric constants eN following the EV-GGA formalism along thecrystal axes resolved into three components are listed in Table 4. Wehave estimated the average value of zero frequency dielectricconstant using the relation

eð0Þ ¼ 13ðexxð0Þþeyyð0Þþezzð0ÞÞ ð7Þ

From this relation, we have obtained e(0) equal to 14.47.Unfortunately, there is no experimental polarized zero frequencydielectric constant available to compare our results with it. Theseresults clearly indicate the anisotropy in the optical properties ofIn6Se7. In order to estimate the degree of anisotropy, we determinethe ratio Eyyð0Þ=Ezzð0Þ as a measure for the average opticalanisotropy. This ratio is equal to 0.906 in our case, if one considersthe value reported in [33] of the ratio equal to 1.115 in case of SnSe,we can note that the In6Se7 is less anisotropic than SnSe.

Fig. 8(a)–(e) shows the calculated results of the energydependence of absorption coefficient, refractive index, extin-ction coefficient, electron energy-loss spectrum and reflectivity,respectively. In our calculation, we used Gaussian smearing of0.1 eV. The absorption spectrum a(o) shows a very intenseabsorption, which occurs between 0.70 and 17.14 eV due toexcitations of phonons. At low frequency (o¼0), we obtain thefollowing relation between the refractive index n(0) and e1(0):nð0Þ ¼

ffiffiffiffiffiffiffiffiffiffiffie1ð0Þ

p. Our calculated n(0) resolved into the three

polarization directions is given in Table 4. The average value ofn(0) is equal to 3.79. From Fig. 8(e), we notice a strongreflectivity maximum between 1 and 15 eV originated from theinterband transitions. The energy-loss spectrum is related to theenergy loss of a fast electron traversing in the material and isusually large at the plasma energy [34]. The most prominentpeak in the energy-loss spectrum is identified as the Plasmonpeak, and located at 14.68 eV for E:x polarization. This corres-ponds to a rapid decrease of reflectance in Fig. 8(e). In fact, e1¼0(root in e1) can give rise to a plasma resonance. If we consider the

e1(o) spectra for E:x polarization, we find that e1(o) equals tozero at 14.48 eV, which corresponds effectively to the Plasmonpeak in L(o) for the same polarization.

4. Conclusion

This study reports a detailed investigation on the electronic andthe optical properties of In6Se7 compound using the first-principlesFP-LAPW method within EV-GGA. The calculation provides anexcellent description of the band structure where we found a directband gap of about 0.38 eV located at the Z-point in the monoclinicBZ, which is less than the experimental value. The charge densityhas been presented and shows the covalent character of In–Sebonds. Our calculations predicted the ionicity factor value of 0.51 forIn6Se7. We have also presented the dielectric tensor components ofthe binary chalcopyrites In6Se7. Our results for the real and theimaginary parts of the dielectric functions are used to reproduceoptical constants, such as the refractive index. The relations of theoptical properties to the interband transitions are also discussed indetail. The calculation details presented could be useful for furtherexperimental investigations and we hope that can be used to coverthe lack of data for this compound.

References

[1] K. Ando, A. Katsui, Thin Solids Films 76 (1981) 141.[2] M. Digiulio, G. Micocci, A. Rizzo, A. Tepore, J. Appl. Phys. 54 (1983) 5839.[3] J.F. Sanchez-Royo, A. Segura, O. Lang, A. Chevy, L. Roa, Thin Solids Films 307

(1997) 283.[4] C. Julien, E. Hatzikraniotis, K. Kambas, A. Chevy, Mater. Res. Bull. 20 (1985) 287.[5] C.J. Smith, C.W. Lowe, J. Appl. Phys. 66 (1989) 5102.[6] H. Lee, D.H. Kang, Mater. Sci. Eng. B: Solid State Mater. Adv. Technol 119

(2005) 196.[7] A. Hirohata, J.S. Modera, G.P. Berera, Thin Solids Films 510 (2006) 247.[8] G. Minocci, A. Tepore, Sol. Energy Mater. Sol. Cells 22 (1991) 215;

G. Minocci, A. Tepore, Sol. Energy Mater. Sol. Cells 26 (1992) 159;G. Minocci, A. Tepore, Phys. Status Solidi A 148 (1995) 431.

[9] G. Slamova, A. liseev, Russ. J. Inorg. Chem. 10 (1963) 1161.[10] T. Guliev, Z. Medvedeva, Russ. J. Inorg. Chem. 7 (1965) 829.[11] K. Imai, K. Suzuki, T. Haga, Y. Hasegwa, Y. Abe, J. Cryst. Growth 54 (1981) 501.[12] T. Massalski, in: CD-Rom, Binary Phase Diagrams, 2nd ed., ASM International,

1996.[13] J.C. Tednac, G.P. Vassiblev, B. Daovchi, J. Rachidi, G. Brun, Crys. Rech. Technol

4 (1977) 611.[14] M. Epple, M. Panthofer, R. Walther, H.J. Deiseroth, Z. Kristallogr. 215 (2000)

445.[15] Y. Igasaki, H. Yamauchi, S. Okamura, J. Cryst. Growth 112 (1991) 797.[16] J.H.C. Hogg, Acta Cryst. B27 (1971) 1630.[17] R. Walther, H.J. Deiseroth, Z. Kristallogr. 210 (1995) 359.[18] S.M. Kitsa, N.P. Gavaleshko, Isv. Akad. Nauk SSSR, Neorg. Mater. 14 (1978)

955.[19] A.F. El Deeb, H.S. Metwally, H.A. Shehata, J. Phys. D: Appl. Phys. 41 (2008)

125305.[20] J. Sandino, G. Gordillo, Surf. Rev. Lett. 9 (2002) 1687.[21] P. Hoenberg, W. Khon, Phys. Rev. B 136 (1964) 864.[22] O.K. Andersen, Phys. Rev. B 12 (1975) 3060.[23] D.D. Koelling, G.O. Arbman, J. Phys. F 5 (1975) 2041.[24] P. Balha, K. Schwartz, G. Madsen, Comput. Phys. Commun. 147 (2006) 71.[25] E. Engel, S.H. Vosko, Phys. Rev. B 47 (1993) 13164.[26] F.D. Murnaghan, Proc. Natl. Acad. Sci. USA 30 (1944) 244.[27] K. Unger, H. Newman, Phys. Status Solidi (b) 64 (1974) 117.[28] Sadao Adachi, Properties of Group-IV, III–V and II–VI semiconductors, Wiley

Series in Materials for Electronic and Optoelectronic Applications, (2005).[29] C. Ambrosh-Draxl, J.O. Sofo, Comput. Phys. Commun. 175 (2006) 1.[30] P.E. Blochl, O. Jepsen, O.K. Andersen, Phys. Rev. B 49 (1994) 16223.[31] A. Delin, A.O. Eriksson, R. Ahuja, B. Johansson, M.S. Brooks, T. Gasche, Phys.

Rev. B 54 (1996) 1673.[32] S. Saha, T.P. Sinha, Phys. Rev. B 62 (2000) 8828.[33] Z. Nabi, A. Kellou, S. Mecabih, A. Khalfi, N. Benosman, Mater. Sci. Eng. B 98

(2003) 104.[34] P. Nozieres, Phys. Rev. Lett. 8 (1959) 1.[35] A.A. Al-Ghamdi, JKAU: Sci. 19 (2007) 157.[36] N.P. Gavaleshko, M.S. Kitsa, A.I. Savchuk, R.N. Simchuk, Fiz. Tekh.

Poluprovodn. 14 (1980) 1390;N.P. Gavaleshko, M.S. Kitsa, A.I. Savchuk, R.N. Simchuk, Sov. Phys. Semicond.14 (1980) 822 English Transl.

[37] J. Sandino, G. Gordillo, Surf. Rev. Lett. 9 (2002) 1687.