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Excitonic effects in the optical properties of alkaline earth chalcogenides from first-principles

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2015 Phys. Scr. 90 085802

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Excitonic effects in the optical propertiesof alkaline earth chalcogenides fromfirst-principles calculations

Hajar Nejatipour and Mehrdad Dadsetani1

Department of Physics, Lorestan University, Khoramabad, Iran

E-mail: nejatipour.h@lu.ac.ir, dadsetani.m@lu.ac.ir and dadsetani_m@yahoo.com

Received 13 December 2014, revised 16 April 2015Accepted for publication 13 May 2015Published 16 June 2015

AbstractThis paper studies excitonic effects in the optical properties of alkaline earth chalcogenides(AECs) by solving the equation of motion of the two-particle Green function, the Bethe–Salpeterequation (BSE). On the basis of quasi-particle states obtained by the GW approximation,(BSE+GW), the solution of BSE improves agreement with experiments. In these compounds,the main excitonic structures were reproduced appropriately. In the optical absorption spectra ofAECs, the main excitonic structures originate in the direct transitions at X and Γ symmetrypoints, as confirmed by the experiments. In addition to real and imaginary parts of the dielectricfunctions, excitonic effects were studied in the electron energy loss functions of AECs.Moreover, the G0W0 approximation was used in order to determine the energy band gaps ofAECs. This showed that except for MgO and BaO, the other AECs under study have indirectband gaps from Γ to X.

Keywords: alkaline earth chalcogenides (AEC), optical properties, Bethe–Salpeter equation,excitonic effects, G0W0 approximation

(Some figures may appear in colour only in the online journal)

1. Introduction

Consisting of elements in the IIa–VIb groups of the periodictable, alkaline earth chalcogenide (AEC) compounds arehighly ionic materials with mostly sodium chloride crystalstructure. The pure and doped AECs form widely usedsemiconductors in the electronic, opto-electronic and mag-neto-optical applications [1–8]. They are promising materialsin direct and potential industrial applications such as lumi-nescence and infrared sensitive devices [2, 3]. There are manystudies on the structural, electronic and optical properties ofthese structures [9–14]. For these compounds, some proper-ties have also been reported such as wide electronic band gaps[10–12] and the structural phase transition to CsCl structureunder pressure [15]. They are closed-shell compounds withmostly indirect electronic band gaps. Due to the difficultgrowth of their single crystals, the precise electronic and in

particular optical properties of AECs had been matters ofdebate. In some theoretical studies, their energy band gapshave been reported to be indirect from Γ to X point [16, 17].However, some other works have predicated the direct gaps[18]. Their electronic structure traits, such as somewhat wideband gaps, have caused them to have strong excitonic struc-tures in their optical absorption responses. As one of the firstexperimental works on the optical properties of AECs,Kaneko et al [18, 19] conducted comprehensive studies. Themain structures in the reflectivity spectra were associated withthe strong excitonic effects which were due to the directelectronic transitions.

In the random phase approximation (RPA) level of cal-culations [20], many researchers have theoretically studiedoptical properties of AECs [21–24]. In our previous studies, wecalculated electronic band structures, energy band gaps andoptical properties of AECs in different generalized gradientapproximations (GGA) [25–30]. Compared to the Perdew–Burke–Ernzerhof-GGA [31], for barium chalcogenides (BaS,

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Phys. Scr. 90 (2015) 085802 (16pp) doi:10.1088/0031-8949/90/8/085802

1 Author to whom any correspondence should be addressed.

0031-8949/15/085802+16$33.00 © 2015 The Royal Swedish Academy of Sciences Printed in the UK1

BaSe and BaTe), the use of the Engel–Vosko–GGA func-tional [32] led to the improvement of band gap values [26]. Inaddition, we calculated and discussed the optical properties ofBa chalcogenides by means of the density functional theory(DFT) [33] and RPA, on the basis of the full potential linearaugmented plane wave (FPLAPW) [26]. We attributed themain peaks in ε2(ω), the imaginary part of the dielectricfunction, to the transition from chalcogen p as the initial stateto the mainly alkaline earth d and also s as the final states. Inanother paper based on the same theory used for Ba chalco-genides, we studied the optical absorption spectra of calcium(Ca) chalcogenides [29]. The main structures originating fromthe electronic band to band transitions were reproducedproperly in the imaginary parts of the dielectric functions.Moreover, we calculated and interpreted the low loss regionin the energy loss functions. In addition, we studied theelectronic and the full range of the optical properties ofstrontium chalcogenides with a similar theoreticalapproach [30].

Excitations are not in a well agreement with the experi-mental spectra and this is a well-known character of Kohn–Sham DFT (KS-DFT). In particular, the excitonic structureswhich originated from the coupling of an electron and a holeleft by the electron transition do not appear in the RPA levelof calculations. There are a few theoretical studies on theexcitonic effects in the optical properties of AECs. Con-sidering the electron–hole (e–h) interactions, Poncé et al [34]have studied the optical responses of CaO and CaS structures.In the computational absorption spectra, relatively qualitativeagreements have been acquired as compared to the experi-ments. However, some quantitative discrepancies can be seenin the results of Poncé et al. By means of the solution ofBethe–Salpeter equation (BSE), Riefer et al [35] have studiedthe imaginary part of the dielectric function of CaO in a 20 eVenergy range. They have acquired a good agreement with theexperiment [36], including the reproduction of excitoniceffects originating from direct transitions and the overalldispersion of the optical absorption spectrum. Shwetha andKanchana [37] have studied the optical properties of theoxides of Mg, Ca, Sr and Ba without solving the BSE and inthe framework of time dependent DFT by means of thebootstrap kernel for the exchange-correlation approximation[38]. In their report, there is no detailed discussion about theexcitonic structures, except that there is a shift to lowerenergy region from Mg to Ba, indicating a decrease trend inthe band gaps. However, the overall shapes of their compu-tational spectra are not match with the experiments. In addi-tion, in ab initio many-body calculations, Schleife et al [39]have dealt with the quasi-particle (QP) band structure andoptical properties of MgO in an energy range of 0–30 eV. Aconvincing agreement with the experiment [40] has beenobtained, including a pronounced peak close to the absorptionedge and some peaks beyond the absorption edge originatingfrom transitions between valence and conduction bands.Energy loss function of MgO also has been discussed and itsfeatures have been attributed to the electronic excitations andplasmonic losses of the sp valence electrons.

To our knowledge, no research has focused on the opticalproperties of AECs, apart from the studies mentioned above,in an advanced method like the many body perturbationtheory (MBPT) [41]. It can be added that the energy lossspectra of AECs, including the excitonic effects, have notbeen reported elsewhere. A significant difference between thepresent manuscript and the two many-body works pointedabove [35, 39] is in the energy range where optical spectrawere computed. In order to detect the excitonic effects in theoptical properties of AECs in the vicinity of the optical gaps,and for a better comparison with the experimental opticalabsorption spectra reported in [18], we concentrate on theexcitonic features in an energy range near the optical gaps.Therefore, the aim of the present paper is to provide a com-prehensive study on the excitonic effects in the opticalproperties of AECs in vicinity of the optical gaps. By solvingthe BSE and by means of the QP corrections, the real andimaginary parts of the dielectric functions and the energy lossspectra of MY structures (M=Mg, Ca, Sr, Ba and Y=O, S, Se,Te) are calculated and discussed on the basis of the FPLAPWmethod. Accordingly, the results presented here can behelpful for the use of the all-electron full-potential many bodymethods. Using the many body G0W0 [42], moreover, wecalculated the electronic band gaps of AECs. In order todetermine, whether the band gaps are direct or indirect, weused the both one-particle KS and many body G0W0

approximations.The format of present paper is as follows: first, compu-

tational methods are described so that they can include thee–h coupling in the optical response. Then, electronic bandstructures of the rocksalt phases of AECs were discussed inthe KS and GW procedures. Finally, there will be a com-prehensive study for the optical spectra of MY structures. Ourfocus is on the excitonic effects in their optical properties bysolving the BSE [43]. With adding the GW corrections, theBSE+GW level of calculations will be applied, and thechanges will be examined.

2. Computational details

2.1. Calculation parameters

We used the exciting-code [44] for electronic band structurecalculations and optical properties in the RPA and in solvingthe BSE [43]. The electronic and optical properties calculatedrelativistically are based on the FPLAPW method in which noshape approximation is made on potential or the electroniccharge density. Basically, this scheme divides space into non-overlapping atomic spheres centered on the atoms and aninterstitial region. Localized electrons are described by atomicwave functions whereas the wave functions of the valenceelectrons are expanded into atomic wave functions within theatomic sphere and into plane waves in the interstitial region.The semi-core states are treated with the inclusion of localorbitals in the basis. The exchange-correlation potentialwithin the GGA has been calculated by means of the schemedeveloped by Perdew, Burke and Ernzerhof [31] for the

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Phys. Scr. 90 (2015) 085802 H Nejatipour and M Dadsetani

ground state calculations. The muffin-tin radii have been fixedat 2.3 a.u. for M and at 2.1 a.u. for Y atoms. The interstitialplane wave vector cut off, KMax, is chosen in a way thatRMTKMax equals 8. The maximum l quantum number for thewavefunction expansion inside the atomic spheres was con-fined to lMax = 8. The GMax parameter was taken to be14 Bohr−1. To calculate the optical dipole matrix elements inthe dielectric functions, Brillouin zone (BZ) integrationswithin self-consistency cycles were performed via a tetra-hedron method. The convergence with respect to the numberof k-points was tested. For all the structures, the opticalspectra were converged with a 6 × 6 × 6 k-mesh in irreducibleBZ. A lifetime broadening of 0.1 eV was applied so thatoptical spectra can be obtained in agreement with theexperiments.

2.2. The e–h interactions

Despite recent improvements, there are still problems in usingthe DFT for the proper description of the exact band gaps, inparticular transition states. The RPA does not explain theelectronic transitions including the coupling of the e–h andthe formation of bound e–h pairs. In addition to the creationof excitonic structures within energy gap region, the e–h pairssometimes redistribute the oscillator strength in the whole ofoptical spectra. This causes that the RPA to give the wrongoscillator strengths in optical absorption spectra. Therefore,the use of the MBPT is justified, particularly for the linear-response regime of infinite systems in which it is very usefulto compare the linear response to experimental data. By sol-ving the BSE for the e–h two particle Green’s function in theframework of advanced methods, the calculations of opticalabsorption spectra of some inorganic compounds such assolid argon [45], Si [46], and some organic materials [47, 48]have recently led to an amazing agreement with theirexperimental data. To this end, the exciting-code, which is afull-potential all-electron package, applies the families of thelinearized augmented plane wave (LAPW), and it uses thesemethods. In this code, the equation of motion for two-particleGreen function is solved on the basis of the FPLAPWmethod. All the information about the two-particle excitationsin a many-body system is included in an effective eigenvalueequation [49]:

H A E A . (1)v c k

vck v c k v c kj j

vckj

,eff∑ =

′ ′ ′′ ′ ′ ′ ′ ′

The eigenenergies E j and eigenvectors Avckj represent the

excitation energy and the coupling coefficient of the jth e–hpair, respectively. In equation (1), the subscripts denotevalence states by v, conduction states by c and a vector in theirreducible BZ by k. The effective e–h kernel, Heff, describingthe all interactions in the optical processes, contains threeinteracting contributions, as follows:

H H H c H , (2)vck v c k vck v c k vck v c k x vck v c kx

,eff

,diag

,dir

,= + +′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′

where Hdiag is the differences of QP energies. The considera-tion of only the first term in the right hand side of equation (2)corresponds to the independent particle approximation. The

second term is the Coulomb attraction between electron andhole and the third one, the exchange term, is responsible forthe splitting between the spin-singlet (cx= 2) and spin-triplet(cx= 0) excitons.

Finally, using the relation between the two-particle cor-relation function derived from BSE and the polarization, themacroscopic dielectric function is obtained, including thelocal field effects (LFE) as well as e–h correlation effects:

( )

Im Avk p ck

E E

E

( )8

, (3)

ii

j vckvckj i

ck vk

j

22

∑ ∑ε ω πΩ

δ ω

=< >

−

× −

where <vk p cki∣ ∣ > is the matrix element of the momentumoperator component pi between the valence and conductionbands, Ω is the crystal volume, and Evk and Eck represent thecorrected Kohn–Sham v and c states energies, respectively.The consideration of QP energies and states in equation (3),obtained by GW approximation leads to a higher level ofcalculations (BSE +GW). In some cases, in addition tocorrecting the peak positions, the BSE+GW level modifiesoscillator strengths in the optical responses.

The excitonic effects in the dielectric functions of allstructures were converged by including all the valence bandsand the 16 conduction states. It is worth mentioning thatindirect transitions and the spin–orbit coupling are notincluded in the theory used in this paper.

3. Results and discussion

3.1. Electronic band gaps

Given the dependency of direct band gaps at high symmetrypoints on the lattice constants, we optimized the structuralproperties of all the AECs in order to get the best results inoptical calculations. Table 1 shows that all parameters are inexcellent agreements with the experimental reports of therocksalt phases of AECs [50–62]. As it can be seen, thechanges in lattice parameters due to the substitution of chal-cogen atom for another chalcogen are more sensitive than thesubstitution of the alkaline earth atom for the other one.Moreover, the changes from oxides to sulfides are larger thanthose from sulfides to selenides or tellurides. In addition, thelarger lattice constants are correlated to the heavierchalcogens.

Since theoretical and experimental studies on the bandgaps of AECs have contradictory reports, we used both one-particle KS-GGA and many body G0W0 approximations [42]in order to determine the exact values of the band gaps and inorder to determine which one is direct and which one isindirect. Figure 1 shows band structures, and table 2 showsthe energy band gap values of AECs in the rocksalt phases.Table 2 compares the energy band gaps to the existingexperimental data at some symmetry points (these points havereceived contradictory reports). The minimum band gapvalues have been represented by bold numbers. Figure 2, in

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Phys. Scr. 90 (2015) 085802 H Nejatipour and M Dadsetani

turn, compares the computational (obtained by both GGA andG0W0 approximations) and experimental band gap values ofAECs. Compared to the KS-GGA, the G0W0 approximationconsiderably decreases the differences between computationaland experimental gaps. One should bear in mind that one-particle KS gaps and many body G0W0 ones, i.e. electronicgaps, can be compared to the experimental gaps from thephotoemission experiments, but one cannot compare opticalgaps with the electronic ones because optical gaps are rele-vant to the physical phenomena including the e–h couplingand the creation of the exciton. It is worthwhile to mentionthat at the normal conditions, MgTe crystallizes in the wurt-zite structure [63], and to the best of our knowledge, there areno experimental measurements of the band gaps of MgSe andMgTe in the rocksalt crystal structure. The energy band gapof MgTe in its stable phase, wurtzite, has been reported to be3.49 eV [64]. For its rocksalt phase, our calculations show anextremely low (0.464 eV) indirect energy band gap, inaccordance with the other theoretical studies [65, 66]. TheG0W0 approximation results in an increase in the band gap toup to 0.963 eV. In both KS and G0W0 approximations, directgap values of MgTe at X and Γ are very close.

Our results show some interesting points, for example: (i)the heavier chalcogens (and AEs) give rise to smaller bandgaps. (ii) While moving from oxides to sulfides, band gapvalues of AECs show a sharp decrease, moving from thesulfides to the selenides and then to the tellurides indicates asmoother decrease (iii) in both one-particle KS and manybody G0W0 procedures, AECs have indirect band gaps fromΓ to Χ, except for MgO and BaO (MgO at Γ and BaO at Χ).Whereas experimental studies of the electronic structure ofCaO show a direct band gap at X [18] or Γ [67], our studyindicates that CaO is a wide band gap semiconductor with anindirect gap from Γ to X. (iv) The overall dispersion of all theAEC band structures are nearly the same. (v) The G0W0

approximation shifts the conduction bands to higher energies,and it also modifies the valence and especially the deeper

electronic states to lower energies, in particular, for AE oxidesand Ba chalcogenides. This shows that the BSE spectra with arigid scissor shift can be different from the BSE spectra withthe QP eigenvalues (BSE +GW level). The former is based onthe solving of the BSE using the KS-eigenvectorsand -eigenvalues shifted by the difference between KS andQP gaps, whereas the latter comes from the solution of theBSE with the QP eigenvectors and eigenvalues. This will beused as an important point in the calculation of absorptionspectra from BSE, discussed below. (vi) In some cases of theAECs, particularly in SrO, BaS and BaSe, the differencebetween the direct and indirect band gaps is small. There arecontradictory reports whether the band gaps of AECs aredirect or indirect. Whereas some theoretical studies havereported that the gaps are direct for CaS [67, 68] and SrS [67],others [69–71] have been reported that they are indirect fromΓ to Χ. Our results confirm the indirect band gaps for thesetwo structures. Since both our KS-DFT and many body G0W0

electronic structure calculations are based on an all-electronfull potential LAPW method, the reports presented here arereliable. (vii) As table 2 indicates, although the G0W0

approximation considerably corrects the direct and indirectband gap values, it underestimates the real band gaps reportedfrom photoemission experiments. This means that the real QPband gaps are larger than our calculated G0W0. This can bethe result of using the GGA in the ground state calculations.

3.2. Optical properties

Studying the coupling between an electron and a hole left bythe excitation of the electron is possible in the framework ofthe many body theories. The equation of motion of the two-particle Green’s function, BSE, includes both the e–h Cou-lomb attraction and the exchange effects (LFE) in the exci-tation process. Despite the fact that the band gaps are indirectin most AECs, the lack of the indirect transitions in our cal-culations cannot be a critical problem. Indirect transitions are

Table 1. Calculated lattice parameters (Å) of AECs from GGA-PBE, compared to experiments.

Mg Ca Sr Ba

This work Exp. This work Exp. This work Exp. This work Exp.

O 4.26 4.25a 4.84 4.80b 5.19 5.16c 5.58 5.53d

S 5.24 5.20e 5.72 5.69c 6.06 6.02f 6.43 6.39g

Se 5.51 5.47h 5.96 5.91c 6.30 6.24i 6.66 6.60c

Te 5.98 6.02j 6.40 6.35k 6.72 6.66l 7.07 7.00m

a

Reference [50].b

Reference [54].c

Reference [55].d

Reference [60].e

Reference [51].f

Reference [57].g

Reference [61].h

Reference [52].i

Reference [58].j

Reference [53].k

Reference [56].l

Reference [59].m

Reference [62].

4

Phys. Scr. 90 (2015) 085802 H Nejatipour and M Dadsetani

not allowed from Γ to X in the AECs due to the parity con-siderations. In the following subsections, optical properties ofAECs are calculated by solving the BSE, and the results arediscussed. With the addition of GW corrections, the BSE+GW level of calculations is applied and the changes arestudied:

3.2.1. Mg chalcogenides. Figure 3 represents the imaginaryparts of the dielectric functions, ε2(ω), of Mg chalcogenides.Clearly, excitonic structures have appeared below thecalculated band gaps. All the spectra obtained by the BSEhave been shifted with the energy difference between thetheoretical KS and GW gaps. In ε2(ω) of MgO, a strong

Figure 1. Band structures of the AECs from KS-GGA (blue lines) and G0W0 (pink lines) approximations.

5

Phys. Scr. 90 (2015) 085802 H Nejatipour and M Dadsetani

excitonic structure has occurred at 3.81 eV. With a rigidscissor shift equal to 2.327 (the difference between the KSand QP energy gaps), it will occur at 6.137 (figure 3, reddashed line). Its high oscillator strength is attributed to astrong direct excitonic transition. The experimental report ofthe reflection spectrum of MgO measured at 77.3 K in theenergy range 7.6–7.9 eV shows that the optical edge in ε2(ω)of MgO is dominated by the two strong excitonic structures at7.692 and 7.765 eV (displayed at the bottom of figure 3) [72].They are related to the splitting of the valence band at Γ point.The differences between the theoretical and experimentalresults refer to the two main points: the first one, i.e., thedifference between peak positions, refers to the neglect of theGW correction in solving the BSE. The second one is due to

the lack of the spin–orbit splitting effects in our calculations.The main difference between experimental and computationalspectra is in the peak positions which can be rathercompensated by adding the GW to the BSE. Using the QPcorrections from the GW approximation in the calculation ofequation (3), we calculated the absorption spectra on a higherlevel (BSE+GW) in order to correct the difference betweenpeak positions. This means that v and c are QP states and Evk

and Eck are QP energies. The results are presented with theblack solid lines in figure 3. As it can be seen, the overallshapes of the absorption spectra of MgO are the same in bothBSE and BSE+GW levels. The BSE+GW level slightlycorrects intensities with respect to the BSE. In addition, theexcitonic feature in MgO occurs at 6.32 eV calculated by

Table 2. Calculated energy band gaps (eV) of AECs from KS (GGA-PBE) and G0W0 approximations, compared to experiments. Theminimum band gap values are bold numbers.

GGA G0W0 Exp.

Structure Γ–Χ Χ–Χ Γ–Γ Γ–Χ Χ–Χ Γ–Γ Γ–Χ Χ–Χ Γ–Γ

MgO 9.146 10.481 4.773 11.264 12.93 7.10 — — 7.7a

MgS 2.711 3.464 4.316 4.044 5.944 4.793 4.48 — —

MgSe 1.767 3.464 2.387 2.682 4.681 3.505 — — —

MgTe 0.464 2.223 2.098 0.963 3.015 3.034 — — —

CaO 3.669 4.040 4.705 5.81 6.321 6.042 — 6.79c 7.03d

CaS 2.450 3.227 3.927 4.13 5.126 5.134 4.434 5.343c 5.80c

CaSe 2.035 2.968 3.371 3.44 4.613 4.507 3.85 4.898c 5.0e

CaTe 1.591 2.650 3.000 2.657 3.995 4.041 — — —

SrO 3.341 3.437 3.924 4.862 5.016 5.167 5.727 5.793c 6.08c

SrS 2.490 2.952 3.680 3.823 4.432 4.897 4.32 4.831c 5.387c

SrSe 2.268 2.866 2.834 3.414 4.147 4.007 3.813 4.475c 4.570c

SrTe 1.816 2.525 2.732 2.677 3.593 3.745 3.4 — —

BaO 2.496 2.320 4.322 3.685 3.391 5.800 — 3.985 8.30c

BaS 2.118 2.444 3.698 3.33 3.565 4.931 3.806 3.941c 5.229c

BaSe 1.986 2.183 3.134 2.990 3.268 4.196 3.421 3.658c 4.556c

BaTe 1.616 1.968 2.988 2.329 2.797 4.068 3.08 — —

a

Reference [67].b

Reference [75].c

Reference [18].d

Reference [76].e

Reference [77].f

Reference [69].g

Reference [24].h

Reference [78].

Figure 2. A comparison between computational (GGA and G0W0) and the existing experimental band gap values (eV).

6

Phys. Scr. 90 (2015) 085802 H Nejatipour and M Dadsetani

means of the BSE+GW. Despite that a scissor shift is appliedin the BSE, the difference in the peak positions in the BSEand BSE+GW level shows that the use of QP corrections hasalso redistributed the oscillator strengths. As mentionedabove, the remaining difference between the theoretical andexperimental peak positions is due to the use of GGA in theground state calculations. The reproduced excitonic structurein ε2(ω) of MgO is in agreement with the many-bodycalculation by Schleife et al [39].

In MgS, excitonic structure is at 3.152 eV. With a rigidscissor shift equal to 1.333, it will occur at 4.439 eV (figure 3).The diffused absorption spectrum of MgS measured at 80 and293 K by Yamashita has an absorption edge near 4.6 eV,corresponding to the optical band gap (at the bottom offigure 3) [73]. It exhibits two main peaks at 5.21 and 5.49 eV.From a different perspective, a dip at 5.34 eV at 80 K (5.18 at293 K) in the diffuse absorption spectrum can be a sign of astrong exciton in the reflection spectrum. Despite the fact thatthere is no report on the behavior of ε2(ω) of MgS, it can bededuced that ε2(ω) has an excitonic structure with its peakposition being near to its correspondent in the reflectionspectrum, according to Kramer–Kronig relations. Figure 3shows that the computational excitonic peak in the BSE+GWhas shifted to the 4.48 eV, along with a modification in itsintensity. A small difference in the peak positions reveals that

the BSE+GW level in MgS slightly leads to the redistribu-tion of the oscillator strength.

To our knowledge, there are no experimental data on theoptical properties of rocksalt MgSe and MgTe. Like MgO andMgS, the computational ε2(ω) of MgSe in BSE has a strongexcitonic structure with a higher oscillator strength at0.956 eV. With a 0.915 eV rigid scissor correction, it shiftsto 1.871 eV (figure3, red dashed line). Like the sulfide andselenide of Mg, MgTe has an excitonic structure occurring at0.811 eV, which has been shifted to 1.310 eV (figure3, reddashed line). In addition to the red shift of the absorption edgefrom MgO to MgTe, with a decreasing trend of band gaps,excitonic effects become weaker in the optical responses. Inaddition to the modification of intensities, the BSE+GWlevel for MgSe and MgTe leads to a shift in the peak positionsto 3.277 and 2.931 eV, respectively (figure 3, black solidlines). Moreover, figure 3 shows that the BSE+GWcorrection has the greatest effects on the absorption spectrumof MgTe among the spectra in terms of shifting and theredistribution of optical spectra.

Furthermore, figure 4 shows the real parts of thedielectric functions and the energy loss functions of Mgchalcogenides, calculated in the BSE+GW level. Thedielectric constants of MgO, MgS, MgSe and MgTe are2.61, 4.45, 8.16 and 11.46, respectively. The experimentaldielectric constant of MgO is 2.95. The first peaks in the

Figure 3. The imaginary parts of the dielectric functions of Mg chalcogenides in BSE (red dashed lines) and BSE+GW (black solid lines)levels, compared to the existing experiments.

7

Phys. Scr. 90 (2015) 085802 H Nejatipour and M Dadsetani

energy loss functions originate from the energy loss due to thedirect excitonic transitions at X and Γ, as described above.Compared to the imaginary parts of the dielectric functions,peak positions of the energy loss are located at higherenergies. In the energy loss spectrum of MgO, an excitonicfeature has occurred at 6.48 eV, corresponding to theexcitonic peak in ε2(ω) which occurred at 6.32 eV. Despitethe fact that the spin–orbit splitting effect has been neglected,the dispersion of the real part of the dielectric function inMgO is in a good agreement with the existing experimentaldata [72].

3.2.2. Ca chalcogenides. Figure 5 portrays the imaginaryparts of the dielectric functions of Ca chalcogenides. Themain excitonic peaks have truly been reproduced. Comparedto our previous RPA calculations for CaY structures [29], thepresent study has better results in an acceptable conformitywith the experiments [18]. This shows that the excitoniceffects play a significant role in the optical properties of thestructures under study. With a 2.141 eV scissor shift, the main

peaks in CaO (A and B at 3.506 and 3.61, respectively) willbe at 5.647 and 5.751 eV (red dashed line). The experimentalones have been reported at 6.815 and 6.91 eV. They areexcitonic structures assigned to the direct transitions at Xsymmetry point. A 0.104 eV difference between the tworeproduced peaks, A and B, is in good agreement with the0.095 eV of the experiment. In CaS, all peaks A, D and E arewell reproduced. The experimental peak positions are at5.273, 5.74 and 5.82 eV. Peak A at 2.73 (D and E at 3.17 and3.42) is attributed to the direct transition at X (Γ) point. Witha shift equal to 1.68, it is reproduced at 4.41 (4.85 and 5.1,respectively) eV. In the case of CaSe, additionally,computational and experimental spectra are in an excellentagreement. The main peaks, A and B, have been reproducedat 2.607 and 2.99 eV. Their experimental correspondents areat 4.828 and 5.056 eV. With a 1.405 eV shift, they occur at4.012 and 4.395 eV. There is no experimental data for opticalproperties of CaTe. Due to the overall agreement between theother computational CaY, the overall shape of ε2(ω) of CaTecan be reasonable. The QP energy gap of the CaTe was foundto be 2.657 eV without any experimental correspondent. The

Figure 4. The real parts of the dielectric functions (left) and the energy loss functions (right) of Mg chalcogenides in the BSE+GW level.

8

Phys. Scr. 90 (2015) 085802 H Nejatipour and M Dadsetani

two main peaks of ε2(ω) in CaTe, A and B, have occurred at2.313 and 2.585 eV, while they occur at 3.379 and 3.651 eVwith the 1.066 eV shift.

Obviously, the agreement of our BSE spectra with theexperiment is far better than the RPA results calculated in ourprevious studies on all the Ca chalcogenides [29]. Black linesin figure 5 show the optical absorption spectra of CaYstructures in the BSE+GW level. In addition to the peakpositions, the oscillator strengths have been modified. Themain peaks A and B are reproduced in the BSE+GW in ε2(ω)of CaO, with peak B being a broad structure not appearing inthe spectrum. In the energy range near the optical gap, theseresults are in agreement with the many-body calculations byRiefer et al [35], including the reproduction of the excitonicfeatures associated to the vertical optical transitions. Com-pared to the BSE, in addition, the overall broadening of themain profile has decreased, and it is in more agreement with

that of experiment. There is a difference (0.538 eV) in peakpositions due to the underestimation of G0W0 band gapcompared to the true QP band gap or the electronicexperimental one. Instead, the overall shape of the ε2(ω) inCaS is better in the BSE+GW than that of the BSE, inaddition to the reproduction of main peaks. In spite of thechanges in the intensities of peaks A and B in the BSE+GWfor CaSe, the energy difference has decreased between them,which is in agreement with the experiment. Moreover, theoscillator strength has partly been redistributed. As for CaTe,the two main peaks of A and B have appeared in theBSE+GW level, as it is the case for the BSE level, while theintensities decrease. One can see an explicit red shift of theabsorption edges when he/she moves from CaO to CaTe,which is a sign of the decrease in the energy gaps.

Figure 6 shows both the real parts of the dielectric andthe energy loss functions of CaY structures in the BSE+GW

Figure 5. The imaginary parts of the dielectric functions of Ca chalcogenides in BSE (red dashed lines) and BSE+GW (black solid lines)levels, compared to experiments [18].

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Phys. Scr. 90 (2015) 085802 H Nejatipour and M Dadsetani

level. The dielectric constants of CaY structures are 2.85,3.87, 4.40 and 5.0, for Y=O, S, Se and Te, respectively. Theyare in a better agreement with the experiments than our RPAreport [29]. Their experimental correspondents are 3.27, 4.24and 4.48 for CaO, CaS and CaSe, respectively [74]. Theexcitonic structures originating from the direct excitonictransitions are obvious in the energy loss spectra. As anexample, in the energy loss spectrum of CaSe, the peaksoccurring at 4.057 and 4.650 eV are attributed to the directtransitions at 3.970 and 4.254 eV which appear at theimaginary part of the dielectric function. Due to the decreaseof energy band gaps from CaO to CaTe, the structures displaya red shift in the loss functions.

3.2.3. Sr chalcogenides. Optical properties of Srchalcogenides were also calculated by means of solving theBSE. Figure 7 shows optical absorption spectra as well as theexperimental results [18]. The unconformity is a little greater

than that of the other AECs between the computational andexperimental absorption spectra of Sr chalcogenides. Despitethe reproducing of excitonic features, the overall dispersionsof the spectra do not match the experiments. The splittingbetween peaks A and B arise from spin–orbit splitting of thevalence band in the experimental absorption spectra of SrO,SrS and SrSe. In addition, the exceeding of peak D (Γexciton) from peak B (X exciton) is a sign of this splitting inSrSe. In order to correct the peak positions and the dispersionof spectra, the BSE+GW level is used. Results representedwith black solid lines in figure 7 show that the overallbehavior of the spectra has changed in BSE+GW level.Peaks A, D and E were reproduced with different intensitiesfrom those of the BSE. As expected, peak B does notappeared due to the lack of spin–orbit effects in this study. Inthe case of SrO, the BSE+GW reproduces the main peaks A,B (and E), with a decrease in their energy difference in thecomparison with the BSE, in a more conformity with the

Figure 6. The real parts of the dielectric functions (left) and the energy loss functions (right) of Ca chalcogenides in the BSE+GW level.

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Phys. Scr. 90 (2015) 085802 H Nejatipour and M Dadsetani

experiment [18]. The spectra in the BSE+GW level have ablue shift in other SrY structures with respect to the BSEspectra, except SrS.

Figure 8 displays ε1(ω) and the energy loss spectra of Srchalcogenides. In SrY, dielectric constants are 2.87, 3.71, 3.96and 4.46 for Y=O, S, Se and Te, respectively. Theexperimental dielectric constants are 3.35, 4.09, 4.33 and4.91 [74]. In the energy loss function of SrO, there are twomain peaks at 4.67 and 4.96 eV and a shoulder at 5.102 eV,attributable to the respective excitonic transitions in ε2(ω),which occur at 4.59 and 4.836 eV and a shoulder whichoccurs at 5.113 eV. Similar interpretations can be used for theother Sr chalcogenides.

3.2.4. Ba chalcogenides. Finally, optical properties of Bachalcogenides were calculated. From Sr chalcogenides to Baones, an improvement appears in the spectra, again. In

figure 9, the imaginary parts of the dielectric function of BaYstructures, computed by solving the BSE have been displayed(red dashed lines). In BaO, peaks labeled as A, B and C in theexperiment are present in the BSE spectrum. As alreadymentioned, because of the spin–orbit coupling, peak C ofBaO occurs prior to the other two features in the experimentalspectrum. The origin of these features is direct transitions at Xpoint. Peak A and B (C) occurring at 1.793 and 2.321 (2.689)eV, respectively, are corrected to 3.006 and 3.534 (3.902) eVby means of the 1.213 eV scissor shift. Experimental peakpositions of A and B (C) have been reported at 4.043 and4.241 eV (3.912), respectively. The 0.528 eV differencebetween A and B features is far from 0.2 eV reported in theexperiment [18]. The differences mentioned above can becorrected by means of the BSE+GW, as shown in figure 9(black solid lines). The two spectra of BaO, obtained by theBSE and BSE+GW, are different in both peak positions and

Figure 7. The imaginary parts of the dielectric functions of Sr chalcogenides in BSE (red dashed lines) and BSE+GW (black solid lines)levels, compared to experiments [18].

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Phys. Scr. 90 (2015) 085802 H Nejatipour and M Dadsetani

oscillator strengths. The BSE+GW level defines animprovement, as compared to BSE. The optical absorptionedge, intensities and oscillator strengths have been modifiedin the BSE+GW. In the BSE+GW, the energy differencedecreases to 0.326 eV between peaks A and B. In theBSE+GW level, the main peaks including A, B and C havebeen reproduced at 3.88, 4.19 and 4.51 eV, respectively. Thisshows that the real QP band gap is larger than our calculatedG0W0. This can be the result of using the GGA in the groundstate calculation, as already implied. The more differenceobserved in the peak positions of A and C can originate fromthe neglecting of the spin–orbit coupling and also from theprobable existence of oxygen vacancies in the experimentalsample, with oxygen vacancies highly affecting the opticalresponse.

In the case of BaS, the main excitonic structures arepresent in the theoretical spectra (figure 9, red dashed line).Peaks A, B and C (D and E) are related to the directtransitions at Χ point (Γ point). With a rigid scissor shift,excitonic structures A, B, C, D and E have occurred at 2.95,

3.632, 4.22, 4.335 and 4.471 eV, respectively. Their corre-spondents in the experiments are at 3.868, 4.010, 4.46, 5.166and 5.30 eV [18]. In order to compensate for thesedifferences, BSE+GW was used. Figure 9 shows theBSE+GW results. As seen heretofore, an improvement canalso be seen in the BaS spectrum (the black line), as comparedto BSE (the red line). Peak positions in BSE+GW are 3.069(A), 3.432 (B), 3.87 (C), 3.976 (D) and 4.180 (E) eV. The0.142 (0.134) eV experimental energy difference between Aand B (D and E) has been obtained at 0.363 (0.204) eV inBSE+GW.

A similar treatment is observed in the case of the selenideand telluride of barium. In both compounds, the mainexcitonic structures including A, B, C, D and E, have beenreproduced appropriately. The use of BSE+GW somewhatgives rise to a redistribution of the oscillator strengths andmore conformity, as compared to the BSE level. Excitonicstructures have obviously been reproduced in the BaYstructures, which were absent in our previous RPA level ofcalculation of the optical responses in Ba chalcogenides [26].

Figure 8. The real parts of the dielectric functions (left) and the energy loss functions (right) of the Sr chalcogenides in the BSE+GW level.

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Phys. Scr. 90 (2015) 085802 H Nejatipour and M Dadsetani

Figure 10 presents the real parts of the dielectricfunctions in BSE+GW level as well as the energy lossspectra. Computational (experimental [74]) dielectric con-stants are 2.80 (3.68), 3.66 (4.26), 4.0 (4.48) and 4.54 (4.71)in BaO, BaS, BaSe and BaTe, respectively. As it has beendescribed above, the successive excitonic features in theenergy loss spectra are attributed to their respective excitonictransitions in the imaginary parts of the dielectric functions ofBa chalcogenides. In the case of BaO, for example, the firstthree structures in the energy loss spectrum of BaO, occurringat 3.836, 4.19 and 4.462 eV can be assigned to the mainstructures of A, B and C in ε2(ω), originating from the directtransitions at X point (occurring at 3.82, 4.13 and 4.47,respectively).

We can summary the main points concerning theelectronic and optical calculations of AECs, including theexcitonic effects as fallow: (i) except MgO and BaO whichhave direct band gaps, the other AECs are indirect band gap

semiconductors. (ii) The calculation of optical responses ofAECs by solving the BSE on the basis of KS states leads tothe reproduction of main excitonic structures, with thedifferences in their peak positions in energy (iii) the use ofthe solution of BSE on the basis of QP corrections modifiesthe peak positions and the intensities accompanied by theredistribution of the oscillator strengths in the opticalabsorption spectra (iv) when the band gap values of AECsare compared and contrasted, a red shift can be seen from Mgto Ba, which reveals a decreasing trend in energy band gaps.Moreover, as expected, a similar behavior is observed ingoing from the oxide to telluride of one AE atom. (v) For allthe cases, due to the use of the GGA in the ground statecalculations, there is a difference between the GW andexperimental band gaps. This gives rise to the differencebetween the excitonic peak positions in the theory andexperiments. (vi) As described in the experimental workreported by Kaneko et al [18], the surfaces of the AEC

Figure 9. The imaginary parts of the dielectric functions of Ba chalcogenides in BSE (red dashed lines) and BSE+GW (black solid lines)levels, compared to experiments [18].

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Phys. Scr. 90 (2015) 085802 H Nejatipour and M Dadsetani

samples actively adsorb the ambient gas molecules. There-fore, the existing differences between experimental andtheoretical optical spectra can be attributed to this absorbing.Furthermore, both defects and coupling with the phonons playimportant roles in the optical results from the experiments.Since the AECs are highly ionic compounds, in addition, theionic screening can be important in optical responses ofAECs, which is not included in our study. Therefore, thesepoints must be considered when comparing the computationaland experimental results.

4. Conclusions

We studied excitonic structures in the AECs,MY (M=Mg, Ca,Sr, Ba and Y=O, S, Se, Te) by solving the BSE for the two-particle Green function. Our results confirm that the excitoniceffects play a significant role in the optical properties of the

structures under study. The excitonic structures originate fromthe direct transitions at high symmetry points X and Γ. Mgchalcogenides show strong excitonic structures in their opticalabsorption spectra. The main excitonic peaks have truly beenreproduced in calcium chalcogenides. For all the Ca chalco-genides, agreement of our BSE spectra with the experiment isfar better than the RPA results calculated in our previousstudies. Despite the reproduction of the excitonic structures inthe case of Sr chalcogenides, the overall dispersions ofabsorption spectra are a little different from the existingexperimental reports. In addition, excitonic structures whichwere absent in the previous RPA calculations of the opticalresponses in Ba chalcogenides have been reproduced in theBaY structures. Furthermore, the BSE was solved by means ofthe quasi-state and quasi-energy corrections for the calcula-tion of optical absorption spectra. Altogether, with respect tothe BSE based on the KS states and energies, the considera-tion of the QP corrections in the BSE, i.e. BSE +GW, led to a

Figure 10. The real parts of the dielectric functions (left) and the energy loss functions (right) of Ba chalcogenides in the BSE+GW level.

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Phys. Scr. 90 (2015) 085802 H Nejatipour and M Dadsetani

more conformity with the experimental results. Significantlymodifying the peak positions and intensities of the absorptionspectra, the BSE+GW gave rise to the redistribution ofoscillator strengths. Compared to previous RPA results,dielectric constants obtained by BSE+GW have betteragreements with the experiments. In addition, the excitonicstructures were reproduced in the energy loss functions ofAECs. Compared to the previous RPA calculations of AECs,at any rate, our results have a remarkable improvement andacceptable conformity with the experiments. Compared to theexperiments, some differences in the optical absorptionspectra arise from the well-known underestimation of GGAfor the exchange-correlation functional in the ground statecalculations and also the lack of spin–orbit splitting, in thisstudy. In addition, experimental conditions such as tempera-ture, defects, atom vacancies and the reaction of the surfaceswith the ambient gases can also be the causes of somedifferences.

Furthermore, in order to determine whether the band gapsof the AECs are direct or indirect, the electronic band struc-tures of MY structures were calculated by means of bothKohn–Sham and many-body G0W0 approximations. ExceptMgO and BaO which have direct band gaps at Γ and X,respectively, other AECs are semiconductors with indirectband gaps at Γ to X.

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