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Solid State Sciences 18 (2013) 24e35

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Solid State Sciences

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DFT þ U study of the structural and electronic properties of theferromagnetic and antiferromagnetic ordering in the PbS-basedternary alloys Pb1�xEuxS (x ¼ 0.25, 0.50, 0.75 and 1)

Kin Mun Wong a,1, S.M. Alay-e-Abbas b,c,1, A. Shaukat c, Yong Lei a,*a Institut für Physik & IMN MacroNano� (ZIK), Technische Universität Ilmenau, Prof. Schmidt-Str. 26, Ilmenau 98693, GermanybDepartment of Physics, GC University Faisalabad, Allama Iqbal Road, Faisalabad 38000, PakistancDepartment of Physics, University of Sargodha, 40100 Sargodha, Pakistan

a r t i c l e i n f o

Article history:Received 14 June 2012Received in revised form21 November 2012Accepted 13 December 2012Available online 3 January 2013

Keywords:AlloysCorrelated calculationsOptimization of EC functionalElectronic structureMagnetic ordering

* Corresponding author. Tel.: þ49 3677693748.E-mail addresses: km2002wong@yahoo.com.sg

ilmenau.de (Y. Lei).1 Equal contribution.

1293-2558/$ e see front matter � 2012 Elsevier Mashttp://dx.doi.org/10.1016/j.solidstatesciences.2012.12.0

a b s t r a c t

We use first-principles full-potential method to study the structural and electronic properties of theferromagnetic (FM) and antiferromagnetic (AF) ordering in the PbS-based ternary alloys Pb1�xEuxS(0 � x � 1). It is shown that the generalized gradient approximation (GGA) functional of DFT in itsstandard form is not sufficient for obtaining the correct physical interpretation of the binary PbS and EuScompounds and, therefore, the calculations have been extended by considering spin-orbit coupling (SOC)in the Pb atoms and by treating strongly localized f electrons of Eu by the self-interaction correction (SIC)scheme. The Wu and Cohen (WC) generalized gradient approximation (GGA) functional has been usedfor structural optimization, whereas the GGA parameterization scheme formulated by Engel and Vosko(EV) has also been utilized for calculating the electronic properties of all the compounds. The exper-imental value of Coulomb parameter (Uf-expt.) within the SIC provide good description for the structuralproperties of all compounds but fails to predict an accurate EuS energy band structure. A comparison ofthe results obtained using the WC GGA þ SOC þ U and EV GGA þ SOC þ U shows that the later delivermuch better band structure profiles of the ferromagnetic EuS and non-magnetic PbS. The ternary alloysconstructed by substituting Eu for the Pb atoms in the rocksalt PbS have been used for computing thestructural and electronic properties of FM and AF ordering in the Pb1�xEuxS. The stability of the alloyswith respect to two magnetic orders has been analyzed and the effect of FM and AF coupling on theelectronic properties of the alloys are carefully studied. In particular, the variation of the degree ofcontribution from the various electronic energy states from the different elements in the alloys has beenelucidated in terms of the total and partial density of states for all the composition ranges. Importantly,our results predict that the ternary Pb1�xEuxS alloys have a direct band gap nature and AF is the preferredmagnetic order.

� 2012 Elsevier Masson SAS. All rights reserved.

1. Introduction

Alloying narrow band gap semiconductors with magnetic ornon-magnetic (NM) ions is a common practice [1,2]. The substitu-tion of transitionmetals and rare-earthmagnetic ions likeMnþ2 andEuþ2 in lead chalcogenides not only gives rise to local magneticmoments due to unfilled 3d and 4f electron shells that result intechnologically important materials for potential uses in spintronic

(K.M. Wong), yong.lei@tu-

son SAS. All rights reserved.08

devices [3,4], it also provides an opportunity for tuning the energyband gap (Eg) in PbS, PbSe and PbTe lattice from 0.334 to 2.22 eV [5].Apart from having isomorphic crystallization in the NaCl-type (B1)crystal structure (space group # 225, Fm3m), the sulfides of Pb andEu show distinct traits in their electronic, optical and magneticproperties. As regards the electronic band structure, PbS has a nar-row direct fundamental band gap (Eg) ¼ 0.29 eV located at the Lsymmetry point of the Brillouin zone [6], while EuS has an indirect(G � X) Eg ¼ 1.64 eV at ambient conditions [7]. The large values ofthe dielectric constant of lead chalcogenides and their uniquevibrational, thermoelectric, thermodynamic and optical propertiesmake them technologically important materials for use in devicessuch as diode lasers, long-wavelength imaging devices and ther-mophotovoltaic energy converters [8e11]. In contrast, the main

K.M. Wong et al. / Solid State Sciences 18 (2013) 24e35 25

interest in Eu chalcogenides has been due to the wide variation ofthe electronic and magnetic properties in these materials. Impor-tantly, the colossal magnetoresistance observed in Eu chalcogenidesand their magneto-optical behavior has earned them a promisingcandidature for spintronic devices, magneto-optical modulators, asmaterial for fast addressable memory system in computers andmagnetic field activated electronic switches [12e15].

Since the development of first mid-infrared pen junction laserbased on Pb1�xSnxTe [16], continuous efforts have been made forfabricating IVeVI based lasers. Importantly, the heterojunction la-sers grown on GaAs substrate were a major step forward thatfacilitated lasing at room temperature by providing an emissionpower of several milliwatts [17]. In this regards, various growthtechniques have been employed for layered epitaxial growth thatinclude liquid phase epitaxy (LPE), vapor phase epitaxy (VPE),metal organic chemical vapor deposition (MOCVD), and molecularbeam epitaxy (MBE) [18]. Among these methods, MBE has provento be most suitable for the development of quantum well (QW)lasers as it assists precise and accurate control of layer compositionand thickness that is required for successful fabrication of micro-and nano-scale devices. On the other hand, newly developedtemplate based surface nano patterning techniques [19,20] for thesynthesis of large-scale ordered arrays of nanostructures [21] andnanodevices ordered arrays of nanostructures [21] and nano-devices [22]. However, the template based techniques are still to beexplored for their possible usage in the fabrication of nanosheets[23], superlattices and QWs. The low-dimensional structures con-sisting of layers of IVeVI DMS with Eu and their (NM) counterpart(e.g. PbX/Pb1�xEuxX, X: S, Se and Te) superlattice and multiplequantum wells have been successfully grown on BaF2 (111) sub-strate using MBE [24]. The wider tuning range of IVeVI based laseras compared to quantum cascade lasers has earned them a renewedresearch interest [25] in the form of MBE grown vertical cavitysurface emitting laser (VECSEL) on Si substrate [26,27].

The stable crystallization of EuS compounds in the B1 crystalstructure caused by the existence of Euþ2 oxidation state witha slightly reduced 4f configuration is distinct as compared to otherrare earth binary compounds [28]. Moreover, stable EuS exists inthe ferromagnetic (FM) state having TC ¼ 70 K which can beenhanced to TC ¼ 290 K under the application of pressure [29].A number of experimental investigations have been carried out fora variety of physical properties of bulk [30,31], thin films [32] andnanocrystals [33] of Pb1�xEuxS, however, theoretical account of thestructural, electronic and magnetic properties is still lacking in theliterature, to the best of our knowledge. Importantly, the preferredanti-ferromagnetic (AF) ordering in the ternary Pb1�xEuxS DMSs[34] as compared to the binary FM EuS, the very small values ofexperimentally determined AF exchange constants among the Eu2þ

ions in lead salts [31,33] and the tendency toward Eu2þ clusteri-zation in Pb1�xEuxS [31] requires a deeper physical insight of thestructural stability of the Eu substituted lead sulfide and the mag-netic mechanism involved therein. To this end, the advances inquantum chemical calculation have not only enabled researchers topredict the possibility of stable crystallization of materials but hasalso paved the way for computing certain electronic and magneticproperties of condensed matter which are difficult or evenimpossible to obtain using current experimental techniques.Therefore, aggravated by the lack of theoretical data on Pb1�xEuxS,we have performed an ab-initio investigation using the highly ac-curate all-electron full-potential linear-augmented-plane-waveplus local orbital (FP-LAPW þ lo) method for the structural andelectronic properties of the ferromagnetic (FM) and antiferromag-netic (AF) ordering in the PbS-based ternary alloys Pb1�xEuxS.

In the present study, we first address the challenges faced byfrequently used DFT functionals in obtaining accurate structural

properties and band structure profiles of binary PbS and EuS. Af-terwards, we investigate the mechanism of the magnetism in theternary alloys (Pb1�xEuxS) for some selected high concentrationranges of Eu-doped PbS in terms of FM and AF. We show that thespin-orbit coupling (SOC) in the heavy Pb atom is responsible forthe relativistic effects and the strongly correlated Eu-4f electronsplay important roles in the magnetic ordering and electronic pro-files of the ternary compounds. These effects require one to gobeyond standard DFT calculations by incorporating the onsiteCoulomb interaction (Uf) of Eu-4f states by modifying the gener-alized gradient approximation (GGA) parameterization schemesand also require the inclusion of SOC for Pb atoms. The applicabilityand performance of two GGA functionals for the electronic struc-ture calculations have been investigated and the results have beencompared with available theoretical and experimental data.

2. Computational details

The FP-LAPW þ lo method as implemented in the WIEN2k code[35] has been used in this study for total energy calculations of thenon-spin-polarized (NS), FM and AF Pb1�xEuxS alloys. For structuraloptimization, exchange-correlation potential of the KohneShamequations has been obtained from the WueCohen (WC) [36]GGA. The reason for choosing the WC GGA for structural opti-mization is twofold: first, the small lattice mismatch (w0.5%) be-tween PbS and EuS [32] and secondly, the fourth-order gradientexpansion of exchange energy functional available in the WC GGAprovide better approximation for computing the structural prop-erties as compared to the other GGA schemes [36]. On the otherhand, since standard GGA functionals are not flexible enough toaccurately determine the total exchange-correlation energy and itscharge derivative, the GGA parameterization scheme formulated byEngel and Vosko (EV) [37] has also been used, for electronicproperties only, as it improves results of quantities such as theelectronic band gap that depend on the energy eigenvalue (in EVGGA this is achieved by producing a more accurate exchange-correlation potential at the expense of less accurate exchange en-ergy) [38].Within a semi-relativistic approximation, self-consistenttreatment of the valence states is performed, while the core statesare treated self-consistently and fully relativistically relaxed inspherical approximation. The expansion of potential, charge den-sity and basis functions is carried out within the radius of muffin-tin (MT) spheres along with spherical harmonic functions witha cut-off of Imax ¼ 10 and with Fourier series in the interstitial re-gion. The parameter Kmax ¼ 7/RMT is used for finding matrix size,where Kmax is the plane wave cut-off and RMT denotes the averageof all atomicMT radii. For Pb1�xEuxS alloys, the values ofMT spheresradii have been taken as 2.8 a.u., 2.8 a.u. and 2.2 a.u. for Pb, Eu and S,respectively. For the FM calculations a 12 � 12 � 12 MonkhorstePack k-mesh has been used, while for the AF calculationsa 11 �11 � 5 k-mesh has been used in the irreducible wedge of theBrillouin zone since we have considered the AF propagation in the[001] direction of the FM cells which requires construction of1 � 1 � 2 supercells. For convergence, both the MT radius and thenumber of k-points are adjusted to above given values and theprocess of energy and charge convergence is iterated till accuracy isachieved up to 0.00001 Ry and 0.0001 e, respectively.

In order to appropriately treat the strong correlation between 4felectrons of Eu, standard density functional theory (DFT) has beenextended by using self-interaction correction (SIC) approach pro-posed by Anisimov et al. [39], which restores the discontinuity inthe one-electron potential to the exact density functional. Initially,the X-ray photoemission spectroscopy measurements’ [40] value ofCoulomb parameter for elemental Eu (Uf-expt. ¼ 10 eV) has beenused for the DFTþ Uf calculations at x¼ 0.25, 0.50, 0.75 and 1. Since

K.M. Wong et al. / Solid State Sciences 18 (2013) 24e3526

the band gap (Eg) in all EuX compounds is formed between thefilled 4f states in the valence band (VB) and empty 5d states in theconduction bands (CB) of Eu, for calculating electronic properties,the correlation effect for the 5d-electrons (i.e. Ud) of Eu has alsobeen taken into account, while Uf and/or Ud have been tuned asparameters to get Eg close to experimental value. In these calcula-tions SOC has been considered for the Pb atoms as it is necessary forobtaining a correct description of the sep splitting [30,41] in theband structure of lead chalcogenides, whereas SOC has been shownto have no effects on EuS band structure profile [42]. As the mainmotivation of this work is to arrive at an accurate first-principlesdescription of the structural and electronic properties of ternaryPb1�xEuxS alloys and its parent binary compounds PbS and EuS, theresults presented in Section 3 are structured as follows: In x 3.1 weanalyze the structural and electronic properties of the binaryparent compounds PbS and EuS obtained using various modifica-tions of WC and EV GGA functional. The electronic structure cal-culations are performed to compute the necessary data and theeffects of Coulomb parameter U for Eu atom and SOC for Pb atomare examined. In x 3.2 we first discuss the structural properties andstability of the FM and AFmagnetic ordering in the Pb1�xEuxS alloysat x ¼ 0.25, 0.50 and 0.75 and then give a detailed account of theelectronic properties of these materials in terms of electronic bandstructure and density of states.

3. Results and discussion

3.1. Structural and electronic properties of PbS and EuS

The parent binary compounds, PbS and EuS, have been modeledin the B1 structure and the DFT calculations are performed keepingin view the complexity associated with the binary and ternarycompounds. In earlier theoretical studies researchers have facedsome serious challenges while using ab-initio techniques for thelead europium chalcogenide alloys. One reason for this is that thenarrow band gaps of Pb chalcogenides are difficult to achievethrough LDA and GGA parameterization schemes [43]. Moreover,the SOC in Pb atom give rise to relativistic effects in Pb chalcogenidebased ternary materials. In addition to that, the strongly localized fstates of Eu atoms require one to go beyond standard DFT func-tionals. To address these issues we have performed NS and spin-polarized DFT calculations for full structural optimization of PbSusing the WC GGA. We observe that for binary lead sulfide, theinclusion of spin-polarization has no noticeable effect on all theground state structural properties of this NM compound. Our cal-culated lattice constant and bulkmodulus of PbS obtained using theWC GGA functional are 5.9112 �A and 61.8127 GPa, respectively

Table 1Calculated equilibrium lattice parameters, bulk modulus, heat of formation, the differenparameters for ternary Pb1�xEuxS alloys, binary EuS and NM PbS. Results for x¼ 0.25, 0.50GGA has been used.

Composition (x) FMa0 (�A)

AFa0 (�A)

FMB (GPa)

AB

0 This work 5.9112 5.9112 61.8127 6Experiment 5.9360a 5.9360a 62.8000c e

0.25 This work 5.9214 5.9265 60.0586 50.50 This work 5.9251 5.9351 58.3415 50.75 This work 5.9332 5.9434 55.7224 51 This work 5.9434 5.9489 54.7564 5

Experiment 5.9680b e 55.5000d e

a Ref. [44].b Ref. [7].c Ref. [6].d Ref. [45].e Ref. [46].f Ref. [47].

(Table 1). As expected from the fourth order gradient expansion ofcharge density in WC GGA, our estimated value of PbS latticeconstant is in between earlier reported values obtained using LDAand GGA functionals [8] which are underestimated and over-estimated, respectively, and amounts to a small mismatch (0.44%)with the experimental lattice constant at 300 K (5.9360 �A) [44].Similarly, our calculated value of lead sulfide bulk modulus is inbetter agreement with experiment as compared to other theoret-ical results [8,45].

In the case of EuS, we first performed standard DFT calculationusing WC GGA functional (for both non-spin-polarized and spin-polarized cases) that resulted in lattice constants which are muchsmaller when compared to the experimental lattice constant of EuS5.9680 �A [7]. Furthermore, the band structures of EuS obtainedusing both WC and EV GGA functionals without considering theCoulomb’s potential (U) show a metallic nature which is contra-dictory to experimental observation that EuS is magnetic semi-conductor. Only after the incorporation of Uf-expt. (¼10 eV) in theWC GGA for FM EuS, we were able to obtain lattice constant(5.9434 �A) and bulk modulus (54.7564 GPa) which are in goodagreement with experimental data (see Table 1) [7]. Moreover, as Uf

influences only the f states of Eu, quite predictably the f states aresplit into occupied and unoccupied states for spin-up and spin-down channels, respectively, giving a band gap around EF whichis discussed below inmore details. Since it is important to know thepreferred magnetic ordering in a magnetic compounds, we havealso performed volume optimization of EuS in AF ordering andobtained the energy difference between the two competing mag-netic orders using the equationDE¼ E(AF)� E(FM). A positive valueof DE (0.14 eV) confirms that B1 phase of EuS is stable in the FMordering. Additionally, the magnetic transition temperature of EuSis very low which implies that the exchange interactions in thesecompounds are weak. Each Eu atom has 12 nearest-neighbors (NN)and 6 next-nearest-neighbors in the FCC EuS. Since the NNN areconnected through S bridges, their interactions J2 are dominated bythe superexchange. On the other hand, the NN interact via smallerexchange coupling J1 that involves direct EueEu exchange. Thepreferred magnetic order can be deduced from the sign of J1 þ J2. Tocompute J1 and J2 for EuS (and subsequently for Pb1�xEuxS (atx ¼ 0.25, 0.50 and 0.75) we map the calculated minimum totalenergies to the Heisenberg Hamiltonian

H ¼ �J1X

<i;j>

ei!� ej

!� J2X

�i;j[

ei!�ej! (1)

where <i,j>and �i,j[ indicate summation of over NN and NNN,respectively, and e! are the spin direction unit vectors. For the FM

ce in minimum total energies of FM and AF ordering and the magnetic exchange, 0.75 and 1.0 have been obtained usingWC GGA þUf-expt.(¼10 eV), while for PbSWC

F(GPa)

FMDH (eV)

AFDH (eV)

DEmeV

J1meV

J2meV

1.7805 �1.0921 �1.0921 e e e

�1.0622e e e e e

9.9190 �1.8832 �1.9923 �650 �81.250 �54.1678.0306 �2.8521 �2.9264 �740 �92.504 �61.676.1421 �3.4574 �3.7004 �910 �113.758 �75.8344.7247 �4.5436 �4.3984 140 17.50 11.667

�4.6777f e e e e

K.M. Wong et al. / Solid State Sciences 18 (2013) 24e35 27

and AF cases considered in this study the classical energies corre-spond to E(FM)¼ E0 � 6J1 �3J2 and E(AF)¼ E0 þ 2J1 �3J2, where E0is the reference energy. The exchange parameters listed in Table 1give a positive J1 þ J2 for EuS that again confirms its FM stability.

As regards the electronic properties of parent binary compounds,the electronic structure calculations for PbS have been performedusing WC GGA, EV GGA and EV GGA þ SOC. On the other hand, forthe binary EuS we have utilizedWC GGAþUf-expt., EV GGAþUf-expt.,EV GGAþUf andWCGGAþUdþ Uf (where values of Ud andUf weretuned to obtain experimental Eg) functional. In Fig. 1 some selectedband structure diagrams of the parent binary compounds are pre-sented. As the spin direction (i.e. spin-up ([) and spin-down (Y))gives a spin-polarized band structure only for the magnetic EuS,both spin-up and spin-down band structures of FM EuS are shown

Fig. 1. Electronic band structures of PbS (top panel) obtained using EV GGA functional withospin-up ([) and spin-down (Y) channels obtained using WC GGA þ Uf-expt. (¼10 eV) (midd

in Fig. 1 which have been obtained usingWC GGAþUf-expt. (¼10 eV)and EV GGA þ Uf (¼6 eV). At this stage it is worth pointing out thatfor all subsequent electronic properties calculated using standard aswell as extended EV GGA functional, we have used the lattice pa-rameters computed using the WC GGA þ Uf-expt. which are listed inTable 1.

In case of PbS, the direct energy band gaps computed using WCand EV GGA functional located at L symmetry point have beenestimated to be 0.19 eV and 1.11 eV, respectively, which are smallerand larger, respectively, when compared with experimental valueEexptg ¼ 0:278 eV [48]. Since our results show that inclusion of SOCcauses the Eg of lead chalcogenides to decrease by producinga better sep splitting in conformity with experimental observations,we find that only EV GGA þ SOC band gap value (Eg ¼ 0.29 eV) is in

ut and with the inclusion of SOC. The spin-polarized band structure of EuS projected forle panel) and EV GGA þ Uf (¼6 eV) (bottom panel).

K.M. Wong et al. / Solid State Sciences 18 (2013) 24e3528

good agreement with the experiment. The sep splitting (afterincorporating SOC) in the conduction and valence bands of PbS isalso clearly visible in Fig. 1 (top right panel). Fig. 2 shows the totaland partial density of states for PbS obtained usingWCGGA, EV GGAand EV GGAþ SOC. It can be clearly seen that in case ofWC GGA thesimple form of this functional is unable to predict correct band order(i.e. the positioning of Pb s and p like states in valence and con-duction bands) of the lead sulfide. The domination of the top ofvalence band around the EF by the Pb-6p states gives rise to a socalled “negative” band gap which is contradictory to experimentand has been thoroughly investigated theoretically in some recent

Fig. 2. Total and partial electronic density of states for NS PbS (left panel) obtained using WCand for FM EuS (right panel) obtained using WC GGA þ Ud (¼8 eV) þ Uf (¼6.5 eV) (black solid(green dashed line) projected for spin-up ([) and spin-down (Y) channels (For interpretationof this article.).

studies [8,43]. In contrast to the results of WC GGA, the resultsobtained using EV GGA functional both with and without the in-clusion of SOC produce correct band order around the EF showingthat the top of the valence band is dominated by Pb-6s and thebottom of conduction band is predominantly made up of the Pb-6pstates (in Fig. 2). Without the incorporation of SOC, the Pb-6s andPb-6p states are hybridized. It can, therefore, be concluded that theEV GGA þ SOC for relativistic effects caused by Pb atoms yields notonly Eg which is in good agreement with experimental value, it alsoproduces the correct band order and sep splitting in Pb containingcompounds compared to other DFT functionals.

GGA (black solid line), EV GGA (red dotted line) and EV GGA þ SOC (green dashed line)line), WC GGA þ Ud (¼3.5 eV) þ Uf (¼10 eV) (red dotted line) and EV GGA þ Uf (¼6 eV)of the references to color in this figure legend, the reader is referred to the web version

K.M. Wong et al. / Solid State Sciences 18 (2013) 24e35 29

In case of EuS, our computed indirect Eg between X � G sym-metry points obtained by usingWC GGAþUf-expt. and EV GGAþ Uf-

expt. are found to be underestimated (1.23 eV) and overestimated(2.41 eV), respectively. Here it is worth mentioning that the largerEg values for both binary PbS and EuS obtained by using EV GGA area direct consequence of increase in energy levels along L, X and G

symmetry points providing a more accurate exchange-correlationpotential at the expense of less accurate exchange energy forboth binary and ternary compounds [49,50]. In order to achieveaccurate electronic band structure profile of EuS we have system-atically implemented various modifications of WC and EV GGA þ Ufunctionals by confirming that: (a) The experimental indirect bandgap (Eexptg ¼ 1.64 eV) of EuS should be produced by the choice offunctional [7], (b) in the spin-up band structure the top of thevalence band should be dominated by the Eu-4f states and thebottom of the conduction band should be dominated by Eu-5d [28]and (c) the S-3p states should appear below the Eu-4f states in thevalence band in the spin-up band structure [28]. In case ofWC GGA,based on the fact that Uf-expt. already underestimates the exper-imental band gap, we adopted optimization procedure similar tothat described in Ref. [13] by choosing Ud and Uf as parameters andvarying them between 2 and 11 eV. On the other hand, in additionto the above optimization procedure for Ud and Uf for investigatingthe EV GGA þ U band structure profile of EuS, we also decreased Uffrom its experimental value in regular steps of 0.5 eV while keepingUd ¼ 0.0 eV. This additional U optimization procedure was adoptedbased on the fact that inclusion of Uf-expt. in EV GGA, unlike the caseof WC GGA þ Uf-expt., already overestimates the experimental bandgap and the S-3peEu-4f splitting in the VB. Some selected values ofthe band gap, positions of states making up the bottom of the CBand the top of the VB and the S-3peEu-4f splitting in the VB arelisted in Table 2. In all spin-down band structures obtained usingeither of the extended GGA functional, the Eu-4f states are com-pletely missing in the valence band and only appear above 10 eV inthe CB.

As already mentioned, the Eg of EuS obtained using the WC GGAfunctional can be tuned to the Eexptg either by using Ud < Uf orUd > Uf, however, for the case of WC GGA þ Ud (¼3.5 eV) þ Uf(¼10 eV) from the DOS plots shown in Fig. 2 and Table 2, it can beseen that the Eu-4f states are shifted below the S-3p states in thevalence band which is contradictory to experiment. On the otherhand,WC GGAþUd (¼8 eV)þ Uf (¼6.5 eV) does succeed in yieldinga better Eu-4f and S-3p splitting and a good approximation to the

Table 2Comparison of the band gap, Eg (eV), states making up the bottom of CB, statesmaking up the top of VB and the S-3peEu-4f splitting inside the VB for EuS obtainedusing WC GGA þ Ud þ Uf and EV GGA þ Ud þ Uf.

Functional Ud (eV) Uf (eV) Eg (eV) Bottomof CB

Topof VB

S-3peEu-4fsplitting inVB (eV)

WC GGA 0 11 1.24 Eu-5d S-3p 0.160 10 1.23 Eu-5d Eu-4f 0.372 10 1.54 Eu-5d Eu-4f 0.273.5 10 1.63 Eu-5d S-3p 0.195 5 0.95 Eu-5d Eu-4f 1.718 6.5 1.62 Eu-5d Eu-4f 0.96

EV GGA 0 11 2.52 Eu-5d S-3p �0.810 10 2.41 Eu-5d S-3p 0.150 8 2.11 Eu-5d S-3p 0.750 7 1.87 Eu-5d Eu-4f 1.120 6 1.62 Eu-5d Eu-4f 1.532 10 2.76 Eu-5d S-3p �0.573.5 10 2.93 Eu-5d S-3p �0.685 5 2.19 Eu-5d Eu-4f 1.548 6.5 3.02 Eu-5d Eu-4f 0.73

experimental Eg for EuS. However, when compared to the results ofEV GGAþUf (¼6 eV), it is evident from Table 2 that the Eu-4f and S-3p splitting is higher and the Eu-5d states appear to dominate thelower CB edge more prominently. In the spin-up band structure ofEuS obtained using EV GGA þ Uf (¼6 eV) (Fig. 1), the presence ofshallow Eu-4f states can be seen below the Fermi level mixed witha minor contribution of the S-3p states (see Fig. 2), which is inaccordance with previous calculations for europium chalcogenides[51]. Hence, from the above results, it can be concluded that theelectronic properties of binary PbS and EuS computed using the EVGGA functional provide a better band structure profile of the binarycompounds provided that the SOC for Pb and the appropriate valueof Uf (¼6 eV) for Eu are taken into consideration. Therefore, for theresults of Pb1�xEuxS alloys presented in the following sections, wehave obtained the structural properties using the WC GGA þ Uf-

expt. þ SOC. However for the electronic properties of the alloysunder study the electronic band structures and the DOS have beencomputed using the EV GGA þ Uf (¼6 eV) þ SOC.

3.2. Ternary alloys Pb1�xEuxS

3.2.1. Structural properties and stable magnetic orderingThe ordered structures of the FM Pb1�xEuxS alloys have been

obtained for the compositions x ¼ 0.25, 0.50 and 0.75 by replacingone, two and three Pb atoms with Eu atoms in the PbS unit cell,respectively. In case of FM alloys at x ¼ 0.25 and x ¼ 0.75, theresulting structure is an eight-atom cubic cell (space group # 221,Pm3m), whereas for x ¼ 0.50 a four-atom tetragonal cell (spacegroup # 123, P4/mmm) has been obtained as the smallest orderedstructure. For the three alloy composition values mentioned above,the AF cell is constructed by doubling the corresponding FM cellalong [001] axis (i.e. a 1 � 1 � 2 supercell of the FM cell). For theternary alloy concentrations x ¼ 0.25 and x ¼ 0.75 the AF cellsobtained from FM cells belong to the tetragonal space group 99(P4mm through 1 � 1 � 2 supercell). The equilibrium lattice con-stant (ao) and the bulk modulus (B0) for all concentration valueshave been computed for FM and AF coupling using WC GGA þ Uf-

expt. ¼ 10 eV (presented in Table 1) by fitting the Murnaghanequation of states [52] to the minimum total energy versus volumecurves. In Fig. 3 we present the total energy as a function of unit cellvolume for x ¼ 0.25, 0.50 and 0.75 values of Pb1�xEuxS alloys in theFM and AF magnetic orders. The optimized ground state structuralparameters listed in Table 1 clearly show that the lattice constantincreases whereas the bulk modulus decreases on going from PbSto EuS show linear dependence on concentration x. In Fig. 4 wepresent the variation of the lattice constant and bulk modulus forFM and AF Pb1�xEuxS alloys along with Vegards’ lawwhich requiresthat the alloy formed from two iso-structural compounds shouldshow linear dependence of structural properties on dopant con-centration [53]. It is clear from Fig. 4 that both the lattice constantand bulk modulus deviate slightly from the Vegard’s law. In case ofFM alloys the lattice constant at x ¼ 0.25 is larger than the corre-sponding value predicted by the Vegard’s law, however, for rest ofthe concentration values our calculated lattice constant values arean underestimate to the ones predicted by Vegard’s law. On theother hand, all the AF lattice constants are larger than the onespredicted by Vegard’s law. Interestingly, the lattice constants arelarger whereas bulkmoduli are smaller for the AF casewhen resultsare compared with the FM case. The larger lattice constants in theAF case and their deviation from Vegard’s law are in conformitywith the experimental variation in lattice constant for Pb1�xEuxSalloys over the range 0 � x � 0.20 [32] where a lattice constant of5.9445 �A was reported at x ¼ 0.20. In general, all the alloys con-taining Eu have lattice constants larger than PbS in spite of the factthat Eu has a smaller ionic radius as compared to Pb. The presence

Fig. 3. Volume optimization plots for FM (dotted line) and AF (solid line) coupling for (a) Pb0.75Eu0.25S, (b) Pb0.50Eu0.50S, and (c) Pb0.25Eu0.75S alloys obtained using spin-polarizedDFT þ Uf-expt. calculations using the WC GGA parameterization scheme.

K.M. Wong et al. / Solid State Sciences 18 (2013) 24e3530

of strongly correlated f states of Eu atom causes an increase in thevolume of FCC B1 structure on going from x ¼ 0 to x ¼ 1. Anotherfeature visible from Fig. 3 is that the change in the volume ofPb1�xEuxS alloys having B1 structure is small when x is varied from0 to 1, which is consistent with experimentally reported smalllattice mismatch (da0/a0 ¼ 0.5%) in PbS and EuS [7,44]. This smalllattice mismatch in Pb1�xEuxS alloys is useful for overcoming theproblems faced in Pb1�xEuxTe and Pb1�xCdxS alloys as it providesa path for achieving good crystal films preparation lattice matched

with the substrate and also causes reduction in the Auger recom-bination problem for long-wavelength range [32].

As regards the stability of the FM and AF ordering in thealloys under study, one can see from Fig. 3 that the Pb1�xEuxS alloysprefer AF configuration as compared to the FM. The difference inthe minimum total energies of AF and FM phases (i.e.DE ¼ E(AF) � E(FM)) for x ¼ 0.25, 0.50 and 0.75 is estimated to be�0.65 eV, �0.74 eV and �0.91 eV, respectively, while the negativesign of J1 þ J2 confirms this fact. Moreover, the larger J2 values for

Fig. 4. Calculated (a) lattice constants and (b) bulk modulus of Pb1�xEuxS alloys for FM and AF B1 structure as a function of x, compared with the results obtained using Vegards’ law.The solid line between the data points serves only as a guide for eye.

K.M. Wong et al. / Solid State Sciences 18 (2013) 24e35 31

Pb0.75Eu0.25S and Pb0.50Eu0.50S listed in Table 1 indicates that theNNN superexchange dominated the AF ordering, whereas forPb0.25Eu0.75S the NN EueEu exchange dominates the AF ordering. Inorder to further exploit the stability of the binary PbS and EuS andthe Pb1�xEuxS alloys in the FM and AF B1 structure, we havecomputed the heat of formation (DH) for all the binary and ternarycompounds using the formula:

DH ¼ ETotalðPb1�xEuxSÞ � ð1� xÞEPb � xEEu � ES (2)

where DH is the heat of formation, ETotal(Pb1�xEuxS) is the mini-mum total energy of Pb1�xEuxS alloys at particular concentration x,EPb, EEu and ES are the ground state minimum energies per atom ofthe cubic close-packed Pb, body-centered cubic (bcc) Eu andorthorhombic S, respectively. 1 � x (x) is the relative content of Pb(Eu) in the alloy system which ranges from 0 to 1.0.

For x¼ 0 our calculated value of the heat of formation for B1 PbSis �1.0921 eV which is comparable with the experimental value�1.0622 eV [46]. On the other hand, for B1 FM and AFM EuS (i.e. for

x ¼ 1.0) the calculated heat of formations are �4.5737 eV and�4.3984 eV, respectively, which are also in good agreement withexperiment [47]. From these negative values of DH it is clear thatthe FM EuS is more stable in the B1 structure as compared to the AFEuS which is in conformity with our calculated DE values. Onincreasing the concentration x from 0 to 1.0, we observe anincrease in the heat of formation of the alloys system under study,which suggests that with the increasing Eu content the stability ofthe alloy increases. Furthermore, the AF phase has higher DHvalues which are not surprising since the alloys also have negativeDE. In spite of the fact that it is difficult to draw a quantitativeconclusion regarding the stability of these materials based only onthe heat of formation for the whole concentration range withouttaking into account the entropy of the system under study, it isworth pointing out here that our calculated heat of formationvalues are nonetheless very reasonable as whole concentrationrange of these alloys have negative DH, having largest negativevalues for binary FM EuS at x ¼ 0 and AF Pb1�xEuxS alloys at0.25 � x � 0.75.

K.M. Wong et al. / Solid State Sciences 18 (2013) 24e3532

We have also computed the total magnetic moments (MM) ofthe unit cell of FM and AF alloys at x ¼ 0.25, 0.50, 0.75 and 1 whichwere 7.0 and 0, respectively. It is worth pointing out here that foreuropium chalcogenides the FM Curie temperatures can be sig-nificantly increased by external pressure [54] or chemical doping[55]. However, from our results (the negative values of DH and DE),it is evident that the Pb1�xEuxS (0.25� x� 0.75) materials prefer AFordering as compared to FM which is consistent with experimentsfindings that these alloys belong to the class of AFmaterials [34]. Onthe other hand, PbS and EuS (corresponding to x ¼ 0 and x ¼ 1) areNM and FM, respectively. For all x values of the FM ordering, exceptx ¼ 0, the magnetism is exclusively from the Eu atoms in thePb1�xEuxS alloys which is consistent with the half filled f states ofEuþ2 with an orbital moment L ¼ 0 and spin moment J ¼ S ¼ 7/2.

Fig. 5. Electronic band structures of AF (a) Pb0.75Eu0.25S, (b) Pb0.50Eu0.50S, and (c) Pb0.25Eu0

GGA þ Uf (¼6 eV) þ SOC.

3.2.2. Electronic propertiesSince the Pb1�xEuxS alloy shows AF as the preferred magnetic

order, the AF electronic band structures at x ¼ 0.25, 0.50 and 0.75are presented in Fig. 5 which have been computed by using EVGGA þ Uf (¼6 eV) þ SOC. From Fig. 5 one can see that the funda-mental energy band gap values (obtained from the difference be-tween the top and bottom of the conduction and valence band,respectively) increase with the increasing value of x. Our estimatedband gaps for Pb1�xEuxS AF alloys x ¼ 0.25, 0.50 and 0.75 are0.52 eV, 0.74 eV and 0.91 eV, respectively. Together with the bandgaps of binary PbS and EuS, the variation of EF over the completeconcentration range shows nonlinear increase with x which isconsistent with the experimental [32] and theoretical [43] studiesfor lead europium chlcogenide alloys. It is also visible from Fig. 5

.75S projected for spin-up ([) and spin-down (Y) channels. Results obtained using EV

Fig. 6. PDOS for AF (top) and FM (bottom) Pb0.50Eu0.50S alloys projected for spin-up ([)and spin-down (Y) channels where the Pb-s (solid black line), Pb-p (dotted red line),Eu-d (dashed dark green line), Eu-f (solid light green line) and S-p (dashed blue line)contribution to the DOS around the EF is shown. All results obtained using EV GGA þ Uf

(¼6 eV) þ SOC (For interpretation of the references to color in this figure legend, thereader is referred to the web version of this article.).

K.M. Wong et al. / Solid State Sciences 18 (2013) 24e35 33

that the direct energy band gap for x ¼ 0.25 and 0.75 is located atthe M symmetry point of the cubic cell, while for x ¼ 0.50 it islocated at the X symmetry point of the tetragonal cell. It is worthpointing out here that the directed band gaps located at the cubicMsymmetry point and the tetragonal X symmetry in the alloys cor-respond to the FCC L symmetry point of binary PbS, showing thatthe nature of electronic structure of the alloys around the EF isdominated by the Pb atom. The valence band (VB) of these com-pounds in the energy range 0 to �4 eV is dominated by the pres-ence of S-3p states yielding a direct band gap. This also providesfurther insight into the direct band gap nature of the Pb1�xEuxSalloy (in Fig. 5) which is also similar to the direct band gap asobserved in the electronic band structure of PbS (in Fig. 1), due tothe domination of the S-3p states in the VB and Pb-6p states in theCB. Hence our results predict that the ternary Pb1�xEuxS alloys(0� x� 0.75) have a direct band gap naturewhereas for EuS (whichcorresponds to x ¼ 1), the results indicate that its band gap is in-direct. Another interesting aspect visible in Fig. 5 is that the EFseems to be slightly pushed into the valence band which indicatesthe presence of a considerable hole concentration for the alloyssystem under study and results in shifting the absorption edge ofthese materials to shorter wavelengths according to the Bursteineffect [56].

We also obtained the electronic band structures by using EVGGA þ Uf (¼6 eV) þ SOC for the FM alloys, however, for the sake ofunderstanding the behavior of important Pb-6s, Pb-6p, Eu-5d, Eu-4fand S-3p states in the band structure of FM and AF coupling thepartial density of states around the EF for at x ¼ 0.50 are com-paratively shown in Fig. 6. The two plots clearly indicate that thespin projected density of states both in the case of FM and AFcoupling are almost the same. In both cases, the lower portion ofconduction band is dominated by the Pb-6p states followed byminor contributions coming from the S-3p and Eu-5d states in theconduction band (as observed in Figs. 2 and 6). The UVB shown inFig. 6 is dominated by the S-3p states for both spin-up and spin-down channels of the AF and FM alloys. Consequently, the com-pounds (0.25 � x � 0.75) have the upper valence band (UVB)touching the EF for both spin-up and spin-down cases (Fig. 6). Inspin-up case the Eu-4f states in both AF and FM alloy appear at thetop of valence band with a slight difference that the contribution inFM is slightly pulled away from the EF.

The TDOS and PDOS plots of the Pb1�xEuxS alloys for x ¼ 0.25,0.50 and 0.75 are shown in Fig. 7. For clarification, the TDOS plotsare shown in the right panel in Fig. 7 for both the FM and AF caseswhereas the PDOS plots are shown in the left panel in Fig. 7 for theAF cases only. The TDOS plots consist of the combined contributionsfrom all the atoms in the material (Pb, Eu and S) whereas in the AFPDOS plots, the individual contribution coming from Pb, Eu atom 1(with spin-up), Eu atom 2 (with spin-down) and S atom are shown.The two Eu atoms have opposite spins and their contributions tothe PDOS plots are shown in order to illustrate the effect in the AFordering. It is observed that the opposite spins from the two Euatoms (the dark dashed green line and light solid green line,respectively) have negligible difference in the PDOS plots (Forinterpretation of the references to color in this figure legend, thereader is referred to the web version of this article.)..

For all concentration range in the lowest valence bands occur-ring between �13 eV and �11 eV, it is primarily made up of s statesof S for both spin projections with a minor Pb and Eu character. Thecharacteristics of the lowest valence bands are almost similar forthe whole x regime and are separated from the UVB with a con-siderable difference in the energy ordering and the Pb-6s and S-3pstates lying in-between �8 eV and �6 eV except for x ¼ 1. In orderto retain the atomic like character of the S-3s states, these arelocated at a large depth from the Fermi level and do not attain

conduction band likewidth. The Eu-4f and S-3p states dominate theUVB complex for x ¼ 0.25, 0.50 and 0.75 in the spin-up PDOS, witha minor contribution from Pb-6s states. On the other hand, in thespin-down PDOS one can see that the Eu-4f states do not contributeto UVB and only appear at about 10 eV in the conduction band (CB).The shallow nature of S-3p states in the UVB of PbS results inacquiring these states a large width (w4 eV). The width of thesestates decreases as one goes from x ¼ 0 to x ¼ 1 (from Figs. 2 and 7)due to the introduction of f and d states of Eu in the UVB and CB,respectively. For all the concentration values x the first maxima of

Fig. 7. TDOS (left panel) for Pb1�xEuxS alloys in FM (solid black line) and AF (dotted red line) ordering projected for spin-up ([) and spin-down (Y) channels at x ¼ 0.25, 0.50 and0.75. The PDOS (right panel) for Pb1�xEuxS alloys in AF coupling projected for spin-up ([) and spin-down (Y) at x ¼ 0.25, 0.50 and 0.75. In the PDOS the contribution from Pb atoms(dotted red line), Eu1 (atom number 1 with spin-up, dashed dark green line), Eu2 (atom number 2 with spin-down, solid light green line) and S (dashed blue line) atoms in the TDOSof AF alloys are shown. All results obtained using EV GGA þ Uf (¼6 eV) þ SOC (For interpretation of the references to color in this figure legend, the reader is referred to the webversion of this article.).

K.M. Wong et al. / Solid State Sciences 18 (2013) 24e3534

the UVB are dominated by the S-3p states and Eu-4f states slightlybelow it. The Eu-4f contribution appears to move toward the Fermilevel with increasing Eu concentration. In the CB it is found that Pb-6p states lay closer to the UVB as compared to the Eu-5d and Eu-4fstates for x ¼ 0 to x ¼ 0.75, this causes the domination of the PbSlike behavior of the fundamental Eg resulting in a direct band gapfor these materials. The Eu-5d states appear and dominate theconduction band in an energy window of 2e8 eV above EF for bothspin-up and spin-down channels, however, the contributions of Eu-5d states in the conduction band are positioned higher in energyscale for spin-down channel as compared to the spin-up channel.Conversely, the contributions in the conduction band of AF alloy(right panel of Fig. 7) seem to move slightly up in the energy rangecompared to FM orderingwhich is a direct consequence of the spin-polarization. The spin effects could be observed by the decreasingstarting point of the green curves in Fig. 7. Finally, since all calcu-lations were first performed as spin-polarized and the spin-orbitcoupling was introduced afterwards, we were also able toobserve the changes in the electronic properties of the alloysbrought about by the spin-orbit coupling in the Pb atom. Unlike theEu-d and Eu-f states, in the spin-polarized calculations, the Pb-s andPb-p states seem to be unaffected by the spin directions (i.e. spin-

up and spin-down). After the incorporation of spin-orbit couplingfor the Pb atoms, the Pb-s and Pb-p states are pushed upward anddownward, respectively, on the energy scale. This is similar to theeffects seen in the case of PbS binary compound where sep split-ting is brought about due to the inclusion of SOC. As far as thecontribution of Pb-s and Pb-p states in the spin-up and spin-downchannel is concerned, our results unambiguously show that the twospin directions have no noticeable effects on the positioning of the sand p like states of the Pb atom.

4. Conclusions

Using the accurate FP-LAPW þ lo method, we have investigatedthe structural, electronic, magnetic and optical properties ofPb1�xEuxS ternary alloys based on PbS. The calculations have beenperformed within DFT with additional considerations for relativisticeffects in Pb atom and the on-site Coulomb interaction parameter (U)treatment of the strongly localized 4f electrons of Eu. The WC GGAfunctional extended with experimental U value for Eu atom givesaccurate structural properties. We confirm the experimental andtheoretical stable phases of PbS and EuS and show that the ternaryalloys are stable in AF magnetic ordering. The experimental value of

K.M. Wong et al. / Solid State Sciences 18 (2013) 24e35 35

Coulomb parameter (Uf-expt.) within the SIC does not predict an ac-curate EuS energy band structure. The improved electronic bandstructure profiles of ferromagnetic EuS and non-spin-polarized PbShave been achieved by considering the effects of spin-orbit couplingfor Pb atoms, and by utilizing a suitable U parameter for Eu atoms’ fand d electrons within the WC and EV GGAs. Among the variousmodified forms of the two GGA functional used for the electronicproperties, the EV GGA functional seems to give a better overall per-formance for electronic and optical properties of Pb1�xEuxS alloyswhen the strong correlation among the 4f electrons of Eu atom aretreatedusing aUf¼6eV, and the relativistic effects are included for Pbatom. The fundamental bandgap increases nonlinearly ongoing fromPbStoEuS.TheelectronicDOSshowthataround theEF theconductionband is dominated by the Pb-6p states followed by contributionscoming from the S-3p and Eu-5d states, while the S-3p states domi-nate the VB complex for x ¼ 0.25, 0.50 and 0.75 in the spin-up andspin-down cases. The only prominent difference in the spin-up andspin-downcases is thepresence of shallowEu-4f states in the spin-upband structure which show little upward shift toward EF withincreasing Eu content. In addition, with the trend toward deviceminiaturization, the spin effects due to the Eu dopants from some ofour theoretical results studied in this paper reveal important utiliza-tion and possible optimization of themagnetic properties of the DMSspintronic devices consisting of Eu-doped nanostructures.

Acknowledgments

Financial support from the European Research Council Grant(ThreeDSurface), Volkswagen Stiftung and BMBF (ZIK: 3DNanoDe-vice) is gratefully acknowledged.

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