Exchange Interaction and T in Alkaline-earth-metal-oxide-based DMS without c Magnetic Impurities: First Principle Pseudo-SIC and Monte Carlo Calculation ∗ Van An Dinh, Masayuki Toyoda, Kazunori Sato, and Hiroshi Katayama-Yoshida The Institute of Scientific and Industrial Research, Osaka University, 8-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan. The prospects of half-metallic ferromagnetism being induced by the incorporation of C atoms into alkaline-earth-metal-oxides are investigated by the first principle calculation. The origin of the ferromagnetism is discussed through the calculation of the electronic structure and exchange 8 coupling constant by using the pseudo-potential-like self-interaction-corrected local spin density 0 method. TheCurietemperature(Tc)isalsopredictedbyemployingtheMonteCarlosimulation. It 0 is shown that by taking theelectron self-interaction into account, thehalf-metallic ferromagnetism 2 induced by C in the host materials is more stabilized in comparison with the standard LDA case, n andtheC’s2pelectron statesinthebandgapbecomemorelocalized resultinginthepredominance a oftheshort-rangedexchangeinteraction. Whiletheferromagnetism inMgO1−xCx isstabilizeddue J to the exchange interaction of the 1st-nearest neighbor pairs and might be suppressed by the anti- 5 ferromagneticsuper-exchangeinteractionathigherx,theferromagnetisminCaO1−xCx,SrO1−xCx, 1 andBaO1−xCx isstabilized byboththe1st-and2nd-nearest neighborpairs,andTc monotonously increases with theC concentration. ] i c s Besidestheattemptstodiscoverdilute magneticsemi- In this Letter, we will discuss the origin of the ferro- - conductors (DMSs) by incorporating transition metals magnetismthatisinducedbysubstitutingCforOinfour l r into various materials to realize ferromagnetic DMSs for alkaline-earth-metal-oxides−MgO,CaO,BaO,andSrO t m spintronic devices, recently the ferromagnetism induced −throughthecalculationofnotonlytheelectronicstruc- . bynonmagneticimpuritieshasalsoattractedgreatatten- ture but also the exchange coupling constant and Curie t a tion among both theoreticaland experimental scientists. temperature(Tc). Itshouldbe notedthatthe prediction m A novelclassofmagnetic materialscanbe formedbyin- of ferromagnetism and T in ref. 3 and ref. 11 is based c corporatingnonmagneticimpuritiesorbylatticedefects. on the electronic structure calculation using the density - d With regardto this issue,the magnetisminduced by the functionaltheorywithin the localdensityapproximation n cationvacancy in MgO [1], Ca vacancy in CaO [2, 3], Hf (LDA) and the mean field approximation (MFA) which o c vacancyinHfO2 with Tc exceeding500K[4, 5, 6,7], and cannot be successful for describing many materials, es- by hydrogen in graphite [8, 9] and in carbon nanotubes pecially for strongly correlated systems. The LDA often [ [10] etc. has been reported. It has also been predicted overestimates the hybridization between electron states 1 that magnetism can be induced by nonmagnetic impu- due to the underestimation of the bandgap energies of v rities such as C and N by substituting the O atoms in semiconductors. Moreover, the calculation of T within 9 c various oxides. For example, the ferromagnetism caused theMFAoftenpredictsT withanexcessivelyhighvalue 4 c 2 by N and C in alkaline-earth-metal-oxides [3, 11] and evenif the substitutionalconcentrationislowerthanthe 2 thehalf-metallicferromagnetisminducedbyNinquartz- percolationlimit[15]. Therefore,inordertodescribeand 1. SiO2 [12]. predict the properties of these systems, we need a more accurate method. There have been several attempts to 0 It is suggested that the magnetism might arise in the 8 improve the LDA to overcome these inaccuracies. One host materials if impurities have a finite local magnetic 0 of the most popular methods is LDA+U which improves moment that interacts with each other to form a mag- : the LDA by taking the strong correlationeffects into ac- v netic moment net. In the materials in which the mag- i netism is induced by incorporatingnonmagnetic impuri- count through the screened Coulomb parameter U [16]. X Theotherpopularapproachesaretheself-interactioncor- ties,thesubstitutionalionsmayhaveanonzeromagnetic r rection (SIC) [17] and the recent pseudo-SIC [18] meth- a moment and the 2p-electrons of these ions, rather than ods. SIC methods differ from LDA+U in that LDA+U the 3d-electrons, play an essential role in inducing the uses the additionalparameterU, while the SIC methods magnetisminthehostmaterials. Theyformanimpurity do not require any additional parameter at all. Con- band in the deep bandgap, and ferromagnetism can be sequently, in order to investigate the origin of ferromag- induced if the Fermi level lies in these impurity bands. netisminalkaline-earth-metal-oxide-basedDMSs,weap- Furthermore, recent research also shows the role of the ply the pseudo-SIC method based on the improvement 2p-like impurity band formed by C and N in the stabi- of the MACHIKANEYAMA2000 packagecoded by Akai lization of the ferromagnetism in Ga1−xMnxAs [13] and [19](LDA+SIC)inthecalculationoftheelectronicstruc- In1−xMnxN [14]. tures, and then we use the formula by Liechtenstein et al.[20] to calculate the exchange interaction J between ij two impurities at the ith- and jth-sites in the ferromag- ∗Electronicaddress: [email protected] neticcoherentpotentialapproximationmedium. Finally, 2 FIG. 1: Total DOS (solid line) and C’s 2p - PDOS (dashed line)forCaO0.9C0.1. Theleft-handfigure(a)showsthestan- dardLDAcalculationandtheother(b)showsthepseudo-SIC results. Theupperplanecorrespondstothemajorityspinand lower plane tothe minority spin. we employ the Monte Carlo simulation to estimate T . c FIG. 2: Total DOS (solid line) and C’s 2p - PDOS (dashed The substitution of O with C in alkaline-earth-metal- line) obtained bypseudo-SICcalculation for MgO0.9C0.1 (a), oxides is treated randomly. For convenience, the lattice CaO0.9C0.1 (b), SrO0.9C0.1 (c), and BaO0.9C0.1 (d). constants of alkaline-earth-metal-oxide-based DMSs are fixedtothevaluesofundopedcrystals[21](a=4.123˚A, 4.909 ˚A, 5.160˚A, and 5.520˚A for MgO, CaO, BaO, and SrO, respectively) and no distortion in lattice structure from Fig. 2, all materials in question are half-metallic. is assumed. Throughout the electronic structure calcu- However,theexchangemechanismcausingtheferromag- lations, 624 independent k-sampling points in the first netism is somewhat different. Figure 2(a) illustrates the Brillouin zone are used. The potential form is restricted DOS of MgO0.9C0.1. With the smallest lattice constant, to the muffin-tin type, and muffin-tin radiiare chosenin MgO0.9C0.1 has the largest bandgap energy. The C’s 2p such a way that the ions at lattice sites can touch each states are located near the top of the valence band orig- other. The relativistic effect is also taken into account inated from anion p-states, resulting in the strong hy- by using scalar relativistic approximation. bridization of 2p electron wave functions. The majority In order to compare the density of states (DOS) spin states hybridize with O’s 2p states, leading to the calculated within the standard LDA with that within appearanceoftheimpuritybandthatconnectstothetop LDA+SICweplottheDOSofthetypicalcaseofalkaline- of the valence band and causes a narrowermajority spin earth-metal-oxides, CaO1−xCx, at x = 0.10 in Fig. 1. bandgap. The minority spins create an impurity band The figure on the left-hand side correspondsto the DOS in the bandgap, which includes the Fermi level. This calculated within the standardLDA (Fig. 1(a)), and the impurity band is broadened with a half-width of about figure on the right-hand side illustrates the DOS within 1.5 eV. The exchange splitting in this material approxi- LDA+SIC (Fig. 1(b)). It can be seen that, taking the mates to 2.14 eV and is the smallest compared with the self-interaction of electrons into account, the LDA+SIC three remaining materials. These impurity bands can calculation gives a wider bandgap than the standard be broadened more strongly with increasing C concen- LDA. While the position of the minority spin states of tration, leading to the antiferromagneticsuper-exchange C’s 2p electrons remains unchanged, the majority spin interaction being easy to occur and the compensation states calculated within LDA+SIC shifts about 1.1 eV of majority and minority spins. These mean that the in comparison with the standard LDA one, leading to a ferromagnetismmightbesuppressedathigherCconcen- higherlocalizationofLDA+SIC2pstatesthanLDA.And trations. In addition, our calculations for higher C con- the local magnetic moment of C increases from 1.242µB centrationsshow that the ferromagnetismin MgO1−xCx (LDA) to 1.482µB (LDA+SIC). The exchange splitting is most stabilized at x ≈ 0.10 (also refer to Fig. 5). inthe caseofLDA+SICincreasesapproximately2times Figures 2(b)−2(d) demonstrate the DOS of CaO0.9C0.1, as compared to that in the case of LDA, resulting in the SrO0.9C0.1, and BaO0.9C0.1, respectively. As seen from possibility of the suppression of super-exchange interac- thefigures,thebandgapbecomesnarrowerandthelocal- tion and the enhancement of the ferromagnetic double izationofthe 2pstates becomes strongerwith increasing exchange mechanism. the distance between atoms (or lattice constant). The Figure 2 depicts the DOS of AO1−xCx (A = Mg, Ca, majority spin states are located in the valence band and Sr,Ba)calculatedwithinLDA+SICatx=0.10. Asseen causeasmallbroadeningoftheband. TheFermilevellies 3 FIG. 3: Local magnetic moment of substitutional C atoms in MgO1−xCx, CaO1−xCx, SrO1−xCx, and BaO1−xCx at x = 0.10. intheimpuritybandinducedbyminorityspinswith1/3 of this impurity band being occupied by electrons. The FIG. 4: Exchange coupling constant vs. distance between C exchangesplittinginthesematerialsisapproximately2.3 atoms in units of the lattice constant a in MgO1−xCx (a), eV, but the C’s 2p states in BaO0.9C0.1 is the most lo- CaO1−xCx (b), SrO1−xCx (c), and BaO1−xCx (d) at several concentrations of C (x=0.05,0.10,0.15, and 0.20). calized. This change in the localization with respect to the lattice constant is also indicated by the value of the localmagneticmomentofCwhichisshowninFig.3. As a result, one can expect the ferromagnetism to be stabi- lized by the predominantdouble exchangemechanismin the1st-nearestneighbors,whiletheinteractionsbetween these materials even at higher C concentrations. the2nd−nearestatomseasilyaccurthroughthemediate atoms such as Sr or Ba which have large ionic radii. The exchange interactions Jij in MgO1−xCx, In short, the role of the nearest neighbors in the ex- CaO1−xCx, SrO1−xCx, and BaO1−xCx are shown changeinteractionchangeswiththelatticeconstant. The in Fig. 4. As seen from Fig. 4(a), the exchange inter- higher the lattice constant, the greater is the contribu- action of the 1st-nearest neighbors in MgO1−xCx is tion from the 2nd-nearest neighbors in the induction of the strongest. The interaction strength is considerably themagnetisminthehostmaterials. Theexchangeinter- strong (about 140 meV for 5% of the C concentration) actionisveryshortrangedandthe ferromagneticdouble and most of the contributions come from the nearest exchange mechanism is expected to be predominant for neighbor interaction J01. Except J01 and J04 (at 10% of the C concentration), contributions from other J all materials in question. Except MgO1−xCx, the mag- ij are very small and ignorable. Thus, the exchange netism in CaO1−xCx, SrO1−xCx, and BaO1−xCx can be more stabilized at higher C concentrations. This is also interaction in MgO1−xCx can be considered as a typical consistentwiththeresultsoftheMonteCarlosimulation case of short ranged interactions. In addition, the for the estimation of T . exchange interaction is considerably suppressed as the c C concentration increases. At x = 0.20, the exchange In order to evaluate Tc, we perform the Monte Carlo interaction becomes very small and the magnetism is simulations. The Metropolis algorithm [22] is applied fully suppressed. Interestingly, by increasing the lattice to calculate the thermal average of the magnetization constant, the role of the 2nd-nearest neighbors (J02) M and its powers. Then, the cumulant crossing method becomes more important. For CaO1−xCx (Fig. 4(b)), proposedbyBinder[22]isemployedandthefourthorder J01 is approximately two times smaller than that in cumulant U4 (a linear combination of <M4>/<M2>2) MgO1−xCx, but the contributions J02 from the 2nd- is calculated as a function of temperature for different nearest neighbors becomes important. The dominance cell sizes (14×14×14,16×16×16, and 18×18×18 of the contributions J02 of the 2nd-nearest neighbor conventional fcc cells) to find the universal fix-point at pairs in SrO1−xCx and BaO1−xCx is shown in Fig. 4(c) Tc. We estimate Tc for four values of the substitutional and Fig. 4(d), respectively. In these former materials, C concentrations: x = 0.05,0.10,0.15, and 0.20. The the exchange interactions of the 2nd-nearest neighbors obtainedresultsaredemonstratedinFig.5. Somepoints aremoreimportantthanthe1st-nearestneighbors. This should be clarified here. canbe causedby the sufficiently largedistances between First,asdiscussedaboveforFig.2(a)andFig4(a),the 4 change interaction J01 between the 1st-nearest neighbor pairs considerably decreases with the increasing lattice constant, owing to the contributions J02 from the 2nd- nearestneighbors,T monotonouslyincreaseswithxand c can gain a value higher than room temperature if the C concentration is high enough. However, a question that arises here is the solubility of C in the materials. In conclusion, we have presented the results of the study on the origin of ferromagnetism and predicted T c ofalkaline-earth-metals-oxide-basedDMSswithouttran- sition metal elements. The electronic structures and ex- changecouplingconstantsarecalculatedbyapplyingthe pseudo-SIC approach. The dominant exchange mecha- nismandtheroleofthenearestneighborsintheseDMSs are discussed, and T is also evaluated by employing the c Monte Carlo simulation. In short, some comments can be made as follows: (i) All DMSs in question have a half-metallic ferro- magnetism. The ferromagnetic double exchange mecha- nism is predominantfor all materials in question, except forMgO1−xCx whenthe C concentrationsishigherthan 10%. FIG. 5: Curie temperature Tc vs. substitutional impurity (ii) The exchange interaction in these materials is concentration in MgO1−xCx (solid line), CaO1−xCx (dashed strong but short ranged. While the contributions line), SrO1−xCx (dashed-dot-dot line), and BaO1−xCx come mostly from the 1st-nearest neighbor pairs in (dashed-dot line). The circles, crosses, and triangles denote thecalculated points. MgO1−xCx, the important role played by the 1st- and 2nd-nearestneighborpairsinCaO1−xCx,SrO1−xCx,and BaO1−xCx is comparable. However, while J01 becomes weaker, J02 gets considerably stronger with the lattice ferromagnetism of MgO1−xCx is the most stabilized at constant being larger and becomes the dominant con- x ≈ 0.10 and will be suppressed at higher C concentra- tribution to the stabilization of the ferromagnetism in tionsduetotheanti-ferromagneticsuper-exchangeinter- SrO1−xCx and BaO1−xCx. action and the suppression of the majority and minority (iii) Correspondingly, Tc of CaO1−xCx, SrO1−xCx, spins. Tc of MgO1−xCx (solid line) increases with x in andBaO1−xCx increasesmonotonouslywiththeincreas- the rangeof xfrom0.05to 0.10andhas a peak of108K ing C concentration, while Tc of MgO1−xCx gains the at x ≈ 0.10, and then it sharply drops as a function highestvalueat10%oftheCconcentration,thenitdrops of −x at higher x. As shown in Fig. 2(a), the impu- sharply and tends to reachzero at higher concentrations rity band formed by C’s 2p electrons is strongly broad- due to the anti-ferromagnetic superexchange interaction ened. ThebandwidthW oftheimpuritybandinthegap and the compensation of majority and minority spins. can be much larger than the effective correlation energy U =E(N+1)+E(N−1)−2E(N),andtheStoner’scondi- tionfortheexistenceofmagnetismmightbecontravened Acknowledgments [11, 12]. Correspondingly, the magnetism might be fully suppressed at the C concentrations higher than 16% be- cause of the strong broadening of the impurity bands of This research was partially supported by a Grant-in- both majority and minority spins, resulting in the com- Aid for Scientific Research in Priority Areas “Quantum pensationofspins;therefore,thematerialinquestionhas Simulators and Quantum Design” (No. 17064014) and a nonmagnetic behavior at higher C concentrations. “Semiconductor Nanospintronics,” a Grand-in-Aid for Second,contrarytoMgO1−xCx,theimpuritystatesin Scientific Research for young researchers, JST-CREST, CaO1−xCx,SrO1−xCx,andBaO1−xCx arelocalizedand NEDO-nanotech, the 21st Century COE, and the JSPS thebandwidthsoftheimpuritybandsformedbyminority core-to-core program “Computational Nano-materials spins in the gap are sufficiently small to satisfy Stoner’s Design.” We aregratefulto Prof. H. Akai (Osaka Univ.) condition; hence, the ferromagnetism is stabilized by a for providing us with the MACHIKANEYAMA2000 predominant ferromagnetic double exchange mechanism packageand to Prof. A. Yanase (OsakaUniv.) for many and is more stabilized as x increases. Although the ex- valuable discussions. 5 [1] L. E. Halliburton, D. L. Cowan, W. B. J. Blake and J. [12] V.A.Dinh,K.SatoandH.Katayama-Yoshida: Sol.Stat. E. Wertz: Phys. Rev.B 8 (1973) 1610. Commun. 136 (2005) 1. [2] I. S. Elfimov, S. Yunokiand G. A. Sawazky: Phys. Rev. [13] V. A. Dinh, K. Sato and H. Katayama-Yoshida: Jpn. J. Lett.89 (2002) 216403. Appl. Phys.42 (2003) L888. [3] K. Kenmochi, M. Seikei, K. Sato, A. Yanase and H. [14] V. A. Dinh, K. Sato and H. Katayama-Yoshida: J. Su- Katayama-Yoshida: Jpn.J.Appl.Phys.43(2004)L934. percond 18 (2005) 47. [4] M. Venkatesan, C. B. Fitzgerand and J. M. D. Coye: [15] K. Sato, P. H. Dederichs and J. Katayama-Yoshida: Nature(London) 430 (2004) 630. Phys. Rev.B 70 (2004) 201202. [5] J. 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