PHYSICAL REVIEW B 89, 115129 (2014)

Ferromagnetism in Cr-based 3d-5d double perovskites: Effective model analysis and simulations

Prabuddha SanyalIndia Institute of Technology Roorkee, Century Road, Roorkee, Uttarakhand 247667, India

(Received 7 December 2013; revised manuscript received 6 March 2014; published 26 March 2014)

Ferromagnetism in Cr-based double perovskites is analyzed using an effective model as well as a simulationapproach. Starting from a microscopic model proposed recently for this class of double perovskites, an effectivespin-only model is derived in the limit of large exchange coupling at the B site. Analytic expressions for theresultant exchange coupling are derived in certain limiting cases. The behavior of this exchange is used toprovide a plausible explanation for the enhanced ferromagnetic tendency as well as the enigmatic increase inCurie temperature observed in Cr-based double perovskites in the series Sr2CrWO6, Sr2CrReO6, and Sr2CrOsO6.The superexchange between neighboring B and B ′ sites is found to play a crucial role in both stabilizingferromagnetism (especially in the latter two compounds) as well as increasing Tc.

DOI: 10.1103/PhysRevB.89.115129 PACS number(s): 75.47.Lx, 71.10.Fd, 71.10.Hf, 71.20.Be

I. INTRODUCTION

In recent times, ferromagnetic transition-metal compoundswith high Tc have attracted attention due to their importancein spintronics and other technological applications. Whiledoped rare-earth manganites have been studied for decadesfor their colossal magnetoresistance (CMR) property, no lesswell known are the double perovskites, with a general formulaA2BB ′O6, where A denotes alkaline–rare-earth metals and B

and B ′ denote transition metals. The most well known member,Sr2FeMoO6 (SFMO), has a Tc of 410 K, which is higherthan most manganites [1,2]. Moreover, the half-metallic nature[1,3–6], coupled with the substantial tunneling magnetore-sistance obtained in these compounds at low temperatures,especially when powdered, makes it valuable for spintronicapplications [7–10]. The high Tc implies a strong polarizationeven at room temperatures, enhancing its industrial impor-tance. In an effort to boost Tc even further, researchers haveelectron-doped this compound [4,11,12,13]. Although Tc doesincrease in this process up to some point [12,14], it has beenshown in a recent work that upon overdoping, Tc actuallydecreases and the ferromagnetism becomes unstable [15,16].In fact, it gets replaced by antiferromagnetic phases. Thisconclusion was derived within both the model Hamiltonianapproach [15] and the ab initio approach. Within the modelapproach, it was observed that the effective exchange inter-action between Fe S = 5/2 core spins changes sign as thefilling increases, signaling a crossover from ferromagnetic toantiferromagnetic. Within the ab initio approach, consideringthe Sr2−xLaxFeMoO6 series [16], it was observed that the fer-romagnetic state became progressively unstable as the numberof valence electrons was increased through increased dopingof La, the balance tilting at an electron count of about 2.4.While the La-overdoped regime of this compound has not beeninvestigated in detail as of yet, recent experimental data supportour claims regarding the disappearance of ferromagnetismon electron doping [17]. Hence simple electron doping byA-site cation substitution is not a very promising methodfor increasing Tc beyond a certain point. There is, however,another well-known technique to change the electron filling,i.e., to substitute instead the B ′-site ion. In this case, it is wellknown that in Cr-based double perovskites such as Sr2CrB′O6,as one goes across the period, substituting B ′ in turn as W,Re, and Os, Tc increases progressively, reaching a high of

over 700 K for the Os compound [18–20]. Obviously, theferromagnetism is not becoming unstable in this series ofcompounds, even though the filling is increasing from 1 to2 to 3 as one goes from W to Re to Os, respectively. Onthe contrary, it is becoming more stabilized, as signified bythe increasing Tc. Also, another difference with the La-dopedSFMO series is that while all the members of this series aremetallic, the Sr2CrOsO6 is insulating. In this paper, we probethe role of a superexchange mechanism, in addition to thekinetic energy driven mechanism already prevalent, to accountfor this anomalous stabilization of ferromagnetic behavior.While ab initio and variational approaches had [21] provideda pointer to this superexchange mechanism, in this paperwe conclusively demonstrate its importance using effectiveexchange calculations and direct numerical simulations. Theorganization of the paper is as follows. In Sec. II, we shallbriefly summarize the Hamiltonian as well as the main resultsfor the Sr2−xLaxFeMoO6 series (i.e., cation site substitution),just to serve as a reminder and as a contrast with the Cr-basedseries (B-site substitution). In Sec. III, we provide a briefaccount of the main results of ab initio studies upon thesecompounds, which have been reported in detail earlier. In Sec.IV, we formulate a modified Hamiltonian for this Cr series.We proceed to derive a low-energy, spin-only model from thisfermionic Hamiltonian in Sec. V, and we analyze its magneticproperties. In Section VI, specific details about the analyticalcalculation are presented, while in Section VII, details of thenumerical calculation and the results are discussed. A briefconclusion is given in Section VIII.

II. A-SITE CATION DOPING: BRIEF SUMMARY

A-site cation substitution can be done upon the parentcompound Sr2FeMoO6 using, for example, La, which hasa nominal valence state of 3+, in place of Sr, which hasa valence state of 2+. This would correspond to electrondoping of the system. Ab initio studies on this series ofcompounds [16] using N th-order muffin-tin-orbital (NMTO)[22] downfolding have shown that the relative positions of Feand Mo t2g orbitals remain almost unchanged upon La doping;only the Fermi energy shifts almost like a rigid band picture.Thereupon, total energy calculations using the Vienna Ab initioSimulation Package (VASP) [23] showed that the magnetic

1098-0121/2014/89(11)/115129(6) 115129-1 ©2014 American Physical Society

PRABUDDHA SANYAL PHYSICAL REVIEW B 89, 115129 (2014)

ground state changed from ferromagnetic to antiferromagneticas the electron doping increased. The spin splitting at the Feand Mo site was also obtained by downfolding everythingexcept the t2g orbitals. It was observed that the spin-splittingat the Mo site increased proportional to the filling, so thatthe Stoner I remained essentially constant. For example, inSr2FeMoO6, which has one electron per site, the spin-splittingat the Mo site is about 0.13 eV, while in La2FeMoO6, witha filling of 3, the splitting is 0.37 eV. This showed that themoment at the Mo site is completely induced from the Fespin, and not intrinsic. It was shown in that paper that theHamiltonian which captures this behavior is given by

HFM = εB

∑i∈B

f†iσαfiσα + εB ′

∑i∈B ′

m†iσαmiσα

−tBB ′∑

〈ij〉σ,α

f†iσ,αmjσ,α − tB ′B ′

∑〈ij〉σ,α

m†iσ,αmjσ,α

−tBB

∑〈ij〉σ,α

f†iσ,αfjσ,α + J1

∑i∈B

Si · f†iα �σαβfiβ . (1)

The f ’s refer to the Fe sites and the m’s to the Mo sites. tBB ′ ,tB ′B ′ , and tBB represent the nearest-neighbor B-B ′, second-nearest-neighbor Mo-Mo, and Fe-Fe hoppings, respectively,the largest hopping being given by tBB ′ . σ is the spin indexand α is the orbital index that spans the t2g manifold. Thedifference between the ionic levels, � = εB − εB ′ , defines thecharge-transfer energy. Since among the crystal-field-split d

levels of Fe and Mo, only the relevant t2g orbitals are retained,this gives rise to on-site and hopping matrices of dimension3 × 3. The Si are “classical” (large S) core spins at the B

site, coupled to the itinerant B electrons through a couplingJ1 � tBB ′ . Variants of this two-sublattice Kondo lattice modelhave been considered by several authors [12,24–27] in thecontext of double perovskites.

The ab initio calculations showed that the charge-transferenergy � remains almost fixed upon La doping. Consequently,the doping only corresponds to increasing the Fermi energykeeping the levels fixed, thereby increasing the electronfilling. Thus, the ground state changes from ferromagnetic toantiferromagnetic at a filling of about 2.2–2.4, as shown inRefs. [15,16].

III. B′-SITE SUBSTITUTION: MAIN AB INITIO RESULTS

In the series of compounds Sr2CrB′O6, the B ′ ion, which isin a nominal 5+ valence state, corresponds to a 5d1,5d2,5d3

configuration of W, Re, and Os, respectively. These materialshave been reported to have Tc as high as about 450, 620,and 725 K, with a progressive increase as one moves fromSr2CrWO6 with one valence electron to Sr2CrOsO6 with threevalence electrons. Taking the number of valence electrons asthe sole consideration, the situation of Sr2CrWO6, Sr2CrReO6,and Sr2CrOsO6 is comparable to Sr2FeMoO6, SrLaFeMoO6,and La2FeMoO6. However, unlike the Cr-B ′ (B ′ = W,Re,Os)series, for Sr2−xLaxFeMoO6, the ferromagnetic Tc was foundto decrease with increasing La concentration (i.e., increasingnumber of valence electrons), and finally the antiferromagneticphase takes over the ferromagnetic phase. The Cr-B ′ (B ′ =W,Re,Os) series, however, bears two fundamental differences

compared to the Sr2−xLaxFeMoO6 series. First, the B ′ ionsin Cr-B ′, being 5d transition metals, exhibit significantspin-orbit coupling, which makes these materials suitablefor magneto-optic applications with a large signal, as hasbeen discussed in Ref. [1]. Second, three different chemicalelements, namely W, Re, and Os, are involved in the Cr-B ′series, while for the Sr2−xLaxFeMoO6 series the increasedelectron count is achieved without any changes in the B-B ′sublattice. First-principles calculations using the generalizedgradient approximation (GGA) [28], carried out using VASP

as well as the linear muffin-tin-orbital (LMTO) method, havebeen carried out for the total energy calculations. Thereafter, afew-band, low-energy tight-binding Hamiltonian was derivedusing the NMTO [22] formalism. Details have been reportedelsewhere [21]; here, we recapitulate the main results. It wasobserved that the moment on the B ′ site increased from 0.3μB

for tungsten (W) to 0.81μB for rhenium (Re) to 1.44μB forosmium. The total moment, on the other hand, goes from 2μB

in W to 1μB in Re to 0μB in Os. It is observed that although thetotal moment decreases in steps commensurate with the filling,and the Cr moment remains almost fixed, the moment at the B ′site increases drastically, pointing to the growing localizationof electrons on this site. To probe this issue in more detail, theSr oxygen orbitals as well as the eg orbitals of Cr and B ′ ionswere downfolded using the NMTO formalism, keeping onlythe Cr and B ′ t2g orbitals. It was observed that the spin-splittingat the B ′ site increased from 0.06 eV at the W site to 0.31 eV atthe Re site to 0.53 eV at the Os site. This is obviously a muchfaster increase than the filling would dictate, indicating that theStoner I is itself increasing. If we multiply the splitting for W,i.e., 0.06, by 2 and 3, respectively, then the extra amount mustcorrespond to a different energy scale in the problem. Let usrefer to this energy scale as J2, while the traditional extremelylarge spin-splitting at the B site, or the chromium site, definesthe other exchange energy scale, J1 [16].

IV. HAMILTONIAN

From the above considerations, it is obvious that thereexists a different exchange energy scale from the problem,apart from the spin-splitting at the Cr site J1, which, beingrelated to the Hund coupling, for all practical purposes canbe considered infinite. The increasing localized character ofthe moment on the B ′ site makes superexchange an obviouscandidate for this new energy scale [29]. Hence we haveadded a superexchange term to the Hamiltonian correspondingto superexchange between the t2g spin on the B site andthe electron spin on the B ′ site. Hence, the representativeHamiltonian is given by

H = HFM + J2

∑i∈B ′

Si · m†iα �σαβmiβ. (2)

This spin-fermion Hamiltonian is generally difficult to solveexactly, but some headway in the direction of understandingthe low-energy magnetic behavior can be made if one canobtain a spin-only model from this with an effective exchange.In principle, this should be possible by tracing out the fermiondegrees of freedom, but that is a Herculean task consideringthe multitude of possible configurations. Instead, we considerthe approximate but enlightening procedure of self-consistentrenormalization (SCR) devised by Kumar and Majumdar [30].

115129-2

FERROMAGNETISM IN Cr-BASED 3d-5d DOUBLE . . . PHYSICAL REVIEW B 89, 115129 (2014)

V. DERIVATION OF THE EFFECTIVE SPIN MODEL

In an earlier paper [15], I obtained an effective spin modelfrom the two-sublattice Kondo lattice model [Eq. (1)] appro-priate for the Fe series (Sr2−xLaxFeMoO6) using the procedureof SCR. In the action corresponding to this Hamiltonian, allbut the last term is translationally invariant, hence they can beFourier-transformed and written in terms of the bare Green’sfunction. Thereafter, the molybdenum degrees of freedom canbe integrated out and the iron degrees of freedom traced over,after taking the limit of J1 → ∞, to obtain an effective modelcontaining only the iron spins. One may imagine that the sameprocedure can be followed in this case. There is a fundamentaldifficulty in this case though, due to the fact that there is spin-disorder even on the molybdenum site, making it impossibleto diagonalize and integrate out the molybdenum degrees offreedom. Hence, the entire procedure of thinning of degrees offreedom has to be done in real space. For simplicity, we assumethat tBB = tB ′B ′ = 0, and there is only one orbital per site. Theaction corresponding to the Hamiltonian in Eq. (2) is given by

A =∑iωn

[(iωn − εB)

∑iσ

f†i,n,σ fi,n,σ + (iωn

− εB ′ )∑iσ

m†i,n,σmi,n,σ + tBB ′

∑〈ij〉

(f †i,n,σmj,n,σ + H.c.)

+ J1

∑iαβ

�Si · f†i,n,σ �τα,βfi,n,β+J2

∑i

�Si · m†i,n,σ �σαβmi,n,β

].

(3)

Now, since the spin-orbit coupling at the B ′ site is largefor the 5d elements such as Re and Os compared to the4d elements such as Mo, the on-site spin anisotropy is alsoexpected to be high. Let us choose our axis of quantizationalong this anisotropy axis, whereupon we need only considerthe diagonal components of the last J2 term. Hence, the actioncan be written as

A =∑iωn

[(iωn − εB)

∑iσ

f†i,n,σ fi,n,σ + tBB ′

∑〈ij〉

(f †i,n,σ mj,n,σ

+ H.c.) + J1

∑iαβ

�Si · f†i,n,α �ταβfi,n,β

+∑

i,α,β,δ

m†i,n,αMαβmi,n,β

], (4)

where the matrix M is given by M = (iωn − εB ′)I + J2Szi+δσz.

Hence, first taking the J1 → ∞ limit [27] and then integratingthe B ′ degrees of freedom out, we get the following form forthe action:

A =∑i,n

(iωn − εB)γ †i,nγi,n

−∑〈〈i,j〉〉

t2BB ′

iωn − εB ′ + J2∑

δ Szδ

cosθi

2cos

θj

2γ†i,nγj,n

−∑〈〈i,j〉〉

t2BB ′

iωn − εB ′ − J2∑

δ Szδ

sinθi

2sin

θj

2γ†i,nγj,n, (5)

where γi,n refer to transformed spinless fermion operators atthe iron site while θi refer to the azimuthal angle made by theFe core spins with the z axis. Following usual practice, wehave neglected Berry’s phase degrees of freedom connectedwith the polar angles.

Hence, if we consider total energy U = −〈 ∂A∂β

〉 as before[15], then we get

U =∑〈〈ij〉〉

t2BB ′

[(2iωn − � + J2

∑δ Sz

δ

)(iωn − � + J2

∑δ Sz

δ

)2 cosθi

2cos

θj

2

+(2iωn − � − J2

∑δ Sz

δ

)(iωn − � − J2

∑δ Sz

δ

)2 sinθi

2sin

θj

2

]Gn,ij . (6)

So far, no approximation has been made except that J2 J1 and there is a large anisotropy at the B ′ site. However, toactually calculate the exchange from this expression, one firstneeds to start with some spin background and calculate theiron Green’s function in real space in this background usingthe original Hamiltonian (2). Then, one needs to recalculate thespin background from the effective spin Hamiltonian (6) againusing some technique such as the Monte Carlo method. Thisprocedure, to be repeated until self-consistency is achieved,defines the technique of self-consistent renormalization [30].The entire procedure is, however, numerically expensive anddifficult. In the next section, we shall try instead to calculatethe exchange analytically in a certain simple arrangement ofthe background spins, namely a fully ordered ferromagneticconfiguration. This can act as the first step of the SCRprocedure, which will be taken up in a future work. As weshall see, it will give us an important analytic pointer at theactual magnetic ground states possible in these compounds.

VI. EXCHANGE IN THE ORDERED SPIN BACKGROUND

It should be noticed that unlike our previous SCR for-mulation [15], the action no longer has full spin-rotationalinvariance due to the anisotropy. (Notice that setting J2 = 0recovers the familiar rotationally invariant Anderson-Hasegawa form.) Hence, even if we start with a perfectlyordered spin background with all spins parallel, we mustalso choose an orientation relative to the anisotropy axis. Weconsider three cases. In the following, we replace εB = 0 andεB ′ = � for convenience. Henceforth, J2 will mean zJ2, wherez is the coordination number.

First, we consider the case θi = 0 for all sites. Then,translational invariance is restored, and we can write therelevant terms of the action (5) in momentum space andcalculate the exchange as follows:

A =∑k,n

(iωn − t2

BB ′

iωn − � − J2

)γ†knγkn. (7)

Now we calculate U = −〈 ∂A∂β

〉 to obtain

U =∑kn

t2BB ′(2iωn − � − J2)

(iωn − � − J2)2Gkn. (8)

We can express this internal energy U as a classical spinHamiltonian of the Anderson Hasegawa [31] type and calculate

115129-3

PRABUDDHA SANYAL PHYSICAL REVIEW B 89, 115129 (2014)

the effective exchange [15,16,21]. Evaluating the Matsubarasum, this effective exchange is given by

J effij =

∑k

1

2[Ek+nF (Ek+) + Ek−nF (Ek−)

− (� + J2)nF (� + J2)]e�ik·(�ri−�rj ), (9)

where Ek± = �+J2±√

(�+J2)2+4ε2k

2 . This gives the final expres-sion for the exchange, which has to be evaluated on a squarelattice.

The case of θi = π gives, proceeding in a similar fashion,the same expression as θ = 0. This is not surprising, sinceuniaxial anisotropy does not distinguish between θ = 0 andθ = π . Now, let us consider the case of θi = π

2 ,

U =∑kn

1

2

[ε2k (2iωn − �)

(iωn − �)2+ ε2

k (2iωn − �)

(iωn − �)2

]〈γ †

knγkn〉. (10)

Thus J2 drops out entirely. The expression for the exchangehere would coincide with that for J2 = 0 evaluated earlier[15], which is also not surprising because the anisotropy axishas no component along the perpendicular direction, so thatthe superexchange has no effect on spin configurations alongthis direction. Replacing the Green’s function and evaluatingthe Matsubara sum gives the familiar expression [15] U =∑

k[Ek+nF (Ek+) + Ek−nF (Ek−) − �nF (�)] as before. Thisis to be expected, since the anisotropy, and consequently thesuperexchange, will have no effect in a direction perpendicularto the easy axis.

Finally, we write the exchange for a general θi = θ ,

Jij = t2FM

∑kn

ei�k·( �Ri− �Rj )

[cos2 θ

2 (2iωn − � − J2 cos θ )

(iωn − � − J2 cos θ )2

+ sin2 θ2 (2iωn − � + J2 cos θ )

(iωn − � + J2 cos θ )2

]Gkn, (11)

where

G−1kn = iωn − ε2

k [(iωn − �) + J2 cos2 θ ]

(iωn − �)2 − J 22 cos2 θ

. (12)

VII. RESULTS

The results of the exchange calculations are shown here. Forthe case of θ = 0 (up), the exchange is as shown in Fig. 1. Sincewe are only interested in looking at the ferromagnetic phase,only the dominant nearest-neighbor exchanges are shown. Asexplained earlier, the same graph is obtained for the θ = π

(down) case. For comparison, the J = 0 curve for the same �

is also shown [15]. This may also be thought to be the exchangedata for the θ = π/2 case. Two effects are immediatelyobserved. First, the extent of the ferromagnetic phase in thefilling regime increases for the finite superexchange case,as compared to no superexchange. This is probably why,for an extended filling regime, the ferromagnetic behaviorpersists rather than the antiferromagnetic behavior in the3d-5d compounds. In particular, while the exchange alreadybecomes positive for a filling of n ≈ 0.7 for the J2 = 0case [15] exhibiting antiferromagnetic instability, it continuesto be positive way beyond N = 1 for finite J2. Second,

0 0.5 1 1.5 2 2.5 3N

-0.4

-0.2

0

0.2

0.4

0.6

0.8

DN

N

J2=0J2=1

FIG. 1. (Color online) Exchange vs filling obtained from SCR,compared between cases with and without superexchange.

the magnitude of the negative part is also larger, provingthat the ferromagnetic Tc is enhanced by superexchange.This is probably because the superexchange stabilizes theferrimagnetic state. Another interesting point to note is that theexchange continues to be large and negative until about N = 1,which corresponds to n = 3 in the real double perovskitesdue to t2g degeneracy (e.g., Sr2CrOsO6), while its magnitudediminishes beyond that. The point N = 2 shows a strongisolated antiferromagnetic tendency, possibly due to the fillingof the B ′ level (W, Re, or Os) with opposite spin in thehalf-metallic state. This point corresponds to n = 6 in the realmaterial if the t2g degeneracy is considered [15]. If the filling isincreased further, then the d levels of the B atom (Cr) have tobe filled in both spin channels. Now, there is a large energy gapbetween the B and B ′ levels, and also the magnetic exchangechanges abruptly from J2 to J1, which is one possible causefor the anomaly. The second reason for the stabilization ofantiferromagnetism over the half-metallic state is the fact that

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07T/

t

J2=0 -->

Δ=−2

0 0.5 1 1.5 2 2.5 3N

0

0.5

1

1.5

2

2.5

3

T/t

<-- J2=1

FIG. 2. (Color online) Tc vs N ferromagnetic part of the phasediagram from an 8 × 8 cluster compared between J2 = 0 and1 (� = −2).

115129-4

FERROMAGNETISM IN Cr-BASED 3d-5d DOUBLE . . . PHYSICAL REVIEW B 89, 115129 (2014)

-1 0 1 2E (eV)

0.5

1

1.5

2

DO

S

Ef

FIG. 3. DOS for Sr2CrOsO6; the Fermi energy is at E = 0.

the number of itinerant electrons in both spin channels is thesame at N = 2, since the electrons on the B site contributeonly to the core spin.

Next, to bring these results in perspective, and to establishthem on a sound footing, we also performed extensivenumerical simulations of the original spin-fermion modelHamiltonian, using exact diagonalization coupled with theMonte Carlo technique (ED + MC). Calculations performedin real space on an 8 × 8 cluster are shown in Fig. 2. Weused � = −2 and J2 = 1 as before. It is found that inclusionof the J2 term in the Hamiltonian not only results in atremendous increase in the Curie temperatures, but also inthe proliferation of the ferromagnetic phase to higher fillings.In fact, for these regime of parameters, the ferromagneticphase is found to make substantial entries into those regionsof filling that were reserved earlier for the antiferromagneticphases [15]. However, both effects are far more augmentedthan is suggested by the SCR calculations. Of course, SCR isperformed assuming ordered spin backgrounds appropriate atlow temperatures, and also only one loop of the actual SCRprocess is performed to derive the analytical expressions givenin this paper, hence it should only be considered as a pointerto the actual renormalization effects close to Tc. Interestingly,in the exact numerical simulations, the ferromagnetic lobes atboth low and high filling join together in the case of finite J2,giving a maximum Tc at N = 1 (which corresponds to n = 3considering the t2g degeneracy in the real material). This pointcorresponds to the Os compound, providing an explanation forthe inordinately high Tc of this material. This is consistent withthe enigmatic high Tc observed in the compound Sr2CrOsO6

[20,32–34]. Also, it substantiates the view that this compoundis at the threshold of a magnetic and electronic transition [20].

Thus the mechanism of superexchange coupling betweenB and B ′ sites provides an alternate and more transparentmechanism for the Tc increase, rather than the Hubbard U

mechanism invoked earlier [26,32]. Note that the additionalmoments on the B ′ site were all obtained in the ab initio cal-culations without using any U either. The insulating behaviorof the Os compound can be explained simply in terms of thecharge-transfer gap between the B and B ′ sites (see Fig. 3). TheFermi level appears at a gap corresponding to the fully filledspin-polarized Os levels. This is the so-called “half-insulator,”or the limit of a half-metal coming near a metal-insulatortransition [20]. It should be noted that this charge-transfer gapis almost absent in the Sr2−xLaxFeMoO6 systems, which iswhy they are all metallic compounds. In addition, the gapparameters determined by hopping, such as �, etc., in this La-doped SFMO series remain constant, whereas across the seriesSr2CrWO6, Sr2CrReO6, and Sr2CrOsO6, the gap parameter �

increases [16,21]. Hence, the tendency toward semiconductingbehavior increases across this series, culminating in a fullygapped insulator in the last Os compound.

VIII. SUMMARY AND OUTLOOK

We showed that superexchange plays the main role inenhancing Tc as well as stabilizing ferromagnetism in theCr compounds. Spin-orbit coupling breaks the rotationalsymmetry on the B ′ site, so that different orientations of spinconfigurations have to be considered for the SCR. Amongthe configurations considered, those along the easy axis arefound to increase the effective nearest-neighbor ferromagneticexchange, as well as to contribute to the increase in the fillingextent of the ferromagnetic phase. Both are signatures ofstabilization of the ferromagnetic phase, which results in anincrease in Tc and a continuation of the ferromagnetic phaseto higher fillings, as for Rh and Os compounds. Extensivenumerical simulations are found to reproduce both of thesebehaviors, and they provide a sound backing to the proposedmodel Hamiltonian as a useful one for describing the behaviorof Cr-based 3d-5d double perovskites.

ACKNOWLEDGMENTS

The author gratefully acknowledges discussions withK. Held, M. Randheria, O. Erten, D. D. Sarma, P. Majumdar,H. Das, and T. Saha Dasgupta. The author is grateful toSNBNCBS for providing access to their cluster facilities.

[1] K.-I. Kobayashi, T. Kimura, H. Sawada, K. Terakura, andY. Tokura, Nature (London) 395, 677 (1998).

[2] D. D. Sarma, Curr. Opin. Solid State Mater. Sci. 5, 261(2001).

[3] T. Saitoh, M. Nakatake, A. Kakizaki, H. Nakajima, O.Morimoto, Sh. Xu, Y. Moritomo, N. Hamada, and Y. Aiura,Phys. Rev. B 66, 035112 (2002).

[4] T. Saitoh, M. Nakatake, H. Nakajima, O. Morimoto,A. Kakizaki, Sh. Xu, Y. Moritomo, N. Hamada, and Y. Aiura,J. Electron Spectrosc. Relat. Phenom. 144-147, 601 (2005).

[5] Y. Tomioka, T. Okuda, Y. Okimoto, R. Kumai, K.-I. Kobayashi,and Y. Tokura, Phys. Rev. B 61, 422 (2000).

[6] D. D. Sarma, P. Mahadevan, T. Saha-Dasgupta, S. Ray, andA. Kumar, Phys. Rev. Lett. 85, 2549 (2000).

115129-5

PRABUDDHA SANYAL PHYSICAL REVIEW B 89, 115129 (2014)

[7] B. Garcia Landa et al., Solid State Commun. 110, 435 (1999).[8] B. Martinez, J. Navarro, L. Balcells, and J. Fontcuberta, J. Phys.:

Condens. Matter 12, 10515 (2000).[9] D. D. Sarma, S. Ray, K. Tanaka, M. Kobayashi, A. Fujimori,

P. Sanyal, H. R. Krishnamurthy, and C. Dasgupta, Phys. Rev.Lett. 98, 157205 (2007).

[10] T. Saitoh, M. Nakatake, H. Nakajima, O. Morimoto,A. Kakizaki, Sh. Xu, Y. Moritomo, N. Hamada, and Y. Aiura,Phys. Rev. B 72, 045107 (2005).

[11] A. Kahoul, A. Azizi, S. Colis, D. Stoeffler, R. Moubah,G. Schmerber, C. Leuvrey, and A. Dinia, J. Appl. Phys. 104,123903 (2008).

[12] J. Navarro, C. Frontera, Ll. Balcells, B. Martınez, andJ. Fontcuberta, Phys. Rev. B 64, 092411 (2001).

[13] J. Navarro, J. Fontcuberta, M. Izquierdo, J. Avila, and M. C.Asensio, Phys. Rev. B 70, 054423 (2004).

[14] C. Frontera, D. Rubi, J. Navarro, J. L. Garcıa-Munoz,J. Fontcuberta, and C. Ritter, Phys. Rev. B 68, 012412 (2003).

[15] P. Sanyal and P. Majumdar, Phys. Rev. B 80, 054411 (2009).[16] P. Sanyal, H. Das, and T. Saha-Dasgupta, Phys. Rev. B 80,

224412 (2009).[17] S. Jana, C. Meneghini, P. Sanyal, S. Sarkar, T. Saha-Dasgupta,

O. Karis, and S. Ray, Phys. Rev. B 86, 054433 (2012).[18] J. B. Philipp, P. Majewski, L. Alff, A. Erb, R. Gross, T. Graf,

M. S. Brandt, J. Simon, T. Walther, W. Mader, D. Topwal, andD. D. Sarma, Phys. Rev. B 68, 144431 (2003).

[19] H. Kato et al., Appl. Phys. Lett. 81, 328 (2002).[20] Y. Krockenberger et al., Phys. Rev. B 75, 020404 (2007).[21] H. Das, P. Sanyal, T. Saha-Dasgupta, and D. D. Sarma,

Phys. Rev. B 83, 104418 (2011).[22] O. K. Andersen and T. Saha-Dasgupta, Phys. Rev. B 62, R16219

(2000).

[23] G. Kresse and J. Hafner, Phys. Rev. B 47, 558(R) (1993);G. Kresse and J. Furthmuller, ibid. 54, 11169 (1996).

[24] A. Chattopadhyay and A. J. Millis, Phys. Rev. B 64, 024424(2001).

[25] O. Navarro, E. Carvajal, B. Aguilar, and M. Avignon,Physica B 384, 110 (2006).

[26] L. Brey, M. J. Calderon, S. Das Sarma, and F. Guinea,Phys. Rev. B 74, 094429 (2006).

[27] J. L. Alonso, L. A. Fernandez, F. Guinea, F. Lesmes, andV. Martin-Mayor, Phys. Rev. B 67, 214423 (2003).

[28] J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R.Pederson, D. J. Singh, and C. Fiolhais, Phys. Rev. B 46, 6671(1992); ,48, 4978(E) (1993).

[29] G. Cohen, V. Fleurov, and K. Kikoin, J. Appl. Phys. 101, 09H106(2007).

[30] S. Kumar and P. Majumdar, Eur. Phys. J. B 46, 315 (2005).[31] P. W. Anderson and H. Hasegawa, Phys. Rev. 100, 675

(1955).[32] O. N. Meetei, O. Erten, M. Randheria, N. Trivedi, and

P. Woodward, Phys. Rev. Lett. 110, 087203 (2013).[33] In real double perovskites, due to the threefold t2g degeneracy,

there are a total of nine degrees of freedom per Fe-Mo pair,consisting of three at Fe site and six at Mo site. Hence, tocompare with experiments, our x axes should be multipliedby 3.

[34] We have deliberately considered a larger value of J2 in the sim-ulations than usually found in real Cr-based double perovskitesto underline the strong Tc enhancement and ferromagnet-proliferation effects. Actual values of J2 may leave an windowfor the antiferromagnetic phases as well. The effects of phasecompetition in this Hamiltonian will be considered in a laterwork.

115129-6