Origin of Magnetism and trend in $T_{c}$ in Cr-based double perovskites: Interplay of two driving mechanisms

1.2 MB·English

by  Hena Das

Origin of Magnetism and trend in T in Cr-based double perovskites: Interplay of two
c
driving mechanisms
Hena Das1, Prabuddha Sanyal1, T. Saha-Dasgupta1,∗, D.D. Sarma2,†
1S.N. Bose National Centre for Basic Sciences, Kolkata 700098, India and
2 Solid State and Structural Chemistry Unit, Indian Institute of Science, Bangalore-560012, India
(Dated: January 25, 2011)
Employingfirstprinciplesdensityfunctionalcalculations,togetherwithsolutionofthelow-energy,
model Hamiltonian constructed in a first principles manner, we explored the origin of magnetism
and T trend in Cr-based double perovskite series, Sr CrB(cid:48)O (B(cid:48)=W/Re/Os). Our study shows
c 2 6
that the apparently puzzling T trend in Sr CrB(cid:48)O (B(cid:48)=W/Re/Os) series can be understood in
c 2 6
terms of the interplay of the hybridization driven mechanism and the super-exchange mechanism.
1
1
0 PACSnumbers: 71.20.-b,71.20.Be,75.50.-y
2

n I. INTRODUCTION issue. Though few DFT based studies10 exist for indi-
a
viduals of these compounds, to best of our knowledge,
J
theoretically no comprehensive study exists to address
4 Oxideswithhighmagnetictransitiontemperature(T )
c the origin of growing trend in T within the series.
2 are important for technological advancement. A much c

discussed family of compounds in this connection are
]
ci dAo=uabllkealpineer/ovrasrkeit-eeas,rthwimthetaalsgaennedraBl/Bfo(cid:48)r=mturalansAiti2oBnBm(cid:48)Oe6t-, II. METHODOLOGY
s
- als. The observation1 of large magnetoresistance with
rl a fairly high ferromegnetic (FM) Tc of about 410 K in The DFT calculations were carried out using the
mt Sr2FeMoO6 (SFMO)anditsunusualoriginofmagnetism plane wave pseudopotential method implemented within
brought this family of compounds into the forefront of Vienna Ab-initio Simulation Package(VASP).11 The
t. activity. Since then the question has been, can T be exchange-correlation functionals were approximated by
a boosted even further and if so, is there a systemcatic generalized gradient approximation (GGA).12 We have
m
trend that can be observed and understood. Attempts used projected augmented wave (PAW) potentials13 and
d- have been made to dope SFMO with La to increase the the kinetic energy cut-off for expansion of wavefunctions
n Tc. Though some partial success has been achieved2 fol- usedwas450eV.Reciprocalspaceintegrationshavebeen
o lowing this path, T was found to be boosted much more carried out with a k-space mesh of 6×6×6. For ex-
c
c efficientlybymovingtodifferentchoicesofBandB(cid:48) ions. traction of a few-band, tight-binding Hamiltonian out of
[ Themicroscopicunderstandingofthisincrease,however, full GGA calculation, we have carried out muffin-tin or-
1 has not been achieved. A collection of measured transi- bital (MTO) based NMTO-downfolding calculations.14
v tion temperatures seems to bear some correlation with The reliability of the calculations in two basis sets has
6 the number of valence electrons among different double been cross-checked.
7
perovskitecompounds(seeFig.1inRef.3). Focusingonto
4
Cr-basedseries,namelySr CrB(cid:48)O (B(cid:48)=W/Re/Os),the
4 2 6
familywithspectacularlyhighT ,themeasuredT shows III. RESULTS
. c c
1 arapidincreaseasonemovesfromSr CrWO (SCWO)4
2 6
0
1 with Tc ≈ 450 K to Sr2CrReO6 (SCRO)5 with Tc ≈ 620 A. Examination of Basic Electronic Structure
K to Sr CrOsO (SCOO)6 with T ≈ 725 K. This rapid
1 2 6 c
: increase is curious considering the hybridization driven In order to unravel the origin of magnetism in
v (HD)mechanism7,8ofmagnetismacceptedsofarfordou-
Xi bleperovskites,proposedinthecontextofSFMO,which Sr2CrB(cid:48)O6 series, let us first critically examine the elec-
tronic density of states (DOS) of these compounds. Left
would predict decrease rather than the increase.9. Fur-
r panel of Fig.1 shows the DOS, as obtained in spin-
a thermore, the compound with the highest T in this se-
c polarizedDFTcalculationswithinGGA.Thestatesclose
ries, SCOO, is found to be an insulator making the issue
to Fermi level (E ) are dominated by Cr and B(cid:48) d states
even more interesting. Also, a distinction between oth- F
hybridized with O p states, while the O p dominated
ers like SFMO and La doped SFMO and this series is
states separated from Cr and B(cid:48) d dominated states oc-
a possible role of spin-orbit coupling (SOC) due to the
cupytheenergyrangefarbelowE andSrsandddom-
presence of 5d elements. F
inated state remain far above E . The d states of Cr
F
We have used density functional theory (DFT) based and the B(cid:48) ions are exchange split as well as crystal field
calculations together with exact diagonalization of Cr- split. TheemptyB(cid:48) t statesintheup-spinchannelap-
2g
B(cid:48) model Hamiltonian constructed in a first-principles- pearinbetweenthecrystalfieldsplitCrt ande states,
2g g
derived Wannier function basis, in order to probe this gapedfromE whiletheB(cid:48)statesinthedownspinchan-
F
2
2 Cr-t2g W-t2Cgr-eg mC r = −2.14[−2.15, 0.02]µB Hybridization Off Hybridization On
States (States/eV/atom)--22002 OO-OO-pp--pp Cr-Rt2e0g-0t.21.6g66WCCrr---tRt2t22eggg-t2gCCrCr--ereg-geg mmNmmmNeWO eCRO tt reM M==o===o −m 0m−000.e3..e1.n800n0.12t92t::8 [[[ [...280−31[.620−.0.00,,,2 2[ −0−. 2,1[.0 01].070...1.0,2]] µ7704Bµµ ]].]0BB µ3µµB ]BB µB CCrr--tt22gg 23.1 00..5917 1.900.73 1.305.50RWOORWeess------tttttt222222gggggg 00..53310 .06 00..3482 00..341000..3315 RWRWOOeess------ttttt2t22222gggggg 1.18 20.1 78.0
ensity of 02 O-pCr-t2g Os-t2gCr-eg mmCO rs == −1.24.421[1[.−322.,2 −10, .01.10]2]µµB B
D O-p Os-t2g Cr-eg mO = 0.08[.08, 0.02]µB
-2 0.05 Cr-t2g Net Moment: 0.0 [0.45]µB
-5 -4 -3 -2 -1 0 1 2 3 4 5
Energy (eV)
Figure 2: (Color online) The energy level diagram (upper
panel) and massively downfolded Wannier functions (lower
Figure 1: (Color online) Left Panels: GGA DOS projected panels) for Sr CrB(cid:48)O series. For Wannier function plots,
ontoCrd(blacksolidlines),B(cid:48) d(Red/Greysolidlines)and 2 6
constantvaluesurfaceshavebeenplottedwithtwooppositely
Op(shadedarea). ZerooftheenergyissetatE . Thenum-
F signed lobes colored differently. From left to right in lower
bers within the boxes indicate the Cr t contribution in the
2g panel, the plots correspond to SCWO, SCRO and SCOO re-
bands crossing E , with respect to that of B(cid:48). Right Panel:
F spectively. The numbers in the energy level diagram are in
Calculated net magnetic moment and magnetic moments at
unit of eV.
Cr, B(cid:48) and O sites. The numbers within the bracket denote
the result of GGA+SO calculations, the first entry being the
spinmomentandthesecondentrybeingtheorbitalmoment.
B (cid:48) site following the dehybridization effect between Cr
From top to bottom, the plots correspond to SCWO, SCRO
and B(cid:48).
and SCOO respectively.
B. NMTO-downfolding calculations
nel hybridized with Cr t states, either cross the Fermi
2g
level as in the case of W and Re compounds, or remain In order to further analyse the findings of the elec-
completelyoccupied,asincaseoftheOscompound. Itis tronicstructurecalculations,wehavecarriedoutNMTO
ratherintriguingtonoticethatthehybridizationbetween downfolding calculations, which is engineered to define
Crt andB(cid:48) t inthedownspinchannel, progressively energy-selected, effective Wannier functions by integrat-
2g 2g
getsweakenedinmovingfromWtoRetoOscompound. ingoutdegreesoffreedomthatarenot-of-interest(down-
ThismaybeappreciatedbyconsideringtheCrcontribu- folding). As a first step we downfolded O p, Sr as well
tion measured with respect to B(cid:48) contribution for the as Cr and B(cid:48) e degrees of freedom. This defines an ef-
g
states in the down spin channel close to E . For W, Re fective basis consisting of Cr t and B(cid:48) t states. In
F 2g 2g
and Os compounds, it is found to be 66%, 16 % and 5% the second step, we applied massive downfolding, keep-
respectively. This is caused by the gradual moving down ing only B(cid:48) t degrees of freedom active and downfold-
2g
of the B (cid:48) energy level, in moving from left to right of ing all the rest including Cr t degrees of freedom. On
2g
the periodic table across the same row (W → Re → Os), site matrix elements of the real space Hamiltonian de-
reflecting an increase in the ionic potential experienced fined in the Cr t -B(cid:48) t basis and the massively down-
2g 2g
by the 5d electrons with an increase in nuclear charge. folded basis give the energy level positions before and
Asdiscussedlater,thehoppinginteractionconnectingCr after switching on the hybridization between Cr and B
andB(cid:48)t statesontheotherhandremainssimilaracross (cid:48) states, respectively. Fig.2 summarizes the results for
2g
the series. Analysing the DFT calculated magnetic mo- W, Re and Os compounds. The energy levels in the left
ments, presented in the right panel of Fig.1, we find the and right half of the upper panel, demarked with verti-
secondinterestingobservationthatthemagneticmoment cal grey line, depict the energy level positions obtained
per electron at B(cid:48) site, defined as m/d, where m is the out of downfolded Cr t -B(cid:48) t basis and the massively
2g 2g
calculated moment and d is the valence count at B (cid:48),15 downfolded basis, respectively. The lower panels exhibit
keeps growing from W to Re to Os. The increase in m/d theplotsoftheoneofthet (xy)Wannierfunctionscor-
2g
becomesevenmoreevidenttakingintoaccountthemag- responding to massively downfolded Hamiltonian in the
netic moment at O site, which is small and point to Cr down spin channel. Examination of Fig.2 brings out two
moment for SCWO, small (large) and point to Re (Os) aspects: Firstly, the progressive dehybridization effect,
momentforSCRO(SCOO).Thispromptsustoconclude as discussed in the context of DOS plots, is evident in
thatthereisagrowingintrinsicmomentthatdevelopsat the plots of Wannier functions. The central parts of the
3
Wannier functions are shaped according to B(cid:48) xy sym- moment, due to the uncompensated orbital moment at
metry and the tails of the Wannier functions sitting at B(cid:48) site. Whether SOC has any influence on the trend
neighboring sites are shaped according to O p and Cr withintheT ’s,thereforeneedstobeexplored. Interplay
c
t symmetry. The tails reflecting the hybridization be- of two driving mechanisms in W-Re-Os series, however,
2g
tweentheCrt andB(cid:48) t -Oweakenasonemovesfrom makes it difficult to extract the magnetic exchanges as
2g 2g
WtoRetoOscompound. Asaconsequence,theratioof energy difference between two specific magnetic configu-
renormalized spin splitting to that of the bare splitting rations. The presence of finite, intrinsic moment at B(cid:48)
atB(cid:48) sitereducesdrasticallyfromW(14.5)toRe(3.3)to site, as is the case for Re and Os compounds, makes
Os (2.2). Secondly, considering the level splitting at B(cid:48) the moment at B(cid:48) site frustrated in an antiferromagnetic
site before switching on the hybridization, we find while (AFM) configuration of Cr spins. The finite presence
thesplittingatWisnegligiblysmallconfirmingthenon- of HD mechanism, on the other hand, disfavors stabi-
magnetic character of the B(cid:48) site, those for Re and Os lization of magnetic configurations with majority of B
are found to be ≈ 0.31 eV and ≈ 0.53 eV, respectively. spins aligned parallely to that of the moment at B(cid:48) site.
These values are significantly larger compared to what Admitting these difficulties, we carried out total energy
one would have expected considering d2 valence in Re calculations of the two possible spin configurations, one
and d3 valence in Os compared to d1 valence in case of FM arrangement and another A-type AFM arrangement
W with 0.06 eV splitting, which would have given rise to of Cr spins, with Cr spins between two adjacent planes
splittings of 0.12 eV and 0.18 eV respectively. This con- are antiferromagnetically coupled, and are ferromagnet-
firms the presence of a growing intrinsic, local moment ically coupled in plane. For the AFM calculations, the
at B(cid:48) site as one moves from W to Re to Os, driven by moment at B(cid:48) site was found to be antiparallely (par-
the dehybridization effect. The magnetism in Cr-B(cid:48) se- allely) aligned to the spins of in-plane (out-of-plane) Cr
ries, therefore, needs to be understood as an interplay of sites which are four (two) in number. As expected, the
two mechanisms: HD mechanism as operative in SFMO energy differences between FM and AFM configurations
whichcausesrenormalized,negativespinsplittingwithin arefoundtobepositiveforallcasesprovingFMarrange-
B(cid:48) states that appear in between the exchange split Cr ment of Cr spins to be the stable phase. The values of
t states, and the superexchange (SE) between the mo- the energy difference is found to increase from W to Re
2g
mentatCrsiteandtheintrinsicmomentatB(cid:48)site,which system (from 0.23 eV/formula unit to 0.25 eV/formula
wouldalignthemomentsatCrandB(cid:48) sitesantiparallely. unit), and then decrease from Re to Os system (from
For W, the intrinsic moment being negligible, the mag- 0.25 eV/formula unit to 0.24 eV/formula unit). Intro-
netismisentirelydrivenbyHDmechanism,whileforthe duction of SO interaction though changes the individual
other extreme of Os, the hybridization effect is weak, SE energy differences by about 0.03 eV, the trend remains
having a rather large contribution. The presence of such unaltered. This indicates that although the presence of
intrinsic moment at 5d site is counterintuitive at a first substantialSOCatB(cid:48) siteisimportantforproducingthe
glance. Comparing the situation, with the double per- large magneto-optical signals,17 it plays little role in set-
ovskite Sr ScReO , for which the magnetism has been ting up the trend in T . In order to examine the role of
2 6 c
recently investigated,16 Re was found to possess a rather missing correlation effect in GGA, we have also carried
small intrinsic moment of size 0.013 µ . Our electronic outLDA+UcalculationswithachoiceofUvalueof3eV
B
structurecalculationscarriedoutforSr ScReO findalso atCrsiteand0.8eVatB(cid:48)site. ApplicationoflargerUat
2 6
a similarly small moment (0.03 µ ). The unusual local- CrsiteandrelativelysmalleroneatB(cid:48) sites,asexpected
B
ized aspect of Re or Os, in case of Cr based compounds fromtherelativeband-widths,werefoundtopreservethe
therefore arises due to the relative positioning of the Cr general conclusions intact with more localized character
and B(cid:48) energy levels, which narrows down the width of of d states at B(cid:48) site, as expected.
the B(cid:48) states substantially in the up spin channel. The
development of the intrinsic moment is thus helped by
thedelicateenergylevelstructureresponsiblefortheHD
D. Exact Diagonalization Study of Model
mechanism and would not have been present otherwise.
Hamiltonian
In view of the difficulty in stabilizing the appropri-
C. Total Energy Calculations ate excited state magnetic configuration, the stability of
the FM arrangements of Cr spins may be measured as
The calculations discussed so far, do not include SOC, the energy difference between the FM spin configuration
which may be important. The numbers within the and the paramagnetic (PM) phase. The description of
bracket in right panel of Fig.1 show the individual spin thePMphaseneedsconsiderationofdifferentdisordered
and orbitalmoments aswellas netmomentsas obtained spin configurations andaveragingover alarge number of
in GGA+SO calculations. As expected, the orbital mo- them, whichisalmostimpossiblewithintheDFTframe-
ments are large at B(cid:48) site. Interestingly, we find while work. Suchcalculationsaremucheasiertohandlewithin
the net moment was zero for SCOO without SOC, it is a model Hamiltonian description. The model Hamilto-
the consideration of SOC that gives rise to a non-zero nian, describing the interplay of the HD and SE mecha-
4
nism may be written as, 0.2
(cid:88) (cid:88) 0.15
H = (cid:15)Cri∈Bfi†σαfiσα+(cid:15)B(cid:48)i∈B(cid:48)m†iσαmiσα mula unit) 0.1
−tCB(cid:48) (cid:88) fi†σ,αmjσ,α−tB(cid:48)B(cid:48) (cid:88) m†iσ,αmjσ,α E(eV/for0.005
<ij>σ,α <ij>σ,α ∆
(cid:88) (cid:88) -0.05
−t f† f +J S ·f† (cid:126)σ f
CC iσ,α jσ,α i iα αβ iβ 1 2 3
N
<ij>σ,α i∈Cr
(cid:88)
+J S ·s Figure 3: PM - FM energy differences plotted as a function
2 i j
of valence electron count, as obtained in exact diagonaliza-
i∈Cr,j∈B
tion calculation. The diamond (square) symbols connected
where the f’s and m’s refer to the Cr t2g and B(cid:48) t2g de- by dashed (solid) line correspond to calculations correspond-
grees of freedoms. t , t , t represent the nearest ing to Hamiltonian, without (with) J term.
CB(cid:48) B(cid:48)B(cid:48) CC 2
neighbor Cr-B(cid:48), second nearest neighbor B(cid:48)-B(cid:48) and Cr-
Cr hoppings respectively. σ is the spin index and α is
the orbital index that spans the t manifold. The dif- valence electron count equal to 3 which corresponds to
2g
ference between the ionic levels, ∆ = (cid:15) −(cid:15) , defines Oscompound, FMbecomestotallyunstable, reflectedin
Cr B(cid:48)
the on-site energy difference between Cr t and B(cid:48) t negative sign of the energy difference. We note a rather
2g 2g
levels. s istheintrinsicmomentattheB(cid:48) site. Thefirst, rapid decrease in moving from N=2 case to N=3 case.
j
six terms of the Hamiltonian, represent the HD mecha- This may reflect the special situation of Os compound,
nism,whichconsistofalargecorespinattheCrsite(S ) withCrt3 -B(cid:48) t3 configuration,anidealsuper-exchange
i 2g 2g
andthecouplingbetweenthecorespinandtheitinerant situation with insulating solution, that adds on to the
electron delocalized over the Cr-B(cid:48) network. Variants of general trend. The situation gets dramatically changed
this part has been considered by several authors18–20 in upon inclusion of the growing localized magnetic nature
the context of SFMO. The last term represents the SE oftheB(cid:48) site,asshownbysquaresymbolsinFig.3. Con-
mechanism,thatconsistsofcouplingbetweenCrspinand sidering J2 values, as obtained in GGA calculations, we
the intrinsic moment at B(cid:48) site. The parameters of the findthatPMandFMenergydifference,recoversthecor-
modelHamiltonianareextractedoutofDFTcalculations recttrendinmovingfromWtoRetoOsashasbeenob-
through NMTO downfolding technique of constructing served experimentally. Mapping the PM and FM energy
the real space Hamiltonian in the basis of effective Cr difference to the mean field Tc, one obtains values 870
t and B(cid:48) t degrees of freedom. t , t and t K,1160Kand1450KfortheW,ReandOscompounds
2g 2g CB(cid:48) B(cid:48)B(cid:48) CC
hoppings are found to -0.35 eV, -0.12 eV and -0.08 eV respectively. Althoughthevaluesareoverestimatedcom-
respectively,withlittlevariationwithintheW-Re-Osse- pared to experimental values, presumably due to the fi-
ries. ∆-sshowavaryingtrendwithintheW-Re-Osseries nite size effect of exact diagonalization calculation and
(∆W = -0.66 eV, ∆Re = 0.03 eV, ∆Os = 0.26 eV). The themeanfieldformula,thetrendisverywellreproduced
parameters involving J and J were obtained from the withTRe/TW =1.33andTOs/TRe =1.25,comparedto
2 c c c c
spin-splitting at Cr site and the extra splitting observed experimental estimates of 1.38 and 1.17 respectively.4–6
atB(cid:48) siteascomparedtothatexpectedfromtheelectron
filling effect and the splitting for W compound.
The constructed model is then solved using exact di- IV. CONCLUSION AND DISCUSSION
agonalization on a lattice of dimension 8 × 8 × 8. Cal-
culationshavebeencarriedoutaswellforlatticesofsize In conclusion, we have studied the counter-intuitive
4 × 4 × 4 and 6 × 6 × 6. The trend is found to be T trend in Cr based double perovskites, Sr CrB(cid:48)O
c 2 6
the same as presented for 8 × 8 × 8. Exact diagonal- (B(cid:48)=W/Re/Os). Analysisoftheelectronicandmagnetic
ization was first carried out considering the B(cid:48) site to be propertiesshowsthattheprogressiveenhancementofthe
totally nonmagnetic, i.e. setting the last term to zero, T acrossthe5dseriesshouldbeunderstoodastheinter-
c
whichboilsdowntothesameunderlyingmodelHamilto- play of two driving mechanisms: HD mechanism respon-
nianasthatofSFMO.Theenergydifferencebetweenthe sibleforthenegativespinsplittingatB(cid:48) siteasinSFMO,
PM and FM is found to decrease with increasing num- and SE mechanism. The HD mechanism gets weaker as
ber of valence electron, as shown by diamond symbols one moves along the series from W to Re to Os, due to
in Fig.3. This is exactly the similar trend as found in the increased energy level separation of B(cid:48) from Cr. SE,
a recent calculation on La doped SFMO,21,22 as well as on the other hand, gets stronger in moving from SCWO
in Ref. 9 with FM getting destabilized with increase of to SCRO to SCOO due to the presence of growing in-
valence electrons. This trend of suppression of T upon trinsic moment at B(cid:48) site, following the dehybridization
c
increasingvalenceelectroncountisfurtheramplifieddue effect. The observation of uncompensated moment in
to change in ∆ within the Cr-B(cid:48) series. This variation SCOO arises due to the presence of SO. SCOO, in that
in ∆ in the present series is in contrast to the prescrip- sense, should be thought as a ferrimagnet rather than a
tion given in ref.9 to achieve high T . Upon reaching ferromagnet. With this, we demystify the puzzling T
c c
5
trend in Cr-B(cid:48) double perovskite series. knowledges discussion with L. Alff.
V. ACKNOWLEDGMENT
TSD and DDS acknowledge support of DST, Swar-
najayanti and J.C.Bose fellowships. TSD gratefully ac- ∗ [email protected] † [email protected]
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