Table Of ContentCooperative Effect of Electron Correlation and Spin-Orbit Coupling on the Electronic
and Magnetic Properties of Ba NaOsO
2 6
H. J. Xiang and M.-H. Whangbo
∗
Department of Chemistry, North Carolina State University, Raleigh, North Carolina 27695-8204
(Dated: February 4, 2008)
7
0 TheelectronicandmagneticpropertiesofthecubicdoubleperovskiteBa2NaOsO6wereexamined
byperformingfirst-principlesdensityfunctionaltheorycalculationsandanalyzingspin-orbitcoupled
0
2 statesofanOs7+ (d1)ionatanoctahedralcrystalfield. TheinsulatingbehaviorofBa2NaOsO6was
shown to originate from a cooperative effect of electron correlation and spin-orbit coupling. This
n cooperativeeffectisresponsiblenotonlyfortheabsenceoforbitalorderinginBa2NaOsO6 butalso
a
for a small magnetic moment and a weak magnetic anisotropy in Ba2NaOsO6.
J
8 PACSnumbers: 71.70.Ej,71.27.+a,71.20.-b,75.30.Gw
1
] Oxidesoforbitallydegenerate3dtransitionmetalions
i TABLE I: Comparison of the zero-field magnetic moments
c at octahedral sites exhibit rich electronic and magnetic
s µexp (per FU) with the calculated moments µcalc (per
rl- pbritoapleartnidescahrairsginegdfergormeesthoef ifnreteerdpolmay. bIentwaenenocstpaihne,dorar-l GFUG)A+foSrOtChe+UfermroemthaogdnewtiicthstUaetfef o=f0B.2a2RNyadOas,Ob.6 using the
t crystal field, the d-orbitals of a transition metal ion are
m split into the t2g and eg levels. A 3d perovskite with un- µS(Os) µL(Os) µS µcalc µexp c
at. evenlyfilledeg levels,e.g.,LaMnO3withhigh-spinMn3+ [111] 0.51 -0.35 0.98 0.63 0.19
m (3d4)ions,hasastrongtendency fororbitalordering[1]. [110] 0.51 -0.34 0.98 0.64 0.22
- Incontrast,a3dperovskitewithunevenlyfilledt2g levels, [100] 0.52 -0.30 0.97 0.67 0.18
d e.g.,YTiO3withTi3+ (3d1)ions,hasareducedtendency a µS(Os) and µL(Os) are the spin and orbital magnetic
n for orbital ordering and hence provides opportunities to
moments calculated for the Osatom, respectively, and
o
observe an intricate interplay between the spin and or-
c µS is thecalculated total spin magnetic moment per FU.
bital dynamics [2, 3]. It is an important issue to under-
[ standthe mechanismsthatselectthegroundstateoutof b The moments are in unitsof µB.
1 c Ref. [8]
numerous possible states arising from this competition
v
[2, 3, 4, 5].
0
6 Most studies probing this question have focused on shows that the dominant spin exchange interaction be-
4 3d oxides, and much less is known about whether re- tweenOs7+ ionsisantiferromagnetic(AFM)[7,8]. How-
1
0 lated 4d and 5d oxides can exhibit similar behavior. ever,Ba2NaOsO6undergoesaferromagnetic(FM)order-
Due to a large spatial extension of 5d orbitals, effects ingbelowT =6.8Kwithaverylowmagneticmoment,
7 C
0 of electron-correlation are weaker in 5d oxides than in i.e., 0.2 µB per formula unit (FU) [7, 8].
∼
/ 3d oxides. However,effects of spin-orbit coupling (SOC)
t
a are stronger in 5d oxides than in 3d oxides. Thus, 5d
m
oxides should exhibit a different balance between spin,
- orbital and charge degrees of freedom. In this context,
d
it is of interest to examine the electrical and magnetic
n
o properties of the cubic double perovskite Ba2NaOsO6
c [6, 7, 8]. In this 5d oxide the NaO6 octahedra share Os
v: corners with the OsO6 octahedra of orbitally degenerate
i Os7+ (5d1) ions, and the nearest-neighborOsO6 octahe- O
X
dra run along the [110] direction while the next-nearest- Ba
r
a neighbor OsO6 octahedra run along the [100] direction
(Figure 1). Ba2NaOsO6 presents several puzzling prop- Na
erties. It is an insulator despite the fact that the struc-
tureremainscubicdownto5Kwithoutanydistortionof
the OsO6 octahedra from their regular octahedral shape
[8]. It is unclear what mechanismlifts the orbitaldegen- FIG. 1: (Color online) Structure of cubic double perovskite
eracy of the Os7+ (d1) ions to make Ba2NaOsO6 insu- Ba2NaOsO6.
lating. The magnetic susceptibility of Ba2NaOsO6 be-
tween 75 and 200 K follows a Curie-Weiss law with a To gain insight into the puzzling electronic and mag-
negative Weiss temperature (i.e. θ 10 K), which netic properties of Ba2NaOsO6, we carried out a first-
≈ −
2
principles density functional theory (DFT) electronic mented in the WIEN2k code [9]. The non-overlapping
structure study. In this Letter, we show that the in- muffin-tinsphereradiiof2.50,2.14,1.86,and1.65auare
sulating property of Ba2NaOsO6 originates from a co- used for the Ba, Na, Os and O atoms, respectively. The
operative effect of electron correlation and SOC. This expansioninsphericalharmonicsoftheradialwavefunc-
cooperative effect is responsible not only for the lack of tions were taken up to l = 10. The value of RminK
MT max
structural distortion in Ba2NaOsO6 but also for the low wassetto7.0. ThetotalBrillouinzonewassampledwith
magnetic moment and the weak magnetic anisotropy [8] 125 k-points. For the exchange-correlation energy func-
of Ba2NaOsO6 in the FM state. tional,thegeneralizedgradientapproximation(GGA)by
Perdew, Burke and Ernzerhof [10] was employed. The
Our first principles DFT electronic structure calcu-
SOC was included on the basis of the second-variational
lations were performed by using the full-potential aug-
method usingscalarrelativistic wavefunctions [11]. The
mented plane waves plus local orbital method as imple-
structural parameters of Ba2NaOsO6 were taken from
the experimental values [7].
Thespin-polarizedGGAbandstructurecalculatedfor
(a)
the FM state of Ba2NaOsO6, presented in Figure 2(a),
has the Fermi level crossing both the up- and down-
5
spin t2g bands. This does not agree with the fact that
Ba2NaOsO6 is an insulator [8]. At Γ the three t2g bands
aredegenerate. Theoverallwidthofthet2g bandsisnar-
0 EF row (approximately 1 eV) because the nearest-neighbor
V) O···OdistancebetweenadjacentOsO6 octahedraislong
y(e (3.216 ˚A) compared with the van der Waals distance of
nerg 3.04 ˚A. The exchange splitting of Ba2NaOsO6 is about
E -5 0.34eV,whichisconsiderablysmallerthantypicalvalues
found for 3d magnetic oxides. Because the bandwidth is
small compared with the on-site repulsion U (approxi-
mately 3.3 eV), it was suggested [8] that electron cor-
-10 relation is important in Ba2NaOsO6. The failure of the
traditionalDFTindescribingstronglycorrelatedsystems
is currently remedied by the DFT plus on-site repulsion
W L Γ X W K U method [12, 13, 14]. Thus, we employed the GGA+U
(b) 1 method to see if the insulating property of Ba2NaOsO6
V) canbeexplained. Toavoidadoublecountinginthenon-
Energy(e 0.05 EF ospmhietrtichaelmpualrttipooflaprotteernmtisalp,rwopeoursteiodnaUletfofJ=inUth−eJadadnedd
GGA+U potential. The band structure calculated for
-0.5
W L Γ X W K
(c) 1
V) 0.5 25 TDOS Os PDOS
nergy(e 0 EF 20 OOs PPDDOOSS 04 O PDOS
E-0.5 15
-4
s
-1W L Γ X W K ate 10
(d) of st 5 -0.5 0 0.5 1
eV) 1 ensity 0
y( 0.5 D
g
er -5
En 0 EF
-10
-0.5
W L Γ X W K
-15
-8 -6 -4 -2 0 2 4 6 8
FIG. 2: (Color online) Band structures calculated for Energy (eV)
Ba2NaOsO6 usingdifferentmethods. (a) GGA,(b)GGA+U
with Ueff = 0.2 Ryd, (c) GGA+SOC, (d) GGA+SOC+U FIG. 3: (Color online) DOS of Ba2NaOsO6 obtained from
with Ueff = 0.2 Ryd. In (a) and (b) the solid and dashed the GGA+SOC+U calculation. The inset shows the PDOS
lines refer to the up- and down spin bands, respectively. In plots calculated for the Os and O atom contributions to the
(d) thevalence band top is taken as thezero-energy point. t2g bands.
3
the FM state of Ba2NaOsO6 using the GGA+U method tion on electronic structure, we carried out GGA+SOC
with U = 0.2 Ryd is shown in Figure 2(b), which re- and GGA+SOC+U calculations for the FM state of
eff
veals a larger exchange splitting (about 0.7 eV). How- Ba2NaOsO6 with the spin quantization taken along the
ever, there is no band splitting at Γ, and the dispersion [111], [110] and [100] directions. The stability depen-
characteristics of the t2g bands are almost the same as dence of the FM state on the spin quantization direc-
thoseoftheGGAcalculation. OurGGA+Ucalculations tion is negligible in the GGA+SOC calculations, but is
with larger U values (up to 0.5 Ryd) did not change not negligible in the GGA+SOC+U calculations. In the
eff
the generalpicture described above. Since Os is a heavy latter calculations with U = 0.2 Ryd, the FM states
eff
element, SOC is expected to play an important role in withthe[110]and[100]quantizationsarelessstablethan
Ba2NaOsO6. To see the effect of SOC on the electronic that with the [111] quantization by 1.9 and 16 meV per
structure of Ba2NaOsO6, we performed GGA+SOC cal- FU, respectively. The spin, orbital and total moments
culations. Figure 2(c) shows the band structure calcu- calculated for the FM state of Ba2NaOsO6 using the
lated for the FM state using the GGA+SOC method GGA+SOC+U method with U = 0.2 Ryd are sum-
eff
with the spin quantization taken along the [111] direc- marized in Table I. For each of the [111], [110] and [100]
tion. The up- and down-spin t2g bands are both split quantizations, the total moment per FU is calculated to
into three non-degenerate bands at Γ. However, these be 0.65µ ,whichissmallerthanthespin-onlyvalueof
B
∼
bandsoverlapwitheachother leadingtoa metallic state 1µ duetothefactthattheorbitalmomentof 0.33µ
B B
∼
for Ba2NaOsO6. is opposite to the spin moment in direction. Neverthe-
As described above, the GGA, GGA+U, and less, the calculated values are still large compared with
GGA+SOC methods all fail to reproduce the insulating the zero-field moments ( 0.2µB/FU) determined from
∼
state for Ba2NaOsO6. Nevertheless, we note that elec- the magnetization study of Ba2NaOsO6 [7, 8].
tron correlation enhances the exchange splitting, while The bandgapopening atthe Fermilevel, the moment
SOC splits the t2g bands. This suggests that a com- reduction and the slight magnetocrystalline anisotropy
bined effect of electron correlation and SOC might in- in Ba2NaOsO6 can be accounted for by considering the
duceabandgapinboththeup-anddown-spint2g bands. effect of SOC on the t2g orbitals of an Os7+ ion. In the
Therefore,wecarriedoutGGA+SOC+Ucalculationsfor second variational approach for spin-orbit coupling [11],
Ba2NaOsO6. The band structure calculated for the FM thescalar-relativisticpartoftheHamiltonianisdiagonal-
statewiththespinquantizationalongthe[111]direction ized on a basis adopted for each of the spin projections
ispresentedinFigure2(d),whichshowsthatBa2NaOsO6 separately, and then the full Hamiltonian matrix is con-
hasaninsulatinggapwiththelowest-lyingdown-spint2g structed on the basis of the eigenfunctions obtained in
bandlyingabovetheFermilevel. Theassociateddensity the first step. The spin-orbit part of the Hamiltonian in
of states (DOS) calculated for Ba2NaOsO6 is shown in the Os spheres is then given by
Figure 3. The partial DOS (PDOS) plots for the Os 5d
andO2pstatesrevealthattheOs5dandtheO2pstates Hˆ =λLˆ Sˆ, (1)
so
·
contribute almost equally in the t2g bands. Though not
shown, the PDOS plots calculated for the Os 5d , 5d where the SOC constant λ > 0 for the Os7+ (d1) ion
xz yz
and 5dxy orbitals show that these orbitals contribute with less than half-filled t2g levels. With θ and φ as the
equally to the t2g bands. The Os-O bonding bands of azimuthal and polar angles of the magnetization in the
the e -symmetry occur well below the Fermi level (i.e., rectangular crystal coordinate system, the Lˆ Sˆ term is
g ·
in the energy region between 7 and 5 eV), which in- rewritten as
− −
dicates the presence of a strong covalent bonding in the
OsnteastS-tiOoceogfbafroroBn,uadno2sduN.rsatOcaatsleOc,u6wl.aetTicooonfinsnsidwdeeifrreetdhpeaeFnrfMoArm-sttyeapdteefioAsrFthtMheemstFaaMgte- Lˆ ·Sˆ = S++ˆz1221(SSLˆˆˆ+−z((c−−osLLˆˆθzz+ssiinn12θθLˆ−−+eLLˆˆ−++iφees−−iniiφφθscio+ns22212θθ2Lˆ+−+eLˆLiˆφ−−seeiiniφφθcs)oi(ns222)2θθ2)).
in which the spins have the FM ordering within each Sincetheup-anddown-spint2g bandsareseparateddue
sheet of Os7+ ions parallel to the (001) plane, but have to the exchange splitting, one can neglect interactions
theAFMorderingbetweenadjacentsheetsparalleltothe between the up- and down-spin states under the SOC to
(001) plane. Our GGA+SOC+U calculation shows that a first order approximation. This allows one to consider
theA-typeAFMstateisconsiderablylessstablethanthe onlytheup-spint2g bandsusingthedegenerateperturba-
FMstate(by267meV/FU),whichisconsistentwiththe tiontheory,whichrequirestheconstructionofthematrix
observationthat Ba2NaOsO6 undergoesan FM ordering elements iHˆso j (i,j =dxy,dyz,dxz) [15]. In this case,
h | | i
below 6.8 K [8]. In the remainder of this work we will only the operators of the first line of Eq. 2 give rise to
consider only the electronic structures calculated for the nonzero matrix elements. In evaluating these matrix el-
FM state of Ba2NaOsO6. ements, it is convenient to rewrite the angular parts of
To examine the effect of the spin quantization direc- the d , d and d orbitals in terms of the spherical
xy yz xz
4
harmonics as d-orbitalscontributeequallytothestateΨ1 forthe[111]
spin quantization, but unequally for the [110] and [100]
dxy = √−2i(Y22−Y2−2) spin quantizations. The inter-octahedron hopping inte-
dyz = √i2(Y21+Y2−1) (3) gral is nonzero along the directions of nearest-neighbor
dxz = √−12(Y21−Y2−1). OzesrOo6aloocntgahtehderadi(ree.cgt.i,on[1s1o1f] nanexdt-[1n1ea0r])esbtu-nteiisghpbraocrtOicsaOlly6
Using these functions, the matrix representation octahedra (e.g., [100]). For the [100] spin quantization,
iHˆso j is found as the Ψ1 leveldoes notprovidehopping alongthe [011]di-
h | | i
rection because it has no d orbital contribution. Con-
yz
0 sinθsinφ sinθcosφ sequently, the spin-orbit coupled band associated with
−
i¯hλ/2 sinθsinφ 0 cosθ . (4) Ψ1 should be higher in energy for the [100] quantization
− than for the [111] and [110] quantizations. This in part
sinθcosφ cosθ 0
− explainswhythe[100]quantizationleadstoahigherelec-
Upon diagonalizing this matrix, we obtain the eigen- tronic energy than do the [111] and [110] quantizations.
values of the three spin-orbit coupled states, namely,
E1 = ¯hλ/2, E2 =0, E3 =h¯λ/2. The associated eigen- In summary, the insulating behavior of Ba2NaOsO6 is
functio−ns Ψ1,Ψ2,Ψ3 are given by caused by a novel cooperative effect of electron correla-
tionandSOC,whichopensabandgapattheFermilevel
Ψ1 = √22(i[scionsθφdx+yc+os(θissiinnφφ)−dco]sθcosφ)dyz athnedohrebnitcaellryemdeogveensearadtreivOinsg7+fo(drc1e)fioornso.rbTithaelsomrdaelrlimngago-f
xz
Ψ2 = −+√22(i[scionsθφdxyc−os(θissiinnφφ)+dco]sθcosφ)dyz (5) ntheeticocmcuopmieedntspofinB-oar2bNita-OcosuOp6ledariuspes-spfrionmt2tghebafnadctgtihvaets
− xz riseto anorbitalmomentthatis inopposite directionto
Ψ3 = cosθdxy+sinθcosφdyz +sinθsinφdxz. the spin moment.
For these three states Ψ1, Ψ2, and Ψ3, the orbital mo- The research was supported by the Office of Basic
mentsLalongthe spinquantizationdirectionare 1µ ,
− B Energy Sciences, Division of Materials Sciences, U. S.
0 and 1µ , respectively, according to their eigenvalues
B Department of Energy, under Grant No. DE-FG02-
and the SOC operator Hˆ = λLˆ Sˆ given that S =
so · 86ER45259.
1/2. Since the spin-orbit coupled state Ψ1 is occupied,
we obtain a negative orbital moment. The moment of
0.35µ from the GGA+SOC+U calculation is consid-
B
−
erably smaller in magnitude than 1µ . This is not
B
−
surprising because the above analysis neglected the fact ∗ Correspondingauthor.E-mail: mike [email protected]
thatthe t2g orbitalsofanOsO6 octahedronarenotpure [1] W.-G. Yin et al.,Phys. Rev.Lett. 96, 116405 (2006).
5d , 5d and 5d orbitals of the Os atom, but are [2] C. Ulrich et al.,Phys.Rev. Lett.89, 167202 (2002).
xz yz xy
[3] R. Schmitz et al., Ann.Phys. 14, 626 (2005).
givenbytheirlinearcombinationswiththe2porbitalsof
[4] G. Khaliullin and S. Okamoto, Phys. Rev. Lett. 89,
the surroundingO atoms. Given that the Os 5d and the
167201 (2002).
O 2p states contribute almost equally in the t2g bands [5] E. Pavarini et al.,Phys. Rev.Lett. 92, 176403 (2004).
(Figure 3), an orbital moment of approximately 0.5µB [6] A. W. Sleight et al.,Inorg. Chem. 1, 245 (1962).
−
should be expected from the occupation of the up-spin [7] K. E. Stitzer et al.,Solid State Sci. 4, 311 (2002).
t2g band associated with Ψ1. [8] A. S. Erickson et al.,cond-mat/0610385.
The above analysis indicates that SOC splits the t2g [9] P.Blahaetal.,inWIEN2K,AnAugmentedPlaneWave
bands into three subbands regardless of the direction of Plus Local Orbitals Program for Calculating Crystal
Properties, edited by K. Schwarz (Techn. Universit¨at
the magnetization. However, SOC does not affect the
Wien, Austria, 2001).
exchange splitting. It is the on-site repulsion that en- [10] J. P. Perdew et al.,Phys. Rev.Lett. 77, 3865 (1996).
hances the exchange splitting and increases the energy [11] J. Kuneˇs et al., Phys. Rev. B 63, 205111 (2001) and
separation between filled and empty bands within each references therein.
spin channel. That is, a cooperative effect of electron [12] V. I. Anisimov et al., Phys.Rev.B 48, 16929 (1993).
correlation and SOC is essential in opening a band gap [13] Our calculations using the method of Czyzyk and
at the Fermi level for Ba2NaOsO6. Eq. 5 shows that STa.wCaztyzzkyyklaedndeGss.enAt.iaSllaywatotzkthy,ePshaymse. Rreesvu.lBts.49Se,e1:42M11.
the contributions of the d , d and d orbitals to the
xy yz xz (1994).
spin-orbit coupled state Ψ1 depend on the direction of [14] Our calculations using the local density approximation
the magnetization (i.e., θ = 90 and φ = 0 for [100];
◦ ◦ led essentially to the same results except that a larger
θ = 90◦ and φ = 45◦ for [110]; θ = arccos(√3/3) and Ueff (>0.21 Ryd)is needed toopen an insulating gap.
φ=45 for [111]). This is responsible for the weakmag- [15] D. Dai and M.-H. Whangbo, Inorg. Chem. 44, 4407
◦
netic anisotropyobservedfor Ba2NaOsO6 [8]. The three (2005).
