Multiband $d-p$ model and self-doping in the electronic structure of Ba$_2$IrO$_4$

0.51 MB·

by  Krzysztof Rościszewski

Multiband d−p model and self-doping in the electronic structure of Ba IrO
2 4∗
Krzysztof Ro´sciszewski1 and Andrzej M. Ole´s1,2
1Marian Smoluchowski Institute of Physics, Jagiellonian University,
prof. S. L(cid:32)ojasiewicza 11, PL-30348 Krako´w, Poland
2Max-Planck-Institut fu¨r Festko¨rperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany
(Dated: January 18, 2016)
Weintroduceandinvestigatethemultibandd−pmodeldescribingaIrO layer(suchasrealized
4
inBa IrO )whereall34orbitalsperunitcellarepartlyoccupied,i.e.,t ande orbitalsatiridium
2 4 2g g
and2porbitalsatoxygenions. Themodeltakesintoaccountanisotropiciridium-oxygend−pand
oxygen-oxygen p−p hopping processes, crystal-field splittings, spin-orbit coupling, and the on-site
Coulomb interactions, both at iridium and at oxygen ions. We show that the predictions based
6 on assumed idealized ionic configuration (with n = 5+4×6 = 29 electrons per IrO unit) do
0 4
1 not explain well the independent ab initio data and the experimental data for Ba IrO . Instead
2 4
0 we find that the total electron density in the d−p states is smaller, n = 29−x < n (x > 0).
0
2 Whenwefixx=1,thepredictionsforthed−pmodelbecomemorerealisticandweaklyinsulating
n antiferromagnetic ground state with the moments lying within IrO2 planes along (110) direction is
a found, in agreement with experiment and ab initio data. We also show that: (i) holes delocalize
J over the oxygen orbitals and the electron density at iridium ions is enhanced, hence (ii) their e
g
5 orbitals are occupied by more than one electron and have to be included in the multiband d−p
1 model describing iridates.

] PACSnumbers: 71.10.Fd,71.70.Ej,74.70.Pq,75.10.Lp
l
e
-
r I. INTRODUCTION tical conductivity, x-ray absorption measurements, and
t
s first-principles band calculations [15]. Recently it has
.
t Iridates such as Sr IrO and Ba IrO are transition been shown that strong spin-orbit coupling changes rad-
a 2 4 2 4
m metal oxides with party occupied, spatially extended ically the electronic states in Mott insulators [16, 17].
5d orbitals. They belong to strongly correlated sys- Within this limit of an insulator, strong spin-orbit in-
d- tems and are weak antiferromagnetic (AF) insulators, teraction accompanied by large crystal-field effects split
n with a complex competition of local Coulomb interac- t2g orbitals of Ir4+ ions into fully filled manifold with ef-
o tions, Hund’s exchange coupling, crystal-field effects and fective total angular momentum Jeff = 3/2 and singly
c very strong spin-orbit interactions. The debate contin- occupied manifold Jeff = 1/2 (half-filled ground state)
[ ues about whether these systems are better viewed as a [18, 19]. Consequently, tight-binding model calculations
1 realization of: (i) spin-orbit Mott insulators [1], or (ii) were performed using a model for t2g orbitals which in-
v an old idea of a Slater insulator with insulating behavior cludes the above effects. It gives results in agreement
5 resulting from long-range AF order in a correlated elec- withtheangle-resolvedphotoemissiondatafortheoccu-
8 tronic band [2] is also at work in iridates [3, 4], or finally pied electronic bands [13].
9
3 (iii) a mixture of both above scenarios interrelated with The origin of AF order in the ground state is more
0 each other and contributing almost equally [5]. subtle. It was shown by ab initio calculations that the
1. TheelectronicstructureofSr2IrO4 (andtoalesserex- Heisenberg superexchange is the largest low-energy scale
0 tentofBa2IrO4)wasasubjectofnumerousinvestigations but also Ising-like compass interactions contribute [20].
6 [3, 6–8] on a different level of sophistication. The inter- Therefore, individual Ba IrO layers provide a close re-
2 4
1 est in Sr2IrO4 is motivated by its structural similarity alization of the quantum spin-1/2 compass-Heisenberg
: tocuprates[9],andindeedd-wavesuperconductivitywas model [21]. We note that similarly complex structure
v
i predictedinelectrondopedSr2IrO4 bynumericalstudies of the superexchange was established for the honeycomb
X [10]. Recentlymappingofthethree-band(d−p)modelto lattice compound Na TiO , where it takes the form of
2 3
r asinglebandwaspresentedanditwasshownthatfurther Kitaev-Heisenberg model which is under intense discus-
a
neighbor hopping is necessary to describe the difference sion at present [22].
betweenholeandelectrondopediridates[11]. Therecent
However,thereremainsomeseriousdoubts(supported
experimentalevidenceseemstosupportthescenariothat
byexperimentaldata)whethersuchasimplifiedscenario
Ba IrO isacloserealizationofspin-orbitMottinsulator
2 4
focused on t orbitals is indeed fully realistic [23]. One
rather than a Slater system [12, 13], similar to Sr IrO 2g
2 4
hastoadmitthatasimplepictureandamodelassimple
investigated using angle-resolved photoemission [14], op-
as possible are useful tools when it comes to interpreta-
tion of rather complicated experimental data. However,
this is only one level of physical description. On the sec-
∗ ThisworkisdedicatedtoProfessorJo´zefSpa(cid:32)lekontheoccasion ondlevelonerequiresmuchbetterunderstandingofwhat
ofhis70th birthday. is going on. This is the aim of the present paper — we
2
wish to verify to what extent the charge transfer from A. Kinetic energy
Ba ions to IrO units is complete and whether the ionic
4
model could be used to describe the electronic structure The kinetic part is:
ofBa IrO . Wearenotsatisfiedwithanidealisticpicture
ofIr4+2 (wi4theg orbitalsbeingempty)surroundedbyO2− Hkin = (cid:88) (cid:16)tiµ;mνc†iµ,σcmν,σ+H.c.(cid:17), (2.2)
ions and we want to investigate the true charge densities
{iµ;mν},σ
and the magnetic order parameters realized in this sys-
tem. Tothisendweconstructedamultibandd−pmodel where c† stands for the creation of an electron at site
iν,σ
andperformedunrestrictedHartree-Fock(HF)computa- i in an orbital ν with spin σ =↑,↓. The model includes
tions on a finite IrO4 cluster which contains 4×4 Ir ions all5dorbitalstatesν ∈{xy,yz,zx,x2−y2,3z2−r2}per
and the accompanying 4×4×4 oxygen ions — half of Ir atom at site m (in this order), and three 2p orbitals
them located within the same plane as Ir ions, while the ν ∈{p ,p ,p }peroxygenatomatsitei. Whenchoosing
x y z
second half being in out-of-plane (apical) positions. The analternativeandmoreintuitivenotation,onecanwrite
modelinvolves(perasingleIrO4 unit)five5dorbitalsat creationoperatorsd†mν,σforelectroncreationindorbitals
each Ir and 4×3 oxygen 2p orbitals per unit cell, and and p† for p orbitals, as used in Sec. IID.
theseorbitalsareoccupiedby: (i)n =5+4×6electrons, iν,σ
0 The matrix elements t are assumed to be non-
i,µ;m,ν
according to the formal and idealized ionic model, or (ii)
zeroonlyfornearestneighboriridium-oxygenp−dpairs,
alowerelectronnumberbyoneelectron,i.e.,n=4+4×6
andasimilarformulastandsfornearestneighboroxygen-
electrons, according to the realistic ab initio computa-
oxygen p−p pairs (i.e., next nearest hopping elements
tions [24].
are neglected). The geometry of the cluster is identical
The paper is organized as follows. In Sec. II we intro- tothatforRuO layer(asrealizedinSr RuO ),thusthe
4 2 4
ducethemultibandmodelwhichincludesall5dstatesat matrix elements t can be directly adapted from the
i,µ;j,ν
iridium ions and 2p states at oxygen ions. The parame- Appendix of Ref. [25].
ters of the model are specified in Sec. III. The Hartree-
Fock approximation for the Coulomb interactions is ex-
plainedinSec. IV,whileinSec. Vwepresenttheresults B. Crystal-field splittings
of numerical calculations: for the formal ionic model in
VA,andforthemodelwithself-dopinginSec. VB.Here
LetusnowpresentthediagonalpartH ofthed−p
diag
wealsodiscussthejustificationoffiniteself-doping(with
Hamiltonian (2.1). It depends only on electron number
respect to the electron densities in the ionic model) and
operators, and takes into account the effects of crystal
the possible effect of electronic correlations. The paper
fieldandthedifferenceofreferenceorbitalenergies(here
is concluded with a short discussion and summary of the
we employ the electron notation),
main results in Sec. VI.
∆=ε −ε , (2.3)
d p
between d and p orbitals, both for empty states (i.e.,
II. MODEL HAMILTONIAN when neglecting the interaction terms from H , see
int
below). We remark that in principle ε might be dif-
p
ferent for apical oxygens and for in-plane oxygens. A
The d−p Hamiltonian for IrO plane reads,
4
large difference ∼1 eV between these level positions was
considered in ruthenate perovskites [28] as a possibil-
H=H +H +H +H . (2.1)
kin diag so int ity. We have found within the present d−p model (also
in a ruthenate [25]) that such a difference is definitely
It consists of several parts — the different terms in Eq.
too large and would overestimate the difference between
(2.1) stand for the kinetic energy (H ), spin-orbit cou-
kin chargedensities. Qualitatively, themainfactorresponsi-
pling (H ), crystal-field splittings (H ), and the in-
so diag bleforthechargeanisotropybetween2porbitalsatthese
traatomic Coulomb interactions (H ).
int nonequivalent oxygen position is their weaker hybridiza-
In Sec. V we report a study of charge space-
tion with 5d orbitals, while the difference between the
homogeneous solutions. All symmetry-equivalent occu-
corresponding level energies is of secondary importance.
pation numbers, i.e., HF primary order parameters, are
Weputthereferenceorbitalenergyε forIr(5d)states
d
assumed to be the same in different parts of the clus-
tobezero,henceweuseonlyε asaparameterandwrite:
p
ter. Whenstudyingthepossibilityofantiferromagnetism
(cid:88)
therearetwosublatticesandthenumberoforderparam- H = ε p† p
diag p i,µ,σ i,µ,σ
eters doubles. Looking for charge space-homogeneous
i,µ=x,y,z;σ
ground states can be considered to be a consequence (cid:88)
+ f d† d . (2.4)
ofstronglong-rangeinter-ionicelectrostaticinteractions. µ,σ m,µ,σ m,µ,σ
These interactions are not explicitly included in the m,µ=xy,yz,...;σ
model (2.1) but to some extent they are accounted for Here the first sum is restricted to oxygen sites, while the
by the homogeneity assumption. secondonerunsoveriridiumsites. Thevectorcontaining
3
the elements of {f } is while they vanish for Ba IrO (see Fig. 1 in Ref. [13]).
µ,σ 2 4
     
2 0 0
C. Spin-orbit coupling
−1 0 0
−1 0 0
      Formally, spin-orbit part H of the Hamiltonian Eq.
 0 1 0 so
      (2.1) has similar mathematical structure to the kinetic
1  0 0 1
f = D  +D  +D  . (2.5) part H [25–27], with tso elements restricted to
3 1  2 2 0 3 0 kin µ,σ;ν,σ(cid:48)
      single iridium sites,
−1 0 0
     
−1 0 0  
     
 0 1 0 H =(cid:88)H(m)=(cid:88) (cid:88) tso d† d +H.c.,
0 0 1 so so µ,σ;ν,σ(cid:48) mµ,σ mν,σ(cid:48)
 
m m µ(cid:54)=ν;σ,σ
(2.6)
Itincludestheorbitalsplittingsof5dorbitalsatIrionsin where the summation runs only over iridium sites. The
the tetragonal crystal field. The constant D1 serves as a matrix elements are all proportional to spin-orbit cou-
crudeestimateofthesplittingbetweentheorbitalxyand pling strength ζ which is large on Ir sites and usually
the orbital doublet {yz,zx}. The {D2,D3} parameters assumed to be about ζ = 0.4 eV [8]. Just like the hop-
refer to splitting between t2g and eg orbitals at iridium ping elements also the elements tsµo,σ;ν,σ(cid:48) can be directly
and are much larger than D1, see below. adaptedfromRef. [25](fortheexplicitentriesfortso
µ,σ;ν,σ(cid:48)
Jahn-Teller part will be neglected in the Hamiltonian see Eq. (2.4) in Ref. [25]). As we use the basis of
(2.1). The exception is the elongation of bonds between real 5d orbitals (and not the spherical harmonics) sev-
iridium and apical oxygens which could be considered eral spin-orbit elements turn out to be imaginary (thus
as static and global Q Jahn-Teller distortion, but it is our Hamiltonian (2.1) is complex). Note that the conse-
3
simpler to include it by a proper renormalization of the quence of finite spin-orbit coupling is that the total spin
crystal-field splittings. Note that in Sr IrO Jahn-Teller of the system and total zth spin component are not con-
2 4
effects are not negligible due to distortions of octahedra, served quantities.
D. Local Coulomb interactions
The last part of the multiband d−p Hamiltonian, H ≡Hd +Hp , stands for strong local on-site interactions.
int int int
For the d orbitals at iridium sites it reads,
(cid:18) (cid:19)
(cid:88) 1 (cid:88) 5
Hd =U n n + U − J n n
int d m,µ,↑ m,µ,↓ 2 d 2 d,µν m,µ m,ν
m,µ i,µ(cid:54)=ν
(cid:88) (cid:88)
− J S ·S + J d† d† d d . (2.7)
d,µν m,µ m,ν d,µν m,µ,↑ m,µ,↓ m,ν,↓ m,ν,↑
i,µ(cid:54)=ν m,µ(cid:54)=ν
where J is the tensor of on-site inter-orbital exchange elements for d orbitals which can be expressed using Racah
d,µν
parameters B and C [29, 30] (see also Table I given by Horsch in Ref. [31]). The anisotropy between different Hund’s
exchange elements {J } vanishes only for orbitals of the same symmetry, i.e., either in pure t system or in pure
d,µν 2g
e system[32]). Forconvenience,werewriteEq. (2.7)toseparateitintothediagonaltermsinelectrondensities(first
g
line) and the quantum fluctuating part (second line) as follows,
(cid:88) 1 (cid:88) 1 (cid:88)
Hd =U n n + (U −3J )n n + (U −2J )n n
int d mµ,↑ mµ,↓ 2 d d,µν mµ,σ mν,σ 2 d d,µν mµ,σ mν,−σ
m,µ mµ(cid:54)=ν,σ mµ(cid:54)=ν,σ
(cid:88) (cid:88)
− J d† d d† d + J d† d† d d . (2.8)
d,µν mµ↑ mµ,↓ mν↓ mν,↑ d,µν mµ,↑ mµ,↓ mν,↓ mν,↑
m,µ(cid:54)=ν i,µ(cid:54)=ν
The formula for local Coulomb interactions 2p orbitals at oxygen sites, Hp , is analogous,
int
(cid:88) 1 (cid:88) 1 (cid:88)
Hp =U n n + (U −3J ) n n + (U −2J ) n n
int p iµ,↑ iµ,↓ 2 p p iµ,σ iν,σ 2 p p iµ,σ iν,−σ
i,µ i,µ(cid:54)=ν,σ i,µ(cid:54)=ν,σ
(cid:88) (cid:88)
− J p† p p† p + J p† p† p p . (2.9)
p iµ,↑ iµ,↓ iν,↓ iν,↑ p iµ,↑ iµ,↓ iν,↓ iν,↑
i,µ(cid:54)=ν i,µ(cid:54)=ν
4
and is defined by the intraatomic Coulomb element U
p
andHund’sexchangeJ ,asthetensorJ hasidentical
elements J for all off-pdiagonal pair ofp,tµhνe oorbitals of 6 dx2−y2
p
the same symmetry. Here n ≡ p† p and p†
iµ,σ iν,σ iν,σ iµ,σ
operatorsrefertothedensityandelectroncreationwithin
(µ,σ) spin-orbital at site i. 6 d3z2−r2
III. THE HAMILTONIAN PARAMETERS D2
D
3
A. Previous studies and deducing the parameters
The effective d − p model (2.1) requires a choice of
d , d
a number of explicitly included parameters. They may 6 xz yz
D 6
be to a large extent deduced from the previous ab ini- 1 d
xy
tio studies. We have adopted the values of in-plane
hopping elements (pdσ) and (pdπ) used in Ref. [8] for
∆
Sr IrO , and rescaled them (both for in-plane and out-
2 4
of-plane bonds) using Harrison formulas [33] to fit the
bond lengths in the structure of Ba2IrO4 (reported by px,py,pz
Moser et al. [13]). The elements (ppσ) and (ppπ) were Irion Oion
taken directly from Ref. [25] and were rescaled in a sim-
ilar way.
The choice of the Coulomb elements is rather difficult
FIG.1. (Coloronline)Artist’sviewofthe5dorbitalenergies
duetotheirunknownscreening. Therearemanyreliable
split by the elements {D ,D ,D } at Ir ions in the absence
1 2 3
estimations for Ud in effective models featuring only 5d ofspin-orbitcoupling(ζ =0)(left),andthed−psplitting∆
Wannier orbitals at iridium ions (2p oxygen orbitals are Eq. (2.3) between the Ir(5d) and O(2p) orbitals (right).
absent, but effective iridium 5d orbitals are then renor-
malized by hybridization with oxygen 2p orbitals). No
need to say that such parameters would have to be com-
bital Coulomb repulsion U at oxygen ions (in iridates).
pletely different in the framework of the multiband d−p p
In Ref. [39] this parameter is estimated to be within
model. A rather low value of U ∈[1.0,2.0] eV was first
d U ∈ [4.0,6.0] eV. We recall that in cuprates the val-
suggested by Mazin et al. [34], but we argue that a p
ues U = 4.5 eV [40] and U (cid:39) 4.0 eV [41, 42] were
more probable value is U = 3.0 eV. This value (and p p
d considered; note that several other values were also sug-
J = 0.6 eV) were used in local density approximation
d gested (all in the range 3-8 eV), while 6.0 eV was in-
(LDA+U) computations for Na IrO [35, 36] and also
2 3 dicated by some experimental data [43]. We use below
for Ba IrO [13]. A similar value of U = 2.72 eV was
2 4 d U =4.4 eV. For Hund’s coupling J at oxygen ions the
obtainedbyconstrainedrandomphaseapproximationin p p
valuesJ =0.6−0.8eVweresuggested[39],whileGrant
Na IrO [37]. Other estimates are: U (cid:39) 1.9 eV [3] and p
2 3 d and McMahan computations in cuprates yield J = 0.8
U = 2.5 eV [4, 23]. Following this discussion and argu- p
d eV [42]. Following these estimates we use below as well
ments presented in Ref. [4], we decided to fix U = 2.5
d J =0.8 eV.
eV. p
Hund’s exchange elements are less screened than in- TocompletethesetoftheHamiltonianparameterswe
traorbital Coulomb elements and are closer to their must provide estimates for the crystal-field splittings of
atomic values. For Hund’s exchange Jd between two t2g Sec. IIB. These are: D1 = −0.07 eV, D2 = 2.6 eV,
electronsweselectedthevalue0.5eV,inagreementwith and D3 = 2.0 eV, and we adopted them from ab initio
the old semiempirical prescription (J /U (cid:39) 0.2) [38]. computations[24]. The5dorbitaldegeneracyisliftedby
d d
The same value was used in Refs. [8, 34] while slightly these terms and the orbital states at Ir ions for ζ = 0
smaller (Jd = 0.4 eV [13]), very small (Jd = 0.14 eV are shown in Fig. 1. The eg orbitals are much higher
[3]), or larger (Jd = 0.6 eV [35, 36]) values were also thant2g ones, andthehighestorbitalenergyisfoundfor
considered. For the sake of fixing precisely Hund’s cou- x2 −y2 orbital, similar to the situation in cuprates for
pling tensor elements J we use Table I from Ref. the corresdponding 3d orbital [41, 42].
d,µν
[31] and in addition we use a semiempirical formula The charge-transfer gap ∆ (2.3) will be examined as a
C (cid:39) 4B for Racah parameters. With this Ansatz one continuous parameter in the range from 1.0 eV up to 5.0
finds J =3B+C ≈7B for a pure t system [32]. The eV, and the representative situation is shown in Fig. 1.
d 2g
J elements concerning e orbitals are finite and fixed Indeed, oxygen 2p orbital energies are below the Fermi
d,µν g
again using the entries from Table I in Ref. [31]. energy, but hybridization with 5d orbitals is responsible
Unfortunately, not much is known about the intraor- for partial electron transfer towards Ir(5d) orbitals ana-
5
ionic model with 6 electrons occupying 2p levels of each
TABLE I. Parameters of the Hamiltonian (2.1) (all in eV)
oxygen and 5 electrons occupying 5d levels of each Ir ion
used in the HF calculations. For the hopping integrals we
can serve only as a rather crude starting view for the
presentonlyrepresentativein-planeSlaterintegrals(pdπ)and
electronic structure of Ba IrO . For the sake of conve-
(ppπ). Out-of-plane integrals are obtained by applying Har- 2 4
nience let us introduce the notion of self-doping x for
rison scaling [33]. The charge-transfer gap, ∆ = ε −ε , is
d p
examined in the range ∆∈[1.0,5.0] eV. a single IrO4 unit: with respect to the idealized for-
mal ionic model where we have x = 0; instead for the
Ud Jd Up Jp ζ D1 real compound we shall consider finite self-doping value
2.5 0.5 4.4 0.8 0.43 −0.07 x = 1.0, i.e., one electron less per IrO unit (according
4
D D (pdσ) (pdπ) (ppσ) (ppπ) to Katukuri [24] x ∼ 0.8; however we shall use x = 1
2 3
instead because our cluster is rather small and while the
2.6 2.0 −1.69 0.78 0.55 −0.14
self-doping x = 1 translates well into integer total elec-
tron number in the cluster, other fractional values of x
would not.)
lyzedbelowinSec. V.Wehavetaken∆asafreeparam-
eterasitsprecisevalueisunknown. Thereareonlyafew
values for ∆ in the literature, and these data are incon-
IV. THE UNRESTRICTED HARTREE-FOCK
clusive. In Refs. [6, 8] ∆ = 3.3 eV was used. According
APPROXIMATION
to ab initio computations [24] of charge-transfer excita-
tions, the value of ∆ is about 2.0 eV or slightly larger —
A. The self-consistent Hartree-Fock problem
this value however is not a bare parameter but includes
spin-orbit coupling plus strong correlation effects; thus
the bare value could be significantly larger. In Ref. [44] We use the unrestricted HF approximation to investi-
the value of ∆=2.0 eV is given for Na2IrO3. gate the IrO4 cluster (with cyclic boundary conditions).
Thetechnicalimplementationisthesameasdescribedin
Finally,thespin-orbitcouplingstrengthζ onIrsitesis
Refs. [25,26,46]. Namely,thelocalCoulombinteraction
large — it modifies the orbital states and mixes the spin
HamiltonianH isreplacedbymeanfieldtermsderived
states at iridium ions [16]. It is usually assumed to be int
0.4 eV [8]. A more accurate value is 0.43 eV [19, 24] and in HF approximation. The averages (cid:104)d†iµ,σdiν,σ(cid:48)(cid:105) (which
we use it in the present study. appear within this treatment) can be treated as order
All the parameters we use in the calculations below parameters(thereisasimilarsetoforderparametersfor
are collected in Table I. Note that during computations oxygens). Forthenumericalcalculationssomeinitialval-
we are setting the value of ε to be zero as the reference ues(someeducatedguess)havetobeassignedtothemto
d
energy. Note that the value of parameter (pdσ) which is start the search for a self-consistent solution of the HF
involved in hopping processes from t to e orbitals is equations. During HF iterations the order parameters
2g g
−1.69 eV, see Appendix in Ref. [25], and the splitting are recalculated self-consistently until convergence.
betweent ande orbitalsis2.0−2.6eV.Thissuggests
2g g
that the expectation that e levels are empty is unrealis-
g
tic. B. Hartree-Fock calculations
We are interested in charge-homogeneous solutions, in
B. Previous views and ab initio studies particular homogeneity concerns primary order parame-
ters, (cid:104)d† d (cid:105). with µ = ν, i.e., the electron densities
iµ,σ iν,σ
(occupations). Toobtainunbiasedresults, wehavestud-
There are several views on the electronic structure of
iedseveraldifferenttypesoforder,andcomparedtheen-
Ba IrO intheliterature. Frequentlyitisbeingassumed
2 4
ergiesoftheself-consistentlyfoundsolutionstoestablish
that e orbitals are empty in Ba IrO . In the present
g 2 4
thegroundstate. Consequently,duringthecomputations
paper we make an attempt to find the electron densi-
the following scenarios were studied:
ties in e orbitals and in 2p oxygen orbitals. We are
g
motivated by the ab initio computations performed on
• nonmagnetic phase, the x,y,z spin components at
a small cluster with embedding [24]. According to Mul-
all atoms were set to be zero;
likanpopulationanalysisthe5d-shellchargeonIrionsis
about6.5e,theeffectiveionicchargewithin2porbitalsis
• ferromagnetic(FM)phasewithspinsalignedalong
about 5.2e on in-plane oxygens and about 5.6e on apical
(1,1,0) direction;
oxygens [45]. Note however that the direct mapping of
these entries to the d−p model can not be perfect as ab • FMphasewithspinsalignedalong(1,0,0)direction
initio computations include in addition oxygen valence (parallel to the a axis);
s-orbitals, and also s, p and f-orbitals at iridium ions
(absent in the d−p model). • FM phase with spins (at all atoms) aligned along
Nonetheless, it seems clear that the formal idealized (0,0,1) direction (parallel to the c axis);
6
Thetotalnumberofdifferentorderparametersislarge
and results in a rather slow convergence of the HF pro-
cedure. Thestandardremedyforpoorconvergenceisso-
called dumping technique or a more sophisticated (pos-
sibly better) quantum chemistry technique called level
shifting [47] (more details on the level-shifting technique
as applied to the d−p multiband model may be found
in Ref. [25]). We have used both techniques in the
present calculations. HF convergence criteria were the
following conditions fulfilled simultaneously for the two
consecutive iterations: (i) relative energy change should
be smaller than 0.2×10−6; (ii) absolute charge change
shouldbesmallerthan0.5×10−3e;(iii)absolutemagneti-
zationchange(perion)shouldbesmallerthan0.5×10−3.
Performingcomputationswithdumpingorwithdifferent
levelshifts,alsowithdifferentstartingconditions(initial
charge and magnetic densities) we completed many runs
for each set of Hamiltonian parameters. The compar-
isons done afterwards lead us to believe that a practical
accuracy of our HF solutions (on convergence) is about
1-2 meV.
V. NUMERICAL RESULTS
Numerical studies of the multiband d−p model (2.1)
FIG. 2. (Color online) Antiferromagnetic spin alignment require not only the parameters which were fixed in Sec.
along (1,1,0) direction in a 4×4 IrO cluster. The arrows
4 III, but also an assumption concerning the total elec-
represent magnetic moments within an IrO plane with the
2 tron number per unit cell. We consider below two differ-
structure of CuO plane in cuprates (apical oxygens are not
2 ent scenarios: (i) the idealistic formal ionic model with
shown),withredarrowsforiridiumions,andbluearrowsfor
n =5+4×6=29 electrons per IrO unit, and (ii) the
in-plane oxygen ions; the spin magnitudes are arbitrary and 0 4
only the directions are relevant. Vertical and horizontal lines realisticmodelwithasmallertotalnumberofn=29−x
are only guides to the eye — they indicate IrO units for a electrons,whereweconsideronlyoneself-doping namely
4
better visibility of individual symmetry and unit cells within x = 1.0 [24]. Thereby we concentrate on the most im-
the cluster. portant results obtained for realistic values of Coulomb
interactions within the framework of these two different
scenarios.
• AF phase with spins (at all atoms) aligned along
(1,1,0) direction according to the pattern shown in
Fig. 2;
A. Formal ionic model (self-doping x=0)
• AF phase with spins (at all atoms) aligned along
Taking the formal ionic model as a starting configu-
(1,0,0) direction (like in Fig. 2 but with spins ro-
ration for the HF iterations, we assume that each IrO
tated by 45 degrees); 4
unit has a negative charge Q = 29e (the two Ba2+ ions
are considered only as donors of 4e to the IrO unit).
• AF phase with spins (at all atoms) aligned along 4
The initial charge distribution assumed on start of HF
(0,0,1) direction.
iterations quickly redistributes to reach the uniform and
• in addition, all the magnetic phases were studied stable distribution during the iteration process (this is
again with an additional constraint that the mag- achievedduetofinited−phybridization). Theemerging
netic moments at oxygen ions vanish, i.e., for non- ground states are shown in Table II: for the ionic model
magnetic oxygens. with x = 0 in the top part, and for the realistic model
with self-doping x = 1 (for the discussion see the next
Let us immediately comment on the the last scenario subsection) in the bottom part. From the data in the
(magnetic iridiums and nonmagnetic oxygens): surpris- top part of Table II it follows that strong magnetic de-
ingly, it yields too high HF energies and therefore it is generacy occurs for the resulting magnetic ground states
never realized in the ground state. Therefore, we con- obtainedinHFapproximation. Itisremarkablethatthe
clude that also oxygen ions contribute to the magnetic values of charge densities as well as average magnetic
order by double exchange mechanism. moments and average angular momenta are almost the
7
TABLEII.ResultsoftheHFcalculationsforthed−pmodel(2.1)concerningthemagneticorderanddegeneracyofHFground
state for: ionic model without self-doping (x = 0) (top), and realistic model at self-doping x = 1.0 (bottom), as obtained for
increasing charge-transfer gap, ∆ (2.3). The quantities presented are: HOMO-LUMO gap G, the average energy of spin-orbit
term (2.6) per a single iridium ion (cid:104)SO(cid:105), HF energy per one IrO unit E . The obtained states are: AF1 — AF state with
4 HF
moment alignment along (1,1,0) direction; AF2 — AF state with moment alignment along (1,0,0) direction, i.e., along the a
axis; FM1 and FM2 are FM states with the numbers 1 and 2 having the same meaning like for AF states. The numbers in
brackets (last column) are in (meV) and denote the energetic distance to the true (best) HF ground state. Note that some
low-energystateswithadifferentmagneticordercouldnotbedetectedduringcomputations(denotedby? inthelastcolumn).
NotealsothatallthesestatesarewellinsulatingduetosizableHOMO-LUMOgapsG. Incontrast,thegroundstatesobtained
for the realistic model (x = 1.0) have very small or small HOMO-LUMO gaps G, and are weakly insulating or even close to
conducting.
model x ∆(eV) G (eV) (cid:104)SO(cid:105) (eV) E (eV) ground state excited states
HF
ionic model 0.0 1.0 0.12 −0.36 141.675 AF1 ?
1.5 0.29 −0.41 130.964 AF2 AF1 (2); FM1 (2); FM2 (2)
2.0 0.36 −0.43 120.093 AF1 AF2 (0); FM1 (2); FM2 (2)
2.5 0.41 −0.46 109.159 AF1 AF2 (0); FM1 (0); FM2 (0)
3.0 0.45 −0.49 98.162 AF1 FM1 (0); AF2 (1); ?
3.5 0.46 −0.54 87.094 AF1 FM1 (4); FM2 (3); ?
4.0 0.40 −0.59 75.952 AF1 AF2 (5); FM1 (5); FM2 (5)
4.5 0.31 −0.64 64.731 AF1 FM1 (3); FM2 (3); AF2 (5)
5.0 0.29 −0.68 53.436 AF1 FM1 (0); FM2 (7); AF2 (7)
realistic model 1.0 1.0 0.028 −0.41 128.974 AF1 AF2 (0); FM1 (1)
1.5 0.032 −0.43 118.613 AF1 AF2 (2); ?
2.0 0.042 −0.45 108.197 AF1 AF2 (0); FM1 (0); FM2 (0)
2.5 0.064 −0.48 97.723 AF1 FM1 (30); ?
3.0 0.154 −0.58 87.134 AF1 FM1 (1); ?
3.5 0.091 −0.61 76.444 AF1 FM1 (0); ?
4.0 0.019 −0.62 65.663 FM1 AF1 (3); ?
4.5 0.129 −0.67 54.846 AF2 FM2 (0); AF1 (2)
5.0 0.231 −0.68 43.852 AF2 FM1 (4.5); AF1 (4.5)
same for degenerate FM and AF HF states. On the one B. Realistic model with self-doping x=1.0
hand, the experimentally observed antiferromagnet with
momentsalignedalong(1,1,0)directionisconfirmed,but
In the second scenario we follow the ab initio results
on the other hand its energy is degenerated with other
of Katukuri et al. [24] and we assume a reduced total
magnetic phases.
numberofelectronsperIrO unit. Takingthetotalelec-
4
Finally,wepresentchargeandmagnetizationdensities tron number n = 29−x with x = 1.0 this corresponds
obtained for the ionic model, see Fig. 3. As expected, to self-doping by one hole. The obtained ground states
increasing ∆ reduces the charge density within 5d Ir or- andtheircharacteristicsareshowninthebottompartof
bitals due to weaker hybridization. Notably, there is a Table II while the corresponding charge and magnetiza-
considerable electron transfer to e orbitals which con- tion densities are displayed in Fig. 4. Altogether, taking
g
taintypicallymorethanonee electronperIrion. More a representative value of ∆ = 3.0 eV one fnds enhanced
g
precisely, the eg electron density decreases with increas- electron density in eg orbitals and all magnetic moments
ing∆from∼1.7to∼1.0intheinvestigatedrangeof∆. larger by a factor close to 2 than those obtained in the
Also charge transfer between 5d and 2p orbitals is found ionic model, cf. Figs. 3 and 4.
for increasing ∆, and the charge densities increase for From the lower part of Table II we conclude that AF1
bothnonequivalentoxygenpositions,in-planeandapical. isthetruegroundstateforabroadrangeof∆∈[1.0,3.5]
The density is closer to (cid:104)n (cid:105) = 6 obtained in the ionic eV. For higher values of ∆ the identification of the true
p
pictureforapicalpositions,wherethehybridizationplays ground state is difficult due to strong degeneracy of the
aminorrole. However,themostimportantfeatureisthe lowestenergystatesobtainedfromHFiterationsforvar-
observed disagreement of the results of Fig. 3. with the ious magnetization distributions. Moreover, very small
availableab initio data[24]. Thechargeoccupationsand HOMO-LUMO gaps obtained frequently are direct ev-
themagnetisationdensitiesforAF(alignedalong(1,1,0) idence that multiconfiguration HF computations would
direction) HF ground state are shown in Fig. 3. be necessary to establish the ground state and its mag-
8
7.5 7.5
7 7
y y
nsit6.5 nsit6.5
e e
d d
ron 6 ron 6
ct ct
ele5.5 ele5.5
5
5
1 2 3 4 5 1 2 3 4 5
∆ (eV) ∆ (eV)
0.3
0.5
y y
nsit nsit0.4
e0.2 e
d d
on on 0.3
ati ati
s s
eti0.1 eti0.2
n n
g g
a a0.1
m m
0
0
1 2 3 4 5 1 2 3 4 5
∆ (eV) ∆ (eV)
FIG. 3. (Color online) Electron occupation and spins in the FIG. 4. (Color online) Electron occupations and magnetic
multibandmodelforincreasing∆≡−ε obtainedinthefor- moments in the multiband model for increasing ∆ ≡ −ε ,
p p
malionicmodelwithoutself-doping(x=0). Theinvestigated obtained in the realistic model with self-doping x = 1. The
groundstateisinsulatingwithAForderalignedalong(1,1,0) investigated ground state is weakly insulating with AF order
direction. No significant orbital order was detected. Legend: aligned along (1,1,0) direction. No significant orbital order
data for Ir ions are represented by red lines; data for O ions wasdetected. Legend: dataforIrionsarerepresentedbyred
arerepresentedbybluelines;datapointsshowelectronoccu- lines;dataforOionsarerepresentedbybluelines;datapoints
pationnumbersint orbitals(pluses),all5dorbitals(×),2p showelectronoccupationnumbersint orbitals(pluses),all
2g 2g
orbitalsatin-planeoxygens(circles),and2porbitalsatapical 5d orbitals (×), 2p orbitals at in-plane oxygens (circles), and
oxygens (diamonds). 2p orbitals at apical oxygens (diamonds).
numbers are quite close to the values reported in ab ini-
netic order in a reliable way. However, such multicon-
tio calculations [24]. The magnetization magnitude at
figuration HF computations are prohibitively costly for
iridium ions is also roughly that which was found in the
the considered cluster, both in terms of computer power
experiment [48].
andcomputationtime. Thereforewepresentratherqual-
itative evidence which suggests that the magnetic order
obtainedinthisrangeof∆maycompetewithothermag-
netic states. C. Sensitivity of the results to variation of
Thus both for idealistic ionic (x = 0) and for realistic Hubbard repulsion Ud
model with x = 1 it is difficult to reach definite conclu-
sions about the nature of the true ground state. How- Considerfirsttheelectrondistributionobtainedforthe
ever one bright point are computations for x = 1 and self-doping model, see Table III. First of all, one finds
∆=3.0 eV. Here the HOMO-LUMO gap G of 0.154 eV almost one hole per ion for O oxygen postions in the
1
islargeenoughtobelievethatHF(withoutcorrelations) IrO planes, and roughly half of the hole at apical O
2 3
performs satisfactorily. Moreover, this value is close to positions. This demonstrates a considerable hole delo-
that reported in Ref. [13], and the electron occupation calization over oxygen orbitals. Furthermore, also e or-
g
9
TABLE III. Average electron densities (cid:104)n(cid:105), average magne- TABLE IV. Average electron densities (cid:104)n(cid:105), average mag-
tization densities (cid:104)Sα(cid:105), and average orbital momenta (cid:104)Lα(cid:105), netizationdensities(cid:104)Sα(cid:105),andaverageorbitalmomenta(cid:104)Lα(cid:105),
with α = x,y,z, as obtained for x = 1 and ∆ = 3.0 eV withα=x,y,z,asobtainedfordifferentatomsatself-doping
for AF1 ground state. Average charge densities are close to x=1 and for two different sets of Hamiltonian parameters:
those reported by an-initio studies [24]. HOMO-LUMO gap — Top part — for large ∆ = 5.0 eV, E = 38.7174 eV,
HF
G=0.154eV(weakinsulator). Altogethertheobtaineddata HOMO-LUMOgapG=0.138eV;thevaluesoforderparam-
agreequitewellwiththeexperimentalresults. Legend: O — eters are almost the same for both (energy degenerate) FM
1
inplaneoxygen;O —apicaloxygen;Ir—iridiumion. Note andAFgroundstates(forAF1phasetheyalternatebetween
3
thatforAF1statethesignsofthe(cid:104)Sα(cid:105)and(cid:104)Lα(cid:105)components the two sublattices); parameters: U =2.0 eV, J =0.4 eV.
d d
alternate between sublattices. — Lower part — for small ∆=1.5 eV, E =124.8861 eV,
HF
HOMO-LUMO gap G = 0.124 eV; the ground state is FM
atom (cid:104)n(cid:105) (cid:104)Sx(cid:105) (cid:104)Sy(cid:105) (cid:104)Sz(cid:105) (cid:104)Lx(cid:105) (cid:104)Ly(cid:105) (cid:104)Lz(cid:105)
(FM1) but the energy of nearly degenerate AF1 state is only
Ir 6.65 0.152 0.152 0.0 −0.18 0.22 0.0 by 50 meV higher; parameters: U =3.0 eV, J =0.6 eV.
d d
O 5.14 0.21 0.21 0.0 0.03 0.03 0.0
1 atom (cid:104)n(cid:105) (cid:104)Sx(cid:105) (cid:104)Sy(cid:105) (cid:104)Sz(cid:105) (cid:104)Lx(cid:105) (cid:104)Ly(cid:105) (cid:104)Lz(cid:105)
O 5.54 0.06 0.06 0.0 −0.01 −0.01 0.0
3
AF1 ground state
Ir 6.59 0.162 0.162 0.0 −0.20 −0.22 0.0
O 5.16 0.20 0.20 0.0 0.03 0.03 0.0
1
bitalsarepartlyfilled,asonefindstheelectronnumbers:
O 5.54 0.06 0.06 0.0 0.0 0.0 0.0
n =5.42andn =1.24peroneIrion. Largeelectron 3
t2g eg
density in e orbitals may be explained by a stronger σ- FM1 ground state
g
bondbetweenx2−y2 orbitalsandO(2p )orbitalswithin Ir 6.58 0.169 0.169 0.0 −0.20 −0.25 0.0
σ
theIrO2 planes. Thesenumberssuggestthateveninthe O1 5.18 0.19 0.19 0.0 0.03 0.03 0.0
effectivemodelswhichfeatureonlyIr(5d)orbitalsalsoeg O3 5.53 0.07 0.07 0.0 0.0 0.0 0.0
orbitals have to be included.
The order parameters presented in Table III suggest
that physical description of the IrO plane in the frame-
4 thecaseoflargerU =3.0eV,onefinds: n =5.40and
workoftherealisticmodel(withself-dopingx=1),with d t2g
n =1.18. Note that these densities are also quite close
charge occupations close to the ab initio data, and with eg
to the ones obtained for the parameters given in Table I.
HamiltonianparametersfromTableIfitstheexperimen-
To summarize, we remark that FM and AF ground
tal data reasonably well. However, a natural question
states with magnetic moments aligned along (1,1,0) di-
arises: Wasthisachievedbyaccidentoraretheobtained
rectionareveryclosetoeachother. Nonmagneticground
results generic (at least in some limited sense) ? To get
state and states with the moments aligned along (0,0,1)
a better insight into the problem we performed numer-
direction are much higher in energy, thus they can be
ousadditionalcomputationsfordifferentsetsofHamilto-
safely excluded.
nianparameters(wecouldnotaffordthefullstudyofthe
phasediagramasthenumberofparametersinvolvedand
alreadydiscussedtechnicalissuesmakethecomputations
rather expensive). D. Can magnetic moments in IrO plane be
4
Theresultsofthesecomputationsindicatethattheob- aligned along z axis?
tained results are indeed generic and the selection of the
parametersisquiterepresentative. Togiveanexplicitex- Thenextquestiononeshouldtrytoansweristhemag-
ample let us consider two additional sets of Hamiltonian netic anisotropy. Why almost all magnetic states are
parameters: the first set with U =2.0 eV and J =0.4 alignedparalleltotheplane, andwhataboutalignments
d d
eV with weaker correlations at iridium ions, and the sec- (if any) along the z axis? In fact, there are also ground
ond set with stronger correlations for U = 3.0 eV and states with magnetic moments aligned along the z axis
d
J = 0.6 eV. All other Hamiltonian parameters remain (perpendicular to the iridiums-oxygen plane). They oc-
d
unchanged (note that we keep the ratio J /U = 0.2). cur both for the idealistic ionic model, (i.e., for x = 0)
d d
Wemakescansvarying∆forrealisticscenariowhenself- andfortherealisticmodel(withself-dopingx=1). The
doping x = 1 is selected. We made a search for and ac- conditionsforthemtoappeararethefollowing: largeU
d
cepted results which give the occupation numbers close and large ∆ (and what follows large charge occupations
toabinitio results,i.e.,then +n electronoccupation onoxygens,closeto6e). Herewealsopresentanexample
t2g eg
on Ir close to 6.5. The collected results are presented in in Table V.
Table IV (compare with Table III). The electron densities obtained for large values of ∆
We emphasize that not only the electron densities and andthemagneticstateswithmagneticorientedalongthe
and magnetic moments are similar for the two sets of zspinaxisshowamuchbetterholelocalizationatIrions.
parametersused inTable IV,butalsothepopulationsof TheholedensitiesatO andO oxygenpositionsarere-
1 3
t and e orbitals at It ions. For the first choice with duced by more than a half with respect to their values
g g
U = 2.0 eV, one finds: n = 5.38 and n = 1.22. In obtained for the states presented in Tables III and IV.
d t2g eg
10
FM order).
TABLEV. Averagechargeoccupations(cid:104)n(cid:105),averagemagnetic
In contrast, interatomic correlations only involve exci-
spin components (cid:104)Sα(cid:105), and average orbital momenta (cid:104)Lα(cid:105),
tionsfromdifferent pairsofions[49]. Theconsequenceis
with α = x,y,z, obtained for different ions the AF ground
that, naively speaking, the number of the occupied elec-
state with magnetic moments aligned along the z axis. Note
tron pairs of the singlet-type (and located on different
that the energy of this AF state is degenerate with that of
FM state (with moments also aligned along the z axis). The ions) is much bigger for AF ground state. Each such
Hamiltonian parameters are: U = 3.0 eV, J = 0.6 eV, occupied singlet-pair (in AF) during CI computations is
d d
∆=5.0 eV, and x=1.0. annihilated (while the virtual singlet-pair is created at
the same time) giving substantial contribution to the to-
atom (cid:104)n(cid:105) (cid:104)Sx(cid:105) (cid:104)Sy(cid:105) (cid:104)Sz(cid:105) (cid:104)Lx(cid:105) (cid:104)Ly(cid:105) (cid:104)Lz(cid:105)
tal singlet correlation energy. The number of occupied
Ir 5.41 0.0 0.0 0.78 0.0 0.0 0.43 triplet-pairs of the electrons is smaller and the triplet
O1 5.52 0.0 0.0 0.01 0.0 0.0 0.04 excitionsgivemuchsmallercorrelationcontributionany-
O 5.72 0.0 0.0 0.00 0.0 0.0 0.03 way (we remind that for the Hamiltonian (2.1) singlet
3
excitations are probably dominant).
For the FM ground state we can apply a similar rea-
soning. Now, the number of occupied triplet pairs of the
With more holes within 5d orbitals, also electron densi-
electrons (on different ions) is bigger. However, the an-
tiesarereducedton =4.41andn =1.00,butagain
t2g eg
nihilation of each such pair gives rise to a much smaller
the density of e electrons is large and also in this case
g
triplet contribution to the total correlation energy.
it is necessary to include all five 5d orbitals at Ir ions.
Insummary,onecanexpectthatthetotalcorrelations
It might be tempting to speculate that for a ground
contributiontothetotalgroundstateenergyismainlyof
state antiferromagnetism aligned along (1,1,0) or (1,0,0)
thesinglet-typeandthatthecorrelationscontributionto
direction results from the optimization of kinetic hop-
the total energy is enhanced for AF ground state while
pings parallel to the iridium-oxygen plane and this hap-
being weaker for FM ground state. Thus one expects
pens only for substantial hole occupations on oxygen or-
that AF order should win in the ground state.
bitals. On the other hand, when charge densities within
2p oxygen orbitals become much closer to six (no holes),
thisblockseffectivelong-rangehoppingandonlythenthe
possibilityofmagnetizationalongthezaxismaybecome VI. DISCUSSION AND SUMMARY
an option.
We have shown that the HF computational results for
theassumedidealizedionicconfigurationdonotdescribe
E. Do correlations favor antiferromagnetic over properly the experimental data for Ba IrO for almost
2 4
ferromagnetic order ? all studied values of ∆. Namely, the charge on iridium
ionsis7e-7.5e, i.e., itissignificantlyhigherthan6.5e(as
Theenergy-degenerategroundstates: FMandAF(see suggested by ab initio [24]) and the HOMO-LUMO gap
Table II) are a puzzle. The question arises which one is G is also too large, so the system is a good insulator.
the true ground state. On the HF level this can not be Only for very large ∆ = 5.0 eV the agreement with ab
answered but only after one includes electronic correla- initio calculations (charges on iridium and also on oxy-
tion one can get the final answer. In general the task gens) improves but still the HOMO-LUMO gap is twice
to compute the correlations is very difficult and costly. largerthan0.14eVexpectedfromtheexperimentaldata
There is however a simple (though not rigorous) intu- [12,13]. Notethatlargeinsulatinggapwasfoundinstead
itive approach which suggests that the AF order should in Na IrO [35].
2 3
be favored in the ground state. In contrast, the realistic model with self-doping x=1
Namely let us consider configuration-interaction com- is doing much better. The charges on iridium are gen-
putationsusedinquantumchemistrytoderivethecorre- erally smaller (in comparison to the x = 0 case) and in
lationenergy,asdefinedforinstanceinRefs. [49,50]. Let particularfor∆=3.0eVthechargeoccupationsonboth
us limit ourselves only to double-excitions from the oc- iridium and oxygens are very close to those indicated by
cupied to virtual states. For the Hamiltonian (2.1) with ab initio [24] and the ground state: (i) is antiferromag-
the parameters from Table I (i.e., with large U in the netic aligned along (1,1,0); (ii) is weakly insulating (not
d
interaction part of the Hamiltonian) it is expected that far from being metal; G = 0.15 eV, i.e., the value indi-
singletexcitationsaredominantandthetripletexcitions cated by experimental data [12, 13]); (iii) has magnetic
are probably much less important. momentswithmagnitudebeingclosetotheexperimental
Now consider intraatomic correlations (each of the findings [12, 13]; (iv) is characterized by some traces of
double excitions is coming out from occupied levels on weak orbital order.
somesingleion[49]). Thesearestrictlylocal,i.e.,forAF We emphasize that the present treatment of the spin-
or FM ground state they look the same and most prob- orbit interaction is sufficient to break the SU(2) symme-
ably give the same contribution to the total correlation try in the spin space and to realize the experimentally
energy. (Thus they do not differentiate between AF and observed type of antiferromagnetic phase. Further justi-