A hybrid-exchange density-functional theory study of the electronic structure of $\mathrm{MnV}_2\mathrm{O}_4$: Exotic orbital ordering in the cubic structure

0.3 MB·English

by  Wei Wu

A hybrid-exchange density-functional theory study of the electronic structure of
MnV O : Exotic orbital ordering in the cubic structure
2 4
∗
Wei Wu
Department of Electronic and Electrical Engineering and London Centre for Nanotechnology,
University College London, Gower Street, London, WC1E 6BT, United Kingdom
TheelectronicstructuresofthecubicandtetragonalMnV2O4 havebeenstudiedbyusinghybrid-
exchangedensityfunctionaltheory. Thecomputedelectronicstructureofthetetragonalphaseshows
ananti-ferroorbitalorderingonVsitesandaferrimagneticgroundstate(thespinsonVandMnare
5 anti-aligned). These results are in a good agreement with the previous theoretical result obtained
1
from the local-density approximation+U methods [S. Sarkar, et. al., Phys. Rev. Lett. 102, 216405
0
(2009)]. Moreover, the electronic structure, especially the projected density of states of the cubic
2
phasehasbeenpredictedwithagoodagreementwiththerecentsoftx-rayspectroscopyexperiment.

r Similartothetetragonalphase,thespinsonVandMninthecubicstructurefavouraferrimagnetic
p
configuration. Most interesting is that the computed charge densities of the spin-carrying orbitals
A
onVinthecubicphaseshowanexoticorbitalordering,i.e.,aferro-orbitalorderingalong[110]but
an anti-ferro-orbital ordering along [110].
3
1
PACSnumbers: 71.15.Mb,75.47.Lx,75.50.Gg,75.25.Dk

]
l
e
I. INTRODUCTION be thought of as a prototype of TMO-based metamate-
-
r rials or even quantum metamaterials [11, 14]. Another
t
s unique point for vanadium spinels is that they are t2g-
t. Many fascinating phenomena in condensed-matter active, in sharp contrast to most of the known TMOs
a physics were discovered in transition-metal oxides showing OO. In this type of compound, the V3+ ions on
m
(TMOs) and their closely related compounds such as theBsitesformapyrochlorelattice. Thetwod-electrons
d- the first high transition temperature superconductivity on V occupying two t2g orbitals lead to a total spin of
n inYBaCuO3 [1,2]. Animportantorderingphenomenon, S =1, resulting in a t2g-activeOO.The OO in this type
o Orbital ordering (OO) has a long history in the physics of compounds is further complicated by their intrinsic
c of TMOs, back to the 1930s,when the conceptof charge geometrical frustration, which played an important role
[ orderinginFe3O4 wasproposed[3]. OOoccurswhenthe in OO. In 1939, the charge ordering on the B-site py-

orbital degeneracy is lifted essentially by the interaction rochlore lattice was proposed to cause a sharp increase
4
v with the lattice environment [4, 5]. The best-known ex- of the electrical resistivity when cooling below ∼ 120 K
2 ample of OO is the Jahn-Teller (JT) distortion observed in Fe3O4 (the ’Verwey’ transition) [3]. Anderson was
2 in LaMnO3 [6]. Thereinthe eg degenerate manifold (dz2 the first to realise the intimate relationship between the
3 anddx2−y2)ina cubicenvironmentisfurther splittedby proton ordering in the ice, the Verwey charge ordering,
0 the JT distortion owing to partially filling of electrons. and the ordering of Ising spins [15]. Regardless of the
0 The underlying physics in OO has been well accounted specific material types, geometrical frustration plays a
.
1 for by the so-called Kugel-Khomskiimodel [7]. Most ex- crucial role to determine the spin or charge configura-
0 perimental and theoretical works so far have been fo- tions,byminimisingexchangeorCoulombenergies. This
5 cused on the system with partially filled eg orbitals, i.e., will in turn trigger many interesting phenomena such as
1
: eg active. This type of orbital occupancy can lead to orbital-glass and orbital-ice states [16–18]. In addition,
v the strongest JT effect because the related d-orbitals di- themuchfasterresponseofelectronchargesascompared
Xi rectly point to ligands and strongly hybridise with the to electronspins will find many potential applications in
2p-orbitals of the anion. advanced electronics such as orbitronics [19].
r
a
Recently much attention has been paid to vanadium
spinels AB2O4 (A = Mg, Co, Fe, Mn, and Cd, etc and MnV2O4 (See Fig.1), in which the Mn2+ ion is in a d5
B=V) [8–13]owingtotheir fascinatingspinandorbital half-filled high-spin configuration (S = 5), first experi-
2
orderingsaccompaniedbycomplexstructuraltransitions encesamagnetictransitiontothecollinearferrimagnetic
atlowtemperature. Theinterplaybetweenstructure,or- state at T = 56 K, and then a structural distortion at a
bital, and spin implies that the underlying physics could slightly lower temperature T =53 K, along with a tran-
beveryinteresting,andthattheremightbelargeapplica- sition to the non-collinear ferrimagnetic state [20, 21].
tionpotentialinthesematerials. Forexample,artificially Theobservedmagnetic orderingbelowT =56K isferri-
tuning structure by applying externalmagnetic field can magnetic, i.e., the magnetic moments on Mn2+ and V3+
are anti-aligned. In the structural transition, the sym-
metry is lowered from cubic to tetragonal. The OO of
the ground state in the tetragonal MnV2O4 structure
∗Electronicaddress: [email protected] hasbeen showntheoretically[12]as anA-type antiferro-
2
orbital ordering with a propagationvector of (002). ing DFT and hybrid-exchange functional PBE0 as im-
plemented in the CRYSTAL 09 code [29]. The crystal
structuresofthe cubic (spacegroupFd3m)andtetrago-
nal (space group I41/amd) MnV2O4 experimentally de-
termined in Ref.[11] have been adopted here to perform
all the calculations. The basis sets for the atomic or-
bitals centred on the Mn[30], V[31], and O[32] atoms,
whicharedesignedforsolid-statecompounds,wereused.
The Monkhorst-Pack samplings [33] of reciprocal space
were carried out choosing a grid of shrinking factor to
be 6 × 6 × 6 (6 × 6 × 5) in order for a consistency
with the ratios among reciprocal lattice parameters for
the cubic (tetragonal) MnV2O4. The truncation of the
Coulomb and exchange series in direct space was con-
trolled by setting the Gaussian overlap tolerance crite-
ria to 10−6,10−6,10−6,10−6, and 10−12 [29]. The self-
consistent field (SCF) procedure was convergedto a tol-
erance of 10−6 a.u. per unit cell (p.u.c). To accelerate
FIG. 1: (Color online.) the conventional cells in the cubic
(left) and tetragonal (right) MnV2O4 crystal structures are convergenceoftheSCFprocess,allthe calculationshave
shown. Mn is depicted as small yellow ball, V as large green beenconvergedbyusingalinearmixingofFockmatrices
ball, and O as large red ball. by 30%.
Electronic exchange and correlation are described us-
Previously,theelectronicstructureandmagneticprop- ing the PBE0 hybrid functional [25], free of any empir-
erties of the tetragonal MnV2O4 have been investigated ical or adjustable parameter. The advantages of PBE0
theoreticallyusingLDA(local-densityapproximation)+ include a partial elimination of the self-interaction er-
U method[12],whereanorbitalorderingamongV3+ was ror and balancing the tendencies to delocalize and lo-
proposed, along with a non-collinear magnetic ordering. calize wave-functions by mixing a quarter of Fock ex-
Following this first-principles calculation, an analytical change with that from a generalized gradient approxi-
modelling [22] has further confirmed that an antiferro- mation (GGA) exchange functional [25]. The broken-
orbital ordering exists in the tetragonal MnV2O4. On symmetry method [34] is used to localize collinear op-
the other hand, the electronic structure (especially OO) posite electron spins on atoms in order to describe the
and magnetic properties of the cubic MnV2O4 are yet anti-ferromagnetic state.
to be studied in detail. It might be worthwhile to (i)
employ a computational method without any adjustable
parameters,(ii)takeintoaccounttheelectroncorrelation III. RESULTS AND DISCUSSIONS
properlyand(iii)comparetheelectronicstructuresofthe
cubic and tetragonal MnV2O4. Hybrid-exchange func- A. Projected densities of states
tionaltheory(HDFT),inwhichtheexactexchangeishy-
bridisedwithgeneralisedgradientapproximation(GGA) The projected densities of states (PDOS) of the cubic
functionaltolocalised-electrons,hasperformedwellfora and tetragonal MnV2O4 structures for the AFM config-
wide range of inorganic and organic compounds [23, 24]. uration (the spins on Mn and V are anti-ligned) have
Inthispaper,HDFTwithPBE0functional[25]hasbeen beenshowninFig.2. Thezeroenergyisalignedwiththe
usedtostudytheelectronicstructureandmagneticprop- valence band maximum (VBM).
erties of the cubic and tetragonal MnV2O4. The results In the cubic and tetragonal structures, Mn 3d PDOS
presentedherearenotonlyinagoodagreementwiththe
(Fig.2a and b) shows that the five d-orbitals including
previoustheoreticalandrecentexperimentalstudies,but
dx2−y2, dz2, dxz, dyz, and dxy are singly occupied, thus
alsopointtoanexoticOOstatethatmixesferro-orbital- giving a spin-5. From the analysis of Mulliken spin den-
and anti-ferro-orbital orderings. The rest of the discus- 2
sities and V 3d PDOS (Fig.2c and d), the two electrons
sion is organised as the following: in §II computational
occupyingt2g statescanbe identifiedasthe originofthe
details are introduced, in §III the calculation results are
OObothinthecubicandtetragonalMnV2O4. Inthecu-
presented and discussed, and in §IV some general con-
bic phase, the molecular orbitals delocalized among the
clusions are drawn.
three t2g states are occupied, whereas in the tetragonal
phase, only d and d are involved. The O 2p PDOS
xz yz
(Fig.2e) for the cubic phase are much more delocalized
II. COMPUTATIONAL DETAILS than those for the tetragonal(Fig.2f); this might be due
to the elongation of the lattice vector a and b in the
The calculations of the electronic structures of the tetragonal structure. The O 2p PDOS are particularly
tetragonal and cubic MnV2O4 were carried out by us- dominant at ∼ 5 eV below the VBM for both the cubic
3
Cubic Tetragonal
Cubic MnV2O4 AFM Mn-3d-orbital PDOS Tetragonal MnV2O4 AFM Mn-3d-orbital PDOS
Density of states (states/eV/cell)-11-050500000 dddddxzxyx2zzy2-y2 Density of states (states/eV/cell)-22000 dddddxzxyx22zzy-y2
-10 -5 0 5 10 -10 -5 0 5 10
Energy (eV) Energy (eV)
(a) (b)
Cubic MnVO AFM V-3d-orbital PDOS Tetragonal MnVO AFM V-3d-orbital PDOS
2 4 2 4
of states (states/eV/cell)115050000 dddddxzxyx2zzy2-y2 of states (states/eV/cell)12468000000 dddddxzxyx2zzy2-y2
y y 0
Densit-50 Densit-20
-10 -5 0 5 10 -10 -5 0 5 10
Energy (eV) Energy (eV)
(c) (d)
Cubic MnVO AFM O-2p-orbital PDOS Tetragonal MnVO O-2p AFM PDOS
2 4 2 4
of states (states/eV/cell)500 222pppxyz of states (states/eV/cell)100 222pppxyz
y y
nsit-50 nsit-10
De De
-10 -5 0 5 10 -10 -5 0 5 10
Energy (eV) Energy (eV)
(e) (f)
Cubic MnVO AFM PDOS Tetragonal MnVO AFM PDOS
2 4 2 4
V/cell)300 MV n3 d3d V/cell)200 MV-n3-d3d
y of states (states/e-1120000000 O 2p y of states (states/e1000 O-2p
Densit-200 Densit
-10 -5 0 5 10 -100-10 -5 0 5 10
Energy (eV) Energy (eV)
(g) (h)
FIG.2: (Coloronline.) ThePDOSofthe3d-orbitalsonMn,the3d-orbitalsonV,andthe2p-orbitalsonOofthecubic(left)
and tetragonal (right) MnV2O4 are shown. The PDOS onto dx2−y2 is in green, dz2 in red, dxz in blue, dyz in purple, and dxy
in black, respectively. For 2p-orbitals, the PDOS onto 2px is in green, 2py in red, and 2pz in black, respectively. In (g) and
(h),the total PDOS onto Mn d-orbitals is in red,V d-orbitals in green, and O 2p-orbitals in black, respectively.
4
and tetragonalstructures, which overlapwith the rather theotherhand,theexchangeinteractionbetweentheMn
weakMn3dandV3dPDOS.TheO2porbitalsinvolved and V spins can also be estimated by using the super-
here will give rise to the so-calledoxygenbonding states exchange formalism, where t can be quantified in the
that feature the hybridisation between the d-orbitals on PDOS, which is ∼ 0.5 eV for Mn and ∼ 1 eV for V. By
transitionmetalsandp-orbitalsonOatoms. Inaddition, using these hopping integrals, the exchange interaction
it can be observed by comparing the spin-up and spin- can be estimated as tMntV (U = UMn+UV), which is ∼59
U 2
down PDOS, that the O 2p states have been strongly meV. This is in the same order as that calculated from
spin-polarized by the magnetic moments on Mn and V. the total energy differences aforementioned (here an av-
The PDOS onto the V sites have also been compared erage value is taken for on-site Coulomb interaction). A
with the previous results obtained by the LDA+U [12], detailed discussion about the exchange interaction, tak-
which suggests that there is a qualitative agreement be- ing into account all the d-orbitals on both Mn and V
tween them except that the DFT band-gap computed would be great, but beyond the scope of this paper.
here (∼ 3 eV) is larger than the previously computed
one (∼ 1.1 eV) [12] and the observed (∼ 1.1 eV)[36].
The computed O 2p PDOS are in good agreement with C. Orbital ordering
the intensities measured in the recent K-edge x-ray ab-
sorption and emission spectra; the combination of the
The OO in the cubic and tetragonal MnV2O4 can be
theoretical work and experiments will be published in a
illustratedbythespindensitiesonV(seeFig.3)–thedif-
forthcoming paper [37].
ferencebetweenthe chargedensities ofspin-upandspin-
Thecomputedcrystal-fieldsplittingbetweent2gandeg downorbitals(essentiallythechargedensitiesofthespin-
statesofMn2+andV3+inthecubicMnV2O4canberead carry orbitals). In the tetragonal structure, the relative
from Fig.2a, which is ∼1 eV. On the other hand, in the
orientationrotationofneighbouringorbitalsisillustrated
tetragonalMnV2O4,forbothMn2+ andV3+,dx2−y2 and by the red arrows in Fig.3b; this is consistent with the
d are closely aligned while the other three 3d-orbitals
xy previous calculations presented in Ref.[12] in which an
overlap well, as shown in Fig.2b. This picture is consis-
anti-ferro-orbital ordering (AFOO) has been predicted.
tent with the previous calculations reported in Ref.[12].
The orbital orientations for the nearest-neighbouring
The crystal-field splitting between these two groups is
(NN) orbitals are perpendicular to each other (defined
∼ 1 eV. The on-site Coulomb interaction U can be ap-
as AFOO).Insharpcontrast,forthe cubic structure the
proximatedbyobservingthegapbetweenlowerHubbard
NN orbtibals (labeled by X and Y in Fig.3a) are orga-
d-bands and uppers ones (∼ 5 eV for Mn site, and ∼ 7
nized in an exotic way, in which the orbitals on one of
eV for V), which is much larger than the crystal-field
the four symmetry-inequivalent V atoms in the unit cell
splitting computed here. This will result in a high-spin
hasadifferentorbitalorientation(labeledbyY),perpen-
state for Mn2+ and V3+ ions, which is consistent with
dicular to the others (labeled by X). The orbital orien-
the previous experiments [11].
tations corresponding to Fig.3a are further illustrated in
Fig.3d, e,andf,whicharethe viewsofthe spindensities
from the lattice direction [100] (~a), [010] (~b), and [001]
B. Exchange interactions (~c), respectively. This leads to an exotic OO: a ferro-
OO (FOO)along[110](blue arrowin Fig.3), whereasan
The magnetic structure could be much more compli- AFOO along [110] (red arrow in Fig.3). This OO can
cated owing to the exchange interaction between the probably be attributed to (i) the crystal field environ-
nearest-neighbouring Mn (V) atoms [13] and the spin- ment formed by neighbouring oxygen atoms and (ii) the
orbit coupling (SOC), which will cause the non-collinear Coulombinteractionbetweend-orbitals. Thisinteresting
magnetic ordering. However, this topic is beyond the OOhasbeenfurtherconfirmedbythecalculationswitha
scope of this paper that is focused on the electronic differenttypeofexchange-correlationfunctional,B3LYP
structure and the collinear magnetic ground state. The [35], which relies on the empirical parameters for mixing
Mulliken spin densities on the Mn and V sites for exactexchange and GGA exchangefunctional. The spin
the cubic (tetragonal) MnV2O4 are 4.8 (4.7)µB and densities on V predicted by using B3LYP functional are
−2.0 (−1.9)µ , respectively. They are close to the ex- showninFig.3c,inwhichFOOisalong[110]andAFOO
B
pected values, i.e. 5µ for Mn and −2µ for V (anti- [110] (all the atoms are made partially transparent in
B B
aligned). For the cubic (tetragonal) structure, the com- order to illustrate OO).
putedtotalenergiesforaconventionalcellwiththespins To see the orbitals involved in the OO of the cubic
on Mn and V aligned (ferromagnetic) is higher than the MnV2O4moreclearly,theatomicorbitalcompositionsof
anti-aligned(AFM)by937(754)meV.Thisinturngives thebands(seeFig.4)attheΓ-pointthatcontributemost
an approximate exchange interaction of ∼ 38 meV (di- tothetwopeaksat−0.26eVand−0.64eVintheVspin-
vided by the factor of 4×SMn×SV=20, where 4 is the down PDOS (see Fig.2c) have been investigated. The
number of Mn-V pairs in the unit cell for the tetragonal bandsVB−8andVB−9contributethe mosttothepeak
structure). Similarly we can estimate the exchange in- at −0.26 eV, whereas VB−19 and VB−20 to the one at
teractionforthe cubic structure,whichis ∼23meV. On −0.64eV.Althoughthe atomiccoefficientsford-orbitals
5
FIG. 3: (Colour on line.) The spin densities on V of the AFM state in the conventional unit cell of the cubic (a) and
tetragonal (b) structures of MnV2O4 are shown. In the tetragonal an anti-ferro-orbital ordering is found. In contrast, in the
cubicstructure,therotationoftheorbitalorientationillustratesanexoticOO,inwhichanferro-orbitalorderingisalong[110],
while anti-ferro-orbital ordering along [110]. This exotic OO is further confirmed by the calculation from B3LYP functional;
the resulting spin densities are shown in (c). The B3LYP spin-density calculation is performed in a unit cell, but shown in a
conventional cell. The views from [100] (~a), [010] (~b), [001] (~c) lattice directions are shown in (d), (e), and (f), respectively.
The isovalue is chosen as 0.04e/˚A3.
are slightly different among V atoms, approximately the
mostdominantlinearcombinationsofd-orbitalsforeach
V atoms contained in these bands read, φVB−8 =0.2|dxzi+0.1|dyzi+0.2|dxyi, (1)
φVB−9 =−0.1|dxzi+0.2|dyzi−0.2|dxyi, (2)
φVB−19 =0.2|dxzi−0.1|dyzi−0.2|dxyi, (3)
φVB−20 =0.2|dxzi+0.2|dyzi+0.1|dxyi. (4)
The corresponding spherical harmonic functions have
been plotted in Fig.5 to see the contributions of d-
orbitals. From the perspectives of linear combination
of atomic orbitals (LCAO), the strong variations of the
orbital orientation illustrated in Fig.3 and Fig.5, result
from the minimisation of the total energy respective to
atomic orbital coefficients. On the other hand, the pro-
cess of energy-minisation can be seen as the orbital re-
orientation driven by Coulomb and crystal-field interac-
tions.
There are two concepts that are closely related to this
exotic orbital ordering, including orbital glass and ice
states. Theorbitalglassstateisformedbytherandomly
orientatedorbitals,whichresultfromJahn-Tellerdistor-
tion and orbital frustrations in a similar manner to spin
frustrations [16, 17]. In light that orbital glass states
have been observed in the spinel compounds FeCr2S4
and LuFe2O4 [16, 17] (which has similar chemical for-
mula to a spinel, but a different structure), this is not
FIG. 4: (Colour on line.) The spin-down band structure completely unexpectable in the cubic MnV2O4. There-
(from -0.8 to 0 eV) of the AFM configuration in the cubic fore, these calculations could suggest that at T = 53
MnV2O4 is shown. Note that the two red arrows refer to K, there is not only a structural transition, but also an
VB-8 and VB-9, and VB-19 and VB-20, respectively. orbital-orderingtransitionfromtheexoticOOillustrated
6
(a) (b) (c) (d)
FIG.5: (Colouronline.) Thelinearcombinationofharmonicsphericalfunctionsareshown. 2Yxz+Yyz+2Yxy (corresponding
toeq.(1))areshownin(a),−Yxz+2Yyz−2Yxy (correspondingtoeq.(2))in(b),2Yxz−Yyz−2Yxy (correspondingtoeq.(3))in
(c), and 2Yxz+2Yyz+Yxy (corresponding to eq.(4)) in (d). Notice that the linear combination illustrated in (c) changes the
orientation of orbitals significantly away from their original ones, which is closely related to theexotic OO shown in Fig.3a.
inFig.3toanti-ferroorbitalordering[12], whenlowering todifferentelectrontransportproperties. Moreadvanced
temperature. In addition, Chern and the collaborators theoreticalmethodssuchasdynamicalmean-fieldtheory
have presented an ice model consisting of triplet orbital [38] is also needed to further understand the electronic
variables (such as p-orbitals), which is however closely structure of MnV2O4. In the ground state the spins on
related to orbital-drivenmany-body phenomena in opti- Mn and V are predicted to be anti-aligned, suggesting a
cal lattice [18]. The analogue of spin-interacting Hamil- ferrimagnetic state for both structures, which is in good
tonian, an orbital-exchange model describing Coulomb agreementwithprevioustheoreticalandexperimentalre-
interactions has been used therein. The exotic ordering sults. The lower and upper Hubbard bands of Mn and
presented here is closely related to the newly proposed V d-electrons have been shown clearly below and above
concept of the orbital ice [18], similar to the spin ice. the band gap. The on-site Coulomb interaction can be
estimatedby using the gapbetween the lowerandupper
Hubbard bands, which is ∼ 7 eV for V and ∼ 5 eV for
IV. CONCLUSION Mn, respectively. The transfer integral responsible for
the delocalization is in the order of 1 eV, which should
The electronic structures of the cubic and tetrago- be attributed to a combination of the first-order direct
nal MnV2O4 structures have been computed by using hopping between Mn and V and the second-order effect
hybrid-exchange density functional theory PBE0 and via O atoms.
B3LYP. The calculated PDOS of the tetragonal struc-
turehasbeeninagoodagreementwithprevioustheoret-
ical results. The most important finding here is that the V. ACKNOWLEDGEMENT
charge densities of spin-carrying orbitals suggest a pos-
sible existence of an exotic orbital ordering: FOO along Iwouldliketoacknowledgethestimulatingdiscussions
[110] and AFOO [110] in the cubic MnV2O4. The theo- with Dr. Leonardo Bernasconi, Dr. Bo Chen, and Dr.
reticalpredictionpresentedherecouldbevalidatedinthe Jude Laverock. I also thank Mr. Barry Searle for tech-
future experiment. Forexample,the differentOOsalong nical supports and Rutherford Appleton Laboratory for
those two lattice orientations aforementioned could lead computational resources.
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