Charge ordering in the spinels AlV$_2$O$_4$ and LiV$_2$O$_4$

0.18 MB·

by  Y. Z. Zhang

Charge ordering in the spinels AlV O and LiV O
2 4 2 4
Y. Z. Zhang1, P. Fulde1, P. Thalmeier2 and A. Yaresko1
1Max-Planck-Institut fu¨r Physik komplexer Systeme, N¨othnitzer Straße 38 01187 Dresden, Germany
2Max-Planck-Institut fu¨r Chemische Physik fester Stoffe, N¨othnitzer Straße 40 01187 Dresden, Germany
(February 2, 2008)
Wedevelopamicroscopictheoryforthechargeordering(CO)transitionsinthespinelsAlV2O4
5
and LiV2O4 (under pressure). The high degeneracy of CO states is lifted by a coupling to the
0
rhombohedral lattice deformations which favors transition to a CO state with inequivalent V(1)
0
and V(2) sites forming Kagom´e and trigonal planes respectively. We construct an extended Hub-
2
bard type model including a deformation potential which is treated in unrestricted Hartree Fock

n approximation and describes correctly the observed first-order CO transition. We also discuss the
a
influence of associated orbital order. Furthermore we suggest that due to different band fillings
J
AlV2O4 shouldremainmetallic whileLiV2O4 underpressureshouldbecomeasemiconductorwhen
8
charge disproportionation sets in.
1

PACS: 71.30.+h; 71.70.-d; 71.10.Fd; 71.20.Be
]
l
e
-
It was Wigner [1] who first pointed out that electrons rochlore lattice. While in AlV O a charge order phase
r 2 4
t form a lattice when the mutual Coulombrepulsiondom- transition at approximately 700 K has been observed
s
. inates the kinetic energy gain from delocalization. He [9] at ambient pressure, in LiV O electronic charge or-
t 2 4
a considered an electron gas with a positive uniform back- ders only under hydrostatic pressure of approximately 6
m
ground (jellium). Within that model a lattice will form GPa [11]. In both cases structural changes are associ-
- only when the electron concentration is very low. De- ated with charge ordering. In AlV O the lattice dis-
d 2 4
tailed calculations have determined the critical concen- tortions change the cubic high-temperature phase into a
n
o tration below which lattice formation or charge ordering lattice with alternating layers of Kagom´e and triangular
c takesplace anda value ofr ≃35a wasfound[2]. Here planes. They are stacked in [111] direction and formed
c B
[
rc is the critical value of the average distance between by nonequivalent V(1) and V(2) sites. The rhombohe-

1 electrons which must be exceeded and aB is Bohr’s ra- dral lattice deformation causes an increase in the V(1)-
v dius. The conditions for charge ordering of electrons are O(1) and V(1)-O(2) bond lengths and a decrease in the
8
much better when the uniform, positive background is V(2)-O(1) bond length. The changes in the distances
0
replaced by a lattice of positive ions. Depending on the V(1)-V(1)denotedbyL andV(1)-V(2)byL belowthe
4 1 2
1 overlap of atomic wavefunctions on neighbouring sites phase transition temperature TCO define a deformation
0 the energy gain due to electron delocalization can be ǫ(T)=(L2−L1)/L. L is the V-V distance in the undis-
5 rather small and a dominance of Coulomb repulsion is torted case. At T = 300 K this value was found to be
0 more likely. For a more detailed description see, e.g. [3]. ǫ(300K)=0.016. Below T the averagevalence of the
CO
/
at A good example is Yb4As3 which exhibits charge order Vionschangesfrom+2.5inthe high-temperaturephase
(CO) and for which a microscopic theory was provided to +2.5-δ at the V(1) sites and to +2.5+3δ at the V(2)
m
[4]. Here Yb 4f holes gain a small energy only by de- sites. This is schematically shown in Fig. 1.
-
d localizing via As 4p hybridization. Another well known
V(1)
n example is magnetite Fe O , a spinel structure. Verwey O(2)
o and Haayman [5] found3a m4 etal-insulator phase transi- V(1) O(1) (cid:68)
c tion to occur which they attributed to a charge ordering V(2) V(1)
:
v oftheFe2+ andFe3+ sitesonthepyrochloresublatticeof O(1) V(1)
z
Xi thespinellattice(B-sites). AccordingtoVerwey[5]inthe V(1) y
chargeorderedstate the corner-sharingtetrahedraform- V(2) x O(1)
r
AlLi AlLi
a ing the pyrochlore structure are occupied by two Fe2+ a a
V(2) [111] V(1) 1g V(2) 1g
andtwoFe3+ ionseach. Thisrule,(tetrahedronrule)was e' e'
(a) g (b) g
mostclearlyformulatedbyAnderson[6]andallowsforan
FIG.1. (a)Sublattice of the V ions. In the presence of a
exponentialnumber ofdifferentconfigurations. Recently
distortion the V(1) sites form a Kagom´e and the V(2) sites
this rule has been questioned by LDA+U calculation [7] a triangular lattice. (b) AlV2O4 and LiV2O4 in the atomic
whichseeminglygivegoodresultsfortheobservedcharge limit. Shown are the splittings of the t2g orbitals, their oc-
disproportionations [8] when the low-temperature struc- cupations, the spin vectors and the angle α with respect to
ture is put into the calculations. the[111] plane.
In this investigation we consider AlV O [9] and Also associated with the charge-order transition are
2 4
LiV O [10,11]. In both systems the V ions form a py- a small but steep increase in the resistivity and a pro-
2 4
1
nounced decrease of the magnetic susceptibility. In term H describes the on-site Coulomb and exchange
int
LiV O the lattice distortion connected with the charge interactions U and J among the t electrons. The last
2 4 2g
order under pressure seems to be similar to that of term contains the Coulomb repulsion V of an electron
AlV O as powder X-ray diffraction indicate [11]. How- with those on the six neighbouring sites. Finally H
2 4 ep
ever the structural parameters have not yet been deter- describes the coupling to lattice distortions. The defor-
mined. Duetothedifferentfillingofthet bandAlV O mation potential is denoted by ∆, it is due to a shift in
2g 2 4
seems to remaina semimetalora smallgapsemiconduc- the orbital energies of the V sites caused by changes in
tor while LiV O is becoming an insulator with a steep the oxygen positions relative to the vanadium positions.
2 4
increaseintheresistivitybelowthephasetransitiontem- While the energy shift is positive for the V(1) sites it is
perature T . negative for the V(2) sites.
CO
ThefirstthoughtistoperformforAlV O andLiV O The elastic constant K refers to the c mode and de-
2 4 2 4 44
under pressure LDA+U calculation of the type reported scribes the energy due to the rhombohedral lattice de-
for Fe O (see [7]). We have done such calculations for formation. It is reasonable to assume that like in Fe O
3 4 3 4
AlV O . However, although the LDA+U approach was [12] and Yb As [13] only the c mode is strongly cou-
2 4 4 3 44
shown to provide a realistic description of the electronic pled with the charge disproportionation. We can give
structure of magnetic insulators, its application to para- at least approximate values for all parameters except V
magnets is less justified because calculations can only and ∆. Their ratio will be fixed by the CO transition
be performed for a magnetically ordered state. In addi- temperature while keeping the constraint V≪U. For the
tion,LDA+Uistoocrudeanapproximationtoreproduce on-siteCoulomb-andexchangeintegralswetakeU =3.0
thelowenergyexcitationspectrumofstronglycorrelated eV and J = 1.0 eV which are values commonly used for
metals and, thus, can hardly be used to study the high vanadiumoxides[14]. Bandstructurecalculationswhich
temperaturemetallicphaseofAlV O andthechangeof we have performed demonstrate that the hopping ma-
2 4
the electronic structure upon heating through the phase trix elements tνν′ between different orbitals ν 6= ν′ are
µµ′
transition. For LiV O LDA+U calculations are not yet negligible. For simplicity we first ignore them and omit
2 4
possible since the CO crystalstructure under pressure is ν,ν′. Then t = −tµµ′(l,l′) when l, l′ are n.n. and t′ for
not known. Therefore we proceed here differently. We n.n.n.. We will take into account the orbital dependent
want to provide a microscopic Hamiltonian in order to hopping matrix elements coming froma tight-binding fit
identifythephysicalprocesswhichresultsintheobserved ofLDAcalculationsasorbitalorderisincluded. Further-
structuralphasetransitionandtheaccompanyingcharge more the c elastic constant is not known for AlV O
44 2 4
order while enforcing a proper nonmagnetic state. The orLiV O ,whilecomputationalmethodsforitsabinitio
2 4
price are adjustable parameters entering the theory for calculationinthecaseofmaterialswithstrongelectronic
which no ab initio values are available. From standard correlatations are not mature enough. Therefore we use
band-structurecalculationsitisknownthattheFermien- a representative value c(0)/Ω = 6.1 · 1011 erg/cm3 for
44
ergy EF of V 3d electrons lies within the t2g bandwhich AlV2O4 whereΩisthevolumeofthecubicunitcellwith
is well separated from the remaining bands. Therefore a lattice constant of a = 5.844 ˚A. This value is close
we only include t2g electrons in the model defined by to the experimental value for Fe3O4 which has also the
spinel structure. This leads to K ≃ 1.1·102eV. The
H =H +H +H , with (1)
0 int e−p deformation potential ∆ is not known but is commonly
H0 = tνµνµ′′(l,l′)c+lµνσcl′µ′ν′σ of the order of the band width. For convenience we in-
hlµX,l′µ′ihhlµ,l′Xµ′iiνν′σ troduce the dimensionless coupling constant λ=∆2/Kt
andlatticedistortionδ =ǫ∆/t. FromLDAcalculations
L
Hint = {(U +2J) nlµν↑nlµν↓+U nlµνσnlµν′σ the bandwidth is 8t=2.7 eV, and therefore a reasonable
Xlµ Xν νX>ν′ value is λt = ∆2/K = 1eV. This means ∆ = 10.5eV
V whichis twicethe valueofYb As [4]. We beginbycon-
+(U −J) nlµνσnlµν′σ}+ 2 nlµνσnl′µ′ν′σ′ sidering H only. By transform4 ing3 to Bloch states a+
νX>ν′ hlµ,l′µX′iνν′σσ′ 0 kξνσ
we obtain
1
H =ε∆ δ − (1−δ ) n +K ε2
ep µ,1 µ,1 lµνσ
Xlνσ Xµ (cid:18) 3 (cid:19) Xl H0 = ǫξ(k)a+kξνσakξνσ ,with
kXξνσ
Here H0 describes the kinetic energy. The electron cre- ǫ1,2(k)=2(t−t′) (2)
ationandannihilationoperatorsarespecified by fourin-
dices, i.e., for the unit cell (denoted by l), the sublat- ǫ3,4(k)=−2t 1∓ 1+ηk +2t′ 1−2ηk±2 1+ηk
tice (µ = 1,2,3,4), the t orbital (ν = d ,d ,d ), h p i h p i
2g xy yz zx
and the spin (σ =↑,↓). The brackets h...i and hh...ii in- Here ηk = (coskxcosky +coskycoskz+coskzcoskx).
dicate a summation over nearest-neighbour (n.n.) and WithfourV ionsperunitcell,threet orbitalsandtwo
2g
next-nearest-neighbour (n.n.n.) sites respectively. The spin directions there are altogether 24 bands, of which
2
twelve are dispersionless and degenerate (ǫ1,2). In ad- lnΞ=3 ln 1+e−β(ǫσξ(k)−µ) (5)
dition there are two sixfold degenerate dispersive bands Xkξσ h i
present (ǫ ). The band structure has been considered
3,4
before[15]andwerefertothatreferenceformoredetails. with β = 1/k T. The chemical potential µ is adjusted
B
TheFermienergyliesinaregionofhighdensityofstates to yield the correct filling, i.e., N = 1 ∂ lnΞ. The en-
(DOS) N(E) which depends sensitively on the ratio t′/t. ergy bands ǫσ(k) depend now onespinβσ∂µfor which the
ξ
Nextwediscusstheeffectofthedeformation-potential
[111] direction is chosen as quantization axis. They are
couplingdescribedby H . Whenitissufficiently strong
ep computedbydiagonalizingtheHamiltonian(1)inmean-
it may lead to charge order. However, there is no gap
field approximationand by self-consistent determination
opening but only a sharp decrease of DOS around E ,
F ofn andm from(∂F/∂n ) =0and(∂F/∂m ) =0,
i.e., the system remains metallic. When we include the µ µ µ µ µ µ
respectively at various temperatures T.
U and V terms in mean-field approximation,assuming a
With the magnetization pattern described above and
paramagnetic state, the results remain qualitatively un-
shown in Fig. 1 (b) we must transform the nearest-
changed. A strong on-site U term suppresses CO while
neighbour hopping matrix elements so that the different
increasing inter-site V term will induce CO. Again there
spindirectionsareaccountedfor. Welabeltheelectronic
isnogapopeningatE butadecreaseofDOSattheCO
F spin functions of sublattice 1 by |σ i and |σ i and those
transition for all t′. We note that in the present case a 1 1
referring to sublattice 2, 3, and 4 by |σ i and |σ i. We
n.n.nhoppingt′supportsCObecauseitincreasesN(E ), µ µ
F then apply the transformation to a local spin quantiza-
while it prevents CO in the half-filled square lattice [16].
tion axis
Since here t′ << t has little effect on CO we neglect it
fmoraItnsiiomonupralilccoiantlygc.u[l1a1t1io]ndsirwecetiaosns.umTehiashyoiemldosgetnheeoufoslldowefionrg- (cid:18)||σσµµii(cid:19)=(cid:18)ceo−siγθ2µsinθ2 −eiγµcsoinsθ22θ (cid:19)(cid:18)||σσ11ii (cid:19) (6)
relationbetweenthelatticedistortionandchargedispro-
portionation where θ = π2 +α and γµ = 0,23π,43π for µ=2, 3, 4. All
the interactiontermsaswellasthe electron-phononcou-
δ =λ(n −n )/2 (3)
L 2 1 pling term remain invariant under this transformation.
where n1 = νσhnl1νσi is the occupation of a V(2) site The angle α is adjusted such that µmµ=0 for all dis-
while n2 = nP3 = n4 with nµ = νσhnlµνσi is the occu- tortions. P
3.0
paItnionanoyf acVas(e1,) tshitee.mean-fieldPanalysis of the model 9 n2=n3=n4 EF
2.6
Hamiltonian (1) leads to a second order phase transi- V/t=1.59
ttihoeneaxspefruinmcetniotnaloofbtseemrvpateiroantuorfe.a sTtrhoisngisfiarsttoodrddserwCitOh n(V/t)(cid:80)12..82 DOS / u.c. V/t=1.60 (CO)
transition. Webelievethatthisistheeffectofchargecor-
0
relations caused by large U and V. To include this effect 1.4 -5 E/t 0
in a qualitative way, we have treated the model within n
1
anunrestrictedHartree-Fockcalculationbybreakingthe 1.0
1.0 1.4 1.8 V/t 2.2 2.6 3.0
magneticsymmetrybutpreservingtheconstraintofzero
FIG. 2. Charge disproportionation as function of V/t.
total moment. We assume that the sites µ have an oc-
Thenµ denotetheoccupation numbersofthefoursitesofa
cupation n = hn i and a magnetization m =
µ νσ lµνσ µ tetrahedron. Intheinsetthechangesinthedensityofstates
νσσhnlµνσi. TPhe spins are assumed to be directed to- areshownwhenV/tisjustbelowandabovethecriticalvalue
Pwards the center of the tetrahedron in the undistorted at which charge ordering sets in. At the critical V/t=1.67
phase. In the distorted phase we change the angle α so n1=2.5+3δ and ni=2.5-δ (i=2-4) with a charge dispropor-
that the netmagnetizationremainszero (see Fig.1 (b)). tionation δ≃ 0.25. Here U=3.0eV, J=1.0eV, λt=1.0eV and
Because there are two tetrahedra per unit cell transla- 8t=2.7eV.
tional symmetry is maintained. The free energy is The result for the charge disproportionation at T=0
as function of the electronic nearest-neighbour Coulomb
1 5NU
F =−β lnΞ+Neµ− 12 n2µ repulsion V is shown in Fig. 2. As usual the charge dis-
Xµ proportionation is larger (by a factor of 2.5 here) than
N δ2 theoneobtainedfromavalencebondanalysisofthe dis-
+12(U +4J) m2µ−NV nµnµ′ +N λ (4) torted structure. This is due to the simplified Hamilto-
Xµ Xµ µX=6 µ′ nianwhichcannotdescribethescreeningeffectsofnon-d
where N is the totalnumberofunit cellandN =10N is electrons. The same observation was made for Yb As
e 3 4
the electron number per unit cell. Furthermore Ξ is the [4]. AlsoshownasaninsetisthechangeintheDOSfora
grand canonical partition function V value just below and above the critical value at which
3
charge order does occur. Apparently the DOS changes opening of a gap is in agreement with the behavior of
considerably near E due to charge order. This may ρ(T) as found in Ref. [11].
F
explain the drop in the susceptibility and the small but 30
steep increase in the resistivity in AlV O when charge
2 4
order sets in. The self-consistent field calculations were
done for different temperatures. The phase transition to V u.c.20
e
the distorted phase is found to be of first order. Fix- S /
O
ingthe valueV/t=1.67leadstothe calculatedtransition D
10
temperature is T =660 K in reasonable agreement with
c
the experimental value of Texp=700 K. The phase dia-
c
gramis found to be verysimple. ForU=3 eV,J=1.0eV 0
and8t=2.7eVwefindaphaseboundarylineoftheform FIG. 3. Density of s0tates Eo f(eVL) i3V2O4 in t6he distorted
phase. It is seen that the system is a semiconductor
λt+3V=7.8t. For all λt values above that line the trigo-
with a gap Eg=0.49eV. HereU=3.0eV, J=1.0eV, V=0.7eV,
nal, i.e., distorted phase is found while for values below ∆2/K=1.0eV, t0=-0.53eV, t1=0.15eV
that line the phase remains cubic.
Insummary,wehaveprovidedamicroscopicmodelfor
It is obvious that with a lattice distortion an associ-
the observedcharge order in AlV O and LiV O under
2 4 2 4
ated orbital order will occur [17]. The point symmetry
pressure which is mainly driven by a deformation po-
in the charge ordered state is D and lifts partially the
3d tential coupling to t states. The model accounts cor-
2g
3-fold degeneracy of the t states. This results in one
2g rectly for the observed first order CO transition. We
a orbital and 2-fold degenerate e′ orbitals. Experi-
1g g have also shown that due to a difference in the filling
ments show in the distorted phase an elongationin [111]
factors AlV O should remain metallic in the charge or-
2 4
direction and a corresponding contraction perpendicular
deredphasewhileLiV O shouldbecomeaninsulatoror
2 4
to thatdirection. Thereforethe orbitalenergyofthea
1g semiconductor.
state is expected to be higher than that of the e′ states.
g We thank Drs. G. Venketeswara Pai, T. Maitra, E.
From the behavior in the atomic limit (see Fig. 1 (b))
Runge and Y. Zhou for helpful discussions.
we expect that the system remains gapless in the charge
orderedstate even when orbital order occurs. This is in-
deed what we find. We have adopted hopping matrix el-
ements which reproduce the LDA band structure. They
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2 2 2 3 3 3 4 4 4
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2 4
erage d electron number per V site is 1.5. The atomic mamoto and H. Takagi, J. Phys. Soc. Jpn. 70, 1456
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4
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5