Antiferromagnet CuFeO2
On the Conflicting Pictures of Magnetism for the Frustrated
Triangular Lattice Antiferromagnet CuFeO 2
Myung-Hwan Whangbo*and Dadi Dai
Department of Chemistry,North Carolina State Uni V ersity,Raleigh,North Carolina 27695-8204
Kwang-Soon Lee
Department of Chemistry,The Catholic Uni V ersity of Korea,Bucheon,Gyeonggi-Do,South Korea 422-743
Reinhard K.Kremer
Max-Planck-Institut fu ¨r Festko ¨rperforschung,Heisenbergstrasse 1,D-70569Stuttgart,Germany
Recei V ed No V ember 29,2005.Re V ised Manuscript Recei V ed January 17,2006
The magnetic structures of the triangular lattice antiferromagnet CuFeO 2below 14K are described by an Ising model despite the fact that its high-spin Fe 3+(d 5)ions (S )5/2,L )0)cannot have a uniaxial magnetic moment.To resolve this puzzling picture of magnetism,we estimated the relative strengths of various spin-exchange interactions of CuFeO 2by performing a spin dimer analysis and then determined the relative stabilities of a number of ordered spin states of CuFeO 2.Our calculations show that,in terms of a Heisenberg model,the noncollinear 120°spin arrangement predicted for a triangular lattice antiferromagnet is more stable than the collinear four-sublattice antiferromagnetic structure observed for CuFeO 2below 11K.To find a probable cause for stabilizing the collinear spin alignment along the c axis below 14K,we considered the defect ions Fe 2+and Cu 2+of the CuFeO 2lattice created by oxygen deficiency and oxygen excess,respectively.Our electronic structure analysis suggests that these defect ions generate uniaxial magnetic moments along the c axis and hence induce the surrounding Fe 3+ions to orient their moments along the c axis.
1.Introduction
The crystal structure of the delafossite CuFeO 2consists of FeO 2layers that are made of edge-sharing FeO 6octahedra (Figure 1a);these layers are bridged by Cu atoms to form linear O -Cu -O dumbbells perpendicular to the layers (Figure 1b).1Mo ¨ssbauer studies 2of CuFeO 2show its Fe atoms are present as Fe 3+(d 5)ions,and hence the Cu atoms are present as diamagnetic Cu +(d 10)ions.The studies also show that CuFeO 2becomes antiferromagnetically ordered below 14K.Furthermore,the magnetic susceptibility studies 3of CuFeO 2showed that the Fe 3+(d 5)ions are in a high-spin state,and the antiferromagnetic coupling between the Fe 3+ions is dominant within rather than between FeO 2layers.The triangular lattice of the magnetic Fe 3+ions in each FeO 2layer is an archetypal spin lattice that causes geometric spin frustration.4The frustrated nature of the magnetic properties of CuFeO 2have been the subject of a number of studies.5-11
When the temperature is lowered,CuFeO 2undergoes a phase transition from a paramagnetic state to an incommensurately
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Figure 1.(a)Perspective view of an isolated FeO 2layer made of edge-sharing FeO 6octahedra.(b)Perspective view of two adjacent FeO 2layers linked by O -Cu -O bridges.The Fe,Cu,and O atoms are indicated by red,blue,and white circles.
1268Chem.Mater.2006,18,1268-1274
10.1021/cm052634g CCC:$33.50?2006American Chemical Society
Published on Web 02/08/2006
ordered antiferromagnetic(AF)state at T N1)14K and then to a collinear four-sublattice AF state at T N2)11K.6,7There is a thermal hysteresis in the11K transition,which indicates the phase transition is first-order.When placed in a magnetic field at a temperature below T N2,CuFeO2undergoes several phase transitions if the field is parallel to the c axis(i.e., perpendicular to the FeO2layer)but one phase transition if the field is perpendicular to the c axis.8The occurrence of such ordered magnetic structures has been explained in terms of a two-dimensional(2D)Ising model.6,8-10As recently pointed out by Petrenko et al.,11however,the magnetic properties of CuFeO2above T N1should be explained by a Heisenberg rather than an Ising model.
There are many compelling reasons why an Ising model is not self-consistent for CuFeO2.11This model predicts an anisotropic magnetic susceptibility above a three-dimensional (3D)ordering temperature T N1,but the magnetic susceptibility measured for single crystals of CuFeO2is rather isotropic above T N1.8,11-13An Ising model is relevant for a magnetic system in which each of its spin sites has a uniaxial magnetic moment.However,a high-spin Fe3+ion(S)5/2,L)0)in an octahedral environment cannot provide a uniaxial mag-netic moment,14because its orbital angular momentum is zero.The ordered spin arrangement of a triangular lattice antiferromagnet(TLA)predicted from a Heisenberg model is the(1/3,1/3)superstructure(Figure2a)in which each triangular plaquette has a noncollinear120°spin arrange-ment.4Indeed,this ordered spin structure is observed for the
CrO2layers of the delafossites LiCrO2and CuCrO2.15 Therefore,it is puzzling why each FeO2layer of CuFeO2 below T N2does not adopt this spin arrangement expected for a TLA but instead adopts the collinear four-sublattice AF arrangement(Figure2b)6in which the magnetic moment at each Fe3+site is parallel to the c axis.Recent attempts to explain the unusual magnetic properties of CuFeO2are made on the basis of lattice distortions16and correlated magneto-electric/magnetoelastic phenomena.17An important clue for resolving the conflicting pictures of magnetism for CuFeO2 might lie in the observation that the stability of the collinear four-sublattice AF structure is sensitively affected by a small amount of point defects.12,18Namely,the collinear four-sublattice AF structure of CuFeO2disappears when a small number of nonmagnetic Al3+ions are substituted for Fe3+ (i.e.,2%).18The collinear four-sublattice AF structure is maintained in nonstoichiometric CuFeO2+δsingle crystals with either oxygen deficiency(δ<0)or oxygen excess(δ>0),but the phase-transition temperature T
N2
is increased by oxygen deficiency,whereas it is decreased by oxygen excess.12The oxygen deficiency generates Fe2+ions,and the oxygen excess generates Cu2+ions.It is crucial to understand how the presence of such defect ions can influence the stability of the collinear four-sublattice AF structure.
In the present work,we probe why an Ising model is required for explaining the ordered magnetic structures of CuFeO2below T N1despite the fact that this model is not supported on the basis of electronic structure considerations. Our work is organized as follows:In section2,we estimate the relative strengths of various spin-exchange parameters of CuFeO2that are consistent with its electronic structure. Using these parameters,we calculate in section3the total spin-exchange interaction energies of various3D ordered-spin arrangements of CuFeO2to find that the collinear four-sublattice AF structure of an isolated FeO2layer is less stable than the(1/3,1/3)superstructure in terms of a Heisenberg model.In section4,we examine whether the defect ions Fe2+and Cu2+of the CuFeO2lattice generated by oxygen nonstoichiometry can provide a driving force for the Fe3+ ions to orient their moments along the c axis,thereby stabilizing the collinear four-sublattice AF structure below T N2.The important results of our work are summarized in section5.
(6)Mekata,M.;Yaguchi,N.;Takagi,T.;Sugino,T.;Mitsuda,S.;
Yoshizawa,H.;Hosoito,N.;Shinjo,T.J.Phys.Soc.Jpn.1993,62, 4474.
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M.J.Phys.Soc.Jpn.1994,63,2017.
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T.Physica B1994,201,71.
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A.Phys.Re V.B2000,62,8983.
(11)Petrenko,O.A.;Lees,M.R.;Balakrisnan,G.;de Brion,S.;Chouteau,
G.J.Phys.:Condens.Matter2005,17,2741.
(12)Hasegawa,M.;Batrashevich,M.I.;Zhao,T.-R.;Takei,H.;Goto,T.
Phys.Re V.B2001,63,184437.
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T.;Aruga-Katori,H.J.Phys.Soc.Jpn.2004,73,1442.
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https://www.sodocs.net/doc/4017245100.html,/pdf/cond-mat/0510701.
(18)Terada,N.;Mitsuda,S.;Prokes,K.;Suzuki,O.;Kitazawa,H.;Aruga-
Katori,H.Phys.Re V.B2004,70,
174412.Figure2.Magnetic superstructures of an FeO2layer:(a)noncollinear120°magnetic structure,i.e.,(1/3,1/3)magnetic superstructure,where the small arrow at each spin site represents the spin moment.(b)Collinear four-sublattice antiferromagnetic structure,i.e.,(1/2,1/4)magnetic superstructure, where the spin moments parallel and antiparallel to the c axis are distinguished by circles with two different colors and the rectangular box represents the smallest unit cell.
Conflicting Pictures of Magnetism for CuFeO2Chem.Mater.,Vol.18,No.5,20061269
2.Spin Dimer Analysis
In describing the magnetic properties of CuFeO 2in terms of an Ising model,we have employed three intralayer spin-exchange interactions J 1,J 2,and J 3(Figure 3a)while neglecting the interlayer spin-exchange interaction J 4(Figure 3b).6,9We estimate the relative strengths of these interactions by performing spin dimer analysis on the basis of extended Hu ¨ckel tight-binding (EHTB)electronic structure calcula-tions.19The spin dimers (i.e.,the structural units made of two Fe 3+ions plus the oxygen atoms in their first coordinate spheres)leading to the J 1,J 2,J 3,and J 4interaction are depicted in Figure 4.These interactions are of the superex-change (SE)type involving Fe -O -Fe paths (i.e.,J 1),the super-superexchange (SSE)type involving Fe -O ???O -Fe paths within each FeO 2layer (i.e.,J 2and J 3),and the SSE type involving Fe -O -Cu -O -Fe paths between adjacent FeO 2layers (i.e.,J 4).The important structural parameters describing these spin-exchange paths are listed in Table 1.The spin-exchange parameter J is written as J )J F +J AF ,where the ferromagnetic term J F is positive and the antiferromagnetic term J AF is negative.In general,J F is very
small,so that the trends in the J values are well-approximated by those in the corresponding J AF values.19In spin dimer analysis completed on the basis of EHTB calculations,the strength of an antiferromagnetic interaction between two spin sites is estimated by considering the antiferromagnetic spin exchange parameter J AF 19
where U eff is the effective on-site repulsion that is essentially a constant for a given compound.The ?(?e )2?term is calculated by performing tight-binding electronic structure calculations for a spin dimer.Each Fe 3+ion of CuFeO 2has five singly occupied d-block levels φμ(μ)1-5),i.e.,five magnetic orbitals.Thus,the ?(?e )2?term is given by
where ?e μμis the energy split that results when two magnetic orbitals φμ(μ)1-5)on adjacent spin sites interact (Figure 5).Because U eff is essentially a constant for a given compound,the trend in the J AF values is determined by that in the corresponding ?(?e )2?values.For the spin-exchange interactions J 1-J 4of CuFeO 2,we calculate the corresponding ?(?e )2?values using the spin dimers shown in Figure 4.The atomic parameters of Fe,Cu,and O used for our EHTB calculations 20are summarized in Table 2.Table 1sum-marizes the calculated ?(?e )2?values together with their relative values.
Table 1reveals that the SE interaction J 1is the strongest AF spin-exchange interaction in CuFeO 2.The intralayer SSE interactions J 2and J 3are much weaker than J 1(by factors of approximately 50and 30,respectively).This is in sharp contrast to the observation 8that,in order for the magnetic structures of CuFeO 2below T N2and under magnetic field to be described in terms of a 2D Ising model,J 2and J 3should be quite strong (i.e.,J 2/J 1)0.5and J 3/J 1)0.75).Furthermore,the interlayer SSE interaction J 4is not negli-gible but is comparable in strength to the intralayer SSE interactions J 2and J 3.This is consistent with the finding that CuFeO 2undergoes a 3D ordered magnetic structure below 14K 7and that neutron-scattering experiments show a significant dispersion along the interlayer direction.21For the interlayer interaction J 4,the two O -Cu -O bridges linking the two FeO 6octahedra are crucial because our calculations lead to ?(?e )2?)0in the absence of the Cu atoms in the spin dimer (Figure 4d).The magnetic orbitals of the spin dimer show that the 3d z 2,3d xz ,and 3d yz of the bridging Cu atoms (with the local z axis taken along each O -Cu -O linkage)promote the interlayer SSE interaction J 4.
(19)For recent reviews,see:(a)Whangbo,M.-H.;Koo,H.-J.;Dai,D.J.
Solid State Chem .2003,176,417.(b)Whangbo,M.-H.;Dai,D.;Koo,H.-J.Solid State Sci .2005,7,827.
(20)Our calculations were carried out by employing the SAMOA (Structure
and Molecular Orbital Analyzer)program package:Dai,D.;Ren,J.;Liang,W.;Whangbo,M.-H.SAMOA ;https://www.sodocs.net/doc/4017245100.html,/,2002.
(21)Petrenko,O.A.;F?k,B.;Bull,M.ISIS Experimental Report RB;
11734,2001;ISIS:Chilton,U.K.,2001;https://www.sodocs.net/doc/4017245100.html,/isis2001/
reports.
Figure 3.(a)Spin-exchange paths J 1,J 2,and J 3within an FeO 2layer.(b)Spin-exchange path J 4between adjacent FeO 2layers.
Table 1.Geometrical Parameters and ?(?e )2?Values Associated with
the Spin-Exchange Interactions J 1,J 2,J 3,and J 4of CuFeO 2
interaction
geometrical parameters a
?(?e )2?(meV 2)J 1Fe -O -Fe
1570
(1.00)b
Fe ???Fe )3.035?∠Fe -O -Fe )96.6°J 2
Fe -O ???O -Fe 32(0.020)
Fe ???Fe )5.257?O ???O )2.706?∠Fe -O ???O )99.9°J 3Fe -O ???O -Fe 50(0.032)
Fe ???Fe )6.070?O ???O )3.035?∠Fe -O ???O )138.3°J 4Fe -O -Cu -O -Fe 36(0.023)
Fe ???Fe )5.984?O -Cu )1.830?∠Fe -O -Cu )120.5°∠O -Cu -O )180°
a
All Fe -O )2.033?.b The numbers in parentheses give the relative tendency for antiferromagnetic coupling.
J AF ≈-?(?e )2?
U eff
(1)
?(?e )2
?≈
15
2
∑μ)1
5
(?e μμ)2(2)
1270Chem.Mater.,Vol.18,No.5,2006Whangbo et al.
3.Ordered Magnetic Structure and Total
Spin-Exchange Interaction Energy In this section,we examine the total spin-exchange energies of various ordered magnetic structures of CuFeO 2in terms of a Heisenberg model by using the Freiser method,19b,22in which the total spin-exchange energy of an ordered magnetic structure is calculated under the assump-tions that spins adopt all possible directions in space (i.e.,the classical spin approximation)and that the spin-exchange interactions are isotropic (i.e.,a Heisenberg description).The classical spin approximation is best suited for a magnetic system made of spin-5/2ions such as high-spin Fe 3+ions.In a long-range ordered magnetic state i of a magnetic system,the spin sites μ()1,2,...,m )of its unit cell located at the coordinate origin (i.e.,the lattice vector R )0)have mean spins σμ0.At high temperatures,the spins are com-pletely disordered so that σμ0)0at all spin sites.As the temperature is lowered,an ordered spin arrangement may set in,thereby leading to nonzero σμ0.For a magnetic solid with repeat vectors a ,b ,and c ,the ordered spin arrangement is described by the spin functions σμ(k )
where M is the number of unit cells in the magnetic solid,k
is the wave vector,and R is the direct lattice vector.23The ordered magnetic state ψi (k )(i )1-m )is then expressed as a linear combination of the spin functions σμ(k )
To determine the energy E i (k )of the state ψi (k )and the coefficients C μi (k )(μ)1-m ),one needs to evaluate the spin-exchange interaction energies μν(k )between every two spin functions σμ(k )and σν(k )
and diagonalize the resulting interaction matrix (k )
For a given set of spin-exchange parameters J μν,one can determine the value of k that leads to the lowest energy E m
(22)Freiser,M.J.Phys.Re V .1961,123,2003.
(23)Given the lattice vector written as R )n a a +n b b +n c c ,where n a ,
n b ,and n c are integers,and the wave vector k written as k )x a a *+x b b *+x c c *,where a *,b *,and c *are the reciprocal vectors and x a ,x b ,and x c are dimensionless numbers,the exp(i kR )term of eq 3becomes exp[i 2π(x a n a +x b n b +x c n c )].For convenience,we denote k by showing only its dimensionless components,i.e.,k )(x a ,x b ,x c
).
Figure 4.Spin dimers associated with the spin-exchange paths (a)J 1,(b)J 2,(c)J 3,and (d)J 4
.
Figure 5.Interaction between two magnetic orbitals φμof a spin dimer
leading to the energy split ?e μμ.
Table 2.Exponents i and Valence-Shell Ionization Potentials H Ii of Slater-Type Orbitals i Used for Extended Hu 1ckel Tight-binding
Calculation a atom i H ii (eV) i C b ′i C ′b
Fe 4s -9.10 1.925 1.0Fe 4p -5.32 1.390 1.0Fe 3d -12.6 6.6080.4038 2.6180.7198Cu 4s -11.4 2.151 1.0Cu 4p -6.06 1.370 1.0Cu 3d -14.07.0250.4473 3.0040.6978O 2s -32.3 2.6880.7076 1.6750.3745O
2p
-14.8
3.694
0.3322
1.659
0.7448
a
The diagonal matrix element H ii is defined as ? i |H eff | i ?,where H eff is the effective Hamiltonian.In our calculations of the off-diagonal matrix elements H eff )? i |H eff | j ?,the weighted formula was used.b Contraction coefficients used in the double- Slater-type orbital.
σμ(k ))
1
M
∑R
σμ0exp(i k ?R )(3)
ψi (k ))C 1i (k )σ1(k )+C 2i (k )σ2(k )+...+C mi (k )σm (k )(4) μν(k ))-∑R
J μν(R )exp(i k ?R )(5)
(k ))[
11(k ) 12(k )... 1m (k ) 21
(k )
22(k )... 2m (k )............ m 1(k )
m 2(k )... mm (k )
]
(6)
Conflicting Pictures of Magnetism for CuFeO 2Chem.Mater.,Vol.18,No.5,20061271
of the bandsΕi(k)(i)1,2,...,m),which occurs from the lowest-lying band E1(k).If we denote this particular k point as k m,then the magnetic superstructure is described byψ1-(k m).
Because each unit cell of CuFeO2has three Fe3+ions,there are three spin-basis functionsσμ(k)(μ)1-3)to consider. The pairs(μ-ν)of the spin sites(μ,ν)1-3)leading to the spin-exchange interactions J1,J2,J3,and J4are listed in Table3,whereas the nonzero contributions to the matrix elements μν(k)from the various spin-exchange paths of CuFeO2are summarized in Table4.Thus the nonzero matrix elements μν(k)are given by
Figure6shows a representative E i(k)vs k plot calculated for the wave vector regionΓ-X-M-Γ,whereΓ)(0,0, 0),X)(1/2,0,0),and M)(1/2,1/2,0).For this particular plot,we used J2/J1)J3/J1)J4/J1)0.1to accentuate the characteristic features of the E i(k)vs k plot.The three bands should be closer together than they appear in the plot because the values of J2/J1,J3/J1,and J4/J1should be smaller than 0.1,according to our estimates in the previous section.The ordered spin arrangements of an isolated FeO2layer given by the wave vectorsΓ,X,and M are shown in Figure7. Along the line X-M,the energy minimum occurs at H) (1/2,1/4,0),which gives rise to the four-sublattice AF structure(Figure2b)in each FeO2layer.Along the M-Γline,the minimum energy occurs at K)(1/3,1/3,0),which leads to the(1/3,1/3)superstructure that has the noncollinear three-sublattice120°arrangement(Figure2a)in each FeO2 layer.It is noticed that with the Heisenberg description,the (1/3,1/3)superstructure is considerably more stable than the (1/2,1/4)superstructure.
To probe the primary spin-exchange interactions respon-sible for the formation of the(1/2,1/4)and(1/3,1/3) superstructures,we evaluated the total spin-exchange interac-tion energies of the ordered spin structures given by the wave vector pointsΓ,X,M,H,and K(per FeO2layer per spin site).Our results are summarized in Table5.In using these results,it should be recalled that J1,J2,J3<0,according to our spin dimer analysis in the previous section.Table5 shows that the intralayer interactions J2and J3are essential for the occurrence of the(1/2,1/4)superstructure because if J2)J3)0,the four-sublattice AF structure would become degenerate with the(1/2,0)superstructure at X and the(1/ 2,1/2)superstructure at M.Furthermore,the(1/2,1/4) superstructure can never become more stable than the(1/3, 1/3)superstructure within the scope of a Heisenberg model, because the energy difference between these two structures is determined primarily by the strongest AF interaction J1 that defines the TLA.
Table3.Pairs of Spin Sites(μ-ν)(μ,ν)1,2,3)Leading to the Spin-Exchange Spin interactions J1-J4
(μ-ν)
path within a unit cell between unit cells
J1(1-1),(2-2),(3-3)
J2(1-1),(2-2),(3-3)
J3(1-1),(2-2),(3-3)
J4(1-2),(1-3)(1-2),(1-3),(2-3) Table4.Nonzero Contributions to the Matrix Elements μν(k)from
the Various Spin-Exchange Paths
μνcell a Fe???Fe(?)contribution to μν(k)
11b[-2,-2,0] 6.070-J3exp(-i4πx a-i4πx b) [2,2,0] 6.070-J3exp(i4πx a+i4πx b)
[-2,-1,0] 5.257-J2exp(-i4πx a-i2πx b)
[2,1,0] 5.257-J2exp(i4πx a+i2πx b)
[-2,0,0] 6.070-J3exp(-i4πx a)
[2,0,0] 6.070-J3exp(i4πx a)
[-1,-2,0] 5.257-J2exp(-i2πx a-i4πx b)
[1,2,0] 5.257-J2exp(i2πx a+i4πx b)
[-1,-1,0] 3.035-J1exp(-i2πx a-i2πx b)
[1,1,0] 3.035-J1exp(i2πx a+i2πx b)
[-1,0,0] 3.035-J1exp(-i2πx a)
[1,0,0] 3.035-J1exp(i2πx a)
[-1,1,0] 5.257-J2exp(-i2πx a+i2πx b)
[1,-1,0] 5.257-J2exp(i2πx a-i2πx b)
[0,-2,0] 6.070-J3exp(-i4πx b)
[0,2,0] 6.070-J3exp(i4πx b)
[0,-1,0] 3.035-J1exp(-iπx b)
[0,1,0] 3.035-J1exp(iπx b)
12[-1,-1,0] 5.984-J4exp(-i2πx a-i2πx b) [-1,0,0] 5.984-J4exp(-i2πx a)
[0,0,0] 5.984-J4
13[-1,-1,0] 5.984-J4exp(-i2πx a-i2πx b) [0,-1,0] 5.984-J4exp(-i2πx b)
[0,0,0] 5.984-J4
23[0,-1,1] 5.984-J4exp(-i2πx b+i2πx c) [0,0,1] 5.984-J4exp(i2πx c)
[1,0,1] 5.984-J4exp(i2πx a+i2πx c)
a The cell notation[n a,n b,n c]indicates the unit cell position given by the direct lattice vector R)n a a+n
b b+n
c c.It is use
d to show th
e contribution that occurs between the[0,0,0]and[n a,n b,n c]cells.Thus, the notation[0,0,0]means the contribution that occurs within a unit cell.
b The interactions betweenμ)2andν)2and those betweenμ)3and ν)2are the same as those betweenμ)1andν)1.
11(k))
22
(k))
33
(k))-J
1
{2cos[2π(x a+x b)]+
2cos[2πx
a
]+2cos[2πx
b
]}-J
2
{2cos[2π(2x a+x b)]+
2cos[2π(x
a
+2x
b
)]+2cos[2π(x
a
-x
b
)]}-
J
3
{2cos[4π(x a+x b)]+2cos[4πx a]+2cos[4πx b]}
12(k))-J
4
{exp[-i2π(x a-x b)]+exp[-i2πx a]+1}
13(k))-J
4
{exp[-i2π(x a-x b)]+exp[-i2πx b]+1}
23(k))-J
4
{exp[i2π(x a+x c)]+exp[i2π(-x b+x c)]+
exp[i2πx
c
]}
(7)
Figure6.E i(k)vs k plots calculated for J2/J1)J3/J1)J4/J1)0.1,where
Γ)(0,0,0),X)(1/2,0,0),M)(1/2,1/2,0),H)(1/2,1/4,0),and K
)(1/3,1/3,0).
Table5.Total Spin-Exchange Energies of a Single FeO2Layer for
the Ordered Spin Arrangements Described by Some Special
Positions in the Brillouin Zone
wave vector energy per spin site
Γ)(0,0,0)-3J1-3J2-3J3
X)(1/2,0,0)J1+J2-3J3
M)(1/2,1/2,0)J1+J2-3J3
H)(1/2,1/4,0)J1-J2+J3
K)(1/3,1/3,0)3J1/2-3J2/2+3J3/2
1272Chem.Mater.,Vol.18,No.5,2006Whangbo et al.
4.Magnetic Moments of the Defect Ions Generated by
Oxygen Nonstoichiometry Our analyses in the previous two sections make it clear that a Heisenberg model cannot account for the occurrence of the collinear four-sublattice AF structure in CuFeO 2.Thus,we face a dilemma in understanding the magnetic properties of CuFeO 2;an Ising model is required for explaining the ordered magnetic structures below T N1and under a magnetic field,whereas a Heisenberg model explains the magnetic properties of CuFeO 2above T N1.This conceptual difficulty is resolved if nominally stoichiometric samples of CuFeO 2contain a small quantity of point defect ions that give rise to a uniaxial magnetic moment and hence induce the surrounding Fe 3+ions to orient their magnetic moments along the c axis.(It should be noted that the electrical properties of CuFeO 2single crystals can be either n-type or p-type depending on the preparation methods,24the probable cause of which is oxygen nonstoichiometry.)The most likely defect ions present in the CuFeO 2lattice would be the Fe 2+or Cu 2+ions resulting from oxygen nonstoichiometry.12In the fol-lowing,we explore the plausibility of the above hypothesis.As depicted in Figure 8a,an oxygen atom vacancy in CuFeO 2generates three FeO 5square pyramids.These square pyramids are only slightly different in structure from an ideal square pyramid (i.e.,that resulting from an FeO 6regular octahedron by removing one oxygen atom),because the FeO 6octahedra of CuFeO 2are almost regular in shape.For
simplicity,it will be assumed that the FeO 5square pyramids generated by an oxygen atom vacancy are ideal square pyramids.Our EHTB calculations show that the d-block levels of an ideal FeO 5square pyramid are split as depicted in Figure 8b,where the local z axis is taken along the bond from the Fe to the apical O atom.Thus,as depicted in Figure 8c,the electronic ground state of an ideal FeO 5square pyramid containing a high-spin Fe 2+(d 6)ion is described by a linear combination of two degenerate electron configu-
(24)Dordor,P.;Chaminade,J.-P.;Wichanchai, A.;Marquestaut, E.;
Doumerc,J.-P.;Pouchard,M.;Hagenmuller,P.;Ammar,A.J.Solid State Chem .1988,75,
105.
Figure 7.Ordered magnetic structures of an FeO 2layer given by the wave vector points (a)Γ,(b)X,and (c)
M.
Figure 8.(a)Group of three FeO 5square pyramids surrounding an oxygen atom vacancy (small green circle).(b)d-Block split pattern of an ideal FeO 5square pyramid.(c)Two degenerate electron configurations of an ideal FeO 5square pyramid containing Fe 2+.
Conflicting Pictures of Magnetism for CuFeO 2Chem.Mater.,Vol.18,No.5,20061273
rations,where one set of degenerate orbitals (i.e.,d xz and d yz )has three electrons.In addition,a rotation around the 4-fold rotational axis interconverts the two orbitals.In the case of oxygen excess,some of the Cu +ions in the linear two-coordinate sites would be oxidized to Cu 2+ions.Our EHTB calculations show that the d-block levels of a linear CuO 2dumbbell are split as depicted in Figure 9a,where the local z axis is taken along the O -Cu -O axis.The d z 2level lies slightly lower than the d xz and d yz levels because the mixing of the Cu 4s orbital strongly reduces the sigma antibonding between the Cu 3d z 2and the O 2p z orbitals.The ground electronic state of a CuO 2dumbbell containing a Cu 2+(d 9)ion is described by a linear combination of two degenerate electron configurations (Figure 9b)in which one set of degenerate orbitals (i.e.,d xz and d yz )has three electrons,and a rotation around the O -Cu -O axis interconverts the two orbitals.
It is known 14that a magnetic ion under a crystal field with n -fold (n g 3)rotational symmetry has a uniaxial magnetic moment along the n -fold rotational axis,if the ground state of the ion is described by a linear combination of two degenerate electron configurations in which one set of two degenerate orbitals has three electrons and if the orbitals |L L z >of the ion are eigenfunctions of the associated crystal-field Hamiltonian.These conditions are satisfied for the Fe 2+and Cu 2+ions of the CuFeO 2lattice discussed above.Thus,in the case of oxygen deficiency,the Fe 2+ion located in an ideal FeO 5square pyramid should have a uniaxial magnetic moment parallel to its 4-fold rotational axis.When all three FeO 5square pyramids surrounding an oxygen atom vacancy are occupied by Fe 2+ions,the vector sum of their uniaxial magnetic moments is pointed along the c axis.In the case of oxygen excess,the Cu 2+ion of a CuO 2dumbbell should have a uniaxial magnetic moment along the O -Cu -O axis,i.e.,along the c axis.
The occurrence of uniaxial magnetic moments resulting from either the Fe 2+ions in the case of oxygen deficiency or the Cu 2+ions in the case of oxygen excess would induce the surrounding Fe 3+ions to orient their moments along the c axis,hence stabilizing the collinear four-sublattice AF structure below T N1.It is of interest to consider implications of this hypothesis.The uniaxial magnetic moment associated with the Fe 2+ions lies in the planes of the Fe 3+ions,whereas that associated with the Cu 2+ions lies outside the planes of the Fe 3+ions.Thus,the oxygen deficiency would exert a stronger driving force for the Fe 3+ions to orient their
magnetic moments along the c axis than would the oxygen excess.This explains why the phase-transition temperature T N2is increased by oxygen deficiency (δ<0)but decreased by oxygen excess (δ>0).12Because the collinear four-sublattice AF structure is not observed for CuCrO 2,15one might speculate whether defect ions with uniaxial magnetic moments are absent in CuCrO 2.An oxygen-atom vacancy in CuCrO 2+δ(i.e.,δ<0)would generate Cr 2+(d 4)ions in CrO 5square pyramids.Such a Cr 2+ion cannot generate a uniaxial magnetic moment,because its high-spin ground-state electronic structure does not lead to a degenerate configuration according to the d-block energy levels of Figure 8b.However,the oxygen excess in CuCrO 2+δ(i.e.,δ>0)would create Cu 2+ions with uniaxial magnetic moments,so that the four-sublattice AF structure might be observable in such a case.The collinear four-sublattice AF structure of CuFeO 2is destroyed when only 2%of the Fe 3+ions are replaced with nonmagnetic Al 3+ions.18This can be under-stood if the Al 3+substitution prevents oxygen nonstoichi-ometry.
5.Concluding Remarks
Our spin dimer analysis for CuFeO 2shows that the intralayer interaction J 1defining a TLA is stronger than the intralayer interactions J 2and J 3as well as the interlayer interaction J 4by a factor of approximately 30or greater.A Heisenberg model using these spin-exchange parameters predicts that the collinear four-sublattice AF structure is less stable than the noncollinear 120°spin arrangement predicted for a TLA.Thus we examined whether defect Fe 2+and Cu 2+ions of the CuFeO 2lattice generated by oxygen nonstoichi-ometry are responsible for the collinear spin alignment of the Fe 3+ions along the c axis below T N1.Our electronic structure analysis suggests that these defect ions give rise to magnetic moments pointing along the c axis,and hence provide a driving force for the surrounding Fe 3+ions to orient their moments along the c axis below T N1.As a consequence,an Ising model would be required for describing the magnetic structures of CuFeO 2below T N1despite the fact that the magnetic properties of high-spin Fe 3+ions are not uniaxial.Above T N1,thermal agitation would negate the moment-orienting effect of the defect ions so that a Heisenberg model would be required for describing the magnetic properties of CuFeO 2.For nonstoichiometric CuCrO 2+δsamples,our hypothesis leads to the prediction that the collinear four-sublattice AF structure will be present in the case of oxygen excess (δ>0)but absent in the case of oxygen deficiency (δ<0).The presence of random point defects could also be the reason the phase transition to the collinear four-sublattice AF structure at 11K exhibits thermal hysteresis.Acknowledgment.The research was supported by the Office of Basic Energy Sciences,Division of Materials Sciences,U.S.Department of Energy,under Grant DE-FG02-86ER45259.K.-S.L.thanks The Catholic University of Korea for the 2004Research Fund.R.K.K.thanks W.Schnelle (MPI-CPfS,Dresden)for helpful discussion.
CM052634G
Figure 9.(a)d-Block split pattern of a linear O -Cu -O dumbbell.(c)Two degenerate electron configurations of a linear O -Cu -O dumbbell containing Cu 2+.
1274Chem.Mater.,Vol.18,No.5,2006Whangbo et al.