Exchange Bias Driven by the Dzyaloshinskii-Moriya Interaction and Ferroelectric Polarization at G-Type Antiferromagnetic Perovskite Interfaces
Shuai Dong,1,2,3Kunihiko Yamauchi,4Seiji Yunoki,5,6Rong Yu,1,2Shuhua Liang,1,2Adriana Moreo,1,2J.-M.Liu,3,7
Silvia Picozzi,4and Elbio Dagotto1,2
1Department of Physics and Astronomy,University of Tenne,Knoxville,Tenne37996,USA 2Materials Science and Technology Division,Oak Ridge National Laboratory,Oak Ridge,Tenne32831,USA 3Nanjing National Laboratory of Microstructures,Nanjing University,Nanjing210093,China
4Consiglio Nazionale delle Ricerche-Istituto Nazionale per la Fisica della Materia(CNR-INFM),CASTI Regional Laboratory,
67100L’Aquila,Italy
5Computational Condend Matter Physics Laboratory,RIKEN,Wako,Saitama351-0198,Japan
6CREST,Japan Science and Technology Agency(JST),Kawaguchi,Saitama332-0012,Japan
7International Center for Materials Physics,Chine Academy of Sciences,Shenyang110016,China
(Received15August2009;published14September2009)
Exchange bias is usually rationalized invoking spin pinning effects caud by uncompensated袖珍椰子树
antiferromagnetic interfaces.However,for compensated antiferromagnets other extrinsic factors,such
as interface roughness or spin canting,have to be considered to produce a small uncompensation.As an
alternative,here we propo two(related)possible mechanisms,driven by the intrinsic Dzyaloshinskii-
Moriya interaction and ferroelectric polarization,for the explanation of exchange bias effects in
perovskites with compensated G-type antiferromagnetism.One of the mechanisms is only active when
a multiferroic material is involved and it is controllable by electricfields.
DOI:10.1103/PhysRevLett.103.127201PACS numbers:75.70.Cn,75.10.Hk,75.30.Et,75.80.+q
Introduction.—The exchange bias(EB)effect,charac-terized by a shift of the magnetic hysteresis loops away from the center of symmetry at zero magneticfield,is widely reported to exist in magnetic systems where there is an interface between antiferromagnetic(AFM)and fer-romagnetic(FM)(or ferrimagnetic)materials[1]. Theoretically,the EB is understood as induced by spin pinning effects at the FM/AFM interface.An uncompen-sated AFM interface is usually invoked to illustrate how the pinning may work.The uncompensated AFM spins at the interface are expected to pin the nearest-neighbor (NN)FM spins via the exchange coupling,giving ri to a preferred direction for the FM moments.However,despite its physical appeal,this simple picture is not enough to fully understand veral real EB cas in a variety of magnetic systems.This approach usually predicts an EB larger than measured,and also fails to answer why there is EB in some fully compensated AFM interfaces[2]. Precily for the subtle ca of compensated AFM in-terfaces,extrinsic factors are also often considered,such as interface roughness[3].Spin canting near the interface can also contribute to the EB[4].Other models have also been propod,such as frozen interfacial and domain pinning. Most of the models still need a small‘‘frozen’’uncom-pensation of the AFM moments near the interface,thus remaining under much debate[2].
Recently,remarkable improvements in oxide thin-film techniques have allowed for the growth and ch
aracteriza-tion of complex oxide heterostructures with(near)atomic precision,opening an avenue for the fabrication of multi-functional devices using strongly correlated electronic ma-
terials[5].In this context,EB has been obrved in BiFeO3
(BFO)bad heterostructures[6].More interestingly,the
EB in multiferroic heterostructures is widely believed to be controllable by electricfields.In addition,the EB has also
been obrved in SrRuO3=SrMnO3(SRO/SMO)superlat-tices[7].Considering that both BFO and SMO are well-
known compensated G-type AFM materials(all NN spins
are antiparallel)and that the interfaces are very smooth,the
origin of the EB in the heterostructures remains a puzzle.
水枪大战The purely magnetic interactions framework stemming
from traditional metallic magnetism appears incomplete to deal with the complex physics unveiled in the strongly correlated electronic systems,and to address the practical matter of how to control the EB by electricfields when a multiferroic material is involved.Therefore,new mecha-nisms that emphasize the many simultaneously active de-grees of freedom in correlated electron systems are needed to better understand the interesting effects.
The model.—Here,we propo two(related)mecha-nisms for EB generation in interfaces involving FM/G-AFM perovskites.In the mechanisms,the G-AFM interface can be fully compensated,namely,the tiny uncompensation caud by various uncertain factors is no longer esntial(although it can still exist).Therefore, our propod mechanism is conceptually different from ideas bad on tiny frozen uncompensated AFM moments [6,7].Instead,the interactions between spins and lattice distortions become the key intrinsic driving force for the mechanisms prented below.
Let us start with the spin-spin interaction in perovskites, with a Hamiltonian
H¼
X
h ij i
½J i;j~S iÁ~S jþ~D i;jÁð~S iÂ~S jÞ ;(1)
where J i;j is the standard superexchange(SE)coupling between NN spins;i and j are site indices,and~S are spin vectors.For veral large-spin transition metal cations in perovskites,such as Mn3þand Fe3þ,adopting the widely ud classical approximation is reasonable.In the follow-ing,the normalization j~S j¼1will be ud(the actual magnitude S of the spins can be absorbed in a redefinition of couplings).The cond term is the Dzyaloshinskii-Moriya(DM)interaction,which aris from the spin-orbit coupling[8,9].Since j~D j is much smaller(by2or3orders of magnitude)than J[9],the DM interaction is often neglected.Originally,the DM interaction was introduced to explain the prence of weak ferromagnetism in AFM materials becau the DM term can produce a small spin-canting.Recently,the DM interaction has also been high-lighted as the origin of afinite ferroelectric(FE)polariza-tion(~P)in multiferroic materials with spiral spin order [10].
景海鹏简历
In perovskites,the DM interaction is determined by the oxygen octahedron tilting.Usually,the A-site cations in perovskites are too small to maintain a stable cubic lattice. Then,the oxygen octahedra surrounding the B-site cations will tilt for a clor packing[11].The tilting can be characterized by the
Glazer aÀbÀcþwhere the three letters denote the rotation angle amplitudes about the[100],[010],and[001]axes,respectively;the positive(negative)superscript indicates that the rotations of two neighboring octahedra,along the tilting axis,are in the same(opposite)direction[12].For instance,in the orthorhombic bulk LaMnO3at low tem-perature(T)),the tilting aÀaÀbþreceives the name ‘‘GdFeO3-type distortion’’and it corresponds to rotations around the[110](dominant)and[001](subdominant)axes of the cubic unit cell[13].For the M-O-M bond(M:B-site metal and O:oxygen),this octahedral tilting moves the oxygen anion perpendicularly away from the midpoint between NN metal cations,as shown in Fig.1(a).Since the tilting rotation is collective,the NN oxygens in the same direction(O1and O2in M-O1-M-O2-M)should move in opposite directions,namely,the NN displacements are staggered.
DM-driven EB.—From symmetry argumentations,the ~D
i;j
vector should be perpendicular to the M i-O-M j bond [9],as shown in Fig.1(a).Thus,the~D vectors between NN bonds along the same direction are also staggered,namely
~D
i;iþ1¼À~D iþ1;iþ2.To simplify the discussion,let us
consider the ca where the rotations of NN octahedra along the[100]and[010]axes are in opposite directions, namely aÀbÀcÃ(Ãcan beþ,À,or0).
For simplicity,all spins in the AFM and FM side are assumed to be collinear.However,becau of the different easy magnetic axes or planes for different materials,in general the NN spins are noncollinear at the FM/G-AFM interface[Fig.1(b)].There are two vectors that are stag-
gered:(1)the AFM interface spins~S AFM
i
given byðÀ1Þi~S A, and(2)the~D ij vectors across the interface given by ðÀ1Þi~D,where i denotes the site(or bond)quence at the(001)interface.Combining the two staggered com-
孕妇能不能吃牛肉
ponents~D ij and~S AFM
i
融资融券开户,it is straightforward to obtain a uniform DM effect at the interface:
H interface
DM
¼
X
h ij i
~D
ij
Áð~S FM iÂ~S AFM
j
Þ¼À~h DÁ
X
i
~S FM
i
;(2)
where~S FM denotes the spin at the FM side and i and j only sums over the interface.~h D is the effective magneticfield that points into the direction~DÂ~S A[Figs.1(b)and1(c)]. Note that~h D is uniform and independent of the FM spins’
direction.The combination of~D i;j and~S AFM
j
,namely~h D, can befixed by thefield-cooling process and then assumed to remain frozen at low T during the hysteresis loop measurement[14].Thus,this provides a biasfield caud by the DM interaction which can produce a EB at inter-faces of FM/G-AFM perovskites.
FE-driven EB.—In the previous discussion,the cond term(DM interaction)of Eq.(1)was propod as the microscopic origin of EB in generic FM/G-AFM oxide heterostructures.However,thefirst term(SE)can also contribute to the EB if multiferroic materials are involved in the heterostructure.In ferroelectric(FE)materials,spon-taneous relative displacements between cations and anions induce an electric polarization.Consider the oxygen posi-tions at the interface shown in Fig.2(a):in addition to the previously mentioned staggered displacements,that do
not FIG.1(color online).(a)The(mutually perpendicular)rela-tionship between the M i-O-M j bond,oxygen displacement,and ~D
老年人睡不着觉的解决方法i;j
vector.(b)Sketch of the interface between FM and G-AFM perovskites,including the oxygen octahedral tilting.The stag-gered directions of the~D ij vectors at the interface are marked as in-and out-arrows,while the uniform~h D vectors are also shown near the oxygens.(c)The uniform~h D should be perpendicular to ~S AFM and~D.
induce a finite ~P
,in some multiferroic materials their FE properties can be assumed to be caud by additional displacements of NN oxygens that should all be along the same direction to avoid a global cancellation.Therefore,the bond-angles at the interface can become asymmetric by the simultaneous consideration of the two displacement modes.Since the SE coupling magnitude is dependent on the bond-angle [13],the modulated bond-angles induce an interfacial SE coupling J that is also staggered,with values that are denoted here as J L and J S .Once again,as with the DM-driven EB,two staggered effects (alternating SE couplings at the interface,and alternating spin orientations on the AFM side of the inter-face)compensate each other.By this procedure,it is straightforward to obtain an additional uniform effective field at the interface
H interface SE ¼X h ij i
J i;j ~S FM i Á~S AFM j ¼À~h J ÁX i
~S
FM
i :(3)Here,~h
J ¼À J ~S A is the effective magnetic field,where J ¼ðJ L ÀJ S Þ=2.When an electric field is applied paral-lel to the interface to change the uniform polarization ~P
,the ~h
J will change simultaneously,namely,it is an electric-field-controllable EB which is potentially important to design multiferroics devices.
Both ~h
J and ~h D may have components parallel to the measuring field,although they are perpendicular to each other.Experimentally,by varying the electric-field direc-tion,estimations for the components of ~h
J and ~h D can be obtained parately,since ~h
D is almost independent of the F
E ~P
,in a first-order approximation [10].Discussion.—The basic physical picture related to the propod DM-and FE-driven EB appears clear,but there are veral practical issues that should be addresd.
First,in the derivations above,both mechanisms are independent of the details of the FM spins.Therefore,both mechanisms should be valid for a variety of FM materials such as perovskites [7]or metallic alloys [6].
The only condition needed is that the oxygen octahedra of the interfacial AFM cations must be ,oxygen must bridge the two materials at the interface.
Also,our model should be robust against other tilting
modes.For a general a b c Ãmode,the NN ~D
ij ’s at the interface are not uniform as long as and are not both
simultaneously zero.If this is the ca,a net ~h
D is still induced,with direction and value varying with the mode.For the tilting mode which only rotates along the [001]axis (the a 0a 0c þmode in the perfect tetragonal lattice),the DM
contribution at the (001)interface is zero.In this ca,~h
J will also be zero since the bond-angles are uniform.However,there is evidence that many perovskite films are not perfectly tetragonal [15].
In addition,since the DM coupling is very weak (par-ticularly in nearly tetragonal thin films),it is necessary to check whether the EB that it generates is compatible in
magnitude with the experimental EBs.Considering ~h
D to be only effective at the interface while the external mag-netic field is applied on all FM spins,the maximum EB
(when the measuring field is collinear with ~h
D )can be estimated as:h EB %j ~h D j =d ¼H interface DM
=ðdm Þ,where d is the FM material thickness in unit cells,and m is the magnetic moment of the FM cation.In a first-order ap-proximation,j ~D
i;j j is proportional to the oxygen displace-ment X O :H interface DM % X O
,with the DM coefficient roughly estimated as 1meV = A
[10].A tiny distortion of the M 1-O -M 2bond across the interface,as small as a 1
bend [16],can result in H interface DM
%0:0175meV if the lattice constant is 4A
˚,indeed very weak compared with J which is usually larger than 10meV for perovskites.Assuming typical values d ¼10and m ¼3Bohr magne-tons,the DM-driven EB is 100Oe which is of the same order of magnitude as the experimentally measured EBs in perovskite heterostructures [7].
For perovskite heterostructures involving multiferroics,both ~h
D and ~h J should be considered.The estimated j ~h J j vs the F
E oxygen displacement (X FE ,which is proportional to
the in-plane projection of ~P
)and parametric with X O are shown in Fig.2(b).When both X O and X FE are small,j ~h
J j behaves approximately as JX O X FE ( %3:7–4:0).Thus,j ~h
J j =j ~h D j is estimated to be JX FE = ,which may be larger than 1in BFO.
It is also important to analyze if the DM-and FE-driven EB mechanisms are robust against roughness,which often is appreciable at interfaces,although recent experimental progress in thin-films substantially reduces this extrinsic effect.Since there are veral uncertain factors controlling the interfacial roughness,it is difficult to reach robust conclusions from the theoretical perspective.For this rea-son a simplified analysis will be given here,by assuming that the FM and AFM cations can be mixed near the interface but they not diffu into inner regions,as shown in Fig.3.If the G-AFM spin order is stable enough
and
FIG.2(color online).(a)FE-polarization-driven asymmetric bond angles and modulated normal SE at the interface.A switch
of the FE polarization will also switch ~h
J .(b)The estimated j ~h J j as a function of X FE for different values of X O (from 0to 0.03A
˚).The lattice constant is assumed to be 4A
˚and J i;j is in propor-tional to cos 4ð i;j Þwhere is the bond angle [13].All displace-ments (in units of A
薄公英˚)are assumed to be coplanar for simplicity.
there are no crystal defects at the interface,the ~D
vectors across the interface alway change simultaneously with the corresponding spins in the G-AFM side following the roughness.In other words,the roughness geometry would
not change the combination of ~D
ij and G-AFM spin vec-tors at the interface.Even though the AFM spins can be canted at the roughened regions,this may decrea but will
not cancel the global ~h
D ,as long as there are no parated 180 magnetic domains or ferroelastic walls.Similarly,it can be shown that the FE-driven mechanism will not be canceled by roughness either.Therefore,both the DM-and FE-driven mechanisms for EB should in principle work,even in the prence of weak interface roughness.
Note also that the DM-and FE-driven EB are anisotropic (related to the crystal direction).Ideally,if the measuring
field is applied perpendicular to ~h
D (if no multiferroics are involved),there would be no EB.A possible example is the ca of LaMnO 3=SMO superlattices,in which no EB has been obrved using an in-plane measuring field since
all spins are almost in-plane collinear (thus ~h D is out-of-plane)[17].选做题是什么意思
Finally,we remark that we have tested our argumenta-tions using numerical techniques on a heterostructure [15],and a robust EB in the hysteresis loop was obtained by considering our two mechanisms.In the simulation,spin-canting effects (that can originate from exchange couplings at the interface or magnetic field reorientation)are in-cluded,but they are not found to affect our results qualitatively.
Conclusions.—Here it was propod that both the Dzyaloshinskii-Moriya interaction and the standard super-exchange (the latter active only when multiferroic materi-als that can be controlled by electric fields are involved)could induce the exchange bias phenomenon at FM/G-AFM perovskite oxides interfaces,even when the antifer-romagnetic spins are compensated.The common precon-dition for the existence of the two mechanisms is the prence of oxygen octahedral tiltings at the interface.Our model highlights the interactions between magnetism and lattice distortions,and propos mechanisms to understand the exchange bias in FM/G-AFM oxides heterostructures.
We thank R.Ramesh,P.Yu,L.W.Martin,and M.Huijben for providing experimental results before publica-tion and fruitful discussions.We also thank C.Ederer,J.-S.
Zhou,O.Chmaism,S.Okamoto,and J.Nogue
´s for helpful discussions.Work supported by the NSF (DMR-0706020)and the Division of Materials Science and Engineering,U.S.DOE,under contract with UT-Battelle,LLC.K.Y .and S.P.were supported by the European Rearch Council under the EU 7th Framework Programme (FP7/2007-2013)/ERC grant agreement No.203523.S.Y .was supported by CREST-JST.S.D.and J.M.L.were supported by the 973Projects of China (2006CB921802,2009CB623303)NSFC (50832002).
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[14]Becau of anisotropies,the AFM spins are assumed to be
almost unaffected during the measuring process.The possible tiny canting following the measuring field will not affect our mechanisms.Also,due to the energy bar-riers between the lattice distortion directions,~D
i;j can also be assumed to be frozen at low T .
[15]See EPAPS Document No.E-PRLTAO-103-014939for
supplementary material.For more information on EPAPS,e /pubrvs/epaps.html.
[16]The bond angles are different along different axes.Here
only the bond along the [001]axis contributes to the EB.[17] A.Bhattacharya et al.,Phys.Rev.Lett.100,257203
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FIG.3(color online).Sketch of atomic-scale interface rough-ness.Only the ideal G-AFM spin order is shown by arrows.The
alternation of the ~D
ij vectors across the interface are shown as in-and out-arrows.In addition,in the roughened ca,the D vectors of the (100)and (010)bonds (open/full squares)will also be active for the EB.
