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Manganites at Low Temperatures and Light DopingBand Approach and Percolation

a r X i v :c o n d -m a t /9804328v 1 [c o n d -m a t .s t r -e l ] 30 A p r 1998

Manganites at Low Temperatures and Light Doping:Band Approach and Percolation

Lev P.Gor’kov 1and Vladimir Z.Kresin 2

1

National High Magnetic Field Laboratory,Florida State University,Tallahassee,FL 32310,U.S.A.and https://www.doczj.com/doc/1a8430493.html,ndau

Institute for Theoretical Physics,Russia

2

Lawrence Berkeley Laboratory,University of California,Berkeley,CA 94720

(Submitted )

A tight-band model is employed for the e 2g -orbitals in manganites.It is shown that a large intraatomic Hund’s cou-pling,J H and the resulting double-exchange mechanism lead to antiferromagnetic ordering along one of the cubic axis sta-bilized by the cooperative JT e?ect which further decreases the band energy of electrons.As a result,LaMnO 3is a band insulator built of 2D ferromagnetic layers.The critical con-centration x c ?0.16,for onset of ferromagnetic and metal-lic behavior at low temperatures in La 1?x Sr x MnO 3and the phase transition are treated in the percolation approach.PACS numbers:75.10Lp;75.30.-m;75.50-y

Current e?orts in studies of the “colossal magnetore-sistance”(CMR)in the manganites,R 1?x B x MnO 3(usu-ally,R=La,B=Ca,Sr;see e.g.[1-3]for a review and references)are,naturally,focused on phenomena occur-ring in the vicinity of the metal-insulator transition tem-perature,T*,which for x =0.2?0.3,reaches the room temperature range.There are no doubts that both the Jahn-Teller (JT)distortions and Zener’s double exchange (DE)mechanism [4,5]are two key ingredients of CMR.The transition between the paramagnetic and conduct-ing ferromagnetic phases is treated in [6]as a localization-delocalization “crossover”driven by thermal ?uctuations of the local JT-modes.In this paper we focus on the analysis of phenomena occurring at low T.In this region the approach should be di?erent.Indeed,the two end members of the series,LaMnO 3and CaMnO 3,are both antiferromagnetic (AF)insulators.“Doping”of LaMnO 3by divalent atoms results in metallic conductivity only above some critical concentration,x c ?0.16.

Low temperature phases of manganites are habitually interpreted in terms of localized orbitals [7,8],more often that not in the modi?ed Hubbard model (see.e.g.[9]).Although it is widely accepted that electron-electron cor-relations play the key role in low temperature behavior of manganites,we show that their major properties can be understood within the framework of the band approach,as well,or in percolation terms.

The following facts have to be explained:1)the in-sulating state of the parent LaMnO 3;2)the A-type of antiferromagnetic ordering (alternating ferromagnetic planes);3)the small value of the N′e el temperature,T N ~150K in the low x-range,while the structural changes occur at about 900K;4)the localization of “holes”in-troduced into the Mn 3+-subsystem at x

conductivity sets in at low T .

In the band model the properties from 1)to 4)are brought about by the DE-and JT-mechanisms.We in-terpret the value x c ?0.16in terms of percolation theory:both the cluster approach and the phase separation pic-ture,as suggested ?rst for the cuprates [10],seem to give the same criterion [11].

Let us start from the A -type of the ground state of LaMnO 3.According to our picture,appearance of such a magnetic state is not caused by any exchange interaction between localized spins on di?erent sites.The mutual arrangement of the distorted octahedra and of the core 3

t 2g -spins,S i (S=3/2),is stabilized only by gains in the kinetic band energy of the “conduction”e 2g -electrons.The single electron Hamiltonian is then of the form [6]:

?H = i,δ

?t i,i +δ?J H S i ·?σ+g ?τi ·Q i +J el Q 2i

(1)Here ?t

i,i +δis the nearest neighbor tunneling matrix,de-?ned on some basis of the two-dimensional cubic repre-sentation,e 2g ;Q i are the active local JT-modes with the matrix elements on that electronic basis in terms of a

“pseudospin”matrix,?τ

(see e.g.,in [7];Q i are de?ned as dimensionless parameters).The static JT deforma-tions and the staggered magnetization, S i =(?1)i S of the A -type phase are treated below in the mean ?eld approximation.

To avoid cumbersome expressions,we ?rst discuss the competition between ferro-and antiferromagnetic order for the one band model:

ε(p )=t cos(ap z )+ˉt

(p ⊥)(2)

Assume ?rst that the local (3t 2g )spins are ordered fer-romagnetically.At large J H ?t the low energy band is

shifted down,by ?J H S .The main contribution to the total energy is E H =?J H S n (n is the number of elec-trons per unit cell),while the kinetic energy contribution

due to the polarized electrons is linear in t,ˉt

.Consider the same problem for AF-ordering along the z -axis (with the period 2a ).It is convenient to discuss a more general case:

S i =(± S z ,M x );(S 2z +M 2

x =S 2)

(3)

i.e.the canted AF-structure [8]for the 3t 2g -spins.Solv-ing the periodic electron spectrum problem in a manner

similar to the solution for the two local sites [5],one ob-tains the following four branches:

1

εl(p)?ˉt(p⊥)=

±

2;ψ2=(?1?i?2)/√

2Q0 0exp(iθ)

exp(?iθ)0 (8)

In the standard notations[7]:Q2=Q0sinθ;Q3= Q0cosθ.Here Q0is the magnitude of the JT distor-tions,while the“angles”,θ=0,2π/3,?2π/3correspond to elongations of the octahedron along the z,x,and y-axes,respectively.

With the above in mind,consider the options for pos-sible ground states in manganites.In the case of stoichio-metric LaMnO3there is exactly one e2g-electron(Mn3+) per unit cell.The ferromagnetic?lling-up for two bands will produce a metallic spectrum.AF-ordering along one axis results in the same main contribution,?J H S(at large J H),while the corresponding kinetic energy contri-bution becomes of the order of t2/J H?t.Second order terms in t in Eq.(4)imply that the spectrum is much less dispersive along the z-axis(M x≡0).Planes with the antiparallel core spins become almost isolated.

AF-ordering along any other direction would lack elec-tronic transport along this direction,again shrinking bands.Thus,it is enough to have the AF order along one axis to access the major energy gain,?J H S.How-ever,to stabilize the AF-order along one axis,one needs to further reduce the total energy by the antiferroelastic JT-distortions along the remaining directions.

Note again that after the AF-ordering sets in,say, along the z-axis,the communication between adjacent layers in real space becomes very weak.Therefore,at large J H it is possible to reduce the problem,to main approximation,to the one of two-dimensional(2D)elec-trons.For decoupled layers,electrons inside the layer are ferromagnetically polarized along the3t2g-(core)spins. Again,in this2D-problem there is one electron per(2D) cell.The metallic spectrum simpli?es to the form:ε2D1,2(p)=(A+B)(cos ap x+cos ap y)±

(A?B) (cos ap x+cos ap y)2+

3(cos ap x?cos ap y)2 1/2(9) where

i,k=(x,y,z);A∝

?2(z;x,y)?2(z+a;x,y)are the two overlap integrals be-tween the two sites separated by the lattice constant,a. Introduce now JT-distortions in the layer with the struc-ture vector(π,π)in a plane.If the JT-splitting is strong enough,then instead of(9),the new spectrum will consist of two pairs of bands separated by energy gaps over the entire(new)Brillouin zone,with the low-energy bands fully occupied.This completes the band picture for the insulating A-type ground state in LaMnO3.

For this interpretation one needs:

J H?t,gQ0;gQ0~>t(10) Eq.(10)does not impose any severe limitations on the model parameters.The above physics should be present in any realistic band https://www.doczj.com/doc/1a8430493.html,ually the band cal-culations use the experimental lattice parameters,and it is not straightforward to single out(e.g.in[3])the competition of the e?ects considered above.

Finding the single electron spectrum in presence of the JT-distortions from(8,9),writing down the total energy of the?lled-up bands and subsequent minimization of the total energy,is a straightforward but rather tedious task. In the general case,there is no small parameter.We will consider numerical aspects elsewhere.

Below we take advantage of the transparent physics of the independent2D-layer approximation.First of all, the2D-ferromagnetic ground state in each layer which was obtained in the mean?eld approximation,will be smeared out by?uctuations:ferromagnetism does not exist as a stable thermodynamical phase in2D,un-less there is a weak interlayer coupling,t2/J H,of eq.

(6).This results in a rather low N′e el temperature, T N?150K,in LaMnO3,at t~0.1eV,J H~1eV.

To illustrate the above,consider the“symmetric”model[9],A=B=t/2in(9):

ε(p)=t(cos ap x+cos ap y)≡?t(p)(11) Eq.(11)gives the two degenerate bands.For one polar-ized2D-electron per unit cell,the two Fermi surfaces run along the lines p x±p y=(±π/a).In the presence of the JT-distortion(8)the new spectrum(in the reduced Bril-louin zone,with the structural vector(π/a,π/a))consists of the four branches:εl(p)=±||t(p)|±?|,where ?=|gQ0/2|.At?>2t there are two pairs of energy bands,and the insulating gap sets in at p x=p y=0, equal to??2t.The distortion gQ0=g2/J el is de?ned from the on-site problem.

Turning back to the parameters(J H,J el,g,A,B)of our general model,connections between gQ0,J el and A and B are established by minimization of the total(elec-tronic and elastic)energies with respect to Q and M x (see eq.(5)).The deformations of octahedra(Q0)are known from the structural data(see e.g.,in[3]).One may also expect that B?A:tunneling between Mn-sites takes places through the shared oxygen ions,and the notion that the d-shell size,d,is small compared to the lattice constant,a(d?a),is quite helpful,as oxides of transition metals are concerned[7,13,14].Currently the spin wave spectrum in LaMnO3is available[15].Al-though we postpone the study of the spin wave spectrum for the future,it is worth mentioning that coexistence of the localized3t2g-spins and itinerant e2g-electronic states (which in the AF-environment are not characterized by the spin-projection quantum number),may result in de-viations from an anisotropic Heisenberg model. Finally,we brie?y discuss the concentration depen-dence.At small x the two mechanisms lead to local-ization of a doped hole.The?rst one is directly related to the2D physics.According to(1),interaction of a hole with the JT-modes may decrease the hole’s energy by an amount of order of g2/J el.If its energy is now below the band’s bottom,the hole becomes self-trapped.Two-dimensionality plays the crucial role in that there are no energy barriers for the process[12].If a dimensionless parameter C~g2/J el W(W is a band-width)exceeds a critical value of order of unity,no itinerant states are possible in the2D system.The second mechanism is the Coulomb binding of holes near dopants[8].

At larger x the band approach becomes useless.In-stead,we adopt the point of view of the percolation the-ory.Let us start?rst with the cluster description[11]. In the so-called site problem,consider formation of an in?nite cluster of the neighboring divalent ions.For the simple cubic lattice the critical x cr(s)=0.31.This value is not universal.

It is well known[11]that correlations between sites rapidly reduce x c(s)to the value x c?0.16,which is also a threshold concentration for a continuous percolation problem.Although holes are located near divalent ions (forming the cluster’s skeleton),the hole’s hopping takes place along the adjacent Mn-sites correlated by the fact that the wave function of a single hole is spread over a few interatomic distances tending to suppress JT-distortions and align spins ferromagnetically in its vicinity.The clus-ter’s structure is complicated,however,at x>x c one may expect that locally its properties are close to the ones of the homogeneous metallic ferromagnetic state. The rest of the sample does not conduct.

Another plausible view is that the material may sep-arate into coexisting insulator and metallic phases[10]. In the cluster approach the Coulomb energy is playing the crucial role con?ning holes in the skeleton’s vicinity. In the continuous description the Coulomb energy would limit sizes of the phases’domains.In both approaches the Coulomb energy may be strongly reduced by su?ciently large dielectric constant of the insulating phase and due to metallic screening in the ferromagnetic phase.

With the temperature increase,concentration of the metallic phase decreases:temperature-induced JT-distortions[6]provide stronger localization of“doped”holes.From such a point of view the metal-insulator transition temperature,T?,may also be treated as a per-colation point(at?xed x),and this agrees well with the extreme sensitivity of CMR to applied magnetic?eld. To summarize,the view of LaMnO3as a band insula-tor,is consistent with the main experimental https://www.doczj.com/doc/1a8430493.html,rge intraatomic Hunds’exchange,J H,forces the cubic sys-tem to order antiferromagnetically along one direction because the band energy of electrons can then be e?ec-tively reduced by the JT-instability resulting in the new (tetragonal)lattice(

√2a×2a).At J

H?t,gQ0, the e2g-electrons form almost disconnected2D-layers. This manifests itself in the low N′e el temperature.The percolation analysis can explain the value of the critical concentration,x c?0.16and CMR itself suggesting new interesting physics.

The work was supported(L.P.G.)by the National High Magnetic Field Laboratory through NSF Cooperative Agreement#DMR-9016241and the State of Florida, and(V.Z.K.)by the U.S.O?ce of Naval research under contract#N00014-97-F0006.

[1]H.Kawano et al.,Phys.Rev.B53

,1146 (1996).

[4]G.Zener,Phys.Rev.82

, 675(1955).

[6]https://www.doczj.com/doc/1a8430493.html,lis,B.Shraiman,and R.Mueller,Phys.Rev.

Lett.77

,145(1960).

[8]P.G.de Gennes,Phys.Rev.118

[9]For review,see K.Kugel and D.Khomskii,Sov.Phys.

-Usp.25

,420 (1987).

[11]H.Scher and R.Zallen,J.Chem.Phys.53

, 1446(1983).

[13]J.Coodenough,in Progress in Solid State Chemistry,

v.5,p.145,edited by H.Reiss(Pergamon,

NY,1971).

[14]P.W.Anderson,in Magnetism,v.1,p.75,edited

by G.Rado and H.Suhl(Academic,NY,1963). [15]K.Hirota,N.Kaneko,A.Nishizwa,and Y.Endoh,

J.Phys.Soc.Jpn.65,3763(1996).

Superkids2新灵通少儿英语二级-故事听力稿

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