Electronic and Magnetic Reconstructions in Manganite Superlattices
aa r X i v : . [ c ond - m a t . s t r- e l ] F e b Electronic and Magnetic Reconstructions in Manganite Superlattices
Kalpataru Pradhan and Arno P. Kampf
Center for Electronic Correlations and Magnetism, Theoretical Physics III,Institute of Physics, University of Augsburg, D-86135 Augsburg, Germany (Dated: February 19, 2018)We investigate the electronic reconstruction at the interface between ferromagnetic metallic (FM)and antiferromagnetic insulating (AFI) manganites in superlattices using a two-orbital double-exchange model including superexchange interactions, Jahn-Teller lattice distortions, and long rangeCoulomb interactions. The magnetic and the transport properties critically depend on the thicknessof the AFI layers. We focus on superlattices where the constituent parent manganites have the sameelectron density n = 0 .
6. The induced ferromagnetic moment in the AFI layers decreases mono-tonically with increasing layer width, and the electron-density profile and the magnetic structurein the center of the AFI layer gradually return to the bulk limit. The width of the AFI layers andthe charge-transfer profile at the interfaces control the magnitude of the magnetoresistance and themetal-insulator transition of the FM/AFI superlattices.
I. INTRODUCTION
Correlated electron materials often involve the com-petition between various ordering tendencies of charge,orbital, spin, and lattice degrees of freedom. If fur-ther supplemented by weak disorder cluster coexistence,percolative transport, and colossal response phenomenaemerge. It therefore remains a continuing challengeto understand the functional properties of transition-metal oxides (TMOs) . The physics at the surface ofTMOs is further enriched by atomic and electronic re-constructions and complicated by the lack of inversionsymmetry . When surfaces of two TMOs were joinedtogether to form an interface , new phenomena were dis-covered in the last decade. In some cases the phasesat the interface are not even realized in the bulk of ei-ther of the TMOs which are joined together . Forinstance the discovery of the two-dimensional electronliquid which forms at the interface between the insula-tors LaAlO and SrTiO started a new subfield in theresearch on oxide interfaces. Subsequent experimentsrevealed superconductivity , ferromagnetism , and eventheir coexistence . These unexpected phases at interfacespose fundamental physics questions and simultaneouslybear promises of technological importance for the designof novel materials .The perovskite manganese oxides are a particularlyremarkable example for the mutual coupling of elec-tronic and lattice degrees of freedom . The man-ganites RE − x AE x MnO , where RE and AE denote rareand alkaline earth elements, respectively, are known fortheir colossal magnetoresistance . The various phasesin manganites with charge, orbital, and magnetic orderhave been elaborated for different combination of RE andAE elements and doping regimes x . The recentdevelopment in superlattices has created yet another toolto explore electronic and magnetic phases of manganites.La − x Sr x MnO , a solid solution of LaMnO (LMO)and SrMnO (SMO) exhibits a number of phases de-pending upon the doping concentration x . In additionto A-type at x = 0 and G-type antiferromagnetism at x = 1 also ferromagnetic and C-type antiferromag-netic (AF) phases exist at low temperature in an inter-mediate doping regime. For example La . Sr . MnO is a ferromagnetic metal and La . Sr . MnO is anAF insulator . In manganite superlattices 2m lay-ers of LMO are deposited on m layers of SMO or vice versa . For m = 1, LMO m /SMO m andLMO m /SMO m are the superlattice counterpart of theLa . Sr . MnO and La . Sr . MnO manganites, re-spectively.For LMO m /SMO m superlattices the uniform FM be-havior of the solid solution is recovered for m ≤ m ≥ . Similarly, LMO m /SMO m shows a C-type AF phase for m = 1 and m = 2 that existin the solid solution counterpart La . Sr . MnO . TheN´eel temperature 250 K of the solid solutions increases to320 K for m = 1. For m ≤
2, charge transfer from LMOto SMO may retain properties similar to the solid solu-tions. In these two scenarios, the charge transfer is likelyconfined to 1-3 unit cells. For m >
2, the increasing dis-tance between the interfaces limits the charge transfer tothe near vicinity of the interface. Due to this charge con-finement LMO and SMO regain their respective parentphases away from the interface.Considerable progress has been achieved in describingthe modulated magnetic structure and the MIT inthe LMO/SMO superlattices. The standard two-orbitaldouble exchange model for manganites was implementedto calculate the average electron density for each layerof the LMO/SMO superlattices; away from the interfaceLMO and SMO layers regain their initial electron den-sities. The electron-density profile across the interfaceis determined by electrostatics with an electron transferfrom the LMO to the SMO side. The magnetic-profilefor each layer follows the bulk phase diagram at the cor-responding local density along with a weak proximityeffect .In the initial studies of manganite superlattices the primary intent was to improve the tunnelingmagnetoresistance in a FM/AFI superlattice wherethe constituent parent FM and AFI manganites have thesame electron density, n = 1 − x . FM/AFI superlatticeswith different combinations of manganites (FM = LSMO,LCMO; AFI = PCMO, GdCMO; n = 0 . havebeen studied in recent years. The observed induced fer-romagnetic moment in the AFI layers depends on theirwidth and the induced moment is tunable by an externalmagnetic field . Remarkably the required externalfield is much smaller than the magnetic field requiredto induce ferromagnetism in the parent AFI compound.This results in a large magnetoresistance in the FM/AFIsuperlattices which are therefore candidate materials forcolossal magnetoresistance at room temperature. Man-ganite trilayers can also be used to design an efficientspin valve .In this paper, we have studied in detail the electronicand magnetic reconstructions in FM/AFI superlatticesat electron density n = 0 . .The paper is organized as follows: In Sec. II, we intro-duce the two-orbital model for bulk perovskite mangan-ites. For superlattices we specify the essential modifica-tion by adding long-range Coulomb (LRC) interactionsand briefly present the applied Monte Carlo technique.The parameter space of the FM/AFI manganite super-lattices is discussed in Sec. III. Electronic and magneticreconstructions at manganite interfaces are emphasizedin Sec. IV while Sec. V is devoted to the MIT. Resultswith and without LRC interactions are compared in Sec.VI. In Sec. VII, various combinations of electron-phononcouplings for FM and AFI manganites are considered.The role of disorder at the interface is discussed in Sec.VIII, and conclusions are presented in Sec. IX. II. REFERENCE MODEL FOR MANGANITES
We construct a two-dimensional model Hamiltonianfor manganite superlattices composed of FM regions sep-arated by AFI regions as H = H F M + H AF I + H lrc , (1) where both H F M and H AF I have the same referenceHamiltonian H ref = αβ X h ij i σ t ijαβ c † iασ c jβσ − J H X i S i · σ i + J X h ij i S i · S j − λ X i Q i · τ i + K X i Q i − µ X iασ c † iασ c iασ . (2)Here, c and c † are annihilation and creation operatorsfor itinerant e g electrons and α , β refer to the two Mn- e g orbitals d x − y and d z − r labelled as a and b , re-spectively. The kinetic energy part involves the nearest-neighbor hopping of e g electrons with amplitude t ijαβ ( t xaa = t yaa ≡ t , t xbb = t ybb ≡ t/ t xab = t xba ≡ − t/ √ t yab = t yba ≡ t/ √ , where x and y denote the spatial direc-tions on a square lattice. J H is the Hund ′ s rule couplingbetween t g spins S i and the e g electron spin σ i , and J isthe AF superexchange between the t g spins. λ measuresthe strength of the electron-phonon coupling betweenthe Jahn-Teller (JT) distortions, Q i , Q i and the or-bital pseudo spin operators τ i = P σ ( c † iaσ c ibσ + c † ibσ c iaσ ), τ i = P σ ( c † iaσ c iaσ − c † ibσ c ibσ ). Here Q i and Q i are thedistortions corresponding to the normal vibration modesof the oxygen octahedron that remove the degeneracy ofthe e g levels. The stiffness of the JT modes is denotedby K . The stiffness of the breathing mode distortion( Q i ), which couples to the local electron density is con-siderably larger than K and therefore, in the adiabaticapproximation the coupling to Q i is neglected in theHamiltonian Eq. 2.We treat all t g spins and lattice degrees of freedom asclassical , and measure energies in units of the Mn-Mn hopping t aa = t . In manganites t is approximately0.2-0.5 eV . The estimated value of the Hund ′ s cou-pling is 2 eV , i.e. much larger than t . For this rea-son we further adopt the limit J H → ∞ , for which the e g electron spin perfectly aligns along the local t g spindirection. The infinite Hund ′ s rule coupling then natu-rally leads to redefine the spinless e g electron operator as c iα = cos( θ i / c iα ↑ + sin( θ i / e − iφ i c iα ↓ , where the polarangle θ i and the azimuthal angle φ i determine the t g spin orientation. In terms of the redefined e g electronoperators, the kinetic energy takes the simpler form H kin = − P
2) cos( θ j /
2) + sin( θ i /
2) sin( θ j / e − i ( φ i − φ j ) . Thefactor controls the magnitude of the hopping amplitudesdue to the different orientation of t g spins at sites i and j . We set K = 1 without loss of generality, because if thelattice variable Q is replaced by √ KQ , λ can be simplyrenormalized as λ/ √ K . The length of the t g spins isset to | S i | = 1. In an external magnetic field a Zeemancoupling H mag = − h · P i S i is added to the Hamiltonian.The reference ‘manganite model’ H ref in 2D is con-structed to reproduce the correct sequence of magneticphases in the bulk limit . The different phases thatare captured at low temperature are an orbitally orderedinsulator at x = 0 , a charge and orbital ordered ferro-magnetic insulator at x = 0.25 , a ferromagnetic metalwindow around x = 0.67, the CE charge and orbital or-dered insulator around x = 0.50, a magnetic two dimen-sional A-type AF phase, and a G-type AF phase at x =1.00.The average electron density of the constituent FMand AFI manganites in the FM/AFI superlattices is fixedby choosing the same chemical potential µ throughoutthe superlattice. The LRC interaction between all thecharges, essential to control the amount of charge trans-ferred across the interface is taken into account via a selfconsistent solution of the Coulomb potentials φ i at themean-field level by setting φ i = αt X j = i h n j i − Z j | R i − R j | (3)in the long-range Coulomb part of the Hamiltonian, H lrc = X i φ i n i . (4)It is assumed that all the point charges Z j from the back-ground ions are fixed and confined to the Mn sites. h n j i refers to the e g electron density at the Mn site R j . Thelong-range interactions between the background pointcharges and the e g electrons are thereby combined withthe charge-neutrality condition in the superlattice. Al-ternatively the Coulomb potentials φ i can be determinedself consistently by solving the Poisson equation whichis equivalent to solving the Eq. 3.The Coulomb interaction strength is controlled by theparameter α = e / ǫat where ǫ and a are the dielectricconstant and the lattice parameter, respectively. Thismean-field level set up is the minimal basis to study thecharge transfer across the interface ; it is also com-monly used in the context of semiconductor interfaces .The background dielectric constant ǫ is order of 20 inmanganites . It includes the lattice and the atomicpolarizability contributions only. The dipolar contribu-tion from the charge carrier motion and the associatedscreening is neglected to treat the absolute permittivitiesof FM and AFI manganites on equal footing . For alattice constant a = 4˚ A , t = 0 . ǫ = 20, theapproximate value of α is 0.2. The screening length intwo dimensions (2D) is larger than in three dimensions(3D) . This is taken into account by choosing a largerdielectric constant in the 2D model ansatz. Specificallywe use α = 0 . α values em-ployed earlier in 3D model calculations were in the range0.1 - 1 while a much smaller value of α was usedin 1D . FIG. 1: Color online: 1st row: The z components of the t g spins; 2nd row: the electron-density for each site on a24 ×
24 lattice at T = 0.01. (a) The parent AF insulator, (b)the superlattice with the width w = 11 of the AFI layer, (c)the superlattice with w = 11 in the presence of an externalmagnetic field, h = 0.004.
We apply an exact diagonalization scheme to the itin-erant electron system for each configuration of the back-ground classical variables of the t g spins and the latticedistortions. The classical variables are annealed by start-ing from a random configuration. At each temperaturethe annealing requires at least 2000 system sweeps, byvisiting every site of the lattice sequentially and updatingthe system by a metropolis algorithm. For each systemsweep the computation time scales as O( N ) where N isthe number of lattice sites. With thousands of anneal-ing sweeps at various temperatures this procedure is timeconsuming.We instead resort to a Monte Carlo sampling techniquebased on the ‘traveling cluster approximation’ (TCA) .The TCA uses a moving cluster of size N c constructedaround the site of the Monte Carlo update. The compu-tational cost thereby decreases to O( N N c ) and allows totreat system sizes up to ∼ ×
40, with an 8 × ×
24 were extensively benchmarked andsuccessfully applied in several earlier studies . Ateach system sweep we additionally solve for the Coulombpotentials φ i in H lrc self consistently until the electrondensity at each site is converged. The maximum relativeerror for the convergence of the electron density is set to0.01 at low temperatures while it is relaxed up to 0.03 athigher temperatures. For each fixed set of parameters,the calculations start from ten different initial realiza-tions of the classical variables. All the physical quanti-ties are averaged over the results for these ten startingconfigurations. Q P ( Q ) AFIFM T ρ AFIFM (a) (b)
FIG. 2: Color online: Bulk system at n = 0.6 (a) Distribu-tion function for the lattice distortions P(Q) of the bulk AFI( λ I = 1 .
75) and bulk FM ( λ M = 1 .
50) at T = 0.01. (b) Theresistivity variation with temperature for AFI and FM bulksystems.
III. SUPERLATTICES OF FERROMAGNETICMETALS AND ANTIFERROMAGNETICINSULATORS
Here we analyze specifically superlattices composed ofFM and AFI manganites. Even with the above describedsimplifications in the model Hamiltonian we have to con-tend with manganite states at different doping x , AF su-perexchange strengths J , and different electron-phononcouplings λ . Also the proper choice of relevant parame-ters for the combination of ferromagnetic metals and AFinsulators is not obvious. In addition, due to the slightstructural and chemical mismatch between the parentFM and AFI manganites the parameters at the interfacemay be altered with respect to the bulk values.To constrain the parameter space we select the FM andAFI manganites of equal electron density n = 1 − x . Wechoose n = 0.6 above half filling to address the existingexperiments on manganite superlattices near x = 1 / J = 0 . forboth the FM and the AFI manganites. With the choiceof the electron density and the superexchange couplingswe are left with the crucial parameter λ to differentiatebetween a FM and an AFI phase.Alternatively λ can be fixed and the differentiation be-tween the ferromagnetic metal and the AF insulator isachieved by varying J . This requires a larger value of J in the AF insulator, but it leaves the ambiguity forchoosing modified superexchange couplings at the inter-face. Yet another possibility is to keep each parameter J and λ fixed and to use different hopping amplitudeson the two sides of the interface. Again we are left withthe possible modifications for the hopping amplitude atand across the interface. Since the λ and J values aremeasured in units of t , a smaller t value is equivalent tothe combinations of larger λ and J values or vice versa.Smaller and larger λ values are thereby closely relatedto the larger and smaller bandwidth manganites, respec-tively. -3 -2 -1 0 1 2 3 ω N ( ω ) AFIFM (a) (b) E F FM VBAFI CB E F E F (AFI)=0E F (FM)=0.1 FIG. 3: Color online: Bulk system at n = 0 . λ I = 1 .
75) and FM ( λ M = 1 .
50) at T = 0.01. (b)Schematic energy diagram.
Here we consider two λ values to differentiate betweenthe FM and the AFI manganites. For the parameters J = 0 . n = 0 .
6, the groundstate is FMfor λ ≡ λ M = 1 .
50 while it is an AFI for λ ≡ λ I = 1 . n = 0 . n =0 . . The AFI phase at n = 0 . P ( Q ) = h N P i δ ( Q − | Q i | ) i whereN is the total number of sites, to compare them with theelectron-density variations. The two peaks in Fig.2(a) forthe AFI at T = 0.01 indicate that the distortions havea bimodal distribution. The sites with a large distortionattract more electrons with a density increase to n i ∼ σ ( ω ) = AN X α,β ( n α − n β ) | f αβ | ǫ β − ǫ α δ ( ω − ( ǫ β − ǫ α )) , (5)with A = πe / ¯ ha and n α = θ ( µ − ǫ α ), a is the latticespacing. f αβ denotes the matrix elements of the para-magnetic current operator j x = it P i,σ ( c † i + x,σ c i,σ − h.c. )between eigenstates | ψ α i , | ψ β i , and ǫ α , ǫ β are the corre-sponding eigenenergies. We extract the d.c. conductivityby calculating the ‘average’ conductivity for a small lowfrequency interval ∆ ω defined as σ av (∆ ω ) = 1∆ ω Z ∆ ω σ ( ω )d ω. (6)∆ ω is chosen two to three times larger than the meanfinite-size gap of the system as determined by the ra-tio of the bandwidth and the total number of eigenval-ues. This procedure has been benchmarked in a previous λ Μ = . λ Ι = . λ Μ = . w24 xy FIG. 4: Schematic view of the FM/AFI superlattice on a24 ×
24 lattice. We consider periodic boundary conditions inboth directions. work . Fig.2(b) shows the temperature dependence ofthe resistivity ρ = σ − av in units of ¯ ha/πe . The insulat-ing behavior in the AFI at low temperatures is due tocharge and orbital order, which opens an energy gap inthe spectrum.The density of states (DOS) N ( ω ) = h N P α δ ( ω − ǫ α ) i is shown in Fig.3(a). The center of the gap in the DOSof the AFI state is chosen as the energy zero. With thischoice the Fermi energy of the FM state at the samedensity is at ǫ F = 0 .
1. The DOS is finite for the FMstate at its Fermi level. Fig.3(b) translates the resultsfor the DOS into a schematic energy diagram. When theFM and the AFI manganites are joined together bandbending near the interface is expected to shift electronsfrom the FM to the AFI side. An FM/AFI superlatticeis shown schematically in Fig.4. The width of the middleAFI layer, sandwiched between the FM layers is denotedby w. Periodic boundary conditions are enforced in bothdirections and thereby represent a superlattice structurecomposed of alternating FM and AFI regions.
IV. CHARGE TRANSFER AND MAGNETICRECONSTRUCTION AT THE INTERFACE
As we show below, if the width of the AF regions issmall, they may become ferromagnetic and metallic inthe superlattice structure. The induced magnetizationis restricted to the near vicinity of the interfaces. Thedistance from the interface at which the parent AF insu-lating character is recovered depends upon the parameter α and the electron-phonon couplings λ M and λ I .In this section we start with the specific choice of λ M = 1 . λ I = 1 .
75, and α = 0 .
1, and discuss re-sults for the decrease of the magnetization with increas-ing width of the AFI layers. This decrease is quantifiedby the average z component of the t g spins h S z i I in theAFI layer, where the angular bracket denotes the ther- < C O > I w S I ( ) w < S z > I (a) (b) FIG. 5: Color online: The average z component of t g spins, h S z i I and the average staggered charge order h CO i I (see text)in the AFI layer at T = 0.01. (b) Ferromagnetic structurefactor S I ( ) for different widths of the AFI layer at T = 0.01.The average z component is also re-plotted as the dotted line. mal average combined with an additional average overten different ‘samples’. We also define a measure for thelocal staggered charge order by h CO i I = N I P i ∈ AF I h n i i e i ( π,π ) · r i for the AFI layer; i denotes one of the N I latticesites in the AFI layer with position r i . In Fig.5(a), weplot h S z i I and h CO i I for different widths w. For w ≤ h S z i I is near unity and starts to decreasefor w >
5, while h CO i I remains small for w ≤ >
5. Similarly (see Fig.5(b)) also thelong-wavelength magnetic structure factor S I ( ) startsto decrease beyond w = 5, where S I ( q ) = N I P ij ∈ AF I h S i · S j i e i q · ( r i − r j ) . So the decrease in the magnetizationbeyond a critical width w c is accompanied by emergingcharge order in the AFI layer. The critical width forfixed λ I may change for a different set of parameters λ M and α . This will be discussed in sections VI and VII.The critical width is therefore not directly related to anintrinsic length scale of the parent bulk AFI.The averaged magnetization necessarily does not re-veal any spatial variation transverse to the interface. Inorder to analyze the magnetization profile, we calculatethe h S z ( x ) i for each line of the superlattice with trans-verse coordinate x . Fig.6(a) shows the line averaged h S z ( x ) i vs. line index for w = 7. h S z ( x ) i decreases for x = 12, 13, and 14, i.e. in the center lines of the AFIlayer. So the induced ferromagnetic moment in the AFIlayer is indeed confined to just the two lines nearest tothe interface.In the bulk for λ I = 1 .
75 the AFI phase extends toelectron densities larger than n = 0 . n = 0 . . It istherefore natural to explore the connection between theinduced magnetization in the AFI layer with the electrondensities at the interface. As shown in Fig. 6(b), atthe interface the line averaged electron density h n ( x ) i ofthe FM layer decreases while h n ( x ) i for the AFI layerincreases beyond the initial electron density 0.6. In fact h n ( x ) i at the fully magnetized lines x = 10 and 16 isnearly equal to 0.75.The direction of electron transfer from the FM to theAFI regions was anticipated already in the discussion in Line Index (x) < n ( x ) > Line Index (x) < φ ( x ) > Line Index (x) < Q ( x ) > Line Index (x) < S z ( x ) > (a) (b)(c) (d)w=07 w=07w=07w=07 FIG. 6: Color online: Line averaged (a) z component of the t g spins h S z ( x ) i , (b) electron density h n ( x ) i , (c) LRC potentials h φ ( x ) i , and (d) lattice distortions h Q ( x ) i vs. line index for w= 7. Open and closed symbols are for the FM and the AFIlayers, respectively. The temperature is T = 0.01 Section III. Away from the interface the average elec-tron density gradually returns to the initial electron den-sity n = 0 .
6. In addition to the overall charge transfer, h n ( x ) i is spatially modulated perpendicular to the inter-face. These charge modulations are Friedel-like densityoscillations; their amplitude decreases with the distancefrom the interface. The spatial density variations arenearly symmetric around the central line, x = 13. Aswe have verified a perfect symmetry can be achieved byaveraging over a larger number of samples.For the incommensurate filling n = 0 .
6, the parent AFIis fragmented into small regions with and without chargeand orbital order. High-density sites (see Fig.1(a)) arealways accompanied by large lattice distortions. Simi-larly a one-to-one correspondence between the the elec-tron densities and the lattice distortions exists in the su-perlattice. The line averaged lattice distortions h Q ( x ) i (see Fig.6(c)) are infact larger in the AFI layer. h Q ( x ) i is modulated similarly to the line averaged electron den-sity shown in Fig.6(b).The amount of charge transfer from the FM to theAFI layers is controlled by the LRC interactions. Theline averaged Hartree potential h φ ( x ) i in Fig.6(d) is neg-ative for x ≤ x ≥
22. The average electron densi-ties for those lines is nearly equal to the initial electrondensity n = 0 .
6. The charge transfer is restricted to afew lines near the interface where the averaged poten-tials h φ ( x ) i are positive and thereby counteracts furtherelectron transfer. Since φ i and n i are the self-consistentsolutions of the Poisson equation, the increase or the de-crease of the averaged electronic density has a one-to-one Line Index (x) < n ( x ) > α=0.1α=0 Line Index (x) < n ( x ) > Line Index (x) < n ( x ) > Line Index (x) < n ( x ) > (a) (b)w=07w=03 w=11w=05(c) (d) FIG. 7: Color online: Line averaged electron density h n ( x ) i for (a) w = 7 with ( α = 0 .
1) and without ( α = 0) LRCinteraction, (b) w = 5, (c) w = 3, (d) w = 11. Open and closedsymbols are for the FM and the AFI layers, respectively. Thetemperature is T = 0.01 correspondence to the negative or positive curvature of h φ ( x ) i .For α = 0 .
1, see Fig.7(a), the electron densities areclearly lower (higher) in the AFI (FM) layer as comparedto α = 0. The LRC potentials decrease the electron den-sity in the center of the AFI layer, but the average elec-tron densities for individual lines at x = 12, x = 13, and x = 14 are still enhanced due to the Friedel-like den-sity oscillations. The electron density in the central line x = 13 results from the superposition of the oscillationsoriginating from x = 10 and x = 16. The construc-tive superposition is more pronounced for the AFI layerwidth w = 5 shown in Fig.7(b), for which the center lineis fully magnetized due to the enhanced electron densityand the induced magnetism originating from the neigh-boring magnetized 2+2 interfacial lines. For w = 3, inFig.7(c), the electron density of the central line is nearlyequal to the initial electron density and the charge trans-fer is truly confined to the interface. For w >
7, densityoscillations in the AFI layer disappear due the recoveryof the bulk AFI character (see Fig.7(d) for w = 9).
V. METAL-INSULATOR TRANSITION
In this section we compare the magnetic structure fac-tor S I ( ) in the AFI layer with the resistivity for the su-perlattice for current flow perpendicular to the interface.Fig.8(a) shows the temperature dependence of S I ( ) inthe AFI layer for different layer width w. For all temper-atures, the ferromagnetic structure factor and the onset T S I ( ) w=05w=07w=09w=11 T ρ T ρ T S I ( ) h=0h=0.002h=0.004 (a) (b)(c) (d)w=11 w=11 FIG. 8: Color online: Temperature dependence of (a) theferromagnetic structure factor S I ( ) and (b) the resistivity ρ for different widths of the AFI layer. The dotted lines in(a) and (b) are for the bulk FM. (c) S I ( ) and (d) resistivityvariation with temperature for w = 11 in the presence of theexternal magnetic fields h = 0.002 and 0.004. Insets in (b) and(d) show the resistivity variation in the linear temperaturescale. temperature for ferromagnetism decrease with increasingw. The z component of the t g spins h S zi i and the elec-tron densities for each site in the superlattice are shownin Fig.1 (b) for w = 11. The two lines in the AFI layernearest to the interface are aligned ferromagnetically tothe t g spins on the FM side in accordance with the dis-cussion in Section IV. The magnetic structure of the cen-ter lines is already similar to the bulk AFI phase, and thedensity profile reveals the emergence of local charge or-dering patterns in the AFI layer. The superlattice for w= 11 is therefore likely to be an insulator.Fig.8(b) displays the temperature dependence of thelongitudinal resistivity ρ of the superlattice. ρ for w =11 steeply rises towards low temperatures as expected foran insulator. For w ≤
7, the superlattice is ferromagneticand metallic at low temperatures. For w = 5, the onsettemperature for ferromagnetism in the AFI layer is near0.05, and simultaneously the resistivity start to decrease.The low-temperature resistivity increasesmonotonously with the AFI layer width. An MIToccurs beyond a threshold width similar as in theexperiments by Li et al. who measured the variationsin the resistivity in LCMO/PCMO superlattices fordifferent widths of the PCMO layer. In addition, theinset in Fig.8(b) shows a hump in the resistivity for w= 9 and w = 7 at the ferromagnetic onset temperaturesT ≃ ≤ with theexception of the experimentally observed hump in thebulk FM limit. The temperature dependence of ρ nearthe MIT in bulk FM manganites is tied to the presenceof intrinsic inhomogeneities and disorder . Here wehave not included any disorder in the Hamiltonian; theMIT in the FM/AFI superlattice has a different origin.It is due to the induced onset of ferromagnetism in theAFI layer.The natural next step is to explore the effect of anexternal magnetic field on the MIT in the superlattice.Fig.8(d) shows that the resistivity for w = 11 is loweredby the magnetic field at low temperatures. The connec-tion between the resistivity decrease and the onset of fer-romagnetism in the AFI layer is verified in Fig.8(c). TheMIT for h = 0.004 is at T ≃ ρ ( h ) − ρ (0)] /ρ (0) there-fore changes sign near the onset temperature for ferro-magnetism.At higher temperatures, the t g spins are randomlyoriented. We recall that in the limit J H → ∞ the e g electron spins are perfectly aligned along the local t g spin direction. So the current is equally carried by up-spin and down-spin electrons. In the presence of a weakexternal magnetic field, the number of e g electrons is in-creased in the up-spin channel only in the FM layers andthe tunneling current from the down-spin channel is de-creased. The up-spin channel in the AFI layer remainsunaltered and restricts the possible enhancement of thetunneling current from the up-spin channel. This resultsin an overall increase of the resistivity and a positivemagnetoresistance.At low temperatures in an external magnetic field thet g spins in the AFI layer tend to align in the same di-rection as the t g spins in the FM layers. So the tun-neling current in the down-spin channel decreases butit increases for the up-spin channel. This decreases theoverall resistivity of the superlattice. The correspondingcrossover of the magnetoresistance from negative to pos-itive was indeed reported in La . Ce . MnO /SrTiO -Nb and La . Ca . MnO /SrTi . Nb . O hetero-junctions.The z component of the t g spins and the electrondensities for each site in the superlattice are shown inFig.1(c) for w = 11 and h = 0.004. Ferromagnetic and AFstructures coexist spatially separated in the AFI layer.The ferromagnetic phase in the upper panel is directlyconnected to the charge disordered regions in the lowerpanel. A magnetic field h ≥ T S I ( ) w=03w=07w=09w=11w=13w=15 T ρ w S I ( ) α =0 α =0.1 T S ( ) (a) (b)(c) (d) α =0 AFI FM Bulk α =0 FIG. 9: Color online: For α = 0 (a) S I ( ) for different widthsof the AFI layers at T = 0.01. S I ( ) for α = 0 . S I ( ) and (c) the resistivity for different widths of the AFIlayers. Legends in (b) and (c) are the same. Inset in (c)shows the temperature dependence of the resistivity in thelinear scale. (d) The temperature dependence of ferromag-netic structure factor for the AFI layer, the FM layer for w =11, and the bulk FM. VI. RESULTS WITHOUT LRC INTERACTIONS
In order to analyze the effect of the LRC interactionswe examine the ferromagnetic structure factor S I ( ) for α = 0 in Fig.9(a). The critical width, beyond which S I ( )starts to decrease, is w c = 7. With the LRC potentials, α = 0 .
1, the critical width w c = 5, as discussed in sectionIV, is smaller due to the decrease in the average electrondensity in the AFI layer which is evident in Fig.7(a). Allthe discussions in the previous two sections remain qual-itatively valid also for α = 0, but with a larger criticalwidth w c .In Fig.9(b), we plot the temperature dependence of S I ( ) in the AFI layer for α = 0. Similar to the resultsin Fig.8(a), the onset temperature for ferromagnetismdecreases with increasing w. Fig.9(c) shows the temper-ature dependence of the longitudinal resistivity of the su-perlattice. For w = 13 and w = 15, the low-temperatureresistivity rises by orders of magnitude. The inset ofFig.9(c) zooms into the MIT at intermediate tempera-tures on a linear scale. The humps in ρ (T) decrease andshift to higher temperatures for w < c for ferromagnetism inthe FM layers and the second for the global ferromag- w S I ( ) λ Μ =1.00, λ Ι =1.75 λ Μ =1.30, λ Ι =1.75 λ Μ =1.50, λ Ι =1.75 λ Μ =1.60, λ Ι =1.75λ Μ =1.50, λ Ι =1.80 w ∆ =0.0 ∆ =0.2MDIs (a) (b) λ Μ =1.50, λ Ι =1.75 FIG. 10: Color online: S I ( ) for different combinations of λ M and λ I values at T = 0.01. (b) S I ( ) without (∆ = 0) andwith (∆ = 0 .
2) quenched disorder, and with magneticallydisordered interfaces (MDIs) (see text) for λ M = 1 .
50 and λ I = 1 .
75 at T = 0.01. The dotted line join the three pointsnamely w = 1 (for MDIs), w = 5 (for ∆ = 0 . netism at T c < T c . The rise in the resistivity ρ nearthe hump just below T c is apparently due to the on-set of ferromagnetism in the FM layers. The rise in ρ is expected for the same reason as for positive magne-toresistance discussed in section V. The downturn in ρ results from the onset of global ferromagnetism in thesuperlattices at T c . These two transition temperaturesare also observed in the LCMO/PCMO superlattices .The temperature dependence of the ferromagnetic struc-ture factor in the bulk FM state is included as the dottedline in Fig.9(d). The ferromagnetic onset temperature forthe FM layers in the FM/AFI superlattices is lower thanfor the bulk. VII. VARIATION OF λ M AND λ I In this section we compare S I ( ) for different combi-nations of electron-phonon couplings λ M and λ I . In theprevious sections λ M = 1.5 and λ I = 1.75 were chosen.If λ I is kept fixed and λ M varies between 1.0 and 1.6,the groundstate is ferromagnetic and metallic in the bulklimit at n = 0 .
6. The induced magnetization in the AFIlayer changes little for λ M ≤ . λ M = 1.6 however, S I ( ) is re-duced for w ≥
5. The difference results from the decreasein the electron density in the FM layers nearest to the in-terface as becomes evident from the comparison with thebulk phase at n = 0 . . For λ M ≤ .
5, the groundstateat n = 0 . λ M = 1 .
6. For λ M ≤ .
5, the magne-tization in the interfacial FM layers is not altered, even ifthe electron density decreases towards n = 0 .
5. But for λ M = 1 .
60, ferromagnetic order becomes unstable at theinterfacial lines. The smaller magnetization at the inter-facial FM layers decreases the induced magnetization inthe AFI layer for w ≥ λ I = 1 .
8, for which the AFI layer recovers the AF, chargeordered state at a smaller width w. It is therefore easierto induce a ferromagnetic moment in large-bandwidth(small λ ) manganites. At n = 0 .
67, the bandwidth ofPCMO is the largest among those manganites for whichan AF, charge ordered insulating phase is experimen-tally observed . In fact, in most of the experimentalFM/AFI superlattices at n = 0 .
67 the insulating man-ganite PCMO is used along with a variety of differentFM manganites.
VIII. DISORDER AT THE INTERFACE
The size mismatch of RE and AE elements in the man-ganites RE − x AE x MnO leads to tilting and distortionsof the MnO octahedra and variations in the local elec-tronic parameters. The tilting and distortions of MnO octahedra is generally known as A-type disorder. Herewe have neglected the intrinsic A-type disorder in boththe FM and the AFI manganites. Their interface is moreprone to disorder due to chemical intermixing, latticemismatch, and A-type disorder. We test the effect ofquenched binary disorder in the terminating line of theFM layers at the interface by adding the potential disor-der term by P j ǫ j n j to the Hamiltonian. The sum overj is restricted to the terminating lines of the FM layersand ǫ j is the quenched disorder potential with ǫ j = 0 andvalues ± ∆.In Fig.10(b), we show the induced magnetization inthe AFI layer for ∆ = 0 . . .
2. Themagnetization of the interfacial lines of the FM layers,where ∆ is included, is nevertheless likely to decrease forlarger ∆. Ultimately the magnetization in those interfa-cial lines will be quenched due to spin disorder. In orderto see the effect of the magnetically disordered interfaceson the AFI layer we have fixed randomly oriented spinsin the interfacial lines throughout the Monte Carlo sim-ulations. Indeed, in this extreme case, the induced mag-netization in the AFI layer is very small (see Fig.10(b)),irrespective of the AFI layer width. The magnetizationof the FM layer at the interface is therefore crucial toinduce the ferromagnetic moments in the AFI layer.In the experiments it is apparently not clear whetheror not there are magnetically disordered interfaces onthe FM sides in the FM/AFI superlattices . Thin AFIlayers are likely to be more susceptible to disorder dueto strain effects. The disorder strength ∆ at the inter-face will therefore increase with decreasing the AFI layerwidth; the width dependence of the disorder strength ishowever hard to quantify. If we combine the results inFig.10(b) such that disorder strength is maximum for smaller AFI layers width and decreases thereafter forlarger w (see the dotted line) then the magnetic mo-ment in the AFI layer behaves non-monotonically whichagrees qualitatively with the LSMO/PCMO superlatticeexperiment . But more theoretical work is needed tounderstand this non-monotonic behaviour. IX. CONCLUSIONS
We have investigated the magnetic and electronic prop-erties of manganite superlattices at the specific electrondensity n = 0 . ρ is observed. The hump in ρ is due to the presence oftwo ferromagnetic transition temperatures in the super-lattice. The height of the hump decreases and shifts tohigher temperatures with decreasing AFI layer width.In summary, the width of the AFI layers controls themagnitude of the magnetoresistance and the MIT in theFM/AFI superlattices. Our 2D model Hamiltonian cal-culations provide a basis for explaining the MIT in man-ganite superlattices. The non-monotonic behaviour ofthe induced ferromagnetic moment in the AFI layer istraced to the effects of magnetic disorder at the inter-face. ACKNOWLEDGEMENT
This work was supported by the Deutsche Forschungs-gemeinschaft through TRR80. We acknowledge helpfuldiscussions with Sanjeev Kumar.0 E. Dagotto, Science , 257 (2005). Y. Tokura, Rep. Prog. Phys. , 797 (2006). H. Y. Hwang, Y. Iwasa, M. Kawasaki, B. Keimer, N. Na-gaosa, and Y. Tokura, Nature Mater. , 103 (2012). J. Heber, Nature , 28 (2009). P. Zubko, S. Gariglio, M. Gabay, P. Ghosez, and J. M.Triscone, Ann. Rev. of Condens. Matter Phys. , 141(2011). A. Ohtomo and H. Y. Hwang, Nature , 423 (2004). N. Reyren, S. Thiel, A. D. Caviglia, L. Fitting-Kourkoutis,G. Hammerl, C. Richter, C. W. Schneider, T. Kopp, A.S. Ruetschi, D. Jaccard, M. Gabay, D. A. Muller, J. M.Triscone, and J. Mannhart, Science , 1196 (2007). A. Brinkman, M. Huijben, M. van Zalk, J. Huijben, U.Zeitler, J. C. Maan, W. G. van der Wiel, G. Rijnders, D.H. A. Blank, and H. Hilgenkamp, Nature Mater. , 493(2007). D. A. Dikin, M. Mehta, C. W. Bark, C. M. Folkman, C. B.Eom, and V. Chandrasekhar, Phys. Rev. Lett. , 056802(2011). J. Mannhart and D. G. Schlom, Science , 1607 (2010). Colossal Magnetoresistive Oxides , edited by Y. Tokura(Gordon and Breach, New York, 2000). Colossal Magnetoresistive Manganites , edited by T. Chat-terji (Springer-Verlag, Berlin, 2004). E. Dagotto,
Nanoscale Phase Separation and ColossalMagnetoresistance: The Physics of Manganites and Re-lated Compounds, Springer Series in Solid State Sciences ,Vol. 136 (2005). S. Jin, T. H. Tiefel, M. McCormack, R.A. Fastnacht, R.Ramesh, and L.H. Chen, Science , 413 (1994). E. Dagotto, New J. Phys. , 67 (2005). R. Kajimoto, H. Yoshizawa, Y. Tomioka, and Y. Tokura,Phys. Rev. B , 180402 (2002). E. Wollan and W. Koehler, Phys. Rev. , 545 (1955). J. Hemberger, A. Krimmel, T. Kurz, H. A. Krug vonNidda, V. Yu. Ivanov, A. A. Mukhin, A. M. Balbashov,and A. Loidl, Phys. Rev. B , 094410 (2002). O. Chmaissem, B. Dabrowski, S. Kolesnik, J. Mais, J. D.Jorgensen, and S. Short, Phys. Rev. B , 094431 (2003). A. Bhattacharya, S. J. May, S. G. E. te Velthuis, M. Waru-sawithana, X. Zhai, B. Jiang, J. M. Zuo, M. R. Fitzsim-mons, S. D. Bader, and J. N. Eckstein, Phys. Rev. Lett. , 257203 (2008). C. Adamo, X. Ke, P. Schiffer, A. Soukiassian, M. Waru-sawithana, L. Maritato, and D. G. Schlom, Appl. Phys.Lett. , 112508 (2008). S. J. May, P. J. Ryan, J. L. Robertson, J.-W. Kim, T. S.Santos, E. Karapetrova, J. L. Zarestky, X. Zhai, S. G. E. teVelthuis, J. N. Eckstein, S. D. Bader, and A. Bhattacharya,Nature Mater. , 892 (2009). C. Lin and A. J. Millis, Phys. Rev. B , 184405 (2008). B. R. K. Nanda and S. Satpathy, Phys. Rev. B , 054427(2008). R. Yu, S. Yunoki, S. Dong, and E. Dagotto, Phys. Rev. B , 125115 (2009). L. Brey, Phys. Rev. B , 104423 (2007). S. Dong, R. Yu, S. Yunoki, G. Alvarez, J. M. Liu, and E.Dagotto, Phys. Rev. B , 201102(R) (2008). R. Cheng, K. Li, S. Wang, Z. Chen, C. Xiong, X. Xu, andY. Zhang, Appl. Phys. Lett. , 2475 (1998). I. N. Krivorotov, K. R. Nikolaev, A. Yu. Dobin, A. M.Goldman, and E. Dan Dahlberg, Phys. Rev. Lett. , 5779(2001). H. Li, J. R. Sun, and H. K. Wong, Appl. Phys. Lett. ,628 (2002). M. Jo, M. G. Blamire, D. Ozkaya, and A. K. Petford-Long,J. Phys.: Condens. Matter , 5243 (2003). M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen Van Dau,F. Petroff, P. Etienne, G. Creuzet, A. Friederich, and J.Chazelas, Phys. Rev. Lett. , 2472 (1988). G. Binasch, P. Gr¨unberg, F. Saurenbach, and W. Zinn,Phys. Rev. B , 4828 (1989). D. Niebieskikwiat, M. B. Salomon, L. E. Hueso, N. D.Mathur, and J. A. Borchers, Phys. Rev. Lett. , 247207(2007). G. Lian, Z. Wang, J. Gao, J. Kang, M. Li, and G. Xiong,J. Phys. D , 90 (1999). S. Mukhopadhyay and I. Das, Europhys. Lett. , 27003(2008). D. Niebieskikwiat, L. E. Hueso, N. D. Mathur, and M. B.Salomon, Appl. Phys. Lett. , 123120 (2008). J. Salafranca, M. J. Calderon, and L. Brey, Phys. Rev. B , 014441 (2008). S. Yunoki, T. Hotta, and E. Dagotto, Phys. Rev. Lett. ,3714 (2000). E. Dagotto, T. Hotta, and A. Moreo, Phys. Rep. , 1(2001). S. Kumar, A. P. Kampf, and P. Majumdar, Phys. Rev.Lett. , 176403 (2006). K. Pradhan, A. Mukherjee, and P. Majumdar, Phys. Rev.Lett. , 147206 (2007). E. Dagotto, S. Yunoki, A. L. Malvezzi, A. Moreo, J. Hu,S. Capponi, D. Poilblanc, and N. Furukawa, Phys. Rev. B , 6414 (1998). A. C. Green, Phys. Rev. B , 205110 (2001). Z. Popovic and S. Satpathy, Phys. Rev. Lett. , 197201(2002). Y. Okimoto, T. Katsufuji, T. Ishikawa, A. Urushibara, T.Arima, and Y. Tokura, Phys. Rev. Lett. , 109 (1995). K. Pradhan, A. Mukherjee, and P. Majumdar, Europhys.Lett. , 37007 (2008). S. Okamoto and A. J. Millis, Phys. Rev. B , 075101(2004). S. Yunoki, A. Moreo, E. Dagotto, S. Okamoto, S. S. Kan-charla, and A. Fujimori, Phys. Rev. B , 064532 (2007). Surfaces and interfaces of solids , H. Luth (Springer-Verlag,Berlin; New York, 1993). P. Lunkenheimer, V. Bobnar, A. V. Pronin, A. I. Ritus, A.A. Volkov, and A. Loidl, Phys. Rev. B , 052105 (2002). J. L. Cohn, M. Peterca, and J. J. Neumeier, Phys. Rev. B , 214433 (2004). B. Korenblum and E. I. Rashba, Phys. Rev. Lett. ,096803 (2002). T. Oka and N. Nagaosa, Phys. Rev. Lett. , 266403(2005). S. Kumar and P. Majumdar, Eur. Phys. J. B , 571(2006). T. G. Perring, G. Aeppli, Y. Moritomo, and Y. Tokura,Phys. Rev. Lett. , 3197 (1997). G. D. Mahan,
Quantum Many Particle Physics (PlenumPress, New York, 1990). S. Kumar and P. Majumdar, Eur. Phys. J. B , 237(2005). J. Burgy, M. Mayr, V. Martin-Mayor, A. Moreo, and E.Dagotto, Phys. Rev. Lett. , 277202 (2001). Y. Motome, N. Furukawa, and N. Nagaosa, Phys. Rev.Lett. , 167204 (2003). Z. G. Sheng, W. H. Song, Y. P. Sun, J. R. Sun, and B. G.Shen, Appl. Phys. Lett. , 032501 (2005). T. F. Zhou, G. Li, N. Y. Wang, B. M. Wang, X. G. Li, andY. Chen, Appl. Phys. Lett.88