Abstract
Halfmetallicity in materials has been a subject of extensive research due to its potential for applications in spintronics. Ferromagnetic manganites have been seen as a good candidate and aside from a small minorityspin pocket observed in La_{2−2x}Sr_{1+2x}Mn_{2}O_{7} (x = 0.38), transport measurements show that ferromagnetic manganites essentially behave like half metals. Here we develop robust tightbinding models to describe the electronic band structure of the majority as well as minority spin states of ferromagnetic, spincanted antiferromagnetic and fully antiferromagnetic bilayer manganites. Both the bilayer coupling between the MnO_{2} planes and the mixing of the x^{2} − y^{2} > and 3z^{2} − r^{2} > Mn 3d orbitals play an important role in the subtle behavior of the bilayer splitting. Effects of k_{z} dispersion are included.
Introduction
Manganites^{1,2} have been widely studied because of their remarkable properties of colossal magnetoresistance^{3} and possible halfmetallicity^{4,5}, where electrons of one spin are metallic and those of the opposite spin are insulating. Metals with a high degree of spin polarization at the Fermi level are of great interest for possible applications in spintronics^{6,7}, enabling the processing of data and memory storage via spins instead of conventional methods involving transport of charge.
Manganites are quasitwo dimensional materials with layered structures similar to those of high T_{c} cuprate superconductors. The structure of LaSr_{2}Mn_{2}O_{7} (LSMO) resembles that of the prototypical perovskite mineral CaTiO_{3} and it can be described in terms of a stacking of double layers^{8} of interconnected MnO_{6} octahedra in which Mn atoms sit at the center and oxygen atoms occupy corners of the octahedron. The MnO_{6} octahedra are distorted and a crystalfieldsplitting parameter E_{z} can be used to characterize the splitting between the x^{2} − y^{2} > and the 3z^{2} − r^{2} > 3d levels of the Mn atoms^{9}.
Doped bilayer manganites display a rich phase diagram, which includes a ferromagnetic (FM) phase as well as a more subtle antiferromagnetic (AFM) state where spins are aligned ferromagnetically within the MnO planes, but canted antiferromagnetically between the adjacent MnO planes^{10}. The bilayer coupling plays a key role in stabilizing the FM phase by preserving phase coherence between the neighboring MnO planes. When doping with Sr from x = 0.38 to 0.59, where x is the electronic doping away from halffilling or, equivalently, the ratio of Sr to La, strength of the bilayer coupling decreases due to the canting of spins between the adjacent layers and finally it vanishes in the fully AFM phase. Angleresolved photoemission spectroscopy (ARPES) experiments^{11} show that the ferromagnetic compound (x = 0.38) exhibits a finite bilayer splitting due to interlayer hopping, while the antiferromagnetic compound (x = 0.59) has zero bilayer splitting since the adjacent layers are oppositely spin polarized.
A ferromagnetic calculation on LSMO based on the generalized gradient approximation (GGA)^{11} shows that bands at the Fermi energy (E_{F}) are primarily of e_{g} character^{12} (i.e. Mn 3d x^{2} − y^{2} > and 3z^{2} − r^{2} >) for the majority spins and of t_{2g} character (i.e. Mn 3d xy >) for the minority spins, which is consistent with ARPES results^{13}. Previous comparisons between ARPES and density functional theory (DFT) computations have revealed that the GGA gives a better description than the local spin density approximation (LSDA) and that the LSDA corrected by a Hubbard parameter (LSDA + U) gives an even poorer description of the ARPES data^{11}. The GGA provides a simple but potentially accurate step beyond LSDA which can improve the description of magnetic properties of the 3d electronic shell^{14}.
The metallic conductivity in the FM phase can be explained within the doubleexchange (DE) mechanism^{15}, where e_{g} electrons hop between the Mn sites through hybridization with the oxygen 2p orbitals. While the DE mechanism appears to capture the tendency towards ferromagnetism, the oxygen orbitals must be explicitly included to explain correctly the metal insulator transition at the Curie temperature^{16}.
Since the DFT band structure is found to be that of a nearly halfmetallic ferromagnet with a small minorityspin FS (Fermi surface), most studies in the literature focus only on the majority bands described within simple tightbinding (TB) models^{17}, neglecting the minority bands. Here, we present a more realistic yet transparent TB model which incorporates the bonding and antibonding x^{2} − y^{2} > as well as the 3z^{2} − r^{2} > orbitals, including the minority states as observed via ARPES in the FM^{13} and AFM^{11} states. Recall that in the cuprates there is strong copperoxygen hybridization, but if one is mainly interested in the antibonding band near the Fermi level, one can study an effective, copperonly model. In this spirit, we develop an effective Mnonly model here, which includes the minority bands in order to provide a precise description of the minority electrons in determining the spin polarization at the Fermi level, a key ingredient needed for the design of spintronics devices. We delineate how our model Hamiltonian gives insight into the delicate interplay between the effects of orbital mixing and nesting features, which impact the static susceptibility and drive exotic phase transitions^{18}. Our approach can also allow a precise determination of the occupancy of the minority t_{2g} electrons through an analysis of the experimental FSs.
Results
Band character near E_{F}
In the DFTbased band structure, E_{F} cuts through the majority x^{2} − y^{2} > and 3z^{2} − r^{2} > bands, while there are only small electron pockets in the minority xy > bands. Coupling between the two MnO layers in the FM state produces bonding and antibonding bands, which are directly observed in experiments^{19}. Accordingly, our fitting procedure is based on a combination of four majority and two minority bands in order to accurately capture the nearE_{F} physics of the system.
For the majority e_{g} bands, the strength of bilayer coupling for x^{2} − y^{2} > orbitals is much weaker than that for 3z^{2} − r^{2} > orbitals because the lobes of x^{2} − y^{2} > orbitals lie inplane, while those of 3z^{2} − r^{2} > orbitals point outoftheplane. The bilayer coupling of various orbitals without hybridization can be seen along the Γ(0, 0)X(π, π) line in Figure 1, where the two x^{2} − y^{2} > bands are nearly degenerate and the two 3z^{2} − r^{2} > bands are split with a separation of ≈1.1 eV. Away from the nodal direction, the x^{2} − y^{2} > and 3z^{2} − r^{2} > orbitals hybridize and the splitting of the related bands becomes more complex. Near the M(π, 0) point, the two lowest bands are primarily of x^{2} − y^{2} > character. The mixing with 3z^{2} − r^{2} > increases the splitting to ≈250 meV.
Regarding the t_{2g} minority bands, since the lobes of xy > orbitals lie inplane, strength of the bilayer coupling is small. Unlike x^{2} − y^{2} >, the lobes of xy > are rotated 45° from the MnO direction, so that the hybridization with other bands and the resulting splittings reach their maximum value at the Xpoint.
Tightbinding model: majority spin
Since there is a large exchange splitting, we discuss the majority and minority bands separately. This section presents the TB model for the majority spins, obtained by fitting to the first principles band structure. The four bands near E_{F} are predominantly associated with the eg orbitals of Mn 3d, x^{2} − y^{2} > and 3z^{2} − r^{2} >, so that the minimal TB model involves four orbitals per primitive unit cell. In this connection, it is useful to proceed in steps and accordingly, we first discuss a 2dimensional (2D) model with bilayer splitting, followed by the inclusion of effects of k_{z}dispersion.
For the 2D model, the relevant symmetric (+) and antisymmetric (−) combinations of the orbitals decouple and the 4 × 4 Hamiltonian reduces to two 2 × 2 Hamiltonians, H_{±}, where the basis functions are ψ_{1±} and ψ_{2±} with the subscripts 1 and 2 referring to the x^{2} − y^{2} > and 3z^{2} − r^{2} > orbitals, respectively. The Hamiltonian matrices are
where
c_{i}(αa) = cos(k_{i}αa), i = x, y and α is an integer. t_{ij} are the hopping parameters where t_{11} is the hopping between the x^{2} − y^{2} > orbitals, t_{22} for the 3z^{2} − r^{2} > orbitals and t_{12} between the x^{2} − y^{2} > and 3z^{2} − r^{2} > orbitals. Here the nearest neighbor hopping is denoted by t_{ij}, the next nearest hopping by and the higher order hoppings are denoted by a larger number of primes as superscripts. Note that the two matrices in Eq. 1 are identical except for the last term on the main diagonal, differing only in the sign of the bilayer hopping terms t_{bi}_{1} and . The chemical potential μ is obtained via a least squares fit to the firstprinciples GGA bands.
If the hopping parameters are deduced within the SlaterKoster model^{20}, one would obtain t_{11} = t_{22} = t_{12} = t_{bi}_{2} and . However, we found an improved fit by letting the parameters deviate from these constraints. A number of additional hopping terms were tested, but found to give negligible improvements and discarded. A least squares minimization program was used to obtain the optimized TB parameters, which are listed in Table 1 (2D model).
Values of TB parameters in Table 1 are consistent with previous results on cubic manganites^{17}. It is reasonable that the four nearest neighbor parameters (t_{11}, t_{22}, t_{12} and t_{bi}_{2}) are the largest in absolute magnitude and are the most important fitting parameters. Sign differences between , and control the presence of a closed FS related to 3z^{2} − r^{2} > bands and an open FS from x^{2} − y^{2} > bands, consistent with earlier studies^{18}. TB parameters with small magnitudes (t″ and t′″) involve overlap between more distant neighbors. We emphasize that even though t″ and t′″ are small, they contribute significantly to the overall goodness of the fit. A small value of t_{bi}_{1} reflects weak intralayer interactions between the x^{2} − y^{2} > orbitals due to the orientation of these orbitals. Since the magnitude of the crystal field splitting parameter E_{z} is smaller than that of t_{12}, the hybridization of x^{2} − y^{2} > and 3z^{2} − r^{2} > is significant when H_{12} is nonzero.
Figure 2 compares the model TB bands (open circles) with the corresponding DFT results (solid dots). While the full 2D model is considered in Figure 2a, we also show in Figure 2(b), results of a much simpler TB model that employs only two parameters (E_{z} and t) with t_{11} = t_{22} = t_{12} = t_{bi}_{2}. For the simple model of Figure 2b, the parameter values (t = −0.431 eV, E_{z} = −0.057 eV and μ = 0.616 eV) were obtained via an optimal fit to the firstprinciples bands. It is obvious that the 2D TB model results shown in Figure 2a provide a vastly improved fit compared to the simple two parameter model in Figure 2b. The agreement in Figure 2a between the TB model and the first principles calculations is overall very good and the TB model correctly reproduces salient features of the band structure.
At Γ, the two lowest energy bands are found to be nearly degenerate in both the TB model and the first principles calculations, with a splitting of −2t_{bi}_{1} = 0.044 eV in the TB model. Following these two bands along Γ − X, one finds that the two larger dispersing bands with x^{2} − y^{2} > character have small bilayer splitting due to the small value of t_{bi}_{1}. The two other bands in the same direction are of 3z^{2} − r^{2} > character and exhibit a larger bilayer splitting of −2H_{bi}_{2} = 1.09 eV. Because H_{bi}_{2} contains the nextnearestneighbor hopping terms, the bilayer splitting of 3z^{2} − r^{2} > bands develops an inplane kdependence. As a result, dispersion of the antibonding band is larger than that of the bonding band. Along the Γ − M and X − M directions, H_{12} is nonzero, leading to the mixing of x^{2} − y^{2} > and 3z^{2} − r^{2} > bands. At the Mpoint, H_{12} reaches its maximum value, yielding a complex bilayer splitting of the Van Hove singularities. In other words, the bare bilayer splitting of x^{2} − y^{2} > is ≈50 meV, but hybridization with 3z^{2} − r^{2} > enhances this splitting to ≈290 meV near M in the TB model as follows:
where and C_{±} = 3H_{11} − H_{22} + E_{z} ± t_{bi}_{1}.
Figure 3 compares the 2DTB (open circles) and firstprinciples (dots) FSs. Agreement is seen to be quite good. The three pieces of FS are labeled by ‘1’, ‘2’ and ‘3’. The larger squarish pocket ‘1’ centered at X is a mix of x^{2} − y^{2} > and 3z^{2} − r^{2} >, the smaller squarish pocket ‘3’ around the Γpoint is primarily of 3z^{2} − r^{2} > character and the rounded FS ‘2’ lying between ‘1’ and ‘3’ centered at X is mostly of x^{2} − y^{2} > character. For comparison Figure 3b shows the FS from the simple two parameter TB model of Figure 2b and we see again that this simple model gives a poor representation of the actual FS.
Recall that in the cuprates, there is a small but finite k_{z}dispersion^{21,22,23,24}, which is also the case in the manganites. Since the 3z^{2} − r^{2} > orbitals have lobes pointing out of the plane, the interlayer hoppings are associated with 3z^{2} − r^{2} > bands. In the 3D model, the 4 × 4 Hamiltonian now cannot be reduced to two 2 × 2 Hamiltonians because of the bodycentered crystal structure. The basis functions are x^{2} − y^{2} > and 3z^{2} − r^{2} > for the upper and lower MnO_{2} layers. By including interlayer hopping t_{z} between 3z^{2} − r^{2} > orbitals and the intralayer hopping for 3z^{2} − r^{2} > orbitals, we obtain the Hamiltonian matrix:
where c_{z}(c) = cos(k_{z}c) and c is the lattice constant in the zdirection, which is approximately 5 times larger than the inplane lattice constant a. The parameters obtained by fitting to the DFT bands are listed in Table 1 (3D model). Compared to the 2D model, the bilayer hopping parameters t_{bi}_{1}, t_{bi}_{2} and are significantly modified. t_{22} and E_{z} change by about 30 meV while other terms undergo only slight modifications. Plausible values of parameters are retained in the 3D model.
The effect of k_{z}dispersion in the 3D model can be seen by comparing the FSs at k_{z}c = 0 and k_{z}c = 2π as shown in Figure 4. While FS ‘2’ with mostly x^{2} − y^{2} > character remains unchanged, the FS piece ‘3’ with primarily 3z^{2} − r^{2} > character changes significantly. ‘3’ is squarish at k_{z}c = 0 but becomes smaller and rounded at k_{z}c = 2π (‘3′’). Although ‘1’ contains a significant 3z^{2} − r^{2} > contribution, the effect of k_{z}dispersion on this FS piece is much smaller than on ‘3’. ‘1’ and ‘1′’ match when k_{x}a = π or k_{y}a = π because the interlayer hopping terms t_{z} and have zero contribution due to the dependence in the bodycentered structure. ‘1’ and ‘1′’ almost match when k_{x}a = k_{y}a because t_{bi}_{1} is almost zero. Thus ‘1’ and ‘1′’ can differ only away from the high symmetry kpoints and this piece of the FS is cylinderlike in 3D.
Tightbinding model: minority spin
Due to the large exchange splitting, we only need to consider two bands in the case of minority spins, which are associated with the t_{2g} xy > orbitals of the upper and lower MnO_{2} layers. The 2 × 2 model Hamiltonian given below is diagonal with a bilayer splitting of Δ between the upper and lower xy > bands.
where
Table 2 lists the parameters obtained from fitting firstprinciples band structure. Figure 5 compares the parameterized TB bands (open circles) with the firstprinciples GGA bands (solid dots). The minority spin FSs are overlayed in Figure 3 as triangles and form two small pockets around Γ, as observed also in the ARPES experiments^{13}.
Doping and magnetic structure
We now turn to discuss how the lowenergy electronic structure of the rich variety of magnetic phases displayed by LSMO is captured by our 2D and 3D TB models. Kubota et al.^{10} have shown that the magnetic structure of LSMO is intimately connected with doping and that it can be parameterized in terms of θ_{cant}, the spin canting angle between the neighboring FM planes. The behavior of θ_{cant}, deduced from experiments, shows a FM structure (θ_{cant} = 0°) for 0.32 ≤ x ≤ 0.38, with the value of θ_{cant} becoming finite at x ≈ 0.39 and reaching 180° for x ≥ 0.48^{10}. In the 2D and 3D models discussed above, for doping greater than x = 0.38, the value of E_{F} was found by assuming a rigid band type approximation^{25} where the total number of occupied electrons N is given by N = 2(1 − x) at doping x. Over this doping range the exchange splitting from GGA was taken to be constant since the spins are ferromagnetically aligned in planes and the inplane lattice parameters are not sensitive to doping^{10}. We then invoke the argument of Anderson et al.^{26} that the transfer integral between any two ions depends on cos(θ/2) where θ is the angle between their spins on neighboring layers as the magnetic state changes from FM to AFM. We thus replaced the bilayer TB parameters H_{bi}_{2}, t_{bi}_{1} and Δ by cos(θ_{cant}/2)H_{bi}_{2}, cos(θ_{cant}/2)t_{bi}_{1} and cos(θ_{cant}/2)Δ and for the 3D model t_{z} was also replaced with cos(θ_{cant}/2)t_{z}, using the experimental values of θ_{cant} at the corresponding dopings given by Kubota et al.^{10}.
Table 3 gives values of ΔE_{F} (where ΔE_{F} is measured with respect to E_{F} at x = 0.50 in the FM state), number of minority electrons, Δn, number of majority electrons, 1 − x − Δn, total number of electrons, 1 − x, canting angle, θ_{cant} and the magnetic moment μ_{B}, all per Mn atom for the doping range 0.38–0.59, as obtained within our 2D and 3D models. Table 4 provides the same quantities over this doping range only in the FM state appropriate for saturating magnetic fields. [The doping range used for calculations in Tables 3 and 4 does not include the experimentally observed anomalous FS behavior^{27}.] The magnetic moment μ_{B} per Mn atom, including the contribution of the three occupied t_{2g} orbitals, is given by μ_{B} = 1 − x − 2Δn + 3 and its values are consistent with magnetic Compton experiments^{28,29}. The number of minority electrons, Δn, found in recent ARPES experiments^{13} is also in good agreement with the corresponding values in Table 3. We find that, in comparison to the GGA, the LSDA underestimates the exchange splitting by 20% and thus overestimates the number of minority electrons. On the other hand, the TB parameters based on LSDA and GGA band structures differ only within 1%.
Figure 6a compares the experimental FS for x = 0.38 (FM)^{13}, with the corresponding 2D TB model predictions. Good agreement is seen between theory and experiment for the FS pieces related to the (red line), the antibonding (green line) and the minority pockets (pink and black lines). The bonding holepocket (blue) is invisible at this photon energy due to matrix element effects^{13,19,21,22}. In order to account for the coexistence of metallic and nonmetallic regions for x ≤ 0.38, which has been interpreted as arising from a phase separation into holerich and holepoor regions^{27}, we found it necessary to adjust the doping of the theoretical FS at x = 0.38 to an effective dopping of x = 0.43. Figure 6b shows the x = 0.59^{11} experimental AFM FS, along with the corresponding 2D TB model results. Here also we find good agreement for the bonding and antibonding bands (blue and green lines). The same level of agreement between theory and experiment is also found for the 3D model, which is to be expected since the values in Tables 3 and 4 for the 2D and 3D models are very similar.
Discussion
The doublelayered manganites, La_{2−2x} Sr_{1+2x}Mn_{2}O_{7}, have attracted much attention in recent years as model systems that present a wide range of transport and magnetic properties as a function of temperature, doping and magnetic field. In the FM phase at x = 0.38, the majority t_{2g} electrons of Mn lie well below the Fermi level and are thus quite inert. Therefore, key to the understanding of the manganites is the behavior of the Mn magnetic electrons with e_{g} character (x^{2} − y^{2} > and 3z^{2} − r^{2} >). The results of magnetic Compton experiments^{28} reveal that the FM order weakens when the occupation of the 3z^{2} − r^{2} > majority state decreases. For spintronics applications, it is important to note that the Fermi level in the FM phase lies slightly above the bottom of the minorityspin conduction band, yielding a nearly halfmetallic ferromagnet. The unwanted FSpocket can be reduced in volume by increasing the doping x. However, the Mn spins (aligned ferromagnetically within the MnO planes) become canted antiferromagnetically between the adjacent MnO planes as x increases, leading to a competing AFM order which destroys the FM phase.
In order to understand this interesting phenomenology, we have developed a TB model encompassing both the FM and AFM phases, which correctly captures the lowenergy electronic structure of LSMO using a minimal basis set. The complex bilayer splitting in the majority spins is well reproduced. In particular, the mixing of x^{2} − y^{2} > and 3z^{2} − r^{2} > orbital degrees of freedom is found to be strong and momentum dependent. With inclusion of k_{z} dispersion, the 3D FS including its various pieces is reproduced in substantial detail. Moreover, our model accurately describes the delicate minority FS pocket.
Since the e_{g} mixing has a pronounced effect on the shape of the FS, an accurate model allowing precise parameterization of the band structure is crucially important for modeling transport properties. Such a model would also provide a springboard for further theoretical work on strongly correlated electron systems, including Monte Carlo simulations to uncover the exciting manybody physics of the manganites^{30,31}. Moreover, a precise description of the minority band is needed for the design of efficient spintronics devices. In this way, the TB models discussed in this study would also help develop the applications potential of the manganites.
Methods
The firstprinciples calculations were done using the WIEN2K^{32,33} code. The electronic structure was calculated within the framework of the densityfunctional theory^{34,35} using linearized augmented planewave (LAPW) basis^{36}. Exchangecorrelation effects were treated using the generalized gradient approximation (GGA)^{37}. A rigid band model was invoked for treating doping effects on the electronic structure along the lines of Ref. 25, but we expect our results to be insensitive to a more realistic treatment of doping effects using various approaches^{38,39,40,41}. We used muffintin radius (R_{MT}) of 1.80 Bohr for both O and Mn and 2.5 Bohr for Sr and La. The integrals over the Brillouin zone were performed using a tetrahedron method with a uniform 14 × 14 × 14 kpoint grid. The ARPES experiments were performed on cleaved single crystals at beam lines 7.0.1 and 12.0.1 of the Advanced Light Source, Berkeley.
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Acknowledgements
This research was performed while one of us (M.B.) was on sabbatical leave from Boston University. The work at Northestern University was supported by the US Department of Energy, Office of Science, Basic Energy Sciences contract DEFG0207ER46352 and benefited from the allocation of supercomputer time at NERSC through DOE grant number DEAC0205CH11231 and Northeastern University's Advanced Scientific Computation Center (ASCC). We thank H. Zheng and J. Mitchell for providing the crystals and J.F. Douglas, A. Fedorov, E. Rotenberg and Q. Wang for help with the experiments. The work at the University of Colorado Boulder was supported by the US Department of Energy under grant number DEFG0203ER46066. The Advanced Light Source is supported by the Director, Office of Science, Office of Basic Energy Sciences, of the U.S. Department of Energy under Contract No. DEAC0205CH11231.
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M.B., H.L., C.L., H.H., R.S.M., B.B., Z.S., D.S.D. and A.B. all contributed to the research reported in this study and the writing of the manuscript.
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Baublitz, M., Lane, C., Lin, H. et al. A Minimal tightbinding model for ferromagnetic canted bilayer manganites. Sci Rep 4, 7512 (2015). https://doi.org/10.1038/srep07512
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