Introduction

Manganites1,2 have been widely studied because of their remarkable properties of colossal magnetoresistance3 and possible half-metallicity4,5, 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 spintronics6,7, enabling the processing of data and memory storage via spins instead of conventional methods involving transport of charge.

Manganites are quasi-two dimensional materials with layered structures similar to those of high Tc cuprate superconductors. The structure of LaSr2Mn2O7 (LSMO) resembles that of the prototypical perovskite mineral CaTiO3 and it can be described in terms of a stacking of double layers8 of interconnected MnO6 octahedra in which Mn atoms sit at the center and oxygen atoms occupy corners of the octahedron. The MnO6 octahedra are distorted and a crystal-field-splitting parameter Ez can be used to characterize the splitting between the |x2 − y2 > and the |3z2 − r2 > 3d levels of the Mn atoms9.

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 planes10. 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 half-filling 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. Angle-resolved photoemission spectroscopy (ARPES) experiments11 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 (EF) are primarily of eg character12 (i.e. Mn 3d |x2 − y2 > and |3z2 − r2 >) for the majority spins and of t2g character (i.e. Mn 3d |xy >) for the minority spins, which is consistent with ARPES results13. 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 data11. The GGA provides a simple but potentially accurate step beyond LSDA which can improve the description of magnetic properties of the 3d electronic shell14.

The metallic conductivity in the FM phase can be explained within the double-exchange (DE) mechanism15, where eg 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 temperature16.

Since the DFT band structure is found to be that of a nearly half-metallic ferromagnet with a small minority-spin FS (Fermi surface), most studies in the literature focus only on the majority bands described within simple tight-binding (TB) models17, neglecting the minority bands. Here, we present a more realistic yet transparent TB model which incorporates the bonding and anti-bonding |x2 − y2 > as well as the |3z2 − r2 > orbitals, including the minority states as observed via ARPES in the FM13 and AFM11 states. Recall that in the cuprates there is strong copper-oxygen hybridization, but if one is mainly interested in the antibonding band near the Fermi level, one can study an effective, copper-only model. In this spirit, we develop an effective Mn-only 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 transitions18. Our approach can also allow a precise determination of the occupancy of the minority t2g electrons through an analysis of the experimental FSs.

Results

Band character near EF

In the DFT-based band structure, EF cuts through the majority |x2 − y2 > and |3z2 − r2 > 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 experiments19. Accordingly, our fitting procedure is based on a combination of four majority and two minority bands in order to accurately capture the near-EF physics of the system.

For the majority eg bands, the strength of bilayer coupling for |x2 − y2 > orbitals is much weaker than that for |3z2 − r2 > orbitals because the lobes of |x2 − y2 > orbitals lie in-plane, while those of |3z2 − r2 > orbitals point out-of-the-plane. 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 |x2 − y2 > bands are nearly degenerate and the two |3z2 − r2 > bands are split with a separation of ≈1.1 eV. Away from the nodal direction, the |x2 − y2 > and |3z2 − r2 > 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 |x2 − y2 > character. The mixing with |3z2 − r2 > increases the splitting to ≈250 meV.

Figure 1
figure 1

Majority-spin band structure at kz = 0 in the FM state for x = 0.50.

The color scale identifies the |x2 − y2 > and |3z2 − r2 > characters of various bands.

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Regarding the t2g minority bands, since the lobes of |xy > orbitals lie in-plane, strength of the bilayer coupling is small. Unlike |x2 − y2 >, 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 X-point.

Tight-binding 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 EF are predominantly associated with the eg orbitals of Mn 3d, |x2 − y2 > and |3z2 − r2 >, 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 2-dimensional (2D) model with bilayer splitting, followed by the inclusion of effects of kz-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 ψ1± and ψ2± with the subscripts 1 and 2 referring to the |x2 − y2 > and |3z2 − r2 > orbitals, respectively. The Hamiltonian matrices are

where

ci(αa) = cos(kiαa), i = x, y and α is an integer. tij are the hopping parameters where t11 is the hopping between the |x2 − y2 > orbitals, t22 for the |3z2 − r2 > orbitals and t12 between the |x2 − y2 > and |3z2 − r2 > orbitals. Here the nearest neighbor hopping is denoted by tij, 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 tbi1 and . The chemical potential μ is obtained via a least squares fit to the first-principles GGA bands.

If the hopping parameters are deduced within the Slater-Koster model20, one would obtain t11 = t22 = t12 = tbi2 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).

Table 1 Values of various tight-binding parameters for the majority spin bands (in meV)
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Values of TB parameters in Table 1 are consistent with previous results on cubic manganites17. It is reasonable that the four nearest neighbor parameters (t11, t22, t12 and tbi2) 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 |3z2 − r2 > bands and an open FS from |x2 − y2 > bands, consistent with earlier studies18. 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 tbi1 reflects weak intra-layer interactions between the |x2 − y2 > orbitals due to the orientation of these orbitals. Since the magnitude of the crystal field splitting parameter Ez is smaller than that of t12, the hybridization of |x2 − y2 > and |3z2 − r2 > is significant when H12 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 (Ez and t) with t11 = t22 = t12 = tbi2. For the simple model of Figure 2b, the parameter values (t = −0.431 eV, Ez = −0.057 eV and μ = 0.616 eV) were obtained via an optimal fit to the first-principles 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.

Figure 2
figure 2

(a) Majority-spin TB band structure (open circles) obtained from the 2D model discussed in the text is superimposed on the corresponding first-principles bands in the FM state for x = 0.50 (solid dots); (b) Same as (a), except that the TB bands here (open circles) are based on a simple two parameter model.

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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 −2tbi1 = 0.044 eV in the TB model. Following these two bands along Γ − X, one finds that the two larger dispersing bands with |x2 − y2 > character have small bilayer splitting due to the small value of tbi1. The two other bands in the same direction are of |3z2 − r2 > character and exhibit a larger bilayer splitting of −2Hbi2 = 1.09 eV. Because Hbi2 contains the next-nearest-neighbor hopping terms, the bilayer splitting of |3z2 − r2 > bands develops an in-plane k-dependence. As a result, dispersion of the antibonding band is larger than that of the bonding band. Along the Γ − M and X − M directions, H12 is non-zero, leading to the mixing of |x2 − y2 > and |3z2 − r2 > bands. At the M-point, H12 reaches its maximum value, yielding a complex bilayer splitting of the Van Hove singularities. In other words, the bare bilayer splitting of |x2 − y2 > is ≈50 meV, but hybridization with |3z2 − r2 > enhances this splitting to ≈290 meV near M in the TB model as follows:

where and C± = 3H11 − H22 + Ez ± tbi1.

Figure 3 compares the 2D-TB (open circles) and first-principles (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 |x2 − y2 > and |3z2 − r2 >, the smaller squarish pocket ‘3’ around the Γ-point is primarily of |3z2 − r2 > character and the rounded FS ‘2’ lying between ‘1’ and ‘3’ centered at X is mostly of |x2 − y2 > 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.

Figure 3
figure 3

(a) TB majority-spin Fermi surface based on the bands of Figure 2a (open circles) and the minorityspin Fermi surface from bands of Figure 4 (triangles) are superposed on the first-principles results for x = 0.50 in the FM state (solid dots). (b) Same as (a), except that the TB bands here (open circles) are based on a simple two parameter model.

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Recall that in the cuprates, there is a small but finite kz-dispersion21,22,23,24, which is also the case in the manganites. Since the |3z2 − r2 > orbitals have lobes pointing out of the plane, the interlayer hoppings are associated with |3z2 − r2 > bands. In the 3D model, the 4 × 4 Hamiltonian now cannot be reduced to two 2 × 2 Hamiltonians because of the body-centered crystal structure. The basis functions are |x2 − y2 > and |3z2 − r2 > for the upper and lower MnO2 layers. By including interlayer hopping tz between |3z2 − r2 > orbitals and the intra-layer hopping for |3z2 − r2 > orbitals, we obtain the Hamiltonian matrix:

where cz(c) = cos(kzc) and c is the lattice constant in the z-direction, which is approximately 5 times larger than the in-plane 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 tbi1, tbi2 and are significantly modified. t22 and Ez 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 kz-dispersion in the 3D model can be seen by comparing the FSs at kzc = 0 and kzc = 2π as shown in Figure 4. While FS ‘2’ with mostly |x2 − y2 > character remains unchanged, the FS piece ‘3’ with primarily |3z2 − r2 > character changes significantly. ‘3’ is squarish at kzc = 0 but becomes smaller and rounded at kzc = 2π (‘3′’). Although ‘1’ contains a significant |3z2 − r2 > contribution, the effect of kz-dispersion on this FS piece is much smaller than on ‘3’. ‘1’ and ‘1′’ match when kxa = π or kya = π because the interlayer hopping terms tz and have zero contribution due to the dependence in the body-centered structure. ‘1’ and ‘1′’ almost match when kxa = kya because tbi1 is almost zero. Thus ‘1’ and ‘1′’ can differ only away from the high symmetry k-points and this piece of the FS is cylinder-like in 3D.

Figure 4
figure 4

Majority-spin Fermi surfaces at kzc = 0 (circles) and kzc = 2π (square) based on the 3D TB model in the FM state for x = 0.50, as discussed in the text.

The color scale identifies the |x2 − y2 > and |3z2 − r2 > characters of various bands.

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Tight-binding 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 t2g |xy > orbitals of the upper and lower MnO2 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 first-principles band structure. Figure 5 compares the parameterized TB bands (open circles) with the first-principles 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 experiments13.

Table 2 Values of various tight-binding parameters for the minority-spin bands (in meV)
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Figure 5
figure 5

Minority-spin band structure obtained from the TB 2D model (open circles) discussed in the text is superimposed on the first-principles results at kz = 0 in the FM state for x = 0.50.(solid dots).

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Doping and magnetic structure

We now turn to discuss how the low-energy 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 θcant, the spin canting angle between the neighboring FM planes. The behavior of θcant, deduced from experiments, shows a FM structure (θcant = 0°) for 0.32 ≤ x ≤ 0.38, with the value of θcant becoming finite at x ≈ 0.39 and reaching 180° for x ≥ 0.4810. In the 2D and 3D models discussed above, for doping greater than x = 0.38, the value of EF was found by assuming a rigid band type approximation25 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 in-plane lattice parameters are not sensitive to doping10. 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 Hbi2, tbi1 and Δ by cos(θcant/2)Hbi2, cos(θcant/2)tbi1 and cos(θcant/2)Δ and for the 3D model tz was also replaced with cos(θcant/2)tz, using the experimental values of θcant at the corresponding dopings given by Kubota et al.10.

Table 3 gives values of ΔEF (where ΔEF is measured with respect to EF 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, θcant and the magnetic moment μB, 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 behavior27.] The magnetic moment μB per Mn atom, including the contribution of the three occupied t2g orbitals, is given by μB = 1 − x − 2Δn + 3 and its values are consistent with magnetic Compton experiments28,29. The number of minority electrons, Δn, found in recent ARPES experiments13 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%.

Table 3 Values of various parameters obtained for our 2D and 3D models over the doping range 0.38–0.59 (all per Mn atom): change in EF (ΔEF); number of minority electrons (Δn); number of majority electrons (1 − x − Δn); total number of electrons (1 − x); magnetic moment (μB); and the canting angle (θcant)
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Table 4 Values of various parameters obtained for our 2D and 3D models for the FM state over the doping range 0.38–0.50 (all per Mn atom): change in EF (ΔEF); number of minority electrons (Δn); number of majority electrons (1 − x − Δn); total number of electrons (1 − x); and, the magnetic moment (μB)
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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 anti-bonding (green line) and the minority pockets (pink and black lines). The bonding hole-pocket (blue) is invisible at this photon energy due to matrix element effects13,19,21,22. 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 hole-rich and hole-poor regions27, 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 = 0.5911 experimental AFM FS, along with the corresponding 2D TB model results. Here also we find good agreement for the bonding and anti-bonding 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.

Figure 6
figure 6

(a) Experimental Fermi surfaces of bonding bands for x = 0.3813 (FM) are overlaid with the results of the 2D TB model. Theoretical Fermi surfaces are derived from various orbitals as follows: |3z2 − r2 > (red line); bonding |x2 − y2 > (blue line), which is not seen in experiments due presumably to effects of the ARPES matrix element at this photon energy; anti-bonding |x2 − y2 > (green line); and, the minority pockets (pink and black lines). (b) Experimental Fermi surface at x = 0.5911 (AFM) overlaid with the 2D TB model predictions. Theoretical Fermi surfaces derived from the bonding and anti-bonding |x2 − y2 > bands are shown by blue and green lines, respectively.

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Discussion

The double-layered manganites, La2−2x Sr1+2xMn2O7, 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 t2g 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 eg character (|x2 − y2 > and |3z2 − r2 >). The results of magnetic Compton experiments28 reveal that the FM order weakens when the occupation of the |3z2 − r2 > 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 minority-spin conduction band, yielding a nearly half-metallic ferromagnet. The unwanted FS-pocket 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 low-energy 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 |x2 − y2 > and |3z2 − r2 > orbital degrees of freedom is found to be strong and momentum dependent. With inclusion of kz 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 eg 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 many-body physics of the manganites30,31. 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 first-principles calculations were done using the WIEN2K32,33 code. The electronic structure was calculated within the framework of the density-functional theory34,35 using linearized augmented plane-wave (LAPW) basis36. Exchange-correlation 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 approaches38,39,40,41. We used muffin-tin radius (RMT) 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 k-point 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.