The electronic structure of the high-symmetry perovskite iridate Ba₂IrO₄

Figure 2(a) presents an ARPES constant energy (CE) map of Ba-214, measured at E = −0.1 eV, near the top of the valence band. The map is extracted from a dataset measured at photon energy hν = 155 eV, and T = 130 K. The blue square is the surface Brillouin zone (BZ) corresponding to the crystallographic unit cell of figure 1(b) [20]. The map shows intense round features (α features in the following) at the M points, the corners of the BZ. A second set of features (β features), with fourfold symmetry, is observed at the X points. Both α and β features are repeated in all BZs of the map. A closer inspection reveals also a weaker, round contour (α*) at all Γ points. It will be clear in the following that α* is a signature of band folding into the smaller c(2 × 2) BZ (green square).

Figures 2(b)–(d) show the experimental E versus k_∥ dispersion along high-symmetry lines marked (b), (c) and (d) in figure 2(a). Along MΓM, figure 2(b) shows a prominent band with a maximum at the M point, where it gives rise to the α contour. As discussed below, this band corresponds primarily to Ir states of j_eff = 3/2 character. It merges around −2 eV with a manifold of O 2p-derived states. The same band is seen to disperse downwards along the MXM direction in figure 2(c). A second band, with a maximum at the X point, generates the β contour. In the Mott scenario, it is assigned to the Ir-derived lower-Hubbard band of j_eff = 1/2 character. The maxima of this band are more visible along the XΓX direction in figure 2(d), which also shows a dispersive feature with a maximum at Γ, associated with the α* contour. Figure 2(e) shows a k_x versus k_z CE map for E = −0.4 eV and k_y = π/a. It is extracted from ARPES measurements with photon energies in the range hν = 95–162 eV, assuming an inner potential V₀ = 10 eV. Apart from slight intensity variations with photon energy, the data are essentially independent of k_z. Namely, the absence of wiggling contours indicates that the k_z dispersion at the top of the valence band (VB), and the interplane coupling for these states, are quite small.

Representative spectra for the Γ, M and X points of the BZ are shown in figure 3. They exhibit rather broad peaks, with maxima at −0.37 eV (at Γ), −0.26 eV (at M), and −0.21 eV (at X), which places the VB maximum at the X point. The peak energy at the VB maximum should yield a lower limit for the energy gap, the actual value depending on the separation between the Fermi level and the conduction band minimum, which cannot be accessed by ARPES. However, the peak binding energy at X (0.21 eV) is already larger than the gap value Δ_g ≃ 2E_a ∼ 140 meV, estimated from the activation energy E_a = 70 meV of the electrical resistivity [20]. This discrepancy, and the broad line shapes, suggest unresolved overlapping features in the spectra of figure 3. This hypothesis is supported by the first-principles calculations of section 4.1. It also explains the different peak energies measured at the top of the band at the M point, and at its backfolded replica at Γ, since the underlying components can be differently modulated by matrix elements.

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We now compare the electronic structure of Ba-214 and Sr-214. Figure 4(a) reproduces ARPES data on Sr-214 from [4]. The spectra are measured along the ΓM'X' contour in the c(2 × 2) BZ, indicated by a yellow triangle in the top panel. Our results for Ba-214, extracted from the dataset of figure 2, are shown for the same triangular contour in figure 4(b). Data measured along the same triangular contour in the adjacent c(2 × 2) BZ are also shown in figure 4(c). Two conclusions can be drawn from the figure. Firstly, there is a good overall correspondence between the band structure of the two compounds. Secondly, while the relative intensities of the Ba-214 j_eff = 3/2 and 1/2 bands in figures 4(b) and (c) are different, their dispersions are identical. The triangular contours in figures 4(b) and (c) are equivalent for the c(2 × 2) BZ, but clearly not for the structural BZ (blue square). This shows that—similarly to the case of Sr-214 [4]—the band structure of Ba-214 is folded into the smaller BZ. The smaller intensity of the α* manifold in figure 4(b), compared with that of the α manifold in figure 4(c), is consistent with band folding from a superlattice potential that is substantially weaker than the primary lattice potential [25]. The different intensities in the two contours can be exploited to disentangle the two bands. We find that in Ba-214 the width of the j_eff = 1/2 band (∼ 0.8 eV) is somewhat larger than in Sr-214 (∼ 0.5 eV) [4], and is considerably smaller than the width of the j_eff = 3/2 band (∼ 2.5 eV). The maxima of the j_eff = 1/2 band at X (− 0.21 eV) and of the j_eff = 3/2 band at M (− 0.26 eV) in Ba-214 are shallower than those (− 0.25 eV and, respectively −0.45 eV) of the corresponding bands in Sr-214. Their energy separation is also smaller (0.05 versus 0.2 eV) in Ba-214.

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We now address the issue, raised in the introduction, of the persistence of the energy gap above T_N ∼ 240 K. Previous theoretical and experimental studies [18, 23, 24] have claimed significant Slater-type contributions to the stability of the gap, which should then collapse in the paramagnetic phase. We collected data over a broad temperature range, from well below (120 K) to well above (300 K) T_N. Figure 5 displays temperature-dependent ARPES spectra measured at the Γ and M points of the BZ. The leading edge of the spectra exhibits a trivial thermal broadening, but no indications that the gap closes at T_N. The energy gap appears to be robust even in the absence of long-range magnetic order. Therefore, the ARPES data do not support a Slater picture, at least in its simplest form.

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We performed a local density approximation (LDA) calculation including the on-site Coulomb and SO interactions (LDA + U + SO) [29]. For this purpose, the linear muffin-tin orbital approach in the atomic sphere approximation (Stuttgart LMTO47 code) [30] was used, with crystal structure data taken from [20]. We include Ba(6s,6p,5d), Ir(6s,6p,5d) and O(2s,2p) states in the orbital basis set. Both the ferromagnetic (FM) and AFM configurations were simulated for different sets of on-site Ir 5d Coulomb repulsion U and intra-atomic exchange interaction J_H. The FM configuration is calculated with one Ir atom per unit cell. In order to reproduce the AFM order observed in Ba₂IrO₄, we have used a supercell containing four Ir atoms. The primitive lattice vectors in units of a = 4.03 Å are (0,2,0), (2,0,0) and (0.5,0.5,−1.65). In the AFM configuration, U = 3 eV and J_H = 0.4 eV produced the correct energy gap value of 140 meV. The same values were recently used in [12].

The calculated band structure is illustrated in figure 6(a), superimposed on the ARPES data. Overall, DFT yields many more states than ARPES can individually resolve, which probably explains the absence of sharp quasiparticle features in the spectra. The partial densities of states of figure 6(b) show that Ir 5d and 2p in-plane oxygen states are strongly hybridized at the top of the valence band, while the bottom of the conduction band is mainly formed by Ir 5d electrons. The 2p states of the apical oxygen atoms are confined between −1 and −2 eV, due to a limited overlap with the Ir 5d orbitals. These states have a non-negligible dispersion along the c-axis. The band observed in ARPES around −3 eV corresponds to in-plane 2p oxygen states.

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We now briefly discuss the implications of the DFT results for the magnetic properties of Ba-214. Firstly, we emphasize that the occupied states close to the gap at the Γ and M points strongly depend on the specific magnetic configuration. For instance, in the FM state the top of the valence band is found at E ≃ − 0.7 eV (not shown). Moreover, a FM insulating ground state with the correct energy gap could only be obtained for unreasonably large values of U and J_H. Therefore, the inter-site exchange interaction plays a decisive role in determining the band structure close to the Fermi level.

Within the LDA + U + SO approximation [29] the electronic Hamiltonian matrix and the corresponding occupation matrix of the system can be defined in the {LS} basis (the eigenfunctions of both spin and orbital moment operators) or {jm_j} basis (the total moment operator eigenfunctions). They correspond to LS (Russell–Saunders) or jj coupling schemes. The basis choice becomes important when one calculates the expectation value of the magnetic moment to compare it with experimental data. In this case, either the LS coupling scheme or the jj coupling scheme should be chosen, depending on the strength of the SO coupling. However, when the SO coupling and the intra-atomic exchange interaction are comparable, neither the LS nor the jj scheme are valid, and an intermediate coupling theory should be developed [29]. Practically, it means that the occupation matrix is neither diagonal in the {LS} nor in {jm_j} orbital basis. Such a situation is realized in Ba-214, where the SO strength λ ∼ 0.48 eV and J_H ∼ 0.4 eV.

Within an {LS} eigenstates basis we obtain for the spin and orbital magnetic moments M_S =  0.12 μ_B and M_L =  0.33 μ_B, respectively. The resulting total magnetic moment is therefore M_LS = (2M_S + M_L) = 0.57 μ_B, in reasonable agreement with the value M = 0.36 μ_B from magnetic susceptibility measurements [20]. The total magnetic moment calculated within a {jm_j} basis is equal to M_J = 0.43 μ_B, in better agreement with the experimental value. However, one should note that in both basis sets there are large non-diagonal elements of the occupation matrix that do not contribute to the expectation value of the magnetic moment. Therefore, an intermediate coupling scheme should be used to correctly describe the magnetism of Ba-214. We leave such a consideration for a future investigation.

The performed first-principles calculations have demonstrated a complicated band structure depending on the metal–ligand hybridization, on-site Coulomb interaction, SO coupling, intra- and inter-atomic exchange interactions. The agreement between all-electron LDA + U + SO and experimental ARPES spectra is not enough to determine the exact microscopic mechanisms that are responsible for forming the electronic structure of Ba₂IrO₄. To solve this problem below we propose a tight-binding model with a minimal set of orbitals.

In order to gain more direct insight in the interplay of orbital ordering, SOC and correlation effects, we also performed a model TB calculation, along the lines of  [27, 28]. We included the whole Ir 5d t_2g and e_g manifolds, and a set of effective hopping terms describing the hybridization between the Ir 5d and O 2p electrons, as illustrated in figure 7. The tight binding Hamiltonian is

$$\begin{equation} H=H_{0} + H_{\rm SO}. \end{equation} \tag{ 1 }$$

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H₀ includes the kinetic term T = Σ_k,ν ε_ν(k)c^†_kνc_kν, where c^†_k and c_k are the fermion creation and annihilation operators and ν spans the Ir 5d manifold, and an octahedral CEF parameterized by 10Dq = E(e_g) − E(t_2g). Since we are primarily interested in the occupied states for a comparison with ARPES, we focus on the t_2g levels. The relevant hopping terms are schematically illustrated in figure 7. The form of the ε_ν's is dictated by the symmetry of the system [27]:

$$\begin{eqnarray} \varepsilon_{xy}&=&-2t_{1}({\rm cos}\,k_x+{\rm cos}\,k_y)-2t_2\,{\rm cos}\,k_x{\rm cos}\,k_y-2t_{3}({\rm cos}\,2k_x+{\rm cos}\,2k_y),\nonumber\\ \varepsilon_{xz}&=&-2t_{4}\,{\rm cos}\,k_x-2 t_{5}\,{\rm cos}\,k_y,\\ \varepsilon_{yz}&=&-2t_{5}\,{\rm cos}\,k_x-2 t_{4}\,{\rm cos}\,k_y.\nonumber \end{eqnarray} \tag{ 2 }$$

The SO coupling term for the 5d orbitals is: $H_{\mathrm { SO}}=\lambda _{5{\mathrm { d}}} \vec {L} \cdot \vec {S}$. We introduce electron correlations in the model in a phenomenological way, by imposing AFM order. This is achieved by an additional Zeeman term with an in-plane staggered magnetic field

$$\begin{equation} H_{\rm AFM}=B\sum_{i,\nu} \mathrm{e}^{\mathrm{i}\vec{Q}\cdot\vec{r_i}}\left(c^{\dagger}_{i\nu \uparrow}c_{i\nu \downarrow}+c^{\dagger}_{i\nu \downarrow}c_{i\nu \uparrow}\right), \end{equation} \tag{ 3 }$$

where $\vec {Q}=(\pi ,\pi )$ is the AFM ordering vector, and the sum is over the Ir sites i and the three t_2g orbitals. The effect of H_AF is to fold all bands into the smaller c(2 × 2) BZ, and to open gaps at the AFM zone boundaries.

The band structures produced by the various terms of the hamiltonian are plotted in figure 8(a)–(c), along the same ΓX'M'Γ contour of figures 4(a) and (b). The parameters of the model are summarized in table 2. Figures 8(a')–(c') schematically illustrate the local electronic structure at the Ir site. Figure 8(a) shows the band dispersion in the presence of the octahedral CEF. The empty e_g and the partially filled t_2g manifolds are well separated by 10Dq, and the system is metallic. Adding the SO interaction, in figures 8(b) and (b'), mixes the CEF states. The t_2g states are split into a four-fold degenerate, fully occupied j_eff = 3/2 (blue), and a doubly degenerate, half-filled, j_eff = 1/2 manifold (red), but the system remains metallic. Figures 8(c) and (c') illustrate the further band splitting induced by H_AFM, namely of the j_eff = 1/2 band into m_j = −1/2 and 1/2 subbands, which effectively simulates the opening of a correlation gap between an occupied lower Hubbard band and an empty upper Hubbard band.

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The calculated TB band structure is compared with the ARPES data in figure 9 for the set of parameters of table 2. Although the TB parameters, namely the external magnetic field, should not be taken too literally, they do provide a useful description of the electronic structure. Figures 9(a) and (b) show data along equivalent contours in the first and in the adjacent c(2 × 2) BZs, as in figures 4(b) and (c). Folded bands are indicated by dashed lines. The good agreement with the data substantiates the description of the bands given in section 3, namely the assignment of the top of the valence band to states of j_eff = 1/2 character. The good agreement of the TB model with the experiment is confirmed by a comparison of the experimental and calculated CE maps shown in figures 9(c) and (d) and figures 9(e) and (f), for E = −0.35 eV (c, d) and E = −0.7 eV (e, f). The experimental α (α*) and β features are well reproduced. Of course, the TB model does not yield any information on the spectral weight, and therefore all c(2 × 2) BZs are equivalent.

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