Phase-Sensitive Observation of a Spin-Orbital Mott State in Sr2IrO4

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Science  06 Mar 2009:
Vol. 323, Issue 5919, pp. 1329-1332
DOI: 10.1126/science.1167106


Measurement of the quantum-mechanical phase in quantum matter provides the most direct manifestation of the underlying abstract physics. We used resonant x-ray scattering to probe the relative phases of constituent atomic orbitals in an electronic wave function, which uncovers the unconventional Mott insulating state induced by relativistic spin-orbit coupling in the layered 5d transition metal oxide Sr2IrO4. A selection rule based on intra-atomic interference effects establishes a complex spin-orbital state represented by an effective total angular momentum = 1/2 quantum number, the phase of which can lead to a quantum topological state of matter.

Transition metal oxides (TMOs) with perovskite structure are hosts to many fascinating phenomena, including high-temperature superconductivity (1) and colossal magnetoresistance (2), in which the valence d-electron states are described in terms of crystal-field (CF) states: triply degenerate t2g states (xy, yz, zx) and doubly degenerate eg states (x2y2, 3z2r2). These CF states are all pure real functions, so that when the degeneracy is removed (e.g., by the Jahn-Teller effect), the orbital angular momentum is totally quenched. However, when the spin-orbit coupling (SOC) becomes effective, the CF states are mixed with complex phases, which may partially restore the orbital angular momentum in the t2g manifold. This effect is particularly pronounced in TMOs with heavy 5d elements, where SOC is at least an order of magnitude larger than those of TMOs with 3d elements and can sometimes give rise to unconventional electronic states.

5d TMO Sr2IrO4 is a layered perovskite with low-spin d5 configuration, in which five electrons are accommodated in almost triply degenerate t2g orbitals. Metallic ground states are expected in 5d TMOs because of their characteristic wide bands and small Coulomb interactions as compared with those of 3d TMOs. Sr2IrO4, however, is known to be a magnetic insulator (3, 4). A recent study has shown that the strong SOC inherent to 5d TMOs can induce a Mott instability even in such a weakly correlated electron system (5), resulting in a localized state very different from the well-known spin S = 1/2 state for conventional Mott insulators, proposed to be an effective total angular momentum Jeff = 1/2 state in the strong SOC limit expressed as Math(1) Math where m is the component of Jeff along the quantization axis and σ denotes the spin state. This state derives from the addition of S = 1/2 to the effective orbital angular momentum Leff = 1, which consists of triply degenerate t2g states but acts like the atomic L = 1 state with a minus sign; that is, Leff = –L. As a result, Jeff = 1/2 has orbital moment parallel to spin (6). Note the characteristic equal mixture of xy, yz, and zx orbitals with complex number i involved in one of the factors and the mixed up-and-down spin states (7).

This realization of a Mott insulator with Jeff = 1/2 moment provides a new playground for correlated electron phenomena, because emergent physical properties that arise from it can be drastically different from those of the conventional Mott insulators. A prime example is when Jeff = 1/2 is realized in a honeycomb lattice structure where electrons hopping between Jeff = 1/2 states acquire complex phase; it generates a Berry phase leading to the recent prediction of quantum spin-Hall effect at room temperature (8), and it also leads to the low-energy Hamiltonian of Kitaev model relevant for quantum computing (9). Experimental establishment of the Jeff = 1/2 state is thus an important step toward these physics, and the direct probe of complex phase in the wave function has been awaited. However, it is usually difficult to retrieve the phase information experimentally, because it is always the intensity, the square modulus of the wave function, that is measured; and thus a reference, with which the state under measurement can interfere, is required.

The resonant x-ray scattering (RXS) technique uses resonance effects at an x-ray absorption edge to selectively enhance the signal of interest, and has become a powerful tool for investigating ordering phenomena (10, 11). So far, the emphasis has been seen only in the amplification of the signal. However, the RXS signal contains important information about the phase of the wave function for valence electrons, because RXS results from quantum interference between different scattering paths via intermediate states of a single site. The RXS process is described by the second-order process of electron-photon coupling perturbation, as schematically shown in Fig. 1, and its scattering amplitude fαβ from a single site is expressed under dipole approximation by Math(2) In this process, a photon with energy (ħ)ω is scattered by being virtually absorbed and emitted with polarizations α and β, respectively; and in the course of the process, an electron of mass me makes dipole transitions through position operators Rα and Rβ from and to the initial state i, via all possible intermediate states m, collecting the phase factors associated with the intermediate states, weighted by some factors involving energy differences between the initial and intermediate states (ħ)ωim and the lifetime broadening energy Γ. The interference between various scattering paths is directly reflected in the scattering intensities of the photon, and in this way the valence electronic states can be detected with phase sensitivity. This process can be contrasted with that in x-ray absorption spectroscopy (XAS), which is a first-order process and measures only the amplitudes of the individual paths, or transition probabilities to various valence states.

Fig. 1.

Schematic diagram of the RXS process. The electron makes a trip from the initial to the final state via multiple paths of intermediate states and thereby scatters a photon with initial and final polarization of α and β, respectively. The presence of multiple scattering paths can give rise to interferences among them, which is reflected in the intensity of the scattered photon.

We have applied this technique to explore unconventional electronic states produced by the strong SOC in Sr2IrO4. Sr2IrO4 is an ideal system in which to fully use this technique. The magnetic Bragg diffraction in magnetically ordered Sr2IrO4 comes essentially from scattering by Ir t2g electrons, to which RXS using the L edge (2p→5d) can be applied to examine the electronic states. The wavelength at the L edge of 5d Ir is as short as ∼1 Å, in marked contrast to >10 Å for 3d elements. This short wavelength makes the detection of RXS signals much easier than in 3d TMOs, because there exists essentially no constraint from the wavelength in detecting the magnetic Bragg signal. Moreover, the low-spin 5d5 configuration, a one-hole state, greatly reduces the number of intermediate states and makes the calculation of scattering matrix elements tractable. The excitation to the t2g state completely fills the manifold, and the remaining degrees of freedom reside only in the 2p core holes. Because the intermediate states are all degenerate in this case, the denominator factors involving energies and lifetimes of the intermediate states in Eq. 2 can drop out. A careful analysis of the scattering intensity can show that the wave function given by Eq. 1 represents the ground state in Sr2IrO4 (4).

Figure 2A shows the resonance enhancement of the magnetic reflection (1 0 22) at the L edge of a Sr2IrO4 single crystal (4), overlaid with XAS spectra to show the resonant edges. Whereas there is a huge enhancement of the magnetic reflection by a factor of ∼102 at the L3 edge, the resonance at L2 is small, showing less than 1% of the intensity at L3. The constructive interference at L3 gives a large signal that allows the study of magnetic structure, whereas the destructive interference at the L2 edge hardly contributes to the resonant enhancement.

Fig. 2.

Resonant enhancement of the magnetic reflection (1 0 22) at the L edge. (A) Solid lines are x-ray absorption spectra indicating the presence of Ir L3 (2p3/2) and L2 (2p1/2) edges around 11.22 and 12.83 keV. The dotted red lines represent the intensity of the magnetic (1 0 22) peak (Fig. 3C). Miller indices are defined with respect to the unit cell in Fig. 3A. (B) Calculation of x-ray scattering matrix elements expects equal resonant scattering intensities at L3 and L2 for the S = 1/2 model. For the Jeff = 1/2 model, in contrast, the resonant enhancement occurs only for the L3 edge, and zero enhancement is expected at the L2 edge.

To find out the necessary conditions for the hole state leading to the destructive interference at the L2 edge, we calculate the scattering amplitudes. The most general wave function for the hole state in the t2g manifold involves six basis states, which can be reduced by block-diagonalizing the spin-orbit Hamiltonian as Math(3) With its time-reversed pair, they fully span the t2g subspace. We neglect higher-order corrections such as small residual coupling between t2g and eg manifolds. In the limit of the tetragonal crystal field [Q Ξ E(dxy) – E(dyz,zx)] due to the elongation of octahedra much larger than SOC (λSO) (that is, Q ≫ λSO), the ground state will approach c1 = 1 and c2 = c3 = 0 and become a S = 1/2 Mott insulator, whereas in the other limit of strong SOC, Q ≪ λSO, ci's will all be equal in magnitude, with c1, c2 pure real and c3 pure imaginary as in Eq. 1. Calculation of fαβ for the L2 edge using Eq. 5 gives Math(4) Math Because the magnetic signal comes from the imaginary part of off-diagonal elements (12), the necessary condition for the vanishing intensity is Math(5) This condition places very stringent constraints on the allowed hole state and rules out all single-orbital S = 1/2 models. For the S = 1/2 case, the imaginary part of the off-diagonal elements does not vanish and equal resonant intensities are expected at the L2 and L3 edges (Fig. 2B). Given the constraints in Eq. 5, the scattering intensities at L3 edge are calculated to be Math(6) Math respectively. Thus, taking the phase convention c1 = 1 without loss of generality, the resonant intensity at L3 measures the imaginary part of c3 relative to the real part of c2, which is also a measure of orbital angular momentum. The large enhancement at L3 necessarily implies that the yz orbital is out of phase with the zx orbital, the contrast between L3 and L2 being maximal when the relative phase is π/2. Taking the constraints of Eqs. 5 and 6 together, we conclude that the ground state is very close to the Jeff = 1/2 limit (c1:c2:c3 = 1:1:i), and the minute enhancement at L2 edge shows the smallness of the deviation from the Jeff = 1/2 limit coming from the factors not taken into account (13). The wave function in Eq. 2, representing Jeff = 1/2, indeed gives zero off-diagonal elements in fαβ for L2 and nonzero elements for the L3 edge. This is a direct measurement of phase and provides evidence for the Jeff = 1/2 state in Sr2IrO4.

Having identified the nature of the local moment, we now look at the global magnetic structure using the enhanced signal due to the resonance at the L3 edge. Sr2IrO4 shows a metamagnetic transition below 240 K and, above the metamagnetic critical field HC (≈0.2 T well below 240 K), shows weak ferromagnetism with a saturation moment of ≈0.1 μB/Ir (4). The origin of this field-induced weak ferromagnetism has remained unidentified, because the neutron diffraction data did not show any detectable indication of magnetic ordering (14). Our RXS results indicate that the magnetic structure of Sr2IrO4 is canted antiferromagnetic.

Figure 3A shows the crystal structure containing four IrO2 layers in a unit cell, enlarged by superstructure from the rotational distortion of octahedra (14). Figure 3B shows the magnetic ordering pattern determined from the experiment shown in Fig. 3, C to E. The arrows in Fig. 3B do not represent spins but Jeff = 1/2 moments. In zero field, the magnetic reflections are observed at (1 0 4n+2) and (0 1 4n), which implies that the moments are aligned antiferromagnetically within a layer and the symmetry changes from tetragonal to orthorhombic (Fig. 3C). The canting of the moments yields a nonzero net moment within a layer, which orders in the up-down-down-up antiferromagnetic pattern along the c axis. This is evidenced by the presence of (0 0 odd) peaks shown (Fig. 3D). The width of the peak gives an estimate of interlayer correlation length of 100 c or 400 IrO2 layers. When the magnetic field greater than HC is applied, the peaks at (1 0 4n+2) disappear and new peaks show up at (1 0 odd) (Fig. 3E), which implies that the net moments in the planes are aligned ferromagnetically to produce a macroscopic field. The temperature dependence of the scattering intensity in the weakly ferromagnetic state above HC, shown in Fig. 3F, scales very well with that of the magnetization and confirms again the magnetic nature of the peaks.

Fig. 3.

Magnetic ordering pattern of Sr2IrO4. (A) Layered crystal structure of Sr2IrO4, consisting of a tetragonal unit cell (space group I41/acd) with lattice parameters a ≈ 5.5Å and c ≈ 26Å (4). The blue, red, and purple circles represent Ir, O, and Sr atoms, respectively. (B) Canted antiferromagnetic ordering pattern of Jeff = 1/2 moments (arrows) within IrO2 planes and their stacking pattern along the c axis in zero field and in the weakly ferromagnetic state, determined from the x-ray data shown in (C) to (E) (4). (C and D) L-scan profile of magnetic x-ray diffraction (λ = 1.1Å) along the (1 0 L) and (0 1 L) direction (C) and the (0 0 L) direction (D) at 10 K in zero field. The huge fundamental Bragg peak at (0 0 16) and its background were removed in (D). r.l.u., reciprocal lattice unit. (E) L-scan of magnetic x-ray diffraction (λ = 1.1Å) along the (1 0 L) direction at 10 K in zero field and in the in-plane magnetic field of ≈0.3 T parallel to the plane. (F) The temperature dependence of the intensity of the magnetic (1 0 19) peak (red circles) in the in-plane magnetic field H ≈ 0.3 T. The temperature-dependent magnetization in the in-plane field of 0.5 T is shown by the solid line.

Our study demonstrates that x-rays can be extended to a new level to probe even finer details of magnetic structure. Until now, only the intersite interference effects were used to study ordering phenomena over a length scale of many lattice sites. A quantitative analysis on the interference effects within a single site provides phase information on the constituent wave function of the electron responsible for the magnetism. This technique should find important applications in systems where complex phases give rise to novel physics.

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