Resonant x-ray scattering reveals possible disappearance of magnetic order under hydrostatic pressure in the Kitaev candidate $\gamma$-Li$_2$IrO$_3$

TL;DR

This study uses resonant x-ray scattering to show that applying hydrostatic pressure to $ ext{γ}$-Li$_2$IrO$_3$ causes an abrupt loss of its spiral magnetic order at 1.5 GPa, indicating a pressure-induced transition to a different ground state.

Contribution

First direct observation of the sudden disappearance of magnetic order in $ ext{γ}$-Li$_2$IrO$_3$ under pressure using resonant x-ray scattering, revealing a critical pressure-induced phase transition.

Findings

Magnetic spiral order vanishes abruptly at 1.5 GPa.

No change in lattice symmetry at the transition.

Emergence of a possible nonmagnetic ground state under pressure.

Abstract

Honeycomb iridates such as $\gamma$-Li$_2$IrO$_3$ are argued to realize Kitaev spin-anisotropic magnetic exchange, along with Heisenberg and possibly other couplings. While systems with pure Kitaev interactions are candidates to realize a quantum spin liquid ground state, in $\gamma$-Li$_2$IrO$_3$ it has been shown that the balance of magnetic interactions leads to the incommensurate spiral spin order at ambient pressure below 38 K. We study the fragility of this state in single crystals of $\gamma$-Li$_2$IrO$_3$ using resonant x-ray scattering (RXS) under applied hydrostatic pressures of up to 3.0 GPa. RXS is a direct probe of the underlying electronic order, and we observe the abrupt disappearance of the $q$=(0.57, 0, 0) spiral order at a critical pressure $P_c = 1.5\ $GPa with no accompanying change in the symmetry of the lattice. This dramatic disappearance is in stark contrast with recent studies of $\beta$-Li$_2$IrO$_3$ that show continuous suppression of the spiral order in magnetic field; under pressure, a new and possibly nonmagnetic ground state emerges.

Equations

$$H=\sum _{\begin{array}{c}i,j\\ \gamma \in x,y,z\end{array}}\left(K{S}_i^{\gamma }{S}_j^{\gamma }+J{\vec{S}}_i\cdot {\vec{S}}_j+\dots \right).$$

Full text

*Introduction-*Honeycomb magnetic materials with strong spin-orbit coupling were recently proposed to realize spin-anisotropic “Kitaev” magnetic exchange jackeli_mott_2009 and therefore to host a highly entangled spin-liquid ground state with fractionalized excitations kitaev_anyons_2006 ; witczak_correlated_2014 . Additional interactions (such as Heisenberg or next-nearest-neighbor couplings) that compete with the Kitaev exchange leads to complex landscapes of possible spin orders khaliullin_kitaev-heisenberg_2010 ; singh_relevance_2012 ; lee_theory_2015 ; kimchi_unified_2015 ; winter_challenges_2016 . For example, the layered compounds Na2IrO3 and RuCl3, composed of edge-sharing IrO6 or RuCl6 octahedra, show a “zig-zag” spin texture that onsets below 10-15 K chaloupka_zigzag_2013 ; johnson_monoclinic_2015 ; sears_magnetic_2015 . The three-dimensional (3D) ‘harmonic’ honeycomb $\beta$ and $\gamma$ polytypes of Li2IrO3 modic_realization_2014 ; takayama_hyperhoneycomb_2015 both exhibit an incommensurate spiral order biffin_unconventional_2014 ; biffin_noncoplanar_2014 . These magnetic ground states derive from the balance of Kitaev (K), Heisenberg (J), and other possible couplings between spins $\vec{S}_{i}$ in the Hamiltonian

[TABLE]

Here the $\gamma=x,y,z$ Kitaev exchange directions couple spins that are perpendicular to the planes formed by adjacent IrO6 octahedra, shown for $\gamma$-Li2IrO3 in Fig. 1AB.

Precisely tuning the relative strength of these magnetic interactions remains an outstanding challenge. Recent resonant x-ray scattering (RXS) studies of $\beta$-Li2IrO3 indicate that a magnetic field both suppresses the spiral order and stabilizes a canted zig-zag spin texture ruiz_beta_2016 , while magnetic circular dichroism studies of $\beta$-Li2IrO3 under hydrostatic pressure (and also in applied fields) suggest a disappearance of the magnetic order near 2 GPa. Studies of Sr2IrO4 indicate quenching of the ferromagnetic moments at 17 GPa haskel_pressure_2012 and demonstrate that hydrostatic pressure can be a useful control parameter in these materials. However, to date there have been no direct probes of the spiral (or other non-ferromagnetic) order under pressure in the honeycomb iridates.

An unambiguous probe of electronic orders resolved in $Q$-space is RXS. In this technique, the energy of incident x-rays is tuned to be on resonance with an element’s absorption edge and so it is sensitive to both charge and magnetic order of the valence electrons Hannon . At the Ir $L_{2,3}$-edge, for example, the intermediate states in the scattering process are sensitive to both the spin and orbital character of the 5d hole states. RXS is particularly useful when neutron scattering is rendered unfeasible by small sample sizes or elements with large neutron absorption cross sections like iridium. Thus, the enhanced cross section at the Ir $L_{3}$-resonance along with the large Ewald sphere afforded by 11.215 keV x-rays have made RXS the best suited scattering technique to investigate the ambient pressure magnetic order in the honeycomb iridates $\gamma$-Li2IrO3 biffin_noncoplanar_2014 , $\beta$-Li2IrO3 biffin_unconventional_2014 , and Na2IrO3 Liu_lio_RXS . Conducting RXS at the Ir $L_{3}$-edge under applied pressure is possible in reflection and transmission modes haskel_sio , but both modes are constrained by the geometry of a high pressure apparatus and resulting limitations on the sample geometry and available reciprocal space. Hence magnetic RXS at non-ambient pressures represents an acute experimental challenge.

In this work, we modify the magnetic ground state in the 3D honeycomb iridate $\gamma$-Li2IrO3 by applying hydrostatic pressure and observe the disappearance of the incommensurate spiral order using RXS. In contrast to recent studies that use symmetry-breaking magnetic fields takayama_hyperhoneycomb_2015 ; ruiz_beta_2016 , here we observe disappearance of the spiral order with no observed discontinuity in the lattice structure to the highest pressures measured. The abrupt disappearance of the spiral Bragg peak at a critical pressure $P_{c}=1.5$ GPa signals the transition to a distinct electronic ground state.

*Experimental-*We track the disappearance of the spiral magnetic order in $\gamma$-Li2IrO3 using RXS on oriented single crystal samples. The transmission scattering scheme, shown in Fig. 1C, allows access to the ($H$ 0 0) direction in specular diffraction using horizontally ($\mathbf{\sigma}$) polarized photons in a vertical geometry and with no polarization analysis of the scattered beam. The magnetic scattering intensity, proportional to $|\sum_{i}e^{i\mathbf{Q}\cdot\mathbf{r}_{i}}(\mathbf{\sigma}\times\mathbf{\pi}_{out})\cdot\mathbf{m}_{i}|^{2}$ Hannon , where $\mathbf{m}_{i}$ is the magnetic moment at site $r_{i}$, projects out the component of $\mathbf{m}_{i}$ parallel to k′ (in the a-c scattering plane).

Crystals of $\gamma$-Li2IrO3 were grown as described previously modic_realization_2014 ; ruiz_beta_2016 . Figure 1B shows the intersecting Ir honeycomb layers in one unit cell of the orthorhombic Cccm crystal structure. For the transmission (Laue) scattering geometry, we polished single crystals to 20-30 $\upmu$m thick; the absorption length of x-rays near the Ir L2,3 edges is $\approx 10$ $\upmu$m.

X-ray scattering studies under pressure (up to 5 GPa) and at temperatures between 5-300 K were performed at beamline 4-ID-D of the Advanced Photon Source at Argonne National Lab. Merrill-Bassett type diamond anvil cells (DAC) with 800 $\upmu$m culets were used with stainless steel gaskets of 250 (150) $\upmu$m initial (pre-indented) thicknesses, with 400 $\upmu$m sample chamber holes Feng_High_2010 . The gaskets were loaded with $\gamma$-Li2IrO3 single crystals (cross-sectional area $150\times 100$ $\upmu$m2, polished thickness $25$ $\upmu$m), along with several ruby balls and $40\times 40$ $\upmu$m2 pieces of 12 $\upmu$m thick Ag foil for ambient- and low-temperature pressure calibration Holzapfel_Equations . The pressure medium was a 4:1 methanol:ethanol mixture. After initial loading, the pressure at ambient temperature was monitored using a custom-built optical spectrometer and a Raman system to measure the ruby $R$1 fluorescence peak; the target pressure upon loading was $\sim$0.1 GPa. The pressure at low temperature was determined in situ using Ag powder peaks and the isothermal bulk modulus of Ag at 5 K ($K_{\textrm{Ag}}$=110.85 GPa, $K^{\prime}_{\textrm{Ag}}$=6.0 GPa) Holzapfel_Equations . We estimate a systematic uncertainty of $\pm$0.1 GPa in the pressures quoted below by comparing the estimated pressure from (1 1 1), (2 0 0), and (2 2 0) Ag powder peaks, and from repeated pressure measurements before and after scans.

Two samples were cooled, aligned in the diffractometer, and studied; all measurements reported here were performed at the cryostat base temperature of 4.5$\pm$0.5 K. Pressure was changed in situ using a helium membrane. Cell layout and sample status were checked after loading (see optical image in Fig. 1D, showing diamond-shaped $\gamma$-Li2IrO3 crystal, Ag foil, and ruby spheres in the DAC gasket hole) and monitored in-situ using both x-ray transmission maps (Fig. 1E, obtained using a slit-defined $30\times 30\ \mu$m2 beam). To track the absolute magnetic Bragg peak intensities with pressure, peak areas were normalized to the integrated (4 0 0) rocking curve intensities. The mosaic full-width at half-maximum is 0.05-0.10*∘* for sample 1 between 0-3 GPa, and 0.01*∘* for sample 2 at 2.0 GPa.

*Results-*Based on the restricted scattering geometry imposed by the DAC, we focused our study on (H 0 0) peaks in reciprocal space. In $\gamma$-Li2IrO3, selection rules forbid the (1 0 0) and (3 0 0) lattice peaks, while the structure factor for the (2 0 0) order is strongly suppressed. Figure 2A shows a reciprocal space map of the (H 0 0) axis, where an intense structural (4 0 0) Bragg peak and weaker (2 0 0) peak are both observed. (In the $\beta$-Li2IrO3 polytype the (2 0 0) peak is forbidden.) Non-structural peaks located at (4$\pm$0.57 0 0) correspond to the incommensurate spiral magnetic order biffin_noncoplanar_2014 ; biffin_unconventional_2014 , which at ambient pressure onsets at $T_{\textrm{sp}}=38$ K. The electronic nature of these peaks is confirmed by fixed-$Q$ energy scans (Fig. 2C) showing a strong enhancement of the diffracted intensity near the 11.215 keV Ir L3 resonance, in contrast to the weak background observed away from the spiral order peak at (3.85 0 0) + q that shows an increase of $\sim$10% above the Ir edge due to fluorescence.

Figure 3 shows the pressure dependence of the (4 0 0) Bragg peak (normalized), as well as the spiral order peak. Note that the shift in the 3.3 GPa (4 0 0) scan is a consequence of a large pressure increment; the a-axis lattice parameter evolves linearly with pressure over the entire range studied. Neither the (4 0 0) Bragg peak nor q peak widths change appreciably with increasing pressure, indicating that both the crystal quality and long range nature of the spiral order remain constant. Figure 4D shows the evolution of the a-axis lattice parameter with pressure, along with a linear fit yielding $d\textit{a}/dP=-0.015\textrm{\AA}/\textrm{GPa}$. Assuming an isotropic fractional change in the unit cell dimensions, the T = 4.5 K bulk modulus $K=-\frac{1}{3}\textit{a}(\frac{d\textit{a}}{dP})^{-1}=130\pm 20$ GPa. Recent electronic structure calculations for $\beta$-Li2IrO3 indicate an anisotropic compressibility Kim_Revealing with a more compressible than c; this would increase $K$ relative to the above estimate and be quite comparable to the 150-250 GPa observed in Sr2IrO4 and Sr3Ir2O7 compounds haskel_sio ; zhao_pressure_2014 , and recent studies of $\beta$-Li2IrO3 powders haskel_beta .

Under an applied pressure of $P_{c}=1.5$ GPa, the spiral order peaks are abruptly extinguished as shown in Fig. 2B for peaks at (4 0 0) + q. We scanned the entire accessible range of 2$<$H$<$6 r.l.u. and found no evidence for the incommensurate peaks anywhere in this region. Aside from contraction of the unit cell (shown in Fig. 4D), no structural changes were observed; the symmetry of the lattice appears intact throughout this pressure range. Figure 4A shows the integrated peak intensity for q as a function of applied pressure, measured at 4.5 K incrementally beginning at 0.5 GPa. To within the uncertainty associated with consistent re-alignment of the sample after changing pressure, the peak intensities gradually increase with pressure until abruptly disappearing at $P_{c}$. An (otherwise identical) additional sample also showed no sign of the spiral order at the after cool-down pressure of 2.0 GPa. The incommensurate wavevector q, normalized to the change in lattice constant, decreases with increasing pressure as shown in Fig. 4B; q decreases by 0.3% before the spiral order disappears at $P_{c}$. The linear and continuous decrease observed in q, ending at an apparently irrational fraction, precludes an incommensurate-to-commensurate transition.

*Discussion-*Figure 4C presents a schematic pressure-temperature phase diagram for $\gamma$-Li2IrO3. Ongoing studies of this material indicate paramagnetic behavior with rapidly emerging magnetic anisotropy favoring the b (easy) axis direction modic_realization_2014 ; ruiz_betagamma_2017 at ambient pressure. With increasing pressure and as T$\rightarrow$0, $P_{c}$ likely continues to represent a sharp phase boundary between the spiral magnetic order and the as-yet undetermined high-pressure electronic phase. As there is no change in the lattice symmetry, and no symmetry-breaking field being used to perturb the material, it is unclear what ordered state may exist beyond $P_{c}$, if any. Under an applied magnetic field, a canted zig-zag order appears to be the next competitive magnetic ground state at ambient pressure ruiz_beta_2016 . However the transition observed under applied fields appears to be continuous, in contrast to the transition at $P_{c}$, indicating that a different ground state emerges in this case.

The effect of pressure on the crystal structure and associated Kitaev, Heisenberg, and other exchange couplings, was recently studied theoretically for $\beta$-Li2IrO3 Kim_Revealing . In this closely related polytype, the Kitaev exchange coupling was predicted to decrease with increasing pressure above ambient, disappearing at 5-10 GPa. If similar evolution of the magnetic interactions appears in $\gamma$-Li2IrO3, our results strongly suggest that the spiral magnetic order is stabilized by Kitaev exchange and suppressed as this mechanism weakens. Precise experimental description of structural distortion with increasing pressure in both Li2IrO3 polytypes will allow for a quantitative analysis of how the change in structure serves to push this material closer to or farther from the pure Kitaev limit.

The scale of $P_{c}$ is modest compared to the 17 GPa required to suppress weak ferromagnetism in Sr2IrO4 haskel_pressure_2012 . The pressure roughly corresponds to an energy density of 9 meV/Å3, or 0.08 eV/Ir; while less than both the spin-orbital energy $\lambda_{\textrm{SO}}\sim 0.2-0.5$ eV as well as electronic interaction U$\sim 0.5$ eV that have been reported in 5d iridate materials lagunamarco_orbital_2010 ; haskel_pressure_2012 ; clancy_spin-orbit_2012 , it is beyond the $\sim 1$ meV scale that was proposed to separate $\beta$-Li2IrO3 from a 3D spin liquid state katukuri_vicinity_2016 . Mixing of the (nominally filled) J$=3/2$ manifold of states could serve to disrupt the J${}_{\textrm{eff}}=1/2$ doublet that is crucial to the stability of unconventional magnetic orders in these materials. Such a picture could be investigated quantitatively with high-pressure studies of x-ray absorption (XAS) and magnetic circular dichroism (XMCD) spectroscopies that provide a quantitative probe of the spin and orbital components of the local magnetic moments.

*Summary-*By conducting resonant x-ray scattering studies at the Ir $L_{3}$ edge, we are able to observe the disappearance of the spiral magnetic order in $\gamma$-Li2IrO3 at an applied pressure of 1.5 GPa. This observation provides strong evidence for tunability of the Kitaev, Heisenberg, and other magnetic exchange couplings with applied pressure. Future resonant diffraction studies will be able to incisively address the possibility of complete disappearance of long range magnetic order in the high pressure ground state of this Kitaev candidate material.