Competition between magnetic order and charge localization in Na₂IrO₃ thin crystal devices

Sodium iridate (Na₂IrO₃) and other 5d iridates have been widely studied in recent years as they have been identified as experimental realizations of the Kitaev model, an exactly solvable model that describes a set of spin-1/2 moments in a honeycomb lattice with highly anisotropic exchange interaction and exciting ground states such as spin liquids Chun ; Singh ; Winter . In real materials such as Na₂IrO₃, the existence of Heisenberg and off diagonal interactions benefit the appearance of magnetically ordered ground states over a spin liquid. Recent studies have explored the effect of external pressure and high magnetic fields on the ground states of Na₂IrO₃ 6Proposal ; 7Proposal ; 8Proposal . First principle calculations 6Proposal have found that under high pressure, Na₂IrO₃ goes through successive structural and magnetic phase transitions, some of them zigzag magnetic ordered and some nonmagnetic, all with low energy excitations that can be well described by j_eff=1/2 states and in particular having some phases (with a structure corresponding to space group $P\bar{1}$)6Proposal resembling a gapped spin liquid Kitaev state. Similarly, magnetic torque measurements have found that at magnetic fields up to 60T, long range spin correlations functions decay rapidly pointing to a field-induced quantum spin liquid 8Proposal . In the absence of high pressure or magnetic fields, recent inelastic x-ray scattering measurements have reported a proximate spin liquid regime above the long-range ordering temperature, where fractional excitations are emergent, revealed by spin-spin correlations restricted to nearest neighbor sites, among other factors Ravelli .

Older and newer studies on Na₂IrO₃ have been limited to probes that require bulk crystals, where the insulating character of Na₂IrO₃ has hampered its integration to an electronic device, advantageous for the study and manipulation of quasiparticle excitations, of interest in topological quantum computingKitaev ; Kasahara . Here we report electronic transport measurements on thin crystals of Na₂IrO₃, where electronic transport is ruled by variable range hopping at temperatures above the magnetic ordering transition known for bulk Na₂IrO₃ ($\sim 15$ K) and electric field assisted above a critical applied field. Fits of our data to these two mechanisms allow us to deduce the localization length as well as the density of states at the Fermi level in our Na₂IrO₃ devices. Our Angle Resolved Photoemission Spectroscopy (ARPES) measurements show a non-vanisihing density of states at the Fermi level, consistent with our electronic transport experiments.

Bulk crystals of Na₂IrO₃ were grown mixing elemental Ir ($99.9\%$ purity, BASF) with Na₂CO₃ ($99.9999\%$ purity, Alfa-Aesar) in a $1:1.05$ molar ratio. The mixture was ground for several minutes and pressed into a pellet at approximately $3,000$ psi. The pellet was subsequently warmed in a furnace to $1050\,^{\circ}\mathrm{C}$ and held at this temperature for 48 hours, before being cooled to $900,^{\circ}\mathrm{C}$ over 24 hours and finally furnace-cooled. Single crystals of more than one square millimeter were collected from the surface of the pellet. Crystals were exfoliated onto a Si/SiO₂(280 nm) substrate and immediately coated with e-beam resist, minimizing the exposure to air.

Some wafers with exfoliated crystals were left uncoated and promptly analyzed through Raman spectroscopy. Raman scans were made over a range of 56 $cm^{-1}$ to 800 $cm^{-1}$. Three Raman active modes were observed at 454 $cm^{-1}$, 484 $cm^{-1}$ and 561 $cm^{-1}$ (Figure 1a) consistent with the reported first-order Raman modes of bulk sodium iridate Gupta . Exfoliated crystals were about $\sim$100 nm thick, measured through atomic force microscopy (Figure  1b and c). Electrodes were patterned on the ebeam resist-coated wafers using standard electron beam lithography, previous to the deposition of Ti(4 nm)/Au(140 nm) using electron beam evaporation.

Current-voltage (IV) characteristics were measured (two probe) in a closed cycle cryostat (Oxford Instruments) from 300 K down to 1.5 K. As shown in Figure 2a, at high temperatures the IV curves show an ohmic-like behavior. As the temperature is lowered, the slope of the IV curves decreases indicating an increase of the device’s resistance. At sufficient low temperatures, the IV characteristics become non-linear (Figure 2b). This behavior is consistent with the known Mott-insulating character of Na₂IrO₃. Previous numerical works that consider both Coulomb interactions and spin-orbit coupling Comin have been able to explain the observed insulating gap in Na₂IrO₃Comin ; Sohn considering a Coulomb repulsion U (3eV) that is larger than the effective bandwidth ($\approx$ 1eV), supporting the idea of Na₂IrO₃ being a spin-orbit assisted Mott insulator.

Being an accurate measurement of the conductance of the sample at zero bias non-accessible at low temperatures, we tracked the change of the current across the sample at non-zero bias as a function of the temperature in a range of 3K-120K (see Fig. 3), and compared it to the thermal activation theory, characterized by an Arrhenius law,

where E_A is the activation energy and $k_{B}$ is Boltzmann constant. Figure 3a shows $I$ versus $1/k_{B}T$ on a semi-log plot for different bias voltages. While the data fits in small range of temperatures to a thermal activated behavior (at around 120K), it follows in a more extended range a three-dimensional variable range hopping (VRH) mechanism Mott ($\approx 15K$-120K), as shown in Figure 3b. Mott’s variable range hopping (VRH) law considers 3-dimensional hopping between remote sites whose energy levels happen to be close to the Fermi level $\mu$, as spatial neighboring sites have larger resistances BookES9 .

Typically, the hopping length increases with lowering temperature, therefore the name of variable range hopping BookES9 . In general, Mott’s VRH is described by

with $\nu=1/4$ for three-dimensional VRH and $T_{o}$ the energy scale related to localization length of the charge carriers,

with $a$ the localization length and $g_{0}$ the density of states at the Fermi level. A fit of the data at low bias (50mV) to three-dimensional VRH yields $T_{o}^{1/4}=31K^{1/4}$ (see Fig. 3b), similar to the reported values for heteroepitaxial Na₂IrO₃ thin films Jenderka .

Inspecting more carefully the data in Fig. 3b at different bias voltages, two main effects are visible. First, there is a change in the trend of the curves near $\sim 15K$ (grey line in Figures 3a and b), corresponding to the critical temperature of the long range antiferromagnetically ordered state reported for bulk crystals after magnetic susceptibility and heat capacity measurements Singh2010 ; Liu2011 , neutron and x-ray diffractionYe . The onset of long range magnetic ordering seems here to have an effect on the charge transport mechanism in the thin crystals of Na₂IrO₃, possibly suppressing VRH below the transition temperature. A similar phenomenon has been observed in the temperature dependence of the resistivity of heteroepitaxially grown thin films of Na₂IrO₃ Jenderka . Also, non-equilibrium optical measurements have shown signature of a long range ordered state in photoinduced reflectance measurements Hinton . Second, as the bias voltage increases, $\ln(I)$ vs $T^{-1/4}$ deviates from a linear dependence, indicating also a divergence from a VRH mechanism.

We can understand effect 1 by taking a closer look to VRH. Following the model by Miller and Abrahams BookES4 , the resistivity $R_{ij}$ resulting from hopping over sites ij is given by:

where $r_{ij}$ is the spatial distance to the nearest neighboring site and $\epsilon_{ij}$ correspond to the difference of the hopping sites energies $\epsilon_{i}$ and $\epsilon_{j}$ lying in a narrow band near the Fermi energy, whose width $\epsilon_{o}$ decreases with temperature and depends on the density of states at the Fermi level g_o and localization length $a$,

Hopping between sites depends on both the spatial and energetic separation of the sites. If $2r_{o}/a>>1$ (with $r_{o}$ the average spatial distance to the nearest neighbor empty size), the conduction mechanism is reduced to nearest neighbor hopping. If on the other hand $2r_{o}/a$ is of the order or less than unity, the second term in equation 4 contributes the most to the resistance and hops to sites that are further away in space but closer in energy are favored. Hopping between sites is in this case variable range, which corresponds to the transport mechanism in our thin crystals of Na₂IrO₃ above the magnetic ordering transition. In fact, Mott’s VRH relation (equation 2) is derived by finding the energy $\epsilon_{ij}$ from equation 4 at which the resistivity is minimum. In the case of a magnetic ordered state, previous works have introduced an extension of Mott’s VRH model in which the hopping energy $\epsilon_{ij}$ has an additional term related to the relative magnetization of the hopping sites Wagner . It is however unclear if associating a cost on the hopping energy based on the spin orientation of the hopping sites is valid, as carriers have tendency to align their spin to the localized spin CommentWagner ; DeGennes . Based on our data, we believe that electronic transport in our Na₂IrO₃ devices in the magnetic ordered regime goes beyond Mott’s VRH model.

Effect 2 caused by the bias voltage, can be unveiled by adding the effect of an electric field $E$ to equation 4, obtaining

At high electric field strengths, where $|er_{ij}E|\geq\epsilon_{ij}$, hopping in the direction of the electric field will compensate the difference energy between hopping sites, $\epsilon_{ij}$. Therefore hops between sites that are closer in energy are no longer energetically favorable and the first term in equation 6 dominates. The average distance between hopping sites $r$ is expected to satisfy $|erE|=\epsilon_{ij}$. Taking $\epsilon_{ij}\sim 1/r^{3}g_{o}$ (where $g_{o}$ is the density of states at the Fermi level) the average distance between hopping sites becomes $r\sim(1/eEg_{o})^{1/4}$ giving rise to a field assisted transport mechanism Shklovskii

with $E_{o}$ the characteristic localization field.

Above a critical external electric field, carrier hopping is therefore mediated by the electric field and there is no longer a clear VRH mechanism, as observed for the highest fields in Figure 3b. In fact, there is an electric field assisted transport at high fields that can be observed in the data, which is accentuated at lower temperatures as is shown in Figure 4, where IV curves at different temperatures are represented following relation 7. Below a critical electric field, transport is no longer field assisted. A fit of the data represented in Fig.4 to equation 7 allow us to extract $E_{o}$. In particular, for 20K we obtain $E_{o}^{1/4}=190(V/m)^{1/4}$.

From the definitions of $E_{o}$ (equation 8) and $T_{o}$ (equation 3), we are able to deduce a localization length $a\approx 3nm$ and a density of states $g_{o}\approx 10^{25}/eVm^{3}$ in our Na₂IrO₃ thin crystal devices. Localization length is about 10 times larger than the one reported for heteroepitaxial thin films, calculated using a density of states estimated for other iridates Jenderka . Our analysis provides a value for both the localization length and the density of states in Na₂IrO₃ thin crystals deduced from the same set of data.

Mott’s VRH requires a non-vanishing density of states at the Fermi level, as hopping between sites are concentrated in a band near the Fermi energy with a bandwidth $\epsilon_{o}$ (eqn. 5). Our ARPES measurements confirm this in our samples.

ARPES measurements were performed at Beamline 4.0.3 (MERLIN) at the advanced light source using $90$ eV photons. Vacuum was better than $5\times 10^{-11}$ Torr. The total energy resolution was $20$ meV with angular resolution ($\Delta\theta\leqslant 0.2^{\circ}$). Data was taken at 290K to avoid charging of the sample. A fresh crystal of Na₂IrO₃ (as those used for the electronic device fabrication) was mounted and cleaved in-situ the ARPES chamber. Figure 5a shows the measured electronic band structure along the high symmetry direction $\Gamma$-M with a photon energy of 90eV. A feature near the Fermi energy is visible, as reported in previous ARPES works. Initially, this in-gap feature was identified as a metallic surface state Alidoust but later on it was found through spatially resolved-ARPES measurements Moreschini that it was consequence of quasiparticle formation in one of the possible cleavage planes of Na₂IrO₃. Indeed, the edge sharing IrO₆ octahedra in Na₂IrO₃ form a layered stacking alternating with pure Na layers, which gives rise to both Na and Ir-O terminated surfaces after cleavage. In the former, the Ir-O octahedra being embedded between two Na layers, create a charge transfer from Na to Ir giving rise to a quasiparticle at the Fermi level Moreschini . This band formed at one of the possible terminations in Na₂IrO₃ after cleavage (the Na-terminated) is what we believe mediates the VRH mechanism in our Na₂IrO₃ devices. In the frame of Mott’s VRH, the width of the band responsible for conduction (eqn. 5) at 290K, corresponding to the ARPES measurements, is $\approx$ 90meV, which falls within the quasiparticle observed in Fig. 5. Fig. 5b shows energy distribution curves (EDC) in correspondance of the quasiparticle (in black, at $\Gamma$) and at a larger momentum outisde (in red) that testifies a non-vanishing density of states at the Fermi level, as required by Mott’s VRH.

After cleavage, the Na and O terminated surfaces are approximately 10-40 $\mu m$ large, as demonstrated through $\mu$ARPES Moreschini . Our ARPES data being taken with a spotsize of $\approx 100\mu m$, captures an average of the two terminations. However, the spectral weight measured in the vicinity of the Fermi level is to be attributed to the Na-terminated surface only, since the Ir-O surface has no states in that energy rangeMoreschini . This does not invalidate any of our conclusions and in fact supports them, since this density of states is also representative of the bulk, which we probe with our transport measurements. The band structure characteristic of the Ir-O cleave that lacks of states near the Fermi energy, is instead an ”anomaly” given the absence of the Na layer which causes a difference charge balance between the two topmost layers of the cleave Moreschini . Therefore, the electronic structure as measured by ARPES, both here and in refs. Alidoust, and Moreschini, , does show a small density of states within a small range from the Fermi level.

In conclusion, we have integrated a thin crystal of Na₂IrO₃ into an electronic device, presenting the same Raman active modes as for the bulk crystals. We have observed that the main transport mechanism at temperatures above the formation of an antiferromagnetic long range order ($\approx$15K) is VRH. This observation is supported by ARPES measurements at the high temperature end (290K) that testify a non-vanishing density of states at the Fermi level as required by Mott’s VRH. We have observed that as the system approaches long range order, it deviates from VRH with no evidence of this mechanism below the ordering temperature. Similarly, we have found that in the presence of an external electric field, transport diverges from VRH, becoming field assisted. Contrasting our data to the Mott’s VRH and electric field assisted models allow us to deduce a localization length of $\approx 3nm$ and a density of states of $\approx 10^{25}$/eVm³ in our Na₂IrO₃ thin crystal devices. Our work constitutes a first approach to integrate an exfoliated thin crystal of Na₂IrO₃ into an electronic device, where separate ARPES measurements inform the electronic transport experiments. Using these two experimental techniques independently on similarly prepared samples can unveil important properties in other layered materials.