Terahertz nano-imaging of metal-insulator transition in Cd₂Os₂O₇

Interactions between charge, spin, lattice and/or orbital degree of freedom have been the central topic in the research of meta-to-insulator transition (MIT) in transition metal oxides (TMO) [1,2]. Cd₂Os₂O₇ [3–5] along with R₂Ir₂O₇ (R = Nd, Sm, Eu, Gd, Tb, Dy and Ho) [6,7] belongs to a special group of TMO whose MIT is argued to be driven by the concurrent emergence of the magnetic ordering of their 5d electrons. In Slater's theory [8], a second-order MIT in a magnetic system can be driven by the doubling of the unit cell due to the collinearly aligned opposite spins. Hints for the Slater transition were identified in early experiments on Cd₂Os₂O₇ [5]. However, instead of the collinear ordering, numerous experimental studies [9–15] confirmed that the magnetic ground state of pyrochlore magnetic insulators is all in/all out (AIAO) where 4 spins point inward/outward from the center of the corner-shared tetrahedral. The AIAO ground state in 5d pyrochlore iridates was predicted to support the topological Weyl semimetal phase [16–18] with a gapless surface state [16]. Therefore, a plethora of new states and phenomena [19] are expected in Cd₂Os₂O₇, especially considering the coexistence of the electron correlation. The undistorted AIAO ground state challenges the Slater scenario but can be better described in terms of the Lifshitz-type transitions [11]. The latter scenario involves the continuous shift of conduction and valence band, which was supported by optical [20] and Raman spectroscopic studies [21] of Cd₂Os₂O₇.

Conducting domain walls (DWs) and charged line defects have been predicted in magnetic and antiferromagnetic insulators [22–24]. THz time-domain spectra on untrained and trained pyrochlore iridates hinted a Drude-like conductivity residing in the antiferromagnetic domain walls between AIAO and AOAI domains [25]. Microwave impedance microscopy successfully captured the existence of the conducting DWs in Nd₂Ir₂O₇ and their disappearance under the high magnetic field [26]. Cd₂Os₂O₇, on the other hand, has been reported to exhibit controllable domain structures via cooling under the magnetic field [14]. Far-field optical studies also revealed an anomalous increase of THz optical conductivity from 150 K to 25 K [5]. This motivates the search for similar nano-scale evidence of the conducting DWs in Cd₂Os₂O₇ using low-energy probes.

To explore the nano-scale electrodynamics of the MIT and the DWs on Cd₂Os₂O₇, we utilized a scattering-type THz scanning near-field optical microscope (THz-SNOM) [27–34]. This technique is a combination of atomic force microscopy (AFM) and THz spectroscopy where the AFM tip scatters the local electric field into detectable far-field radiation. A pair of GaAs photoconductive antennas (PCA) is activated by a 780 nm femtosecond fiber laser to generate and detect the THz radiation. The scattered field is demodulated at the first and second harmonic of the tip-tapping frequency which results in a scattering amplitude denoted by $S_{1,2}\propto | \tilde{E}^{\mathrm{NF}}|$. The near-field scattering amplitude carries information on the spatially localized electromagnetic response [35–38]. By probing the scattering amplitude, we overcame the diffraction limit and managed to image the nano-scale textures of the sample, especially the assumed DWs. The SNOM technique has been successfully applied to explore MIT in Mott materials like VO₂ [27,39,40], V₂O₃ [41] and NdNiO₃ [42]. In this experiment, we conducted nano-THz imaging from 230 K to 43 K to track the MIT in Cd₂Os₂O₇ and searched for evidence of the possible DW conductivity.

Here, we focus on the frequency-integrated ("white-light", WL) near-field signal within the 0.5–1.5 THz range. Apart from the 100 cm ⁻¹ (3 THz) and 200 cm ⁻¹ (6 THz) phonons, the optical conductivity of Cd₂Os₂O₇ in the THz range is almost frequency independent [5,20]. As a result, the WL signal is sufficient to track the THz response of MIT in Cd₂Os₂O₇. Besides, high-fidelity WL images can be acquired within a reasonable time scale while the sample surface remains pristine. Therefore, we chose the WL signal instead of the time/frequency domain spectra [43] to present the near-field mapping of the sample surface.

The scattered signal is modulated at the tip oscillation frequency and the demodulation at the first (S₁) and second (S₂) harmonics are collected. The typical signal-to-noise ratio (SNR) of S₁ images (∼30:1) is three times that of S₂ images (∼10:1). In principle, higher-order demodulations better suppress far-field scattering [35] and therefore deliver near-field images with genuine local response and finer resolution. In nano-THz, however, the mismatch between the long wavelength of the THz beam and the short tip shank efficiently suppresses the far-field component [35,38]. Over 90%of the signal in $S_{1}\,(S_{2})$ results from the tip scattering within 500 nm (150 nm) above the sample surface. Therefore, we consider S₁ and S₂ both carry local sample responses at the vicinity of the near-field probe.

We investigated the THz near-field response of high-quality Cd₂Os₂O₇ single crystals as a function of temperature across MIT. The measurement was performed on the as-grown {111} and {100} surfaces [14], on which a 20 nm gold thin film is partially coated (fig. 1(a)). We treated the gold films as good THz reflectors [28] and used them to normalize the signal on Cd₂Os₂O₇ crystals for quantitative analysis [27]. Besides the edge of the gold film, the other prominent topographical feature in fig. 1(b) is the straight diagonal lines across both gold and Cd₂Os₂O₇, presumably arising from the corrugation of the crystal surface. These corrugations are widespread in our samples. In fig. S2 in the Supplementary Material Supplementarymaterial.pdf (SM), we show that the topographic contrast at the corrugation is below 10 nm. Nevertheless, these features impact the near-field signal and yield dark diagonal lines in near-field images (fig. 1(c)). In some regions, these corrugations are too dense (fig. S2 in the SM) to achieve a proper characterization of the near-field response of the crystal. Hence, we specifically chose the region shown in fig. 1(b) and (c), where the corrugation density is low, to perform the systematic near-field study. Smaller areas with uniform signals are chosen to carry out the statistics of the near-field signal for quantitative analysis.

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The temperature dependence of the THz near-field response of Cd₂Os₂O₇ single crystals below the nominal transition temperature $T_{MIT}=225\ \text{K}$ [4,11] is shown in fig. 2. At 220 K, the near-field signal of Cd₂Os₂O₇ is comparable to that on the Au thin film (fig. 2(a) and (b)). The high signal level persists down to 190 K as shown in fig. 2(a) and (b). At lower temperatures, $S_{1}\,(S_{2})$ on Cd₂Os₂O₇ decreases gradually until reaching 92%(86%) of the Au signal level at T = 43 K. We found that the signal is mostly uniform in Cd₂Os₂O₇ at all temperatures, except for the regions impacted by corrugations or defects. No obvious DWs can be observed.

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Quantitative analysis of the temperature dependence of the averaged near-field signal in Cd₂Os₂O₇ is displayed in fig. 3(a) and (b). The histograms of the near-field signal are plotted in fig. 3(c) and (d). A gradual decrease in the near-field signal at T < 190 K is clearly visible. The histogram helps to better understand the nature of the MIT transitions [27,41,42]. In general, a first-order phase transition is characterized by an abrupt change of near-field signals accompanied by the percolation of insulating or metallic phases [27,41,42]. In fig. 3(c) and (d), we instead see a gradual evolution of the near-field signal at THz frequencies, resembling a 2nd-order phase transition [42].

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We used the lightning rod near-field model to fit the data following the procedure described in ref. [44]. Due to the lack of optical conductivity data at THz frequencies, we started with the direct-current (DC) conductivity of Cd₂Os₂O₇ [4]. We assumed a simple Drude response with DC conductivity $\sigma_{0}$ and scattering rate γ at each temperature:

$$\begin{equation} \sigma (\omega )=\frac{\sigma_{0}}{1-i\omega /\gamma}. \end{equation} \tag{ 1 }$$

The DC conductivity $\sigma_{0}$ is adopted from experiments [4] and the scattering rate γ is assumed to be large compared to the investigated frequencies. As a result, $\epsilon_{1}\approx \epsilon_{\inf }$ and $\epsilon_{2}\approx \frac{\sigma_{0}}{\omega }$ (Supplementary Note 3 in the SM) at THz frequencies. This assumption is supported by the optical measurement [20] where the carrier density at the Fermi energy gradually changes due to the shift of the conduction and valence band. The result is indicated as solid lines in fig. 3(a) and (b), which matches perfectly with the experiment result.

We note that there is an apparent inconsistency between the lower "transition temperature" (∼190 K) observed in the near-field experiments and the values reported by transport and far-field experiments. This can be well explained by the signal response function of near-field probes. Cd₂Os₂O₇ has a relatively high conductivity which yields a near-saturated near-field signal at high temperatures above 190 K. Only when the temperature is reduced below 190 K, the conductivity of the material is low enough to yield a good contrast to the metallic phase. In ref. [20], the plasma frequency $\omega_{p}$ extracted from the extended Drude model implies the existence of residual carrier density below the transition temperature T_MIT . This observation is consistent with the result in our experiment that the near-field signal is still high at the lowest temperature (>85% of the RT value).

Cd₂Os₂O₇, like Nd₂Ir₂O₇, is an anti-ferromagnetic insulator at low temperature whose DW is claimed to be conductive [25,26]. To search for the existence of the DWs, we performed nano-THz scans over a wide range of the sample surface. The Cd₂Os₂O₇ single crystal holds stripe-like domains as wide as ∼50 μm on a series of facets detailed in ref. [14] when the crystal is field cooled, while other facets are single domain. The interaction between magnetic field and the magnetization located at the surface and domain boundary of AIAO ordering leads to the migration and annihilation of the DWs [45–47]. Without the external magnetic field, DWs are mainly trapped at crystalline defects [45]. Because of the current challenge in applying the high magnetic field with near-field methods, the chance of discovering the DWs mainly depends on the density of the crystalline defect. Our THz near-field measurements conducted on {111} and {100} Cd₂Os₂O₇ did not demonstrate any clear-cut features that can be ascribed to conductive DWs.

Two reasons can likely explain the absence of the DW contrast. Firstly, magnetic domain walls in pyrochlore iridates are theoretically predicted to host various kinds of quantum phases ranging from metal to insulator [48]. Therefore, Cd₂Os₂O₇, sharing the same crystal structure, can host an insulating DW state. Secondly, the conductivity of Cd₂Os₂O₇ is high enough to generate >85% of near-field contrast of Au even at the insulating state below T_MIT (fig. 3). The high conductivity overshadows the impact of the domain wall conductivity. On the other hand, in Nd₂Ir₂O₇, the far-field THz time-domain spectrum hinted that an appreciable near-field contrast can be observed due to the non-zero imaginary part of conductivity above 0.5 THz. Therefore, our experiment result calls for near-field experiment on Nd₂Ir₂O₇ and other candidates in the family of pyrochlore magnetic insulators with lower conductivity.

In conclusion, we have investigated the metal-to-insulator transition of Cd₂Os₂O₇ single crystals using THz range near-field technology. The evolution of nano-THz signal across the transition is in perfect quantitative arrangement with the DC resistivity measurements and optical measurement in ref. [20]. The near-field signal across the transition is spatially uniform except for extended corrugation steps. Domain walls revealed in x-ray imaging do not produce a distinguishable nano-THz contrast.

Research on "Terahertz nano-imaging of metal-insulator transition in Cd2Os2O7" at Columbia University is supported entirely by the Center on Precision-Assembled Quantum Materials, funded through the US National Science Foundation (NSF) Materials Research Science and Engineering Centers (award No. DMR-2011738). DNB is Moore Investigator in Quantum Materials EPIQS #9455. DNB is the Vannevar Bush Faculty Fellow ONR-VB: N00014-19-1-2630. DNB, RJ and LX conceived the experiments. RJ and LX performed the THz-near-field imaging experiments. RJ constructed the THz near-field device and beam line. HTH and ZH fabricated the Cd₂Os₂O₇ sample. RJ conducted the lightning-rod modeling. ASM, YS, XC and ML provided helpful comments. RJ and DNB wrote the paper with input from all coauthors. DNB supervised the project.

Data availability statement: The data that support the findings of this study are available upon reasonable request from the authors.