Possible Topological Phase Transition in Fe-Vacancy-Ordered $\beta$-Fe$_{4+\delta}$Se$_{5}$ Nanowires

TL;DR

This study investigates a topological phase transition in Fe-vacancy-ordered $eta$-Fe$_{4+ ext{delta}}$Se$_{5}$ nanowires, revealing a first-order metal-insulator transition, colossal magnetoresistance, and potential topological implications related to charge-orbital ordering.

Contribution

It provides the first evidence of a topological transition linked to magnetotransport and charge-orbital ordering in FeSe-based nanowires with Fe-vacancy order.

Findings

First-order metal-insulator transition at ~28 K

Colossal positive magnetoresistance near transition temperature

Transition from 2D-like to 1D-like conduction

Abstract

We studied the electrical transport on $\beta$-Fe$_{4+\delta}$Se$_{5}$ single-crystal nanowires, exhibiting $\sqrt{5}\times\sqrt{5}$ Fe-vacancy order and mixed valence of Fe. We observed a first-order metal-insulator transition of the transition temperature at $\sim$28~K at zero magnetic field. The dielectric relaxation reveals that the transition is related to an energy gap expansion of $\sim$12~meV, involving the charge-orbital ordering. At nearly 28~K, colossal positive magnetoresistance emerges, resulting from the magnetic-field dependent shift of the transition temperature. Through the transition, the magnetotransport behavior transits from two-dimension-like to one-dimension-like conduction. The transition temperature demonstrates anisotropy with the $c$-axis as the preferred orientation in magnetic fields, suggesting the spin-orbital coupling. Our findings demonstrate the novel magnetoresistive transition intimating a topological transition in the Fe-vacancy-ordered $\beta$-Fe$_{4+\delta}$Se$_{5}$ nanowires. The results provide valuable information to better understand the orbital nature and the emergence of superconductivity in FeSe-based materials.

Full text

††preprint: APS/123-QED

The discovery of FeAs-based Kamihara et al. (2008) and FeSe-based Hsu et al. (2008) superconductors created an exciting platform for better understanding the physics of high-temperature superconductivity. The observation of wide-range of transition temperature ($T_{\text{c}}$), with the highest confirmed Cooper-pair formation temperature up to 75 K in monolayer FeSe film Peng et al. (2014) is indeed intriguing and provides a unique opportunity to gain more insight into the origin of high-temperature superconductivity.

The multiple-orbital nature of FeSe, combined with spin and charge degrees of freedom, results in the observation of many interesting phenomena such as nemacity Fernandes et al. (2014); Yu et al. (2018), orbital-selective Mott transition Yi et al. (2013); Yu and Si (2013); Herbrych et al. (2018), and orbital ordering Yi et al. (2017). There are suggestions that the orbital fluctuation may provide a new channel for realizing superconductivity Saito et al. (2010); Kontani and Onari (2010). Direct determination of the correlation between the orbital nature of the low-energy electronic states and the superconducting gap is crucial to understand the superconductivity mechanism of the Fe-based superconductors.

It has been a debate, similar to the cuprate superconductors, whether there exists an antiferromagnetic Mott insulating parent phase for FeSe superconductors. Chen et al. reported the existence of the $\beta$-Fe4Se5 phase with $\sqrt{5}\times\sqrt{5}$ Fe-vacancy order and argued this $\beta$-Fe4Se5 phase to be the parent phase of FeSe superconductors Chen et al. (2014). A theoretical simulation shows that the ground state of $\sqrt{5}\times\sqrt{5}$ Fe-vacancy ordered $\beta$-Fe4Se5 has a pair-checkerboard antiferromagnetic order, and the Fe 3$d$ orbitals govern the low-energy physics Gao et al. (2017). Analogously, recent studies demonstrated unambiguously that the K2Fe4Se5, which exhibits a $\sqrt{5}\times\sqrt{5}$ Fe-vacancy order accompanying with an antiferromagnetic order Bao et al. (2011); Zhao et al. (2012), is the parent phase of the superconductor K2Fe4+xSe5 Wang et al. (2015, 2019). Detailed studies of the Fe vacancy in K2Fe4+xSe5 reveal that its order/disorder is closely associated with superconductivity.

Fe-deficient $\beta$-Fe4Se5 is expected to exhibit mixed-valence, which may show correlations among the orders of charge, spin, and orbital, resembling magnetite Jeng et al. (2004); Huang et al. (2006), which exhibits the Verwey transition around 120 K. The metal-insulator transition and the drop in magnetization are the signatures of the Verwey transition Walz (2002). In crystal-like $\beta$-Fe4Se5 sheets, a metal-insulator transition and a drop in magnetic susceptibility have been observed at low temperature Chen et al. (2014). A more detailed investigation of the Fe-deficient $\beta$-Fe4+δSe5 will provide clear picture of the physics underlying and contribute to better understanding the superconductivity in FeSe. Nonetheless, in spite of the success of synthesizing FeSe mesoscale materials having various Fe-vacancy orders Chang et al. (2012, 2014); Chen et al. (2014), Fe-vacancy-ordered $\beta$-Fe4Se5 is yet lack of sufficient investigation.

In this work, we present the electrical transport of the Fe-deficient $\beta$-Fe4+δSe5 single-crystal nanowires with $\sqrt{5}\times\sqrt{5}$-ordered Fe vacancies and mixed valence of Fe. A first-order metal-insulator (MI) transition with an onset transition temperature at $T$ $\sim$ 28 K at zero magnetic field is observed. We investigate the magnetotransport and the dielectric relaxation to understand the carrier transport character through the transition. Our study demonstrates the possible topological transition in $\sqrt{5}\times\sqrt{5}$-Fe-vacancy-ordered $\beta$-Fe4+δSe5.

The $\beta$-Fe4+δSe5 nanowires were grown by the on-film formation method Chang et al. (2014). The nanowires are typically rectangular strips with few-hundred-nanometer width and several-ten-micrometer in length as shown by the scanning electron microscope (SEM) image in Fig. 1(a). We used the 30-nm-thick Si3N4 membranes to hold the nanowires for the benefit of characterizing the nanowires in situ by the transmission electron microscope (TEM). The electrical leads, patterned by electron beam lithography with 90 ${}^{\circ}\text{C}$ baked PMMA, were made of Au/Ti bilayer (150/30 nm) after the surface oxide layer of the nanowires removed by Ar plasma. The resistances of the nanowires were measured by the four-probe method and lock-in technique with an alternative-current (AC) excitation less than 300 nA. Sample temperature and external magnetic field are controlled by PPMS (physical property measurement system), and the field orientation to the sample is changed by rotating the sample via a rotation insert.

For sample characterization, we used TEM to obtain the selected area electron diffraction (SAED) pattern, the scanning TEM energy dispersive X-ray spectrometer (STEM-EDS) for element analysis, and the electron energy loss spectroscopy (EELS) to determine the Fe valence. The element mappings show uniform distributions of Fe and Se on the nanowire as shown in Fig. S1. The ratio between Fe and Se in this nanowire is 4.48/5 according to EDS analysis. As shown in Fig. 1(b),The SAED pattern reveals that the Fe4+δSe5 nanowire is a single crystal with a tetragonal structure, where the $b$-axis is parallel to the longitudinal axis of the nanowire, the $c$-axis is perpendicular and the $a$-axis is parallel to the wide side of the nanowire. The electron diffraction also clearly shows the super-structure related to the $\sqrt{5}\times\sqrt{5}$ Fe-vacancy order Chen et al. (2014). The $a$-axis lattice constant is estimated to be 3.73 Å. The EELS data in Fig. 1(c) demonstrate the absorption peaks for the Fe L3 edge (Fe2+ at 706.4 eV and Fe3+ at 710.2 eV) and the Fe L2 edge (Fe2+ at 718.4 eV and Fe3+ at 723.5 eV), confirming the existence of mix-valence state of Fe2+ and Fe3+.

We measured the temperature $(T)$ dependence of the resistance $(R)$ with the temperature control rate of 1 K/min and the AC excitation frequency at 37 Hz. Figure 2 shows the $R$ versus $T$ in cooling and warming measured at different magnetic fields $B$ parallel to the $c$-axis. The transition shows clear thermal hysteresis. At $B$ $=$ 0 T, $R$ undergoes a MI transition with the transition onset temperature $T_{\text{MI}}$ $\sim$ 28 K, where $T_{\text{MI}}$ is defined as the temperature at which the thermal hysteresis appears. As temperature decreases, $R$ drastically increases from $\sim$10 k$\Omega$ and reaches a resistance maximum $\sim$0.6 M$\Omega$ at $\sim$21 K, and then decreases to about 0.4 M$\Omega$ at $T_{\text{o}}$, which is defined as the temperature where the thermal hysteresis disappears. The thermal hysteresis of $R(T)$ indicates the transition is first-order, which is further confirmed by the applied magnetic field $B$, which enhances the thermal hysteresis loop. The upper inset in Fig. 2 shows more details of the $R(T)$ at $T$ $<$ 24 K. A dip is found at $T$ $\sim$ 19 K at $B$ $=$ 0 T and shifts to higher temperature with increasing $B$ field. The resistance peak value does not change monotonically with $B$ fields. Notice also that the $R(T)$ plateau at $T$ $<$ 17 K is solid under $B$ fields up to $B$ $=$ 9 T, and reaches a fixed value of $\sim$0.436 M$\Omega$ at 2 K. The lower inset of Fig. 2 displays the phase diagram of the transition between the high-$T$ ($T$ $>$ $T_{\text{MI}}$) and low-$T$ ($T$ $<$ $T_{\text{o}}$) phases. Both $T_{\text{MI}}$ and $T_{\text{o}}$ increase with increasing $B$.

To gain more insight into the nature of the transition, we investigated the AC-excitation frequency $(f)$ dependence of $R(T)$ at zero $B$ field. The data were measured from a different section of the same nanowire for the $B$-field dependent experiment. The $R(T)$ at different $f$ are shown in Fig. 3(a). $R$ decreases with increasing $f$ while being weakly $f$-dependent at $T$ $\sim$ 28 K. As shown in the inset of Fig. 3(a), as $f$ $>$ 37 Hz, $1/R$ shows power-law $f$ dependence, i.e. $1/R$ $\propto$ $f^{\alpha}$ with $\alpha$ $\sim$ 1, for both $T$ $<$ $T_{\text{o}}$ and $T$ $>$ $T_{\text{MI}}$. The power-law spectrum is known as a fingerprint of dielectric relaxation for hopping systems Jonscher (1999); Zvyagin (2006). The distinct weak $f$ dependence at $T$ $\sim$ 28 K, which is around $T_{\text{MI}}$ at zero field, suggesting that it is the critical point of the phase transition. The kinks occur in the transition regime at $f$ $\leq$ 100 Hz may result from the discrete jumps in order parameters, which has also been observed in the MI transition of mesoscale strongly correlated systems Zhai et al. (2006); Sharoni et al. (2008); Uhlír et al. (2016).

Figure 3(b) shows the $f$ dependence of $T_{\text{MI}}$ and $T_{\text{o}}$. When $f$ $>$ 37 Hz, $T_{\text{MI}}$ and $T_{\text{o}}$ shift with the Arrhenius law $f(T_{\text{X}})$ $\propto$ exp$(-E_{\text{a}}/k_{\text{B}}T_{\text{X}})$, where $E_{\text{a}}$ is the activation energy and $T_{\text{X}}$ is $T_{\text{MI}}$ or $T_{\text{o}}$. The Arrhenius law indicates the shift is thermally activated. The extracted $E_{\text{a}}$’s are 32.3 meV and 44.0 meV for $T_{\text{MI}}$ and $T_{\text{o}}$, respectively. The $E_{\text{a}}$ enhancement across the transition with decreasing $T$ is estimated to be ${\Delta}E$ $=$ $E_{\text{a}}(T_{\text{o}})-E_{\text{a}}(T_{\text{MI}})$ $\sim$ 12 meV. The ratio of $T_{\text{MI}}$ to ${\Delta}E$ is $\sim$30K/12meV $=$ $\sim$2.5 K/meV, which is comparable to $\sim$125K/50meV $=$ $\sim$2.5 K/meV, the ratio of $T_{\text{V}}$ to the energy gap change deduced from optical conductivity Pimenov et al. (2005) and photoemission spectrum Schrupp et al. (2005); Park et al. (1997) for single-crystal magnetite. In magnetite studies, the energy gap expansion refers to the increase of $E_{\text{a}}$ when lowering $T$ over the Verwey transition Park et al. (1997); Pimenov et al. (2005); Schrupp et al. (2005); Ramos et al. (2008), and the $E_{\text{a}}$ values extracted from different experimental approaches Kuipers and Brabers (1976); Park et al. (1997); Ramos et al. (2008); Gooth et al. (2014), including the $f$-dependent shift in AC conductance, are comparable. Thus, ${\Delta}E$ observed in the $\beta$-Fe4+δSe5 nanowire could be attributed to an energy gap expansion alike to the energy gap expansion observed in the Verwey transition, intimating that charge-orbital ordering may play an important role in the MI transition Huang et al. (2006); Jeng et al. (2004).

As mentioned earlier, the MI transition can be substantially enhanced by applying magnetic fields, as shown in Fig. 2. Consequently, we decided to carry out detailed magnetotransport measurement. Figures 4(a) and 4(b) show the magnetoresistance $R(B)$ of the nanowire at different temperatures with the $B$-field direction parallel to the $c$-axis. At $T$ well above zero-field transition temperature $T_{\text{MI}}(0\text{T})$, i.e. $T$ $>$ 30 K, $R(B)$ exhibits parabolic-like dependence. At $T$ above but close to $T_{\text{MI}}(0\text{T})$, e.g. $T$ $=$ 30 K, $R(B)$ shows linear $B$-field dependence. As 20 K $<$ $T$ $<$ 26 K, a resistance peak in $R(B)$ emerges owing to the $B$-dependent shift of the $R(T)$ peak, and the position of the peak maximum decreases with lowering $T$. Colossal positive magnetoresistance with ${\Delta}R(B)/R(0\text{T})$ $>$ 10, where ${\Delta}R(B)$ $=$ $R(B)-R(0\text{T})$, is observed in 26 K $<$ $T$ $<$ 34 K. The colossal magnetoresistance is due to the rise of $T_{\text{MI}}$ by applying $B$ fields. Strikingly, $R(B)$ at $T$ $\leq$ 17 K is essentially independent on $B$. We do not find obvious magnetic hysteresis in magnetoresistance, being consistent with the picture of antiferromagnetism in $\sqrt{5}\times\sqrt{5}$ Fe-vacancy-ordered $\beta$-Fe4Se5 Chen et al. (2014); Fang et al. (2016); Gao et al. (2017).

We further investigated the $B$-field orientation dependence of the transition at $B$ $=$ 9 T with the nanowire rotated by an angle $\phi_{a}$ ($\phi_{b}$) with the rotation axis along the $a$-axis ($b$-axis). The $B$-field direction is parallel to the $c$-axis as $\phi_{a}$ ($\phi_{b}$) is zero. The data of $\phi_{a}$ and $\phi_{b}$ dependences were acquired from two different sections of the same nanowire. Figures 5(a) and 5(b) show the $R(T)$ results at $B$ $=$ 9 T at different $\phi_{a}$ and $\phi_{b}$, respectively. As the $B$-field direction changes from parallel to perpendicular to the $c$-axis, the positions of the MI transition and the resistance peak shift to lower $T$. Figures 5(c) and 5(d) show the $T_{\text{MI}}$ ($T_{\text{o}}$) versus $B_{c}$, the $B$-field components parallel to the $c$-axis, where $B_{c}$ $=$ $9\text{T}\times|\text{cos}(\phi_{a})|$ and $B_{c}$ $=$ $9\text{T}\times|\text{cos}(\phi_{a})|$ for different $\phi_{a}$ and $\phi_{b}$ at $B$ $=$ 9 T, respectively. The $T_{\text{MI}}$ ($T_{\text{o}}$) values of different $\phi_{a}$ and $\phi_{b}$ scatter along the trace of $T_{\text{MI}}$ ($T_{\text{o}}$) versus $B_{c}$ $=$ $B$, where the $B$-field direction is parallel to the $c$-axis. In other words, $T_{\text{MI}}$ ($T_{\text{o}}$) is enhanced anisotropically in $B$ fields with the preferred orientation along the $c$-axis. This observation indicates that the transition may be associated with the spin-orbital coupling in the Fe-vacancy-ordered $\beta$-Fe4+δSe5 .

The insets in Fig. 5(a) and 5(b) show ${\Delta}R(B)/R(0\text{T})$ at $T$ $=$ 40 K of different $B$-field orientations as an example to demonstrate the magnetotransport in the high-$T$ phase. When $B$ fields orient along the $c$-axis, ${\Delta}R(B)/R(0\text{T})$ $\propto$ $B^{\gamma}$, where $\gamma$ $=$ 2.7 ($\gamma$ $=$ 2.6) in the inset of Fig. 5(a) (5(b)), which is not a conventional Lorentzian magnetoresistance. ${\Delta}R(B)/R(0\text{T})$ decreases when $B$ fields orient from parallel to perpendicular to the $c$-axis. When $B$ fields are perpendicular to the $c$-axis, $\left|{\Delta}R(B)/R(0\text{T})\right|$ $<$ 0.01 and is weakly $B$-dependent, showing a two-dimensional-like (2D-like) magnetotransport character and indicating the stripe-shaped nanowire as a quasi-2D system. At $T$ $<$ 17 K, $R$ values under applying $B$ fields is rigid regardless of changing $B$-field orientation and magnitude, behaving like one-dimensional (1D) conduction of fully spin polarization Chen et al. (2008, 2012), reminiscent of a Wigner-like phase Wu et al. (2012); Zhang et al. (2013). The $B$-field orientation dependence of $R$ peak value is complicated, as shown in Fig. 5(a) and 5(b). When $B$ fields are parallel to the $a$-axis ($b$-axis), the peak height increases (decreases) with increasing $B$. The $R$ peak emerging in the transition is in the phase coexistence regime of the first-order transition. The peak-like feature is also observed in three-dimensional (3D) topological insulators and is regarded as a sign of the existence of the topological surface state Cai et al. (2018); Wolgast et al. (2013); Luo et al. (2015); Dankert et al. (2018). The high-$T$ 2D-like phase and the low-$T$ 1D-like phase may be different in topology. $R$ peak feature of the nanowire at $T$ $<$ $T_{\text{MI}}$ may reveal the emergence of the topological 1D channels. The magnetoresistive transition suggests a possible topological phase transition.

We studied the electrical transport of tetragonal $\beta$-Fe4+δSe5 nanowires possessing $\sqrt{5}\times\sqrt{5}$ Fe-vacancy ordering and mixed-valance state of Fe2+ and Fe3+. The first-order MI transition of $T_{\text{MI}}$ $\sim$ 28 K observed at zero $B$ field, and a Verwey-like energy gap expansion of $\sim$12 meV, suggesting that charge-orbital ordering may involve the transition, is extracted from the thermal activation of the dielectric relaxation. When applying external $B$ fields, the MI transition shifts toward higher temperatures remarkably, resulting in the colossal positive magnetoresistance, and the thermal hysteresis gets more pronounced. The $B$-dependent rise of $T_{\text{MI}}$ and $T_{\text{o}}$ shows anisotropy with the preferred orientation parallel to the $c$-axis, indicating that the spin-orbital coupling in the nanowire is significant. Based on the results of magnetoresistance, the high-$T$ phase shows 2D-like conduction, and the low-$T$ phase shows the rigidity under external $B$ fields, behaving like fully spin-polarized 1D conduction. The high-$T$ and low-$T$ phases might be topologically distinct, demonstrating the possible topological phase transition in $\sqrt{5}\times\sqrt{5}$ Fe-vacancy-ordered $\beta$-Fe4+δSe5 nanowires. Our results could provide critical information for better understanding the origin of superconductivity in FeSe-based superconductors and the orbital-related physics in the FeSe-based materials.