Dimensional crossover of charge order in IrTe₂ with strong interlayer coupling

Dimensionality in a quasi-two-dimensional (quasi-2D) system works as one of critical physical parameters in determining structural and electronic characteristics, which often induces various phase transitions including superconductivity Xi et al. (2016); Song et al. (2021), charge density waves Xi et al. (2015); Yu et al. (2015); Chen et al. (2022), or symmetry breaking Lin et al. (2017); Ni et al. (2021). Quasi-2D van der Waals (vdW) materials, such as transition metal dichalcogenides (TMDs), feature strong in-plane covalent bonding and weak out-of-plane vdW bonding, which enable tuning their dimensions by adjusting thickness without introducing disorder or defects Ghosh et al. (2010); Jin et al. (2013). Because a typical electronic order in vdW materials is three-dimensional (3D) due to finite interlayer coupling Yang et al. (2012); Jin et al. (2013); Yang et al. (2014), modulating out-of-plane interaction can generate fascinating layer-dependent physical properties particularly when the electronic order is strongly linked to the modulation of interlayer bonding or stacking configuration Lee et al. (2019, 2021).

IrTe₂, one of exotic TMDs showing intertwined charge ordering phases, is associated by the structural phase transition from high temperature trigonal to low temperature monoclinic lattice ($T_{c}\sim 275$ K) with the wave vector of $q_{0}=(1/5,0,1/5)$ Matsumoto et al. (1999); Yang et al. (2012). Due to the intralayer Ir-Ir dimerization competing against the interlayer Te-Te bonding Machida et al. (2013); Joseph et al. (2013); Dai et al. (2014), the so-called cross-layer charge ordering with stripe charge modulations [$q_{n}=(3n+2)^{-1}$] have been observed in the low temperature phase Hsu et al. (2013); Pascut et al. (2014). Furthermore, bulk intrinsic superconductivity emerges when the charge ordering is fully suppressed by chemical doping Yang et al. (2012); Pyon et al. (2012); Ootsuki et al. (2012); Kamitani et al. (2013) or rapid cooling Kim et al. (2016); Oike et al. (2018). In contrast to bulk IrTe₂, the coexistence of superconductivity and stripe charge modulation has recently been observed in mechanically exfoliated flakes with the thickness range of 20–200 nm Park et al. (2021). Such a contrasting observation suggests that the coexisting stripe charge order significantly enhances the out-of-plane coherence length and the coupling strength of superconductivity in thin IrTe₂ flakes. In this respect, IrTe₂ can be a proper model system for studying the role of interlayer coupling in the interplay between charge ordering and superconductivity at different dimensions, which has not been investigated experimentally.

In this article, we have investigated atomically thin IrTe₂ flakes ranging from monolayer to multilayer (up to 20 L) in order to correlate a dimensional crossover of charge order with thickness. Using the Al₂O₃-assisted mechanical exfoliation Deng et al. (2018), we systematically reduce the thickness of IrTe₂ down to monolayer and directly probe structural and electronic properties with low temperature scanning tunneling microscopy and spectroscopy (STM/STS) measurements. We observe a gradual evolution of charge order and an exponentially decay of Coulomb gap, which have been attributed to the subtle interplay between the strong interlayer coupling and substrate-induced disorders. Furthermore, we discover an unusual depression of density of states (DOS) in empty states, which appears at an abrupt lateral interface from 2D to 3D. Our findings demonstrate that the stability of the cross-layer charge ordering and the phase competition with different ordering periods are highly sensitive to the dimensional crossover, highlighting the important role of the interlayer coupling in IrTe₂.

Figures 1(a) and 1(b) show our experimental setup and a typical IrTe₂ flake fabricated by the Al₂O₃-assisted mechanical exfoliation in an inert atmosphere Deng et al. (2018) (see more details in Appendix A). We identify thicknesses of IrTe₂ using optical transmission contrasts, which are highly sensitive to layer numbers of IrTe₂ flakes (see Appendix B). As shown in Figs. 1(b) and 1(c), we confirm the excellent correlation between the optical contrasts and STM height profiles. Surface cleanliness, which is crucial for surface-sensitive STM, is routinely checked by room-temperature STM measurements before cooling. A typical atomic STM image of IrTe₂ monolayer clearly reveals the triangular Te atomic lattice without any significant surface contamination such as vacancies or adsorbates [Fig. 1(d)]. The sample was inserted to a cooled STM stage using a continuous flow cryostat. STM/STS measurements were performed in an ultrahigh-vacuum ($8\times 10^{-9}$ Pa) cryogenic STM ($T_{\rm STM}=88$ K). All STM images were recorded in the constant-current mode with an electrochemically etched W tip. STS measurements were performed by recording $dI/dV$ spectra via a lock-in technique with feedback open, using a modulation frequency of approximately 1 kHz with a root-mean-square modulation amplitude of $V_{\rm mod}=7$ mV.

To investigate thickness-dependent charge orders, we have performed STM measurements on atomically thin IrTe₂ flakes from monolayer (1 L) to multilayer up to 20 L well below the bulk transition temperature. In sharp contrast to the dimerized monolayer and bilayer IrTe₂ grown by molecular beam epitaxy (MBE) on graphene Hwang et al. (2022), we found no signature of charge orders in our mechanically exfoliated monolayer and bilayer IrTe₂ [Figs. 2(a) and 2(b)]. Local stripe charge orders start to emerge from trilayer (3 L) and become dominant above heptalayer (7 L) IrTe₂ [Figs. 2(c)–2(g)]. To quantify the thickness dependence of the stripe charge orders, we analyze the relative peak intensities of fast Fourier transform (FFT) of charge orders with respect to Bragg peaks from atom-resolved STM images [Fig. 2(h)] Pásztor et al. (2017). To improve the reliability, we statistically averaged the relative peak intensities obtained from a sufficient number ($\sim$10) of topographic images on different locations for each thickness. The charge order with the wave vector of $q_{0}=(1/5,0,1/5)$ gradually becomes stronger with increasing thickness above 3 L while the $q=1/3$ charge order exists locally on 3 L and becomes more visible above 5 L. Beyond 20 L, we found no further thickness dependence of the charge order as reported previously Park et al. (2021). This observation strongly suggests that the stripe charge order of few layer IrTe₂ exhibits a gradual evolution with increasing thickness, in contrast to the abrupt metal-insulator transition between MBE-grown monolayer and bilayer IrTe₂ Hwang et al. (2022).

The structural distortion and strain effect from the substrate may give rise to thickness-dependent lattice constants, which may lead to the observed thickness dependence of the charge order. To check this possibility, we have performed the quantitative analysis of the lattice constants on various thicknesses (1–20 L). Within our experimental measurement error ($\pm$0.15 Å), few-layer IrTe₂ on Al₂O₃ does not show significant thickness dependence of the lattice constants (3.65–3.82 Å). Thus, we can exclude that atomic distortion and strain effect directly from the Al₂O₃ substrate on the thickness dependence of the charge order.

To further understand the dimensionality-induced evolution of the charge orders, we systematically probe local DOS of IrTe₂ as a function of thickness by normalizing differential tunneling conductance [Fig. 3(a)]. All spatially averaged STS spectra on IrTe₂ flakes (1–20 L) show metallic behaviors. Such an observation is distinct from that in the MBE-grown monolayer IrTe₂ with a band gap larger than 1 eV Hwang et al. (2022). While the normalized STS data exhibit a similar trend for all thicknesses, a close look reveals that an enhanced DOS appears above 0.3 eV (indicated by triangles) in IrTe₂ flakes thicker than bilayer. This DOS enhancement is originated from the Ir-Ir dimerization in the charge-ordered phase, which is supported by density functional theory calculations Pascut et al. (2014). Thus, this thickness-dependent spectroscopic feature is quite consistent with the observed gradual evolution of charge orders with increasing thickness.

In addition, we observe a significant DOS suppression [gray region in Fig. 3(a)] at the Fermi energy ($E_{\rm F}$). To visualize more clearly the DOS suppression, we subtract quartic polynomial fits [dashed lines in Fig. 3(a)] from the normalized differential conductance spectra. Each fit is obtained from the corresponding spectrum using data with 0.1 V$<|V_{\rm b}|<$0.2 V. The subtraction results in the symmetric ‘V’-shaped DOS suppression that becomes more pronounced with decreasing thickness [Fig. 3(b)]. Such a series of suppressed DOS features are often observed in disordered metallic thin film due to disorder-enhanced Coulomb interaction, known as a Coulomb gap Efros and Shklovskii (1975). This resemblance suggests that the underlying Al₂O₃ substrate causes electronic disorders in IrTe₂, which can lead to the thickness-dependent ‘V’-shaped DOS suppression.

As an atomically thin layer such as graphene monolayer simply follows the contours of the underlying substrates Lui et al. (2009), thin IrTe₂ flakes may also reflect surface corrugation of the underlying Al₂O₃ substrate. To unveil such substrate induced effect, we carefully analyze the surface roughness of 1 L and 2 L IrTe₂ with low-pass filtered STM topographic images. Figures 3(c) and 3(d) show the filtered STM topographic images of Figs. 2(a) and 2(b), respectively, by masking the outer six Bragg points from the atomic lattice. The filtered images clearly show the thickness-dependent surface corrugation [Fig. 3(e)] due to the underlying Al₂O₃. The decrease in roughness with increasing thickness strongly indicates that the structural inhomogeneity is originated from the underlying Al₂O₃, which naturally leads to the thickness-dependent electronic disorder causing the enhanced Coulomb interaction in IrTe₂ flakes.

A quantitative analysis of the Coulomb gaps demonstrates that the gap depth, which is defined by $\Delta dI/dV$ at $E_{\rm F}$ [Fig. 3(b)], exponentially decreases with increasing layer number, where the effective screening layer number is $t_{0}=1.2$ corresponding to only 0.6 nm [Fig. 3(f)]. The similar interlayer screening with an exponentially decaying behavior is also observed in few-layer graphene Lee et al. (2009) and MoS₂ nanoflakes Li et al. (2013) with thicker effective screening layer numbers of 10 and 3.5, respectively. The much smaller screening layer number indicates that IrTe₂ has much high DOS at $E_{\rm F}$ than other vdW materials. Since STM exclusively probes topmost surface layers, the observed exponential decay strongly suggests that metallic IrTe₂ overlayers effectively screen the Coulomb gap feature of the bottom layer, which is directly affected by the underlying Al₂O₃ substrate.

Another intriguing analysis provides a dimensional crossover of the system with the disorder-driven Coulomb gap. By fitting the Coulomb gap equation Efros (1976) to the subtracted $dI/dV$ curves [Fig. 3(b)], we can obtain thickness-dependent dimensionalities. Since the effective energy resolution ($\Delta E$) of STM is given by $\Delta E\approx\sqrt{(3.5k_{\rm B}T)^{2}+(2.5eV_{\rm mod})^{2}}$ Lauhon and Ho (2001), we exclude data points of $|V_{\rm b}|$$<$15.6 mV near $E_{\rm F}$ in the fitting process due to the energy resolution obtained from both thermal broadening ($T$ = 88 K) and a modulation amplitude ($V_{\rm mod}=7$ mV) of a lock-in amplifier in our experiments. For 1 L IrTe₂, the DOS exhibits a linear dependence on bias voltage ($V_{\rm b}$). According to the Coulomb gap equation (DOS $\propto|V_{\rm b}|^{d-1}$), we obtain the dimensionality $d=2.02~{}\pm~{}0.1$ for 1 L. This strongly indicates that 1 L IrTe₂ on Al₂O₃ behaves as genuine 2D with the disorder-induced Coulomb gap, which has mainly been investigated in theoretical approach Lee et al. (2016); Szabo et al. (2020). With increasing thickness, we find that the DOS near $E_{\rm F}$ gradually loses its linear feature due to the strong interlayer coupling. With the strong interlayer interaction, the topmost layer in thicker IrTe₂ can experience the Coulomb gap of the electronically disordered bottom layer, which leads to the resulting intermediate dimensionality between 2D and 3D.

Occasionally, we can find an atomically seamless interface between thin and thick IrTe₂ as shown in Figs. 4(a)–(e). Such an interface would provide an unprecedented platform to investigate an interface between 2D and 3D without abrupt steps on the surface. The atom-resolved STM image [Fig. 4(d)] and the height profile [Fig. 4(e)] across the 20/2 L interface clearly demonstrate that the topmost IrTe₂ layer is seamlessly connected between 20 L and 2 L while the 21/1 L interface shows an 1-nm-high atomic step. As discussed above, the thicker 21 L and 20 L show the charge orders while the thinner 1 L and 2 L do not [Fig. 4(b)]. In addition, the atom-resolved STM image shows an unambiguous charge order with period of $3a_{0}$ near the interface as indicated by triangles in Fig. 4(d), which is somewhat different from the ground states in bulk showing the $q=1/6$ charge order Oh et al. (2013); Dai et al. (2014); Chen et al. (2017).

To understand the electronic property of the intriguing interface, the spatially resolved STS measurement is taken across the interface along a black arrow in Fig. 4(d), which exhibits an apparent DOS suppression near $V_{\rm b}$ = 0.175 V mainly on the 20 L area with width of 10 nm from the interface [Fig. 4(g)]. Such a state near the interface can be qualitatively explained by the strong interlayer coupling. As illustrated in Fig. 4(h), the interlayer coupling delivers electronic properties of edge-terminating atoms (indicated by red circles) to the topmost layer by inducing an exotic DOS suppression only on the 20 L side. On the other hand, the 2 L side does not exhibit such a DOS suppression because its top layer is not connected to any edge-terminating atoms. This finding suggests that the strong interlayer coupling in IrTe₂ also plays a vital role in an atomically seamless interface with abrupt thickness changes.

Lastly, the observation of the ($3\times 1$) charge orders locally near the 2 L region [Fig. 4(d)] contrasts to the absence of the stripe charge order in the isolated bilayer IrTe₂ [Fig. 2(b)]. It can be considered as a proximity effect due to the neighboring 20 L. Remarkably, the signatures of the $q=1/3$ charge order are found nearly $\sim 10$ nm apart from the interface. The long-range proximity effect in the metallic system is quite unusual, which may indicate the inherent instabilities of IrTe₂ against various stripe charge orders. Particularly, the type and stability of the stripe charge order is found to be extremely sensitive to the strain of just $\sim 0.1$% Nicholson et al. (2021). A possible strain near the interface can stabilize the $q=1/3$ stripe charge ordering phase in the otherwise disorder-induced pristine phase. Therefore, this finding highlights the incipient instability of IrTe₂ to charge ordering, which would provide an unprecedented way to create exotic charge-ordered phases by introducing local strains or terminating atomic edges with functional molecules Zhao et al. (2018).

We have fabricated the atomically thin and clean IrTe₂ flakes varying from monolayer to multilayer by utilizing the Al₂O₃-assisted exfoliation method, which can overcome the strong coupling between Te interlayers. Our STM/STS measurements exhibit the gradual evolution of charge orders and the exponentially decaying Coulomb gaps with increasing thickness. Our finding reveals that these phenomena are originated from the interplay between the stripe charge ordering with strong interlayer coupling and the substrate-driven disorder. Furthermore, the induced DOS suppression from edge-terminating atoms is found at the atomically seamless interface via the strong interlayer coupling. Our observation strongly suggests that thinner IrTe₂ layers would have relatively much lower transition temperature of the charge ordering than our measurement temperature. Further experiments are needed to determine the thickness dependence of the transition temperature.