Topological Landscape of Competing Charge Density Waves in 2H-NbSe2

TL;DR

This paper uncovers the topological nature of competing charge density wave structures in 2H-NbSe2, revealing how their entanglement leads to incommensuration and complex domain landscapes, advancing understanding of phase coexistence.

Contribution

It introduces the concept of topological excitations as key to understanding incommensurate CDWs and their phase competition in 2H-NbSe2, a novel perspective in the field.

Findings

Identification of two competing CDW structures in 2H-NbSe2.

Topological excitations entangle these structures, causing incommensuration.

A network of topological excitations explains domain landscape complexity.

Abstract

Despite decades of studies on charge density wave (CDW) of 2H-NbSe2, the origin of its incommensurate CDW ground state has not been understood. We discover that CDW of 2H-NbSe2 is composed of two different, energetically competing, structures. The lateral heterostructures of two CDWs are entangled as topological excitations, which give rise to a CDW phase shift and the incommensuration without a conventional domain wall. A partially melt network of the topological excitations and their vertices explain an unusual landscape of domains. The unconventional topological role of competing phases disclosed here can be widely applied to various incommensuration or phase coexistence phenomena in materials.

Full text

The competition of distinct periodicities or ordered phases are widely found in various of material systems Feng et al. (2015); Kundu et al. (2017); Uehara et al. (1999); Cheong et al. (2002). Periodically ordered phases in spin Geerken et al. (1982), charge Wilson et al. (1975); Moncton et al. (1975, 1977), and orbital degrees of freedom Mori et al. (1998) have also interactions with underlying lattice periodicities, which result in commensuration-incommensuration phenomena and create domain-wall topological excitations Bak (1982); Huang and Cheong (2017). Notable examples are Frenkel-Kontorova solitons in one dimensional systems Frenkel and Kontorova (1938) and Moire structures in van der Waals layers Alden et al. (2013); Woods et al. (2014). In most of the cases, competitions between ordered phases and those between them and underlying lattices have been separately discussed. However, here, we observe a unique case where those two phenomena, the competing phases and the incommensuration are topologically entangled to exhibit an unusual landscape of domains of coexisting phases.

We pick up the charge density wave (CDW) in two dimensional (2D) materials, which is one of the most extensively investigated topics in condensed matter physics together with superconductivity Grüner (1988). For most of 2D CDW phases Withers and Wilson (1986), ground states are commensurate with their atomic lattices, which would transit into incommensurate CDW phases with topological excitations, discommensurations or domain walls carrying a CDW phase shift McMillan (1976), through thermal excitation Fung et al. (1981), doping Yu et al. (2015); Li et al. (2015) or pressure Sipos et al. (2008); Joe et al. (2014). Topological excitations are believed to have crucial roles not only in the superconductivity emerging with the commensurate CDW order suppressed Sipos et al. (2008); Yu et al. (2015); Joe et al. (2014); Li et al. (2015) but also in outstanding functionalities of CDW materials such as the ultrafast switching behavior Stojchevska et al. (2014) and the memrister operation Yoshida et al. (2015). Nevertheless, atomic structures of domain walls are largely unknown Cho et al. (2017) and thus naturally microscopic mechanisms for the emerging superconductivity and functionalities have been elusive.

In one of the most widely investigated model systems of 2D CDW, 2H-NbSe2, the intriguing nature of the incommensurate phase has become even a long term mystery. 2H-NbSe2 undergoes a phase transition to the $(3\times 3)$ CDW state at 33 K Moncton et al. (1975, 1977). While the microscopic origin of the transition is still under debates, the CDW ground state has been reported to be intriguingly incommensurate as confirmed by electron microscopy Eaglesham et al. (1985), nuclear magnetic resonance Ghoshray et al. (2009), and recent scanning tunneling microscopy (STM) Soumyanarayanan et al. (2013) experiments. However, none of these experiments identified the atomic structure of the discommensuration or the domain wall. Moreover, there is still ambiguity in the atomic structure of the commensurate CDW itself. In the early neutron scattering experiment, the CDW state was suggested to have a hexagonal symmetry Moncton et al. (1977) but later the electron microscopy clarified the symmetry as an orthorhombic one Eaglesham et al. (1985). Three types of distinct CDW structures had been suggested in early theories Walker and Jacobs (1981); Littlewood and Rice (1982) and their atomic structures together with possible magnetic orders were discussed by a very recent density functional theory (DFT) calculation Zheng et al. (2018).

In this work, we clarify the unconventional atomistic origin of the incommensurate CDW state in 2H-NbSe2 using STM experiments and DFT calculations. The CDW state of 2H-NbSe2 is divided by domains of two different CDW structures. Our DFT calculations and experimental results disclose atomic structures of the distinct CDW structures and their degenerate energies. The characteristic heterostructure of two CDW domains produces a phase shift for the discommensuration McMillan (1976). Such a topological role of two competing phases explains the unusual domain distribution observed in the experiment. This reveals a distinct mechanism of the CDW discommensuration and a unique case of topologically entangled domain formation, which might be considered not only in CDW systems but also in a wide variety of materials with incommensuration phenomena and competing phases including Moire structures in van der Waals layers Alden et al. (2013); Woods et al. (2014).

Single crystal 2H-NbSe2 is cleaved in high vacuum to obtain clean surfaces and transferred to the ultrahigh-vacuum chamber equipped with a commercial cryogenic STM operating at 4.3 K. STM images were obtained using the constant current mode. Our DFT calculations were performed using the Vienna ab initio simulation package Kresse and Furthmüller (1996) with the projector augmented wave method Blöchl (1994); Kresse and Joubert (1999) and the Perdew-Burke-Ernzerhof exchange correlation functional Perdew et al. (1996). Single-layer structures of NbSe2, inserted between 10 $\AA$ vertical vacuum, were calculated with a $(6\times 6\times 1)$ k-point mesh and a plane wave cut-off energy of 366.5 eV. The residual force criterion to terminate atomic relaxation is 0.01 eV/$\AA$.

The STM topography of 2H-NbSe2 shows clearly the coexistence of two different CDW structures [Fig. 1(a)]. Hexagonally arrayed protrusions represent top Se atoms of NbSe2 and the $(3\times 3)$ CCDW order is clear with periodic enhancements of protrusions. Bright triangular protrusions or Se trimers are conspicuous in one of the CDW domains. The other domain exhibits a structurally distinct CDW state with a single bright Se protrusion per each $(3\times 3)$ unit cell. Figure 1 (b) shows a detailed line profile across two CDW structures; the single big peak and two small ones for the CDW structure with monomer protrusions smoothly transform into two big peak structures which are parts of bright Se trimers. The coexisting CDW states is ubiquitous in our measurements over whole measured surfaces [Fig. 1(c)], different cleavages, crystals and tunneling biases (see Fig. S1 in Supplemental Material Sup ). Moreover, the coexisting structures are noticeable in most of the STM images published previously although they have never been discussed explicitly Soumyanarayanan et al. (2013); Dai et al. (2014). These CDW structures exhibit an intriguing domain structure as shown in Fig. 1(c); the landscape of all-connected monomer and all-isolated trimer domains deviate certainly from a random heterogeneous domain mixture of competing phases. As revealed below, this unusual landscape has a topological origin.

We identify atomic structures of two CDW structures using STM images and DFT calculations [Fig. 2]. As discussed in the previous literature, CDW maxima can locate on cation (Nb), anion (Se), or hollow sites to form different CDW structures Walker and Jacobs (1981); Littlewood and Rice (1982) (see Fig. S2 in Supplemental Material Sup ). The cation-centered (CC) structure has the hexagonal symmetry but the anion-centered (AC) and hollow-centered (HC) structures have the orthorhombic symmetry Walker and Jacobs (1981); Littlewood and Rice (1982). The hexagonal symmetry is not consistent with the electron microscopy experiment Eaglesham et al. (1985) and the CC structures are not favorable in our calculation and comparison to STM images Sup . We thus just consider the latter two structures. In terms of Nb atom distortions relaxed in the corresponding DFT calculations, HC and AC CDW structures have similar units of a large triangle with six Nb atoms and a small triangle of a Nb trimer (Fig. 2). However, the HC structure has Nb triangles centered on the hollow sites while the AC structure’s triangles are centered on Se atoms.

STM simulations of our $(3\times 3)$ CDW structures reproduce well the local topographies. In the HC structure, the bright trimer protrusion of three Se atoms on the Nb trimer is well compared with the STM topography at 100 mV and the bright (dark) triangular protrusion of six (three) Se atoms is in good agreement with the experiment at -100 mV. In contrast, in the AC structure, the brightest protrusion made by the center Se atom of the Nb sextet agrees well with the STM topography at both biases. The CDW ground state would be the HC structure which has a 13 (17) meV energy gain per $(3\times 3)$ unit cell compared with the AC (CC) structure. A very recent DFT study, considering only HC and CC structures, also reported a consistent result Zheng et al. (2018).

With these CDW atomic structures, we can clarify the structural origin of the CDW discommensuration. As a matter of fact, a translation between HC and AC structures cannot be defined due to the absence of a common reference as shown in Figs. 1(b) and 3 but one can define it between two HC domains sandwiching a stripe of an AC domain or vice versa. Figure 3 shows such a structure observed by STM and its schematic atomic structure. This kind of structures is found all over the surface [see Fig. 1(c)]. One of Se atoms nearby the Nb trimer (small blue triangle) in the HC structure and the Se atom at the center of a Nb sextet (large red triangle) in the AC structure are superimposed to connect the domains. This structure is fully relaxed in the DFT calculation and produces the STM image [Fig. 3(c)], which is fully consistent with the observed one [Fig. 3(a)]. The AC domain between two HC domains and two domain boundaries between the HC and AC structures together play the role of the discommensuration, which give rise to the translation of the HC CDW by a single unit cell of the pristine structure $\pm\mathbb{a_{1}}$. This corresponds to the phase shift of $(\mp\frac{2\pi}{3},0,\pm\frac{2\pi}{3})$ McMillan (1976); Jacobs and Walker (1982) in the HC CDW order parameter [Figs. 3(a) and 3(b)].

Among domain walls in CDW materials, those of 2H-NbSe2 is unique. Most of CDW translations by discommensurations have been assumed between the same CDW structures with the translational degeneracy and is composed of well isolated broken CDW clusters. While there are very few examples of CDW domain walls with their structures determine, the recent case in 1T-TaS2 shows such broken CDW clusters clearly Cho et al. (2017). In contrast to such an abrupt local change of structures, the present heterostructure for the CDW translation is a very smoothly connected structure Cho et al. (2017), making it hard to be identified. More contrastingly, while the domain walls of 1T-TaS2 result in strongly localized in-gap states on the broken CDW clusters, the current HC-AC-HC heterostructure does not induce any strong electronic singularity (see Fig. S4 in Supplemental Material Sup ).

In fact, the idea of degenerate CDW structures and their possible role as the discommensuration was introduced as early as 1982 for the case of 2H-TaSe2 Littlewood and Rice (1982). The main idea was that the two competing CDW structures in phase shifting heterostructures (HC-AC-HC in this work) can reduce the energy cost of conventional domain walls. While this idea didn’t find clear experimental evidence in 2H-TaSe2 yet but is fully consistent with what observed here. The only difference is that the previous theory considered only CC and AC domains, neglecting simply the HC structure Littlewood and Rice (1982). The free energy consideration of the previous work does not depend on any structural details of the phases involved Littlewood and Rice (1982) so that it can equally be applied to the present case of HC and AC competing structures Sup .

The discommensuration would ideally form a honeycomb lattice Bak (1982) as shown in the inset of Fig. 1(c) (see the Supplemental Material for a detailed explanation Sup ). In the present system, this should correspond to hexagonal HC domains and the honeycomb network of the AC CDW stripes [Fig. 4(a)]. To form a honeycomb network, three AC CDW stripes should merge and the neighboring HC domains should have proper phase shifts [indicated by the domain numbering in Fig. 1(c)]. If the atomic structure of the HC-AC-HC discommensuration is put into this honeycomb network, the merging point of AC stripes is dictated to have two distinct structures; one with the dislocations (dashed circles) of AC CDW but the neighboring one without [Fig. 4(a) and also see Fig. S5 Supplemental Material Sup ]. These correspond to a domain wall vortex and antivortex as dictated by the Z3 topology Huang and Cheong (2017). They are indeed observed in the STM topography; Fig. 4(b) shows two representative merging points of AC stripes and only one of them shows the dislocations at its core (dashed circles). While the experimentally observed domain distribution in Fig. 1 is largely disordered, it can be closely mapped into the ideal honeycomb network Sup . This network explains naturally why the HC and AC structures have distinct distributions of all isolated and all connected domains, respectively, and indicates the topological nature of the domain distribution as a partially melt network of topological excitations.

In this work, the long standing puzzle, the microscopic nature of the incommensurate CDW of 2H-NbSe2 is thereby solved Sup . The competition between HC and AC structures with a very small energy difference must be closely related to the quantum phase fluctuation observed very recently Feng et al. (2015); Kundu et al. (2017). While the previous work suggested the 1Q phase as the source of the quantum fluctuation with the limited microscopic information, we very rarely observe the 1Q phase and its origin in the inhomogeneous strain Soumyanarayanan et al. (2013) is not easily compatible with the spontaneous quantum fluctuation. The concept of topologically entangled competing phase and the discommensuration without a domain wall may find its application in various different materials systems with incommensuration phenomena.

The present work is supported by Institute for Basic Science (Grant No. IBS-R014-D1). The authors appreciate the discussion with C. Park.