Exotic quantum spin Hall effect and anisotropic spin splitting in carbon based TMC6 (TM = Mo, W) kagome monolayers

Accepted Manuscript Exotic quantum spin Hall effect and anisotropic spin splitting in carbon based TMC6 (TM = Mo, W) kagome monolayers Xinru Li, Ying Dai, Yandong Ma, Qilong Sun, Wei Wei, Baibiao Huang PII:

S0008-6223(16)30744-8

DOI:

10.1016/j.carbon.2016.08.089

Reference:

CARBON 11275

To appear in:

Carbon

Received Date: 25 December 2015 Revised Date:

21 August 2016

Accepted Date: 29 August 2016

Please cite this article as: X. Li, Y. Dai, Y. Ma, Q. Sun, W. Wei, B. Huang, Exotic quantum spin Hall effect and anisotropic spin splitting in carbon based TMC6 (TM = Mo, W) kagome monolayers, Carbon (2016), doi: 10.1016/j.carbon.2016.08.089. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Exotic Quantum Spin Hall Effect and Anisotropic Spin Splitting in Carbon Based TMC6 (TM = Mo, W) Kagome Monolayers

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Xinru Li1, Ying Dai1*, Yandong Ma2, Qilong Sun1, Wei Wei1 and Baibiao Huang1 School of Physics, State Key Laboratory of Crystal Materials, Shandong University, Jinan

250100, People’s Republic of China

Department of Physics and Earth Sciences, Jacobs University Bremen, Campus Ring 1, 28759

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Bremen, Germany Corresponding Author: [email protected]

Abstract

Two dimensional topological insulators with high feasibility and room-temperature

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band gaps are desirable at present. Here, kagome monolayers TMC6 (TM = Mo, W) are systematically investigated by density functional theory and molecular dynamics simulations. TMC6 lattice with one TM layer sandwiched between two trigonal

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carbon layers is proved to be stable. We identify that band inversion occurs in TMC6

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with opened gaps up to 226 meV, making the system suitable for room temperature applications. By analyzing orbital resolved band structures and partial charge densities, we determine that the band inversion is mainly contributed by TM-d orbitals after introducing spin orbit coupling. The nontrivial topological feature is confirmed by direct calculation of Z2 invariant (Z2 = 1) indicating that TMC6 is quantum spin Hall (QSH) insulator. Distinct from previous QSH insulators, there is spin splitting along Γ→K direction (perpendicular to the mirror plane), while there is

ACCEPTED MANUSCRIPT no splitting along Γ→M direction (parallel to the mirror plane). Further investigation unravels that the anisotropic spin splitting could be attributed to the in-plane structural inversion asymmetry. TMC6 kagome monolayers with nontrivial topology and

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anisotropic spin splitting could open up a way for new generation spintronic and electronic devices. I. Introduction

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Currently, two dimensional (2D) materials with intriguing properties are attracting

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great research interests [1-5]. They can be utilized in spintronic devices, photocatalysts, field-effect transistors, quantum computation devices and energy storage [6-8], owing to their intriguing geometrical and electronic properties. Graphene with exceptional mechanical, electronic properties and easy preparation

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process has sparked new areas of interests in 2D graphene-like materials [9-11]. Particularly, it is found that graphene can be converted to a quantum spin Hall (QSH) insulator in an extra low temperature [12]. QSH insulator, also known as 2D

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topological insulator (TI), with insulating bulk but conducting edge states, can carry

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dissipationless spin currents which is realized by the spin orbit coupling (SOC) rather than an external magnetic field in the quantum Hall effect. As the first concept QSH insulator, graphene composed with light carbon element has weaker SOC. However, it does provide a starting point for more QSH insulators in 2D materials with stronger SOC [13]. Subsequently, QSH effect has been verified by transport experiments in several 2D quantum wells in extreme experimental conditions [14, 15]. The crucial obstacle of these QSH insulators for application at room temperature is their tiny band

ACCEPTED MANUSCRIPT gaps. Thus, it is of great importance to search for room temperature 2D TIs. To date, series of large-gap 2D TIs have been investigated, however, further experiments are still awaited to check the feasibility of these 2D TIs, examples including Tl, As, Sb, or

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Bi based 2D materials [16-24], transition metal compound monolayers [25-27], Van der Waals heterostructures [28, 29]. However, all these predicted 2D QSH insulators are inorganic materials which restrict the expansion of 2D TIs.

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Encouragingly, Liu et al. have predicted that a 2D extended organometallic

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triphenyl-metal framework, which is composed of transition metal and carbon atoms, is a new 2D TI with a band gap of 8.6 meV [30]. It opens up a new way for 2D QSH insulators despite that the band gap of this organic framework is tiny. Besides, carbon or carbon-metal based frameworks with diverse configurations and intriguing

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properties have been predicted and some of them have been fabricated [31-33]. Recently, a 2D π-d conjugated polymer Cu-benzenehexathiol with extremely high room temperature conductivity and ambipolar transport has been prepared via a

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liquid-liquid interface reaction [34]. Besides, novel triangular and hexagonal

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alternated covalent carbon frameworks with high surface areas and unique luminescent properties have already been constructed [35]. Furthermore, it is worth to mention that group IV elements based kagome lattices consisted of triangles and hexagons have potential band inversions which could probably cause topological characters [36]. Along with the progress of 2D carbon-metal based polymers, it is our firm conviction that more 2D carbon-metal frameworks with intriguing properties are awaited to be explored. Inspired by previous studies, here comes an epiphany if we

ACCEPTED MANUSCRIPT can construct a carbon-metal kagome lattice that is consisted of carbon atoms and transition metal (TM) atoms with strong SOC. According to our knowledge, TM (TM = Mo, W) based 2D monolayers including transition metal dichalcogenides and

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MXenes have been predicted to be QSH insulators [25, 26]. Moreover, a recent highlighted study indicates that chemical vapor deposition makes transition metal carbides go 2D [37, 38]. In the view of the fact that 2D metal-organic kagome lattices

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with high carrier mobilities, large surface area or exotic topological properties could

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have bright future in the application of electronic devices, photocatalysts or gas storage devices, we intend to develop novel stable metal-carbon frameworks for better application in spintronic, electronic or other devices.

In our present work, a TMC6 (TM = Mo, W) kagome lattice is constructed and

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investigated by density functional theory. Ab initio molecular dynamic simulations and phonon spectrum indicate thermal and dynamical stabilities of TMC6 monolayers. We identify that TMC6 monolayers are room temperature QSH insulators by directly

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computing Z2 = 1 invariant. Exotically, the spin orbit coupling introduces anisotropic

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band splitting along different directions. We suggest that the anisotropic band splitting is attributed to in-plane structural inversion asymmetry (SIA): For k being perpendicular to a mirror plane (along Γ→K), there is a splitting because of the existence of in-plane SIA. While if k lies within a mirror plane of the system (along Γ→M), there is no spin splitting at all because of the absence of in-plane SIA. We construct a novel stable 2D kagome lattice with QSH effects, unravel the mechanism of the anisotropic spin splitting and also propose that an effective way tuning the

ACCEPTED MANUSCRIPT anisotropic spin splitting is to exert external field in particular directions. Our investigations show that TMC6 kagome monolayers could be applied into new generation electronic and spintronic devices.

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II. Computational Methods First-principles calculations are carried out by using the Vienna ab initio simulation package (VASP) code with the generalized gradient approximation (GGA) in the

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parametrization of Perdew-Burke-Ernzerhof (PBE) [39-41]. The 2D system is

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separated from its periodic images by a vacuum slab of 18 Å. The 2D Brillouin zone was sampled by 11×11×1 k-points within the Monkhorst-Pack scheme for structural optimization and by 13×13×1 k-points for electronic structural calculations [42]. The cutoff energy for plane-wave expansion is 500 eV and the geometrical structures have

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been fully optimized until the residual forces on each atom becomes less than 0.02 eV/Å. As TM atoms include heavy d orbital electrons, SOC is taken into account self-consistently. Considering possible underestimation of the band gap within PBE

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functional, the Heyd–Scuseria–Ernzerhof (HSE06) hybrid functional is further

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supplemented [43, 44]. The phonon dispersions are calculated by using DFPT method [45,46]. Thermal stability is also examined with ab initio molecular dynamics (AIMD) by using 3×3 supercells at 300 K and 600 K within each time step of 3 fs after running 1000 steps.

III. Results and discussion 3.1 Geometrical structures and stability for TMC6 (TM = Mo, W) kagome lattice We start our study of transition metal carbide kagome lattice by a detailed

ACCEPTED MANUSCRIPT investigation of their structural properties. The crystal structure of TMC6 (TM = Mo, W) with a hexagonal kagome lattice is shown in Fig 1a. TMC6 is basically a three atomic layer structure with one TM layer sandwiched between two trigonal carbon

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layers. A TM atom bonded with six nearest carbon atoms forms a trigonal bipyramid. There are one TM atom and six carbon atoms with a chemical formula of TMC6 (TM = Mo, W) in each unit cell. Structural parameters of MoC6 and WC6 kagome lattices

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are listed in Tab 1. Results shown that lattice constants, C-C bond lengths and C-TM

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bond lengths of MoC6 and WC6 are rather similar within a small difference of 0.003 Å, respectively. The length of C-C bond in TMC6 is about 1.41 Å, which is quite accordance with the length of C-C bonds (1.42 Å) in graphene, indicating sp2 hybridization between adjacent carbon atoms. The lengths of C-TM bonds are about

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2.1 Å, the value of which is the same with the C-TM bonds in oxygen functionalized MXenes, TM2CO2 (M = W and Mo), which are proved to be stable with one TM atoms surrounded with three O atoms and three C atoms forming a trigonal bipyramid

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[26]. Moreover, the electron localization functions (ELF) of MoC6 monolayer are

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plotted in Fig 1c and d. The ELF is a powerful tool to depict the characteristics of chemical bonding by qualitatively describing the degree of electron localization. The values of the function are set from 0 to 1, where 1 indicates the strong localization of the covalent bonding and 0 corresponds to delocalization. The ELF shows that there are high electron localization regions between two adjacent carbon atoms indicating obvious π electrons in carbon rings. Meanwhile strong TM-C covalent bonds can also be improved by high electron localization between C and TM atoms. Each TM atom

ACCEPTED MANUSCRIPT with six valence electrons is connected to six three-membered carbon rings making TM atom in the trigonal prismatic coordinated position. According to ELF, we consider that each carbon ring has two π electrons and there will be one electron

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transfer from each carbon ring to TM atom. Thus the TM will have two electrons after forming two-electron two-center bond with adjacent C atoms. The reasonable bond lengths and types of C-C and C-TM give proof that the model of TMC6 kagome

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lattice can probably exist.

Figure 1. (a) Side and top view of optimized structures of TMC6 (TM = Mo, W). (b) The enlargement of bonding conditions for MoC6 kagome monolayer, each atom is labeled with Arabic numbers. (c) Election localization functions (ELFs) of C1-C2-C3 surface. (d) ELFs of C1-Mo4-C7 surface. Red isosurface represents the electrons that are highly localized and blue one signifies the electrons with almost no localization.

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Table 1. Lattice constants a (Å), C-C bond lengths (Å) and C-TM bond lengths (Å) for TMC6 (TM = Mo, W) kagome lattice. Structure a (Å) C-C (Å) C-TM (Å) MoC6 4.385 1.408 2.103 WC6 4.382 1.411 2.101

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However, the bond types and lengths in TMC6 monolayers could not give eloquent evidence for their stability. Thus, the emphasis is focused on the examination of their

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thermal and dynamical stabilities by employing phonon spectra analysis and molecular dynamics simulations. Phonon dispersion curves of MoC6 and WC6 kagome monolayers are presented in Fig 2a, respectively. All branches over the entire Brillouin zone have positive frequencies without imaginary part. Two acoustical

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branches are with linear dispersions near the Γ point. Whereas the lowest transverse branch displays a quadratic dispersion near the Γ point. To further check the thermal

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stability of TMC6 monolayers, we adopt molecular dynamics simulations. The snapshots of structures for both MoC6 and WC6 kagome monolayers at the

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temperature of 1000 K are shown in Fig 2b. With each step set to 4 fs, after running 2500 steps, no bond is broken though each lattice shows various degree of distortions. Meanwhile the evolutions of free energy for MoC6 and WC6 during MD simulation at 1000 K for 10 ps are shown in Fig 2c, respectively, which further prove that both MoC6 and WC6 systems are thermodynamically stable at the temperature of 1000 K.

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Figure 2. (a) Phonon band dispersions of MoC6 and WC6 kagome monolayers, respectively. (b) present snapshots of the MD simulation of the structures for MoC6

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and WC6 at the temperature of 1000 K, respectively. (c) Variation of free energies at 10 ps during ab initio molecular dynamics simulations (AIMD) at the temperature of

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1000 K for MoC6 and WC6, respectively.

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3.2 Electronic properties of TMC6 (TM＝ ＝Mo, W) monolayers As the stabilities of TMC6 monolayers are proved to be thermodynamically stable, then we focus on the electronic structures of TMC6 kagome lattice. The band structures without SOC for TMC6 kagome monolayers are shown in Fig 3a and e. Recalling that, when excluding SOC, CBM and VBM are well separated from each other except for the touch point at the Γ point. Magnifying the three dimensional bands around the Fermi level, we can see a clear parabolic band dispersion with the

ACCEPTED MANUSCRIPT Fermi level exactly locating at the Γ point. Inspired by previous studies that TM (TM = Mo, W) based 2D materials could have strong SOC effects, thus, the SOC is included for the calculation of band structures in TMC6 as shown in Fig 3b and f,

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respectively. If SOC is further considered, TMC6 (TM = Mo, W) kagome monolayers become insulators with direct band gaps 69 meV and and 226 meV by PBE functional. Considering that the PBE functional may underestimate the band gaps of

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semiconductors and sometimes misjudge the dispersion of energy structures, we also

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use HSE hybrid functional to ensure the reliability. After the correction by HSE hybrid functional with SOC, the band gaps of MoC6 and WC6 are 223 and 462 meV, respectively. The SOC induced band gaps are quite larger than the room temperature thermal energy (26 meV), implying that the band gaps of TMC6 monolayers could be

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measured feasibly at room temperature. Meanwhile, the band structures without considering SOC is also checked by HSE functional. Results indicate that the parabolic dispersions of VBM and CBM near Fermi level still keep in touch. Besides,

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no magnetism is observed in both MoC6 and WC6 after introducing spin-polarized

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calculation. As previous carbon-metal frameworks always have tiny band gaps, our constructions for TMC6 kagome lattice open up a way for more practical 2D metal-organic materials. In the view of the fact that SOC plays an important role in electronic structures especially in gap opening and band splitting, thus we ponder how is the orbital resolution and whether a band inversion can be introduced by considering SOC.

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Figure 3. Band structures for kagome lattice: (a) MoC6 without SOC, (b) MoC6 with

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SOC, (e) WC6 without SOC, (f) WC6 with SOC. And the corresponding three dimensional band structures near Fermi Level are also shown in (c), (d), (g) and (h), respectively.

3.3 Band inversion and edge states in TMC6 (TM = Mo, W) monolayers

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To understand the band splitting and band gap opening with the involvement of SOC in TMC6 kagome monolayers, further investigation about band evolution and partial

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charge distributions around the Fermi levels at the Γ point are studied. Taking MoC6 as an example, from total and projected partial densities of states in Fig 4a, it is

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indicated that in TMC6 kagome lattice, d orbitals of TM atoms make dominant contributions near the band edges, while the s and p orbitals of C atoms located -1.3 eV below the Fermi level which indicates that C contributes few states near the Fermi level. The enlargements of schematic band structures illustrated in Fig 4b give further insights into the band evolution in MoC6 after introducing SOC. When excluding SOC, CBM and VBM near the Γ point are respectively contributed by dx2-y2 and dxy, which are totally separated from each other except for the touching point at Γ. Thus,

ACCEPTED MANUSCRIPT we put emphasis on orbitals’ assignments of the single Γ point instead of specific directions to analyze the nontrivial band edge evolution. Correspondingly, calculated total charge densities without SOC at Γ near the Fermi level is depicted in Fig 4c,

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which is threefold symmetric because of threefold rotational symmetry. To make the explanation of orbital revolutions go one step further, we separate total charge densities near the Fermi level at Γ into two parts: partial charge densities of CBM (as

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in Fig 4c i) and partial charge densities of VBM (as in Fig 4c ii). It is quite discernible

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that without SOC, the rotation of partial charge densities for CBM (i) are 45 degrees distinguished from that of partial charge densities for VBM (ii), which just exactly represent dx2-y2 and dxy, respectively. However, with the inclusion of SOC, the orbitals of TM-dx2-y2 and TM-dxy are thoroughly mixed in band edges at Γ and the mixture

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proportion of dx2-y2 and dxy are right fifty to fifty. Partial charge densities for CBM and VBM with SOC are also shown in Fig 4c iii and iv, respectively. Obviously, the morphology of partial charge densities for both CBM (iii) and VBM (iv) are spheres

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with the mixtures of dx2-y2 and dxy. The inclusion of SOC would not simply remove

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the orbitals’ degeneration and separate dx2-y2 from dxy at the Γ point. According to Fig 4, the SOC will first mix the dx2-y2 and dxy orbitals and then split them. As a result, both the VBM and CBM near the Γ point are mainly composed of both dx2-y2 and dxy orbitals, which is totally different from the case without SOC. Briefly put, the orbitals revolutions and band features suggest a nontrivial topological phase in TMC6 kagome lattice.

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Figure 4 (a) Total and projected partial densities of states for MoC6 kagome monolayer without SOC and the inset is the enlarged plot around the Fermi level. (b) Schematic of orbital-resolved band structures at Γ near the Fermi level for MoC6 without and with SOC, respectively. (c) Calculated partial charge densities at Γ for CBM and VBM without (i, ii) and with SOC (iii, iv), respectively. Total charge

ACCEPTED MANUSCRIPT densities near Fermi level at Γ without SOC are plotted between i and ii. The isovalue is 0.1 e/Å3.

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Still, further evidence is needed to firmly identify the nontrivial topological phase in TMC6 kagome monolayer. It is well known that the band topology can be characterized by the Z2 invariant. Z2 = 1 indicates a nontrivial band topology while Z2

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= 0 characterizes a trivial band topology. Since TMC6 kagome lattice do not have a

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center of inversion symmetry, parity analysis can not be applied [47]. Here, we use the method in previous studies to directly calculate the Z2 invariants in terms of n-field configuration from first principles results [48-50]. The n-field configuration indicates the properties and distribution of vorticities. The Z2 topological invariant can

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be obtained by summing the n-field over half of the torus formed by G1 and G2 reciprocal space vectors in Brilliouin zone. We thus identify a nontrivial Z2 = 1 invariant for freestanding TMC6 kagome lattice. After investigation of band inversion

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and direct calculation of the Z2 invariant in TMC6 kagome lattice, therefore, we

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confirm that quantum spin Hall effect can be realized in TMC6 metal-carbon monolayers.

3.4 Anisotropic spin splitting in TMC6 kagome monolayers Except for the intriguing band inversion caused by SOC in TMC6 kagome lattice, another noticeable phenomenon is that there is band splitting along specific direction under the influence of SOC. Plenty of previous works unravel that SOC can introduce some potentially intriguing effects. One type of the effect coming from the structure

ACCEPTED MANUSCRIPT inversion asymmetry (SIA) is called Rashba splitting effect. Another type owing to bulk inversion asymmetry (BIA) is known as Dresselhaus splitting effect [51]. In the 2D TMC6 system with the point group C3v, only the in-plane structure inversion

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asymmetry is preserved [52]. The spin degeneracy is removed after considering SOC which leads to a pair of split bands in the momentum space. It is worth to mention that the SOC-splitting degree is of great difference along different directions. To probe the

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insights of the anisotropic spin splitting in TMC6 kagome lattice, we have illustrated

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splitting band structures along Γ→K and Γ→M directions in Fig 5, respectively. Correspondingly, direct and reciprocal lattice vectors, the first Brillouin zone and high symmetrical points are also depicted in Fig 5d. In hexagonal TMC6 lattice, there are three mirror plane perpendicular to the surface. It is clear that Γ→M is parallel to the

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mirror plane of the system, while Γ→K is right perpendicular to the mirror plane. We conclude that, on one side, for k being perpendicular to a mirror plane (along Γ→K), there is a splitting because of the in-plane SIA. On the other side, if k lies within a

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mirror plane of the system (along Γ→M), there is no spin splitting at all as shown in

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Fig 5b due to the lack of in-plane SIA. Our results are right consistent with the previous works in the spin splitting of anisotropic 2D electron gas [53]. It can be interpreted that a breaking of the in-plane inversion symmetry results in an asymmetric in-plane component of the potential gradient. As the manipulation of spin in nanomaterials is one of the key problems in the field of spintronics and electronics, our insights about the relationship between spin splitting and in-plane SIA propose an effective way to manipulate the spin orbital splitting: An external field perpendicular

ACCEPTED MANUSCRIPT to the mirror plane of system can be exerted to increase the band splitting along Γ→K. At the same time, if splitting within mirror plane wanted, an external field parallel to the mirror plane could also be applied to break the in-plane symmetry. Therefore, our

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investigation paves way for the manipulation of the magnitude and direction of the spin splitting in 2D nanomaterials. However, some questions still remain open including that how to evaluate the strength of spin splitting, whether external field

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could tune the splitting regularly. Research efforts are still needed to explore the spin

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splitting in 2D monolayers further both theoretically and experimentally.

Figure 5. Band structures with SOC of MoC6 monolayer along (a) Γ→K and (b) Γ→M directions near the Fermi level. (c) Enlargement of spin splitting along Γ→K direction for the CBM (i) and VBM (ii), respectively. (d) The first Brillouin zone and high symmetrical points in MoC6 kagome lattice. IV. Conclusions Generally, we report the discovery of a novel stable TMC6 (TM = Mo, W) kagome

ACCEPTED MANUSCRIPT lattice with hexagonal symmetry. The stability is firmly detected by combining ab initio molecular dynamics simulations and phonon spectra calculations. We find that both MoC6 and WC6 are promising candidates for 2D QSH insulators with band gaps

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as large as 226 meV by PBE functional, which is suitable for application of room temperature devices. Importantly, the nontrivial topological feature of TMC6 is confirmed by direct calculation of Z2 invariant (Z2 = 1) with the method of n-field

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configuration. Intriguingly, distinguished from previous 2D QSH insulators, there is

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anisotropic spin splitting in TMC6 monolayer. We demonstrate that the anisotropic spin splitting are attributed to in-plane structural inversion asymmetric (SIA). It is unraveled that there is spin splitting along the direction perpendicular to the mirror plane while there is no spin splitting at all along the direction within the mirror plane.

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We also propose that the anisotropic spin splitting can be tuned by exerting external field along specific directions. We believe that the TMC6 kagome lattice with reasonable configuration, thermodynamical stability, exotic QSH effect and

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anisotropic spin splitting could be explored experimentally in the near future and

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further applied into new generation spintronic and electronic devices. Acknowledgments

This work is supported by the National Basic Research Program of China (973 program, 2013CB632401), National Natural Science foundation of China under Grants 21333006, 11374190, 11404187, and 111 Project 297B13029. We also thank the Taishan Scholar Program of Shandong Province. References and Notes

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