Designing Flexible Quantum Spin Hall Insulators through 2D Ordered

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C: Physical Processes in Nanomaterials and Nanostructures

Designing Flexible Quantum Spin Hall Insulators through 2D Ordered Hybrid Transition Metal Carbides Zhongheng Fu, Zhaorui Liu, Dominik Legut, Timothy C. Germann, Chen Si, Shiyu Du, Joseph S. Francisco, and Ruifeng Zhang J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.9b05962 • Publication Date (Web): 01 Aug 2019 Downloaded from pubs.acs.org on August 6, 2019

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The Journal of Physical Chemistry

Designing Flexible Quantum Spin Hall Insulators through 2D Ordered Hybrid Transition Metal Carbides Zhongheng Fu,1,2 Zhaorui Liu,1,2 Dominik Legut,3 Timothy C. Germann,4 Chen Si,1,2 Shiyu Du,5 Joseph S. Francisco,6 and Ruifeng Zhang1,2,* 1

School of Materials Science and Engineering, Beihang University, Beijing 100191, P. R. China

2

Center for Integrated Computational Materials Engineering (International Research Institute

for Multidisciplinary Science) and Key Laboratory of High-Temperature Structural Materials & Coatings Technology (Ministry of Industry and Information Technology), Beihang University, Beijing 100191, P. R. China 3

IT4Innovations, VSB-Technical University of Ostrava, CZ-70800 Ostrava, Czech Republic

4

Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico, 87545,

United States 5

Engineering Laboratory of Specialty Fibers and Nuclear Energy Materials, Ningbo Institute of

Materials Technology and Engineering, Chinese Academy of Sciences, Ningbo, Zhejiang, 315201, P. R. China

ACS Paragon Plus Environment

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The Journal of Physical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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Department of Earth and Environmental Science and Department of Chemistry, University of

Pennsylvania, Philadelphia, PA, 19104, United States


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ABSTRACT

Quantum spin Hall (QSH) insulators have attracted much attention due to their potential application ranging from electronic devices to quantum computing. In general, a large bandgap is regarded as a critical descriptor in designing QSH insulators, however it faces challenges when additional factors such as strain and surface oxidation are involved in practical applications. In this work, taking M′′2M′C2O2 (M′ = Ti, Zr, Hf, M′′ = Mo, W) as a representative, results reveal that two-dimensional ordered transition metal carbides (MXenes) are promising candidates for flexible spintronic devices, which is ascribed to the mechanical flexibility and robust QSH states under straining. Although a large bulk bandgap is shown in M′′2HfC2O2, a strain-induced topological phase transition may limit its flexible application. On the other hand, M′′2TiC2O2 having a smaller gap, its topological nontrivial state survives under straining. When n changes from 0 to 4 in M′′2TinCn+1O2, a topologically nontrivial-trivial phase transition is observed in W2HfnCn+1O2, whereas a topologically nontrivial state remains in Mo2TinCn+1O2. After further screening a variety of promising coatings, it is found that fluorographene may effectively preserve the topologically nontrivial nature of M′′2M′C2O2 with surface oxidation resistance, even under straining, providing a feasible application of M′′2M′C2O2 as flexible QSH insulators.


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1. Introduction Quantum spin Hall (QSH) insulator is a two-dimensional (2D) topological insulator (TI) with a bulk bandgap and gapless helical edge states where electrons propagate counter with opposite spins.1-2 The backscattering of edge states is forbidden by time-reversal symmetry, which stimulates the interest in the fields of dissipationless transport, spintronics, fault-tolerant quantum computing, energy conversion, etc. 1-2 In the design of QSH insulators, a large bulk bandgap is generally considered as a critical quantity to guarantee that it could be operated at elevated temperature.3 When focusing on the practical application of QSH insulators, however, several other critical factors such as surface oxidation and strain must be considered. For instance, experimental results indicated that various chemical reactions could occur at the surface and edge of TIs, resulting in surface oxidation,4 dramatic changes of chemical potentials,5 and ageing effect on surface-transport properties.6 To avoid such surface chemical reactions, a solution is accordingly proposed to build surface coating layers on QSH insulators. On the other hand, the substrates, buffer layers, and/or protective layers are indispensable in using QSH insulators as field effect transistor (FET)7 and spintronics devices,8 and therefore, an internal strain induced by lattice mismatch may fundamentally modify the properties of QSH insulators. In addition, the flexible, stretchable, wearable electronic devices have gathered an increasing attention,9 in which strain is another critical factor in modulating the electronic properties and thus must be considered in a rational design. Motivated by addressing these issues, it is of practical value in search of novel 2D materials with mechanical flexibility and robust topologically nontrivial order under straining to meet the requirements for flexible devices, as well as to design an appropriate coating layer to prevent possible surface reactions.


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The metallic electric conductivity, cation intercalation ability, and hydrophilicity of 2D transition metal carbides and nitrides (MXenes) make them promising as energy storage, catalytic, and electromagnetic interference shielding materials.10 MXenes have a general chemical formula Mn+1XnTx, where M is early transition metal, X is carbon/nitrogen, n ranges from 1-3, and T stands for the surface termination –OH/–F/–O that depends on the selective etching process.10-11 To date, a large number of MXenes, e.g. M2CO2 (M = Mo, W),12 M′′2M′C2O2 (M′ = Ti, Zr, Hf, M′′ = Mo, W),13-15 M′′2M′2C3O2 (M′ = Ti, Zr, Hf, M′′ = Mo, W),15-16 Ti3N2F2,17 and Sc2C(OH)218 have already been predicted as QSH insulators, among which the Mo-based MXenes such as Mo 2CTx,19 Mo2TiC2Tx, and Mo2Ti2C3Tx20 have been synthesized experimentally. Although MXenes have been demonstrated to possess excellent mechanical strength and flexibility,21-24 as proposed in the first section, two issues remain to be resolved for the flexible application of MXenes in the topological fields: i) strain modification on the bulk bandgap and band topology, ii) surface chemical stability. For the first one, Si et al have demonstrated a topological semimetal-insulator transition at a biaxial strain of 3% in Mo 2Ti2C3O2,16 whereas it is unknown whether Mo 2Ti2C3O2 is mechanically stable under straining. As regards the second one, some MXenes-based QSH insulators are thought to be fully oxidized surfaces (e.g., M′′2M′C2O2 and M′′2M′2C2O2) and are stable against oxidization/degradation after the conversion of the surface termination –OH to – O.14, 16 However, such a precondition is in disagreement with the experimental results that the conversion of Ti3C2Tx to anatase TiO2 appeared at edges and other structural defects in spite of a high-quality sample,25-27 limiting the practical application of MXenes-based QSH insulators. To address this issue, a solution scheme has been proposed which involves encapsulating them with impermeable 2D materials;27 nonetheless, the effect of coating layers on the topological properties of MXenes is so far not explored to date, limiting our understanding the physics insight.


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In this work, we systematically explore the effect of strain, metal layer thickness and coating layer on band topology of M′′2M′nCn+1O2 (M′ = Ti, Zr, Hf, M′′ = Mo, W). It is found that M′′2M′C2O2 is a flexible QSH insulator material whose topological properties may be kept under reasonable straining. Although a larger bulk bandgap appears in M′′2HfC2O2 than that of M′′2TiC2O2, the latter remains its topologically nontrivial nature in a wider range of strain. Exploiting the effect of layer thickness on the topological properties of M′′ 2TinCn+1O2 indicates that various Mo2TinCn+1O2 keep a similar topological order under straining, different from a topologically nontrivial-trivial phase transition in W2HfnCn+1O2. By screening, finally an appropriate coating material, fluorographene is shown to be a promising candidate to preserve the nontrivial topological nature as well as to avoid surface oxidation of M′′2M′C2O2, even under straining. These findings should pave the way for designing new flexible electronic devices via newly discovered MXenes.

2. Theoretical Methods Density functional theory (DFT) calculations were performed with plane-wave method using Vienna ab-initio simulation package (VASP).28 The projector augmented wave (PAW) method29 was used with the Perdew, Burke, and Ernzerhof (PBE) exchange-correlation functional.30 A centered 18 × 18 × 1 Monkhorst-Pack k-point meshes and a 600 eV cutoff energy were adopted during the structural optimization, and the force criterion in the optimization was chosen as 10 -3 eV/Å. To avoid the interaction between the layers and theirs images, a more than 12 Å vacuum layer was adopted in supercell construction. Phonon dispersions were calculated based on density functional perturbation theory (DFPT) 31 through Phonopy code.32 Ab-initio molecular dynamics


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(AIMD) simulations were performed for 4 ×4 ×1 supercells at 300 K to assess the thermal stability using a canonical ensemble (NVT) with a Nose−Hoover thermostat.33-34 The energy/stress-strain relationship was derived using our ADAIS code,35 in which the strain ⃑ to the deformed one 𝑇 ⃑ ′ by multiplying a is imposed by transforming the original lattice vector 𝑇 deformation matrix D, ⃑′=𝑇 ⃑ ∙ (𝐼 + 𝜉 ), 𝑇 𝜀𝑥𝑥 where I is a 3 × 3 identity matrix, and 𝜉 is a strain matrix ( 0 0 state, 𝜀 = 𝜀𝑥𝑥 = 𝜀𝑦𝑦 =

𝑎−𝑎0 𝑎0

(1) 0 𝜀𝑦𝑦 0

0 0). For biaxial strain 0

, where a0 and a are the equilibrium and strained lattice constants,

respectively. To avoid the energy saddle points during the relaxation, the subtle ionic distortions were randomly imposed at each step to break the symmetry constraint.23,

36

A thickness

independent in-plane strength  was adopted,24 i.e., 1

𝜕𝐸

𝜕𝐸

𝜎 = 2𝐴 (𝜕𝜀 + 𝜕𝜀 ), 𝑥𝑥

𝑦𝑦

(2)

where A is the area of the basal plane and E is total energy. When focusing on the topological properties, the spin-orbit coupling (SOC) was taken into account self-consistently. The Z2 invariant is used to describe two types of time-reversal invariant band insulators: the ordinary (𝑍2 = 0) insulators that are topologically equivalent to the vacuum, and “topological” (𝑍2 = 1) ones that cannot be adiabatically converted to the vacuum without a bulk gap closure.37 Due to the inversion symmetry in MXenes, two methods are used to calculate the Z2 invariant.


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Method I According to the method proposed by Fu and Kane,38 Z2 can be determined by the parity eigenvalue 𝜁𝑚 (𝑘⃑𝑖 ) = ±1 of all occupied band eigenstates labeled by m up to N at four timereversal invariant momentum (TRIM) points 𝑘⃑𝑖 [(0, 0, 0), (0.5, 0, 0), (0, 0.5, 0), and (0.5, 0.5, 0) points] in the Brillouin zone, that is (−1)𝑍2 = ∏4𝑖=1 𝛿(𝑘⃑𝑖 )

(3)

⃑ 𝛿(𝑘⃑𝑖 ) = ∏𝑁 𝑚=1 𝜁𝑚 (𝑘𝑖 )

(4)

Method II A Wannier function for 2D unit cell 𝑅⃑ is defined as 𝑉 |𝑅⃑𝑛⟩ = (2𝜋)2 ∫𝐵𝑍 𝑑𝑘⃑𝑒 −𝑖𝑘⃑∙(𝑅⃑−𝑟) |𝑢𝑛 (𝑘𝑥 , 𝑘𝑦 )⟩

(5)

where V is the cell volume, and |𝑢𝑛 (𝑘𝑥 , 𝑘𝑦 )⟩ is the periodic part of the Bloch function |𝜓𝑛 (𝑘⃑)⟩. Accordingly, hybrid Wannier charge center (WCC) is defined as the expected value 𝑥̅ 𝑛 = ⟨0𝑛|𝑥̂|0𝑛⟩ of the position operator 𝑥̂ in the state |0𝑛⟩, which describes the Wannier function at 𝑅⃑ = 0, equivalently 𝑖

𝜋

𝑥̅ 𝑛 (𝑘𝑦 ) = 2𝜋 ∫−𝜋 𝑑𝑘𝑥 ⟨𝑢𝑛 (𝑘𝑥 , 𝑘𝑦 )|𝜕𝑘𝑥 |𝑢𝑛 (𝑘𝑥 , 𝑘𝑦 )⟩.

(6)

The gauge-independent electronic polarization 𝑃𝑒 (𝑘𝑦 ) 39 is the summation of the hybrid WCC 𝑥̅ 𝑛 (𝑘𝑦 ), i.e. 𝑃𝑒 (𝑘𝑦 ) = 𝑒 ∑𝑛 𝑥̅ 𝑛 (𝑘𝑦 ),


8

(7)

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where e is the electronic charge. When Hamiltonian 𝐻(𝑘𝑦 ) evolves adiabatically with 𝑘𝑦 changing from 𝑘𝑦1 to 𝑘𝑦2 , as well as the |𝑢𝑛 (𝑘𝑥 , 𝑘𝑦 )⟩ is continuous, the variation of electronic charge40 is expressed as, 1

𝜋

𝑃𝑒 (𝑘𝑦2 ) − 𝑃𝑒 (𝑘𝑦1 ) = 2𝜋 ∫−𝜋 𝑑𝑘𝑥 (𝐴𝑥 (𝑘𝑥 , 𝑘𝑦2 ) − 𝐴𝑥 (𝑘𝑥 , 𝑘𝑦1 )),

(8)

where 𝐴𝑥 (𝑘𝑥 , 𝑘𝑦 ) = 𝑖 ∑𝑛⟨𝑢𝑛 (𝑘𝑥 , 𝑘𝑦 )|𝜕𝑘𝑥 |𝑢𝑛 (𝑘𝑥 , 𝑘𝑦 )⟩ is the Berry connection, in which kx and ky are independent, and therefore a Berry curvature 𝐹𝑥,𝑦 (𝑘𝑥 , 𝑘𝑦 ) is defined as,

𝐹𝑥,𝑦 (𝑘𝑥 , 𝑘𝑦 ) = 𝑖 ∑𝑛 (⟨𝜕𝑘𝑥 𝑢𝑛 (𝑘𝑥 , 𝑘𝑦 )|𝜕𝑘𝑦 𝑢𝑛 (𝑘𝑥 , 𝑘𝑦 )⟩ − ⟨𝜕𝑘𝑦 𝑢𝑛 (𝑘𝑥 , 𝑘𝑦 )|𝜕𝑘𝑥 𝑢𝑛 (𝑘𝑥 , 𝑘𝑦 )⟩). (9) Accordingly, the Equation 8 changes to 𝜋

1

𝑦2

𝑃𝑒 (𝑘𝑦2 ) − 𝑃𝑒 (𝑘𝑦1 ) = 2𝜋 ∫−𝜋 𝑑𝑘𝑥 ∫𝑦1 𝑑𝑘𝑦 𝐹𝑥,𝑦 (𝑘𝑥 , 𝑘𝑦 ).

(10)

Because of the Kramers pairs, it is noted that only half period needs to be considered instead of a full one. According to the suggestions of Fu and Kane,41 the electronic polarization can be split to two parts in a time-reversal invariant system. Considering the Kramers degeneracy, a Bloch state 𝐼 𝐼𝐼 𝑢𝛼,𝑘 can transform to its Kramers partner 𝑢𝛼,−𝑘 after a time inversion operation 𝜃 and a changed 𝑥 𝑥

phase 𝜒𝛼,𝑘𝑥 ,41 i.e., 𝐼 𝐼𝐼 |𝑢𝛼,−𝑘 ⟩ = −𝑒 𝑖𝜒𝛼,𝑘𝑥 𝜃|𝑢𝛼,𝑘 ⟩, 𝑥 𝑥

(11)

𝐼𝐼 𝐼 |𝑢𝛼,−𝑘 ⟩ = 𝑒 𝑖𝜒𝛼,−𝑘𝑥 𝜃|𝑢𝛼,𝑘 ⟩, 𝑥 𝑥

(12)


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in which the 2N bands are labelled by the Kramers pair α = 1, ..., N and elements I and II are labeled by 𝜐 in each pair. The partial polarizations 𝑃𝑒𝐼 (𝑘𝑦 ) and 𝑃𝑒𝐼𝐼 (𝑘𝑦 )41 for both elements are defined as 𝜋

1

𝑃𝑒𝜐 (𝑘𝑦 ) = 2𝜋 ∫−𝜋 𝑑𝑘𝑥 𝐴𝜐𝑥 (𝑘𝑥 , 𝑘𝑦 ),

(13)

and the Berry connection for each element is given by 𝐴𝜐𝑥 (𝑘𝑥 , 𝑘𝑦 ) = 𝑖 ∑𝑛⟨𝑢𝑛𝜐 (𝑘𝑥 , 𝑘𝑦 )|𝜕𝑘𝑥 |𝑢𝑛𝜐 (𝑘𝑥 , 𝑘𝑦 )⟩.

(14)

By summing both partial polarizations, the total electronic polarization 𝑃𝑒 (𝑘𝑦 )39 is expressed by 𝑃𝑒 (𝑘𝑦 ) = 𝑃𝑒𝐼 (𝑘𝑦 ) + 𝑃𝑒𝐼𝐼 (𝑘𝑦 ).

(15)

Similarly, a new time-reversal polarization 𝑃𝜃 (𝑘𝑦 )41 can also be defined through the difference of the both partial polarizations, i.e., 𝑃𝜃 (𝑘𝑦 ) = 𝑃𝑒𝐼 (𝑘𝑦 ) − 𝑃𝑒𝐼𝐼 (𝑘𝑦 ).

(16)

Thereafter, the Z2 invariant41 is finally defined as 𝑇

𝑍2 = 𝑃𝜃 ( ) − 𝑃𝜃 (0)𝑚𝑜𝑑2. 2

(17)

In the present work, the WannierTools package42 was used to calculate the evolution of WCCs for 2D materials. The helical edge states were calculated using the iterative Green’s function43 as implemented in the WannierTools package42 in which the maximally localized Wannier functions (MLWFs) 44 is used in the tight-binding model with d orbitals of transition metal M and p orbitals of C and O as


10

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a basis set. Using the MLWFs, the edge Green’s functions (EGFs) 𝐺𝑒 (𝑘⃑, 𝜔) of the semi-infinite Mo2TiC2O2 and W2HfC2O2 can be constructed, and then the surface spectrum function 𝐴(𝑘⃑, 𝜔) can be obtained from the imaginary part of EGFs 1 𝐴(𝑘⃑, 𝜔) = − π lim+ ImTr𝐺𝑒 (𝑘⃑, 𝜔 + 𝑖𝜂), 𝜂⟶0

(18)

which provides the dependence of energy and momentum on the local density of states (LDOS).

3. Results and Discussion Lattice and electronic structure of M′′2M′C2O2. M′′2 M′C2O2 has a space group symmetry of P3-M1, in which surface oxygen terminations are attributed to the etching process from MAX precursors (a ternary layered ceramic material) to MXene derivatives. Although the experimental results indicate that a mixture of different groups such as –O, –F, and –OH coexisted at the surface of MXenes by nuclear magnetic resonance (NMR) spectroscopy45 and X-ray photoelectron spectroscopy (XPS),46 prior theoretical investigations demonstrated that the distribution of surface functional groups could be effectively modulated by the solution environment,47 in line with the experimental observations.45 Thus, a high-quality M′′2M′C2O2 might be synthesized in the specified solution environment and after post-treatment process since it is thermodynamically stable at ambient conditions. Figure 1a presents the oxygen terminations adsorbed in two sites of the MXenes surface, termed as site I and II. The combination of two sites in the top and bottom surfaces forms three different configurations, namely configuration I (two sites I), II (two sites II) and III (a site I and a site II). The total energy for each M′′2M′C2O2 is listed in Table S1 indicates that the configuration II for all M′′2M′C2O2 is the thermodynamically most stable configuration,


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in agreement with the previous reports.14-15 The absence of imaginary frequency in the phonon dispersions (Figure S1) and a periodic fluctuation of total energy during AIMD simulation (Figure S2) further validate the structural stability of M′′2 M′C2O2 (M′ = Ti, Zr, Hf, M′′ = Mo, W).

Figure 1. (a) The side view of lattice structure of M′′2M′C2O2. (b) The band gap comparison of Mo2TiC2O2 and W2HfC2O2 in GGA, GGA + U and HSE06. The projected band structure of (c) Mo2TiC2O2 and (d) W2HfC2O2. The Fermi level is set to zero energy.

To study the band topology of M′′2M′C2O2, the projected band structures of M′′2M′C2O2 (M′ = Ti, Zr, Hf, M′′ = Mo, W) are calculated, and the results are provided in Figure S3. It is found that the band structures of Mo 2M′C2O2 (M′ = Ti, Zr, Hf) and W2M′C2O2 (M′ = Ti, Zr, Hf) show a similar profile when the same M′ is selected, which indicates that the inner M′ atoms are responsible for the different band shapes of M′′2M′C2O2. Then, two prototypes are chosen, i.e., Mo2TiC2O2 and W2HfC2O2, to meticulously compare their band topologies in Figure 1c-1d. Two


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+ |𝑀′ − 𝑑𝑥𝑦,𝑥 2−𝑦 2 ⟩ states (Parities of the states are marked by +, −) are doubly degenerated at  + ⟩ states at  point is lower point at the Fermi level in the both systems. The energy of |𝑀′ − 𝑑𝑥𝑧,𝑦𝑧

than that of |𝑀′′ − 𝑑𝑧−2 ⟩ state in Mo2TiC2O2, showing an opposite trend in W2HfC2O2. When SOC + is considered, the degeneration of the two |𝑀′ − 𝑑𝑥𝑦,𝑥 2−𝑦 2 ⟩ states are lifted, forming a finite

energy gap at Fermi level. Considering the possible on-site Coulomb interaction, further comparison is made of the results derived from different functionals of normal GGA in Figure S3, GGA plus Hubbard U (GGA + U)48 in Figure S4-S5, and Heyd-Scuseria-Ernzerhof (HSE) screened hybrid functional49 in Figure S6-S7. A series of U values were set for the d electrons of M′′ and M′ from 0 to 5 eV. The calculated band gaps by GGA + U approach are shown in Table S2, and are compared with those by normal GGA and HSE method. Because a nonmagnetic ground state remains regardless of the changed U values, it can be concluded that M′′2M′C2O2 is not potential quantum abnormal Hall (QAH)- or Mott insulators, both of which are magnetic systems. It can be clearly seen in Figure 1b and S5-S7 that the band gaps obtained from GGA + U framework (an on-site Hubbard U value of 4 eV) match well with those from more exact HSE functional, both being comparable to those from GGA one. Notably, W2HfC2O2 (0.38 eV) shows a larger bulk bandgap than Mo 2TiC2O2 (0.11 eV), in agreement with the results reported by Khazaei et al.,15 which might be attributed to larger atomic on-site SOC strengths in the heavier elements according to the formula ΔΓ = 2𝜆 ( is the energy gap,  the SOC strength) in the tightbinding model proposed by Si et al. 14 Based on the inversion symmetry of M′′2 M′C2O2, the topological invariant Z2 was calculated by means of the approach proposed by Fu and Kane.38 It is found that Z2 = 1 is derived for Mo 2TiC2O2 and W2HfC2O2, indicating the nature of QSH + − ′′ insulator because of the band inversion between |𝑀′ − 𝑑𝑥𝑦,𝑥 2−𝑦 2 ⟩ states and |𝑀 − 𝑑𝑧 2 ⟩ states,


13


which is topologically nontrivial for the opposite parities, in consistency with the conclusions obtained by Si et al.14 and Khazaei et al.15 The critical band inversion in M′′2 M′C2O2 stems from the orbital hybridization in specific crystal structures instead of SOC interaction, in a similar manner to that in -GaS, -GaSe,50 ZrTe5,51 square-octagonal WS2,52 MX (M = Zr, Hf; X = Cl, Br, I).53 In addition, it is found that the critical band inversion remains in the band structures calculated by HSE methods (Figure S7), indicating that the robustness of topological properties of M′′2M′C2O2 (M′ = Ti, Zr, Hf, M′′ = Mo, W).

Figure 2. (a) The energy E vs strain  curves and (b) stress  vs strain  curves of M′′2M′C2O2. The top view and side view of electron localization function (ELF) maps of W2HfC2O2 under different straining are presented in the right panels to illustrate the instability mode. The snapshots at three different strain values (c) in equilibrium, (d) at the peak


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stress, and (e) after mechanical instability are selected for comparison. The ELF color scale ranges from 0 (more delocalized electron - blue) to 0.8 (more localized electron - red).

Effect of strain on band topology of M′′2M′C2O2. Before considering the effect of strain on electronic structures of M′′2M′C2O2, it is necessary to define a reasonable strain range to ensure its mechanical stability. Thus, the energy E/stress  – strain  curves under biaxial tension are calculated and shown in Figure 2a-2b. It is seen that the ideal strength (critical strain) of M′′2M′C2O2 follows the order: W2HfC2O2 (54.0 N/m (0.16)) > W2ZrC2O2 (52.3 N/m (0.16)) > W2TiC2O2 (49.3 N/m (0.14)) > Mo 2HfC2O2 (48.5 N/m (0.16)) > Mo2ZrC2O2 (46.9 N/m (0.16)) > Mo2TiC2O2 (45.6 N/m (0.13)). Overall, W2M′C2O2 shows a higher ideal strength than Mo2M′C2O2. Because of the similar lattice structures and valence electron configurations in all the M′′2M′C2O2, the close critical strains/ideal strengths and similar instability modes are obtained for the E/ –  curves of different M′′2M′C2O2. Under an increasing strain, the increasing distance between W atoms leads to a weakening surrounding in-plane delocalized electron gas, but an increasing electrostatic repulsion between C and O atoms owing to an out-of-plane Poisson contraction (see the ELF maps for points A and B in Figure 2c-2d). The electrostatic repulsion between C and O atoms is thought to play a dominant role in the adsorption of oxygen terminations at site I in M 2CO2 (M = Ti, Zr, Hf, V, Nb, and Ta); on the contrary, the in-plane delocalized electron gas among the metal atoms is responsible for the adsorption of oxygen terminations at site II in M 2CO2 (M = Cr, Mo, and W).22 Therefore, O atoms moves from site II to site I under a critical straining (see the ELF maps for point C in Figure 2e), resulting in the mechanical instability. Based on the strain range determined by mechanical calculation, the effect of strain in the two prototypes Mo2TiC2O2 and W2HfC2O2 on the band topology is investigated at strain  = 0 - 15%. + − ′ + ′′ Figure 3a-3b shows the shift of the |𝑀′′ − 𝑑𝑧+2 ⟩, |𝑀′ − 𝑑𝑥𝑦,𝑥 2 −𝑦 2 ⟩, |𝑀 − 𝑑𝑥𝑧,𝑦𝑧 ⟩ and |𝑀 − 𝑑𝑧 2 ⟩


15


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states at  point as an increasing strain in Mo 2TiC2 O2 and W2HfC2O2, respectively, and Figure S8S9 present their detailed projected band structures. Compared with the unchanged topological order in Mo2TiC2O2 (a similar manner in W2TiC2O2 in Figure S8) under straining, an energy level + − crossing between |𝐻𝑓 − 𝑑𝑥𝑦,𝑥 2−𝑦 2 ⟩ and |𝑊 − 𝑑𝑧 2 ⟩ states occurs in W2HfC2O2 (a similar manner

in Mo2HfC2O2 in Figure S9) at strain = 10%. The crossing results in a parity exchange between the occupied and unoccupied states at Γ point, which is responsible for a change of topological invariant Z2, i.e. a topological phase transition illustrated in Figure 3b. To be noted that the replacement of the inner Ti atoms to Hf results in different topological behaviors in M′′ 2M′C2O2 under straining. To further confirm the appearance of the topological phase transition, the topological invariant Z2 in Mo2TiC2O2 and W2HfC2O2 in equilibrium and at strain 

= 10% is

calculated, and the evolution of WCCs is shown in the right panels in Figure 3. A comparison between DFT and MLWFs fitted band structures for Mo 2TiC2O2 and W2HfC2O2 are shown in Figure S10. The odd times crossing of WCCs by an arbitrary horizontal line are shown in Mo2TiC2O2 either in equilibrium or at strain  = 10% (points A and B in Figure 3), validating that Z2 equals to 1, i.e. a characteristic topologically nontrivial phase. A similar trend appears in W2HfC2O2 in equilibrium (point C in Figure 3), whereas the crossing appears in even times during the evolution of WCCs at strain  = 10% (point D in Figure 3), indicating that Z2 equals to 0, confirming that the occurrence of the strain-induced topological phase transition. According to bulk-boundary correspondence,54 another distinctive feature of QSH insulators is the topologically protected edge state. Figure 3 presents the edge band dispersion of Mo2TiC2O2, in which the bands cross the Fermi level once (three times) between  and M point in equilibrium (at strain  = 10%), indicating the topologically nontrivial nature of Mo2TiC2O2 under straining (points A and B in Figure 3). On the contrary, the edge bands in W2HfC2O2 cross over the Fermi level twice at strain


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 = 10% (point D in Figure 3), differing from the edge states at equilibrium, in which the crossing appears three times (point C in Figure 3), demonstrating the appearance of a topological phase transition.

+ − ′ + ′′ Figure 3. Energy levels of |𝑀′′ − 𝑑𝑧+2 ⟩, |𝑀′ − 𝑑𝑥𝑦,𝑥 2 −𝑦2 ⟩, |𝑀 − 𝑑𝑥𝑧,𝑦𝑧 ⟩ and |𝑀 − 𝑑𝑧 2 ⟩ states at  point vs strain 

curves of (a) Mo2TiC2O2 and (b) W2HfC2O2. The evolution of Wannier charge centers and momentum-dependent local density of states of Mo2TiC2O2 and W2HfC2O2 in equilibrium and at strain  = 10% are presented in the right panels, respectively.

Effect of metal layer thickness on band topology of M′′ 2M′nCn+1O2. Because of the critical effect of the inner metal atoms (i.e. Ti and Hf) on the topological order of M′′ 2M′C2O2 under


17


straining, one may wonder how the change of the metal layer thickness (a changed number of inner metal atoms, i.e. n in M′′2M′nCn+1O2) influences the band topology under straining, which is different from the odd−even oscillation of the topological order as the increasing layers (i.e. topologically trivial vs nontrivial phase for even vs odd layers).55 Figure 4a and 4c present the shift + − ′ + ′′ of the |𝑀′′ − 𝑑𝑧+2 ⟩ , |𝑀′ − 𝑑𝑥𝑦,𝑥 2 −𝑦 2 ⟩ , |𝑀 − 𝑑𝑥𝑧,𝑦𝑧 ⟩ and |𝑀 − 𝑑𝑧 2 ⟩ states at  point as an

increasing n from 0 to 4 in Mo 2TinCn+1O2 and W2 HfnCn+1O2, respectively. It should be noted that + + ′′ the |𝑀′ − 𝑑𝑥𝑦,𝑥 2 −𝑦 2 ⟩ states in Mo 2CO2 and W2CO2 correspond to |𝑀 − 𝑑𝑥𝑦,𝑥 2−𝑦 2 ⟩ ones due to

+ the absence of M′. Compared to the band inversion between |𝑀 ′ − 𝑑𝑥𝑦,𝑥 2−𝑦 2 ⟩ states and

+ |𝑀′′ − 𝑑𝑧−2 ⟩ ones in Mo2TinCn+1O2 (n = 0 - 4), a level crossing between |𝑀′ − 𝑑𝑥𝑦,𝑥 2 −𝑦 2 ⟩ and

|𝑀′′ − 𝑑𝑧−2 ⟩ states occurs in W2Hf2C3O2, which results in the value of Z2 changing from 1 to 0, i.e. a topologically nontrivial to trivial phase transition. The shift of the |𝑀′′ − 𝑑𝑧+2 ⟩ , |𝑀′′ − + − ′′ 𝑑𝑥𝑦,𝑥 2 −𝑦 2 ⟩, and |𝑀 − 𝑑𝑧 2 ⟩ states at  point vs strain curves in M′′2CO 2 (M′′ = Mo, W) and the

+ − ′ + ′′ shift of the |𝑀′′ − 𝑑𝑧+2 ⟩ , |𝑀′ − 𝑑𝑥𝑦,𝑥 2−𝑦 2 ⟩, |𝑀 − 𝑑𝑥𝑧,𝑦𝑧 ⟩ and |𝑀 − 𝑑𝑧 2 ⟩ states at  point vs

strain  curves in Mo2TinCn+1O2 (n = 2 - 4) are shown in Figure S11-S14 for comparison. An unchanged topological order remains in Mo 2TinCn+1O2 (n = 0 - 4) and W2CO2 under straining. To further demonstrate the origin of the topological phase transition with the variation of n in W2HfnCn+1O2 (n = 0 - 4), the lattice constant a vs n and the distance of two M′′ layers h vs n of Mo2TinCn+1O2 and W2HfnCn+1O2 (n = 0 - 4) are shown in Figure 4b and 4d, respectively. Compared to the similar linear relationship between h and n in Mo2TinCn+1O2 and W2HfnCn+1O2 (n = 0 - 4), a profound increasing trend of a is observed for W2HfnCn+1O2 (e.g. 2.88 Å in W2CO2 vs 3.07 Å in


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W2Hf2 C3O2) than that for Mo2TinCn+1O2 (e.g. 2.88 Å in Mo 2CO2 vs 2.97 Å in Mo2Hf2C3O2). In order to check whether the topological phase transition in W2HfnCn+1O2 is related to the increasing lattice constant, a compression strain is applied in W2Hf2C3O2 (see Figure S11 and S13); A reduced + − ′′ energy difference between |𝑀 ′ − 𝑑𝑥𝑦,𝑥 2 −𝑦 2 ⟩ and |𝑀 − 𝑑𝑧 2 ⟩ states at  point under increasing

compression indicates that a possible topologically trivial-nontrivial phase transition might occur under a larger straining, which suggests that the increasing lattice constant might play a secondary role on the topologically nontrivial-trivial phase transition.

+ − ′ + ′′ Figure 4. Energy levels of |𝑀′′ − 𝑑𝑧+2 ⟩, |𝑀′ − 𝑑𝑥𝑦,𝑥 2 −𝑦2 ⟩, |𝑀 − 𝑑𝑥𝑧,𝑦𝑧 ⟩ and |𝑀 − 𝑑𝑧 2 ⟩ states at  point vs the + number of inner Ti/Hf n curves of (c) Mo2TinCn+1O2 and (b) W2HfnCn+1O2. Note that the |𝑀′ − 𝑑𝑥𝑦,𝑥 2 −𝑦2 ⟩ states in


19


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+ Mo2CO2 and W2CO2 correspond to |𝑀′′ − 𝑑𝑥𝑦,𝑥 2 −𝑦2 ⟩ for the absence of M′. Lattice constant a and distance between

two M′′ metal layers h vs the inner M′ number n curves of (b) Mo2TinCn+1O2 and (d) W2HfnCn+1O2.

Effect of coating layer on band topology of M′′2M′C2O2. In search of a promising coating layer, a variety of quantum well structures are built with common insulating/semiconducting 2D h-BN, fluorographene (FG), graphane (fully hydrogenated graphene, i.e. HG), transition metal dichalcogenides (TMDs) as coating layers (see Figure 5a or 5c). The lattice mismatch  between M′′2M′C2O2 and coating layers is defined as 𝑎

𝛾 = |𝑎𝐴 − 1|,

(19)

𝐵

where aA and aB are the lattice vectors of M′′2M′C2O2 and coating layers, respectively. By minimizing the lattice mismatch through the variation of lattice periods, the following models are 2

2

2

adopted: 2 × 2 h-BN/ √3 × √3𝑅30° Mo2TiC2O2 ( 2

2

= 1.6%), 2 × 2 h-BN/ √3 × √3𝑅30° 2

W2HfC2O2 (3.8%), 2 × 2 fluorographene/ √3 × √3𝑅30° Mo2TiC2O2 (2.0%), 2 × 2 2

2

2

2

fluorographene/ √3 × √3𝑅30° W2HfC2O2 (0.2%), 2 × 2 graphane/ √3 × √3𝑅30° Mo2TiC2O2 2

2

(0.5%), 2 × 2 graphane/ √3 × √3𝑅30° W2HfC2O2 (2.7%), 1 × 1 MoS2/ 1 × 1 Mo2TiC2O2 (7.4%), 1 × 1 WS2/1 × 1 W2HfC2O2 (5.4%). Figure S15 shows the calculated electronic band structures of all the studied quantum well structures. Differing from the disappeared bulk bandgaps in

h-BN/Mo2TiC2O2

(see

Figure

5b),

h-BN/W2HfC2O2,

graphane/Mo2TiC2O2,

graphane/W2HfC2O2, MoS2/Mo2TiC2O2, and WS2/W2HfC2O2, a bulk bandgap remains in fluorographene/Mo2TiC2O2 (Figure 5d) and fluorographene/W2HfC2O2. A projected band structure shown in Figure 5d indicates further that the critical band inversion could be well preserved, even under a finite strain of 5% (shown in Figure S16), confirming that fluorographene


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might be an appropriate candidate for the coating layers. It is noted additionally that fluorographene was demonstrated as a promise insulator interlayer in solar cells.56

Figure 5. Schematics of quantum well structures consisting of M′′2M′C2O2 and (a) h-BN and (c) fluorographene (FG). The atomic projected band structures of the corresponding structures: (b) h-BN/M′′2M′C2O2/h-BN and (d) fluorographene/M′′2M′C2O2/fluorographene.

Discussion. Since the QSH effect in M′′2M′nCn+1O2 is derived from the band inversion between + + − ′′ ′′ |𝑀′ − 𝑑𝑥𝑦,𝑥 2−𝑦 2 ⟩ states (|𝑀 − 𝑑𝑥𝑦,𝑥 2 −𝑦 2 ⟩ states in M′′2CO2 (M′′ = Mo, W)) and |𝑀 − 𝑑𝑧 2 ⟩

states, the band inversion strength E can be defined as the energy differences between these two states, ∆𝐸 = 𝐸|M′′ −𝑑− ⟩ − 𝐸|𝑀′ −𝑑 + 𝑧2

⟩ 𝑥𝑦,𝑥2 −𝑦2

,

(20)

where a positive (negative) E corresponds to its topologically nontrivial (trivial) nature. The relationship between band inversion strength E and lattice constant a in all the studied systems


21


is shown in Figure 6, in which a linearly decreasing E as increasing a appears in Mo2TinCn+1O2 and W2TinCn+1O2 (n = 1 - 4). As revealed through a sharp increasing a as n in W2TinCn+1O2, a negative E appears in W2TinCn+1O2 (n = 2 - 4), suggesting the connection between E and a. In addition, Mo2TinCn+1O2 can remain its topologically nontrivial nature at strain  = 15%, whose lattice constant a (about 3.4 Å) is far greater than that of topological nontrivial W2Hf2C3O2 (3.07 Å), suggesting that the inner Ti atoms play a more critical role in determining the topological behavior than lattice constant a.

Figure 6. The relationship between band inversion strength E and lattice constant a in the studied MXenes.

In general, the surface functionalization provides an efficient approach to tune the electronic and topological properties of 2D materials. For instance, various functionalized 2D materials such as ethynyl-functionalized stanene (SnC2H),57 amidogen-functionalized stibium (SbNH2), amidogenfunctionalized stibium bismuthum (BiNH2),58 and arsenene oxide (AsO)59 were already proposed as potential QSH insulators. For the oxygen functionalized M′′2M′C2O2, the oxygen atoms react with the dangling bonds consisting of d states of M′′ atoms, resulting in the movement of certain d states away from the Fermi level, thus providing a theoretical foundation for the transition from


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metallic M′′2M′C2 to QSH insulators M′′2M′C2O2. The effect can be attributed to the known orbital filtering effect (OFE), in a similar manner to the mechanism in AsO.59 Compared to the theoretical prediction for other MXenes-based QSH insulator materials Ti3N2 F217 and Sc2C(OH)2,18 the recent preparation of Mo-based MXenes (e.g., Mo2CTx,19 Mo2TiC2Tx, and Mo2Ti2C3Tx20) undoubtedly provides a solid foundation for the practical application of MXenes-based QSH insulators. After high-temperature annealing in an ultrahigh vacuum environment, highly ordered oxygen-functionalized MXene Ti3C2O2 was synthesized,60 indicating that it is feasible to synthesize high-quality Mo2TiC2O2 and Mo2Ti2C3O2 by welldesigned experiments. In addition, the thermoelectric properties of Mo 2TiC2Tx, and Mo2Ti2C3Tx reported in recent experiments61 provides further a perspective to uncover their topological insulating characteristics experimentally in view of a tight connection between thermoelectricity and TIs.62 Nevertheless, several critical issues remain to be addressed for MXenes-based QSH insulators. First, a small bulk bandgap of Mo 2TiC2 O2 (0.09 eV) might limit its practical application at elevated temperature according to an estimation of thermal energy kBT (0.026 eV at 300 K, 0.09 eV at ~1000 K), in which kB is Boltzmann constant and T is temperature. Therefore, W2HfC2O2 with a larger bulk bandgap is more promising for the practical application but it has not been synthesized experimentally thus far. Second, the oxidative degradation of MXenes (e.g., Ti3C2Tx to anatase TiO2) is unavoidable to date.25-27 Although composites consisting of MXenes (Ti3C2Tx) and graphene derivatives (reduced graphene oxide) are reported by several researchers,63-66 it is still challenging to build high-quality heterostructures at the atomic level. Third, despite the observed mechanical flexibility of MXene sheets/paper,61 the mechanical predictions are always based on an ideal defect-free model, which provides an upper limit of strength and flexibility that a real material can achieve. Compared to the insensitivity of topological properties to defects,1 it


23


is well known that defect structures play a vital role in the mechanical properties of materials,67 and therefore it is necessary albeit challenging to consider the effect of defects on the mechanical properties in the design of flexible MXenes-based QSH insulators.

4. Conclusions In summary, by means of a comprehensive investigation on the effect of strain, metal layer thickness, and coating layer on the band topology of various 2D hybrid MXenes, results from this work reveal that some MXenes are promising candidates for flexible electronic devices because of their mechanical strength and flexibility, as well as robust QSH state under straining. The key findings are summarized as below: i) Although a larger band gap appears in M′′2HfC2O2 than that in M′′2TiC2O2, the latter remains its topologically nontrivial nature in a wider strain range, even till its mechanical instability, bringing a novel pathway in designing flexible QSH insulators. ii) Mo2TinCn+1O2 exhibits the same topologically nontrivial order under straining when moving n from 0 to 4, different from a topologically nontrivial-trivial phase transition in W2HfnCn+1O2, which is attributed to the difference of inner metal atoms and more profound lattice expansion in W2HfnCn+1O2. iii) Screening has identified an appropriate coating layer among various 2D materials, fluorographene, which is found to be a promising candidate to preserve the nontrivial topological nature and avoid surface oxidation of M′′2M′C2O2, even under straining, providing another effective solution for the practical application of M′′2M′C2O2 as flexible QSH insulators.


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ASSOCIATED CONTENT Supporting Information Total energies of M′′2M′C2O2 in different configurations, the test of U values, phonon dispersions and AIMD simulation of M′′2M′C2O2, projected band structures of M′′2M′C2O2 in GGA, GGA + U, HSE, the fitting of wannier bands to DFT bands, projected band structures of M′′2M′nCn+1O2 (n = 0 - 4), and atomic projected band structures of Mo 2TiC2O2 and W2HfC2O2 with various coating layers. (PDF)

AUTHOR INFORMATION Corresponding Author *[email protected] ACKNOWLEDGMENT This work is supported by the National Key Research and Development Program of China (No. 2017YFB0702100), National Natural Science Foundation of China (NFSC) with No. 51672015, National Thousand Young Talents Program of China, and Fundamental Research Funds for the Central Universities. D. L. is supported by the European Regional Development Fund at the IT4Innovations National Supercomputing Center

–

Path to

Exascale Project,

No.

CZ.02.1.01/0.0/0.0/16_013/0001791 within the Operational Programme Research, Development and Education and by Czech Science Foundation Project No. 17-27790 and Mobility grant No. 8J18DE004 and SGS No. SP2019/110. We appreciate the support from the key technology of


25


nuclear energy, 2014, CAS Interdisciplinary Innovation Team and ITaP at Purdue University for computing resources. We would also like to thank Prof. G. Kresse for valuable advice for the application of VASP.

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10. Anasori, B.; Lukatskaya, M. R.; Gogotsi, Y., 2D Metal Carbides and Nitrides (MXenes) for Energy Storage. Nat. Rev. Mater. 2017, 2, 16098. 11. Naguib, M.; Mochalin, V. N.; Barsoum, M. W.; Gogotsi, Y., 25th Anniversary Article: MXenes: A New Family of Two-Dimensional Materials. Adv. Mater. 2014, 26, 992-1005. 12. Weng, H.; Ranjbar, A.; Liang, Y.; Song, Z.; Khazaei, M.; Yunoki, S.; Arai, M.; Kawazoe, Y.; Fang, Z.; Dai, X., Large-Gap Two-Dimensional Topological Insulator in Oxygen Functionalized MXenes. Phys. Rev. B 2015, 92, 075436. 13. Li, L., Lattice Dynamics and Electronic Structures of Ti3C2O2 and Mo2TiC2O2 (MXenes): The Effect of Mo Substitution. Comput. Mater. Sci. 2016, 124, 8-14. 14. Si, C.; Jin, K.-H.; Zhou, J.; Sun, Z.; Liu, F., Large-Gap Quantum Spin Hall State in MXenes: d-Band Topological Order in a Triangular Lattice. Nano Lett. 2016, 16, 6584-6591.


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Figure 1. (a) The side view of lattice structure of M′′2M′C2O2. (b) The band gap comparison of Mo2TiC2O2 and W2HfC2O2 in GGA, GGA + U and HSE06. The projected band structure of (c) Mo2TiC2O2 and (d) W2HfC2O2. The Fermi level is set to zero energy. 170x114mm (300 x 300 DPI)


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Figure 2. (a) The energy E vs strain  curves and (b) stress  vs strain  curves of M′′2M′C2O2. The top view and side view of electron localization function (ELF) maps of W2HfC2O2 under different straining are presented in the right panels to illustrate the instability mode. The snapshots at three different strain values (c) in equilibrium, (d) at the peak stress, and (e) after mechanical instability are selected for comparison. The ELF color scale ranges from 0 (more delocalized electron - blue) to 0.8 (more localized electron - red). 132x98mm (300 x 300 DPI)



Figure 3. Energy levels of , , and states at  point vs strain  curves of (a) Mo2TiC2O2 and (b) W2HfC2O2. The evolution of Wannier charge centers and momentum-dependent local density of states of Mo2TiC2O2 and W2HfC2O2 in equilibrium and at strain  = 10% are presented in the right panels, respectively. 141x116mm (300 x 300 DPI)


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Figure 4. Energy levels of , , and states at  point vs the number of inner Ti/Hf n curves of (c) Mo2TinCn+1O2 and (b) W2HfnCn+1O2. Note that the states in Mo2CO2 and W2CO2 correspond to for the absence of M′. Lattice constant a and distance between two M′′ metal layers h vs the inner M′ number n curves of (b) Mo2TinCn+1O2 and (d) W2HfnCn+1O2. 145x115mm (300 x 300 DPI)



Figure 5. Schematics of quantum well structures consisting of M′′2M′C2O2 and (a) h-BN and (c) fluorographene (FG). The atomic projected band structures of the corresponding structures: (b) hBN/M′′2M′C2O2/h-BN and (d) fluorographene/M′′2M′C2O2/fluorographene. 152x102mm (300 x 300 DPI)


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Figure 6. The relationship between band inversion strength E and lattice constant a in the studied MXenes. 85x71mm (300 x 300 DPI)



Table of Contents Image 85x71mm (300 x 300 DPI)


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Designing Flexible Quantum Spin Hall Insulators through 2D Ordered

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