Half-auxetic effect and ferroelasticity in a two-dimensional monolayer TiSe

Two-dimensional (2D) materials exhibit extraordinary physical properties which distinct from those in their three-dimensional counterparts due to the quantum confinement effect [1], and are regarded to have great application potential in the next-generation electronic [2–5], optoelectronic [6, 7], electromagnetic devices [8]. Great efforts have been devoted to the investigation of 2D materials especially in the last decade, thousands of 2D materials were theoretically predicted, and their electronic properties were studied [9].

Despite the great success in the discovery and investigation of 2D functional electronic materials, the mechanical properties of 2D materials are less reported. Common materials undergo a transverse contraction (expansion) when they were stretched (compressed). However, the auxetic materials [10] undergo a transverse contraction (expansion) when compressed (stretched), and are promising candidates in biomedicine, fasteners [10], sensors, national security and defense [11, 12], and other fields due to their largely increased mechanical toughness. The auxetic effect has been reported in a number of 2D materials, such as δ-phosphorene [13], monolayer BP₅ [14], 2D silicon dioxide [15], monolayer CaSi [16], 2D honeycomb structures [17] (graphene, silicene, h-BN, h-GaN, h-SiC and h-BAs). The origin of the auxetic effect in these 2D materials was attributed to pure geometric effect, pure electronic effect, or remain obscure in most of the cases.

Ferroelasticity represents the effect of the mechanical switching between at least two orientation states of a crystal by external stress [18, 19], and ferroelastic materials can be used to construct nonvolatile memory readable/writeable devices [20] at ambient conditions. The extraordinary combination of auxetic effect and ferroelasticity in a single compound makes the 2D material a highly versatile and promising in electromechanical at the nanoscale. However, because auxetic 2D materials and 2D ferroelastic materials are both rare, the discovery of 2D materials coexisting of these two intriguing properties is still limited to several exemplary materials [14, 21–24].

In this study, using density functional theory based first principle calculations, we report that the monolayer TiSe [25, 26] in p2mm-type exhibits both half-auxetic effect [27] and ferroelasticity. Different from the traditional auxetic materials, which have a linear mechanical response to a uniaxial strain close to the equlibrium state, monolayer TiSe undergoes a transverse expansion no matter stretched or compressed under small strain, that is TiSe possesses a half-auxetic effect. This unusual mechanical behavior is elucidated by analyzing both the nearest neighboring Ti–Se interactions and also the next-nearest Ti–Ti interactions under strain. Under a suitable uniaxial tensile (compressive) strain along the short (long) axis direction, a ferroelastic switching occurs with a switching barrier of 0.25 eV. Since monolayer TiSe is dynamically and thermally stable, and its exfoliation energy is low (∼0.025 eV Å⁻²), it provides a promising platform for experimental investigation of the exotic properties and potential applications in nanoscale devices.

The structure and projected density of states calculations were performed by using the Vienna ab initio simulation package (VASP) code [28, 29]. We used the density functional theory (DFT) within the generalized gradient approximation of the Perdew−Burke−Ernzerhof (PBE) functional [30]. An energy cutoff of 350 eV was used for the plane-wave basis set and 15 × 15 × 1 K-points was used to sample the Brillouin zone, 45 × 45 × 1 K-points was used when calculating the density of states. The vacuum length is 20 Å along the z axis. The monolayer TiSe was fully relaxed until the energy was converged to 10⁻⁸ eV and the maximum forces on each atom were below 0.001 eV Å⁻¹. A non-local correction function vdW-DF (optB86b-vdW) [31, 32] was employed to consider the van der Waals interactions. Phonon spectrum was calculated using the DFPT method implemented in Phonopy code [33]. Ab initio molecular dynamics (AIMD) simulations were performed to estimate the thermal stability [34]. The solid-state nudged elastic band (SSNEB) method [35] was used to search the ferroelastic transition pathway.

To calculate the mechanical response of monolayer TiSe under uniaxial strain along x- (or y-) direction, the strain along x- (or y-) direction is fixed, and the strain along the other direction is allowed to relax until equilibrium. As a fundamental mechanical property, Poisson's ratio describes the negative ratio of transverse strain to infinitesimal longitudinal strain [36, 37]. Similarly, we use the equations $-\frac{{\varepsilon }_{y}}{{\varepsilon }_{x}}=-\frac{a{\Delta}y}{b{\Delta}x}$ and $-\frac{{\varepsilon }_{x}}{{\varepsilon }_{y}}=-\frac{b{\Delta}x}{a{\Delta}y}$ to describe of magnitude of mechanical response to a large range of uniaxial strain along x- and y- directions, respectively. Here, a and b are the lattices constants of TiSe along x- and y- directions at equilibrium, Δx and Δy are the lattice constant deviations of TiSe along x- and y- directions under uniaxial strain, respectively.

The layered bulk material of TiSe is obtained from the previous literature [38]. The optimized structure models of 2D TiSe monolayer with the space group p2mm are shown in figure 1(a). The primitive cell is labeled by a black rectangle, and it contains two Ti atoms (gray) and two Se atoms (yellow). Each Ti atom is surrounded by four Se atoms. The relaxed lattice constants of monolayer TiSe are 4.40 Å and 3.57 Å along x-, and y- direction, respectively. Figure 1(b) presents a Brillouin zone marked by the high symmetry k-points. In order to estimate the dynamical stability of monolayer TiSe, we calculated the phonon dispersion as shown in figure 1(c). No imaginary frequency is observed, indicating that monolayer TiSe is dynamically stable. In addition, we also checked thermally stability of monolayer TiSe. The ab initio molecular dynamics (AIMD) simulation results in figure 1(d) clearly shows that after 5 ps there is neither structure reconstruction nor bond breaking at 300 K, implying the monolayer TiSe is thermally stable. To estimate the possibility of monolayer exfoliation from its bulk material, the normalized binding energies was calculated by E_b = (E_monolayer − E_bulk)/S, where E_monolayer and E_bulk are total energies of monolayer and bulk TiSe, and S is the area in the xy-plane. Van der Waals interaction was considered as described in the method part. The computed binding energy of TiSe is around 25 meV Å⁻², indicating the high possibility of monolayer exfoliation from the bulk material.

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Next, we proceed to examine the structural response to uniaxial strain and evaluate the resulting negative ratio of strain, as demonstrated in figure 2. The resultant lateral strain in response to a uniaxial strain in the range of ±7% applied along the x- or y- direction is calculated. Non-linear variation of ɛ_y (or ɛ_x ) to ɛ_x (or ɛ_y ) can be observed. Under uniaxial strain along x- direction, a linear increasing of $-\frac{{\varepsilon }_{y}}{{\varepsilon }_{x}}$ as compressive strain is observed, while a non-linear variation of $-\frac{{\varepsilon }_{y}}{{\varepsilon }_{x}}$ as tensile strain is observed. Under uniaxial strain along y- direction, the mechanical response is different, indicating an anisotropy mechanical properties of TiSe. In this case, under compressive strain, the $-\frac{{\varepsilon }_{x}}{{\varepsilon }_{y}}$ firstly increases as strain and then reaches a platform. Under tensile strain, the curve decreases first, then linearly increases after it reaches a minimum value. A transition from negative to positive is observed at the tensile strain around 4%.

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To understand this novel mechanical behavior under strain, a geometric structure evolution model is constructed as shown in figure 2(c). Here, we only qualitatively analyze the case that an uniaxial strain along x- direction is applied, since the mechanism of geometric evolution under an uniaxial strain along y- direction is similar. Under stretch, the Ti₁ atom moves to the left, while the Ti₂ atom moves to the right. The Se atom moves downward simultaneously due to the strong attractive interactions from Ti₁ and Ti₂ atoms. The downward shift of the Se atom further generates repulsive forces pointing from Se atom to Ti₃ and Ti₄ atoms. Meanwhile, the move of Ti₁ and Ti₂ atoms also induces the increasing of neighboring Ti-Ti distance. Figure 2(d) shows a bonding state below Fermi energy and an antibonding state above Fermi energy, which are contributed by d_xy orbitals of Ti atoms. This phenomenon indicates that the next-nearest interactions cannot be ignored. Overall, Ti₃ atoms undergo two forces, repulsive force F_Se from Se atom, and attractive force F₁₂ from Ti₁ and Ti₂ atoms, as shown in the top-right panel in figure 2(c). If the included angle φ between resultant force and F₁₂ is larger than 90°, an expansion of the lattice constant along y- direction occurs, and the auxetic effect of TiSe thus occurs. Inversely, if the included angle φ is less than 90°, no auxetic effect of TiSe occurs. Similarly, the geometric structure evolution under compression is shown in the bottom-right panel in figure 2(c). Ti₃ atoms undergo an attractive force from Se atom and a repulsive force from Ti₁ and Ti₂ atoms, the $-\frac{{\varepsilon }_{y}}{{\varepsilon }_{x}}$ is also determined by the direction of resultant force F_tot.

Under uniaxial strain along x- direction (or y- direction), both tensile strain and compressive strain lead to the expansion of TiSe in the y- direction (or x- direction) near the equilibrium state. This behavior is different from the typical 3D artificial auxetic materials in which the auxetic effect is totally a geometric effect, and also different from the recently reported 2D auxetic materials which have linear mechanical response to uniaxial strain, at least under small longitudinally compression or stretch. This half-auxetic effect makes TiSe a good platform for the researches of novel functional applications.

There are two stable orientation variants of monolayer TiSe named initial variant I and final variant III in figure 3(a). For the initial variant I, the larger lattice constant lies in the x- direction. After compressing (or stretching) it continuously along the x- (y-) direction, the larger lattice constant may switch to the y- direction, that is the final variant III, where |a'| = |b| and |b'| = |a|. From the analysis of the lattice constant, the final variant III can be obtained by rotating the initial variant I under 90° clockwise rotation. At the same time, during the compression process, there will be a square transition state II with equal lattice constant in the x- and y- directions (|a| = |b|). The ferroelastic switching pathway between variant I and III was studied by using the solid-sate nudged elastic band (ss–NEB) method. As shown in figure 3(b), the intermediate variant connecting initial variant I and final variant III is the transition state II with a symmetry of p4mm, and its lattice constants along the x- and y- directions are equal, a = b. The energy barrier switching from initial variant I to transition state II is about 0.25 eV/atom, which is smaller than that of BP₅ (0.32 eV/atom) [14]. The overall geometric evolution of TiSe under tensile strain along y- direction contains several steps. Firstly, under small strain the lattice constant along x- direction increases due to large repulsive force from the top Se atom to the Ti atoms in the strain direction, resulting in an auxetic effect in TiSe. As further increasing the strain, the geometric structure then contracts along y- direction. From the transition state II to the initial state I and final state III, the geometric structure evolution processes are the same except that the directions are different. Relative to the initial state I of TiSe, the transition state II occurs at a stretch strain around 15% along y- direction.

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Since the surrounding atoms of Se atom along x- and y- directions are the same, the origin of symmetry broken should come from the electronic structures. Figure 4 shows the projected density of states (PDOS) of monolayer TiSe on 3d-ortibals of Ti atom under both ground state and transition state. At the transition state II, the TiSe has a p4mm plane group. The Ti atom and its four neighboring Se atoms form a distorted tetrahedron. In this crystal field, the 3d-orbitals splitting into four groups, d_xz d_yz , d_xy , d_z² , d_{x² -y²} as shown in figure 4(a). The d_xz and d_yz orbitals degenerate in this square lattice, as can be seen that the green and red curves are completely overlapped, and both orbitals are partially filled. When the structure is distorted from the square lattice to a rectangular lattice, the degenerated d_xz and d_yz orbitals split into two non-degenerated ones. Here in the variant I, the d_xz orbital goes down, while the d_yz orbital goes up. Our calculated results reveal that at the transition state II, both the d_yz and d_xz orbitals are occupied by 0.99 electrons. At the group state-variant I, these two orbitals are occupied by 1.22 and 0.81 electrons, respectively. This phenomenon is known as Jahn-Teller distortion, more electrons occupy the lower orbital and less electrons occupy the higher orbital, thus lowers the total electronic energy, TiSe thus prefers the lattice with lower symmetry. Similar ferroelastic transition was theoretically predicted in α-SnO monolayer: from p2mm to p4mm, then to p2mm [39].

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In summary, by performing first-principles calculations, we predicted that p2mm-type TiSe monolayer is dynamically and thermally stable, and has a coexistence of a unique half-auxetic effect and ferroelastic property. The half-auxetic effect originates from the interplay of both the nearest and next-nearest interactions in a buckled geometric structure. Anisotropic mechanical properties along x- and y- directions are observed due to the geometric symmetry broken in these two directions. The ferroelastic switching can be achieved by applying tensile strain along short axis or compressive strain along long axis with a transition barrier of about 0.25 eV/atom. The origin of the ferroelasticity in TiSe is the Jahn–Teller effect, which breaks the degeneracy of partially occupied d_xz and d_yz orbitals of Ti atom, and lowers the total electronic energy. This exotic combination of half-auxetic effect and ferroelasticity makes TiSe promising as multiple functional materials at the nanoscale.

The data that support the findings of this study are available upon reasonable request from the authors.

This work was financially supported by National Nature Science Foundation of China (61888102), National Key Research and Development Projects of China (2016YFA0202300 and 2018YFA0305800), the Strategic Priority Research Program of the Chinese Academy of Sciences (XDB30000000). Computational resources were provided by the National Supercomputing Center in Tianjin.