In-plane Compressive Strain Research Articles

Enhanced Oxygen Evolution Electrocatalysis in Strained A-Site Cation Deficient LaNiO3 Perovskite Thin Films.

As the BO6 octahedral structure in perovskite oxide is strongly linked with electronic behavior, it is actively studied for various fields such as metal-insulator transition, superconductivity, and so on. However, the research about the relationship between water-splitting activity and BO6 structure is largely lacking. Here, we report the oxygen evolution reaction (OER) of LaNiO3 (LNO) by changing the NiO6 structure using compositional change and strain. The 5 atom % La deficiency in LNO resulted in an increase of the Ni-O-Ni bond angle and an expansion of bandwidth, enhancing the charge transfer ability. In-plane compressive strain derives the higher dz2 orbital occupancy, leading to suitable metal-oxygen bond strength for OER. Because of the synergistic effect of A-site deficiency and compressive strain, the overpotential (η) of compressively strained L0.95NO film is reduced to 130 mV at j = 30 μA/cm2 compared with nonstrained LNO (η = 280 mV), indicating a significant enhancement in OER.

Switching of majority charge carriers by Zn doping in NdNiO3 thin films

We have studied the effects of Zn doping on the structural and electronic properties of epitaxial NdNiO3 thin films grown on single-crystal LaAlO3 (001) (LAO) substrates by pulsed laser deposition. The films are deposited in two sets, one with variation in Zn doping, and another with variation in thickness for undoped and 2% Zn doping. The experimental investigations show that Zn occupies Ni-site and that the films are grown with an in-plane compressive strain on LAO. All the films show metal-to-insulator transitions with a thermal hysteresis in the temperature-dependent resistivity curves except 5% Zn-doped film, which remains metallic. The theoretical fits show non-Fermi liquid behaviour, which gets influenced by Zn doping. The Hall resistance measurements clearly show that Zn doping causes injection of holes in the system which affects the electronic properties as follows: i) the metallic conduction increases by two factors just by 0.5% Zn doping whereas, 5% doping completely suppresses the insulating state, ii) a reversal of the sign of Hall coefficient of resistance is observed at low temperature.

Discovery of Real-Space Topological Ferroelectricity in Metallic Transition Metal Phosphides.

Ferroelectric metals-with coexisting ferroelectricity and structural asymmetry-challenge traditional perceptions because free electrons screen electrostatic forces between ions, the driving force of breaking the spatial inversion symmetry. Despite ferroelectric metals having been unveiled one after another, topologically switchable polar objects with metallicity have never been identified so far. Here, the discovery of real-space topological ferroelectricity in metallic and non-centrosymmetric Ni2 P is reported. Protected by the rotation-inversion symmetry operation, it is found that the balanced polarity of alternately stacked polyhedra couples intimately with elemental valence states, which are verified using quantitative electron energy-loss spectroscopy. First-principles calculations reveal that an applied in-plane compressive strain creates a tunable bilinear double-well potential and reverses the polyhedral polarity on a unit-cell scale. The dual roles of nickel cations, including polar displacement inside polyhedral cages and a 3D bonding network, facilitate the coexistence of topological polarity with metallicity. In addition, the switchable in-plane polyhedral polarity gives rise to a spin-orbit-coupling-induced spin texture with large momentum-dependent spin splitting. These findings point out a new direction for exploring valence-polarity-spin correlative interactions via topological ferroelectricity in metallic systems with structural asymmetry.

Novel ferroelectric phase in bulk BaO obtained by application of anisotropic strain

Ferroelectricity is expected to be observed in alkaline earth metal oxides (BaO, MgO, CaO, and SrO) with the application of adequate strain, according to theoretical predictions by Bousquet et al. The resultant polarization, dielectric constant, and piezoelectric response would be comparable to those of typical ferroelectric materials, like perovskites. In yet another recent work, Nascimento et al. explored these theoretical predictions by simulating the behaviour of ultra-thin films of BaO under applied in-plane strain to mimic the effect of epitaxial strain induced by an adequate substrate. Effects of both in-plane compressive strain as well as temperature, varied in the ranges of 0 to − 10% and 100–1500 K, respectively, were explored in the simulations. An antipolar phase was observed for in-plane compressive strain values from 3 to 10% and a temperature range of 100 to 1500 K. However, only a compressive strain value around 9%, typically too high to be obtained by epitaxial strain, was able to produce an ordered single domain. Polarization and ferroelectricity are well known as hard to obtain in ultra-thin films due to their limited thickness. The observed existence of an antipolar phase in ultra-thin films would in principle avoid the problems associated with uncompensated surface charges created by the discontinuity of the normal polarization component at the surface. However, can ferroelectricity be obtained in thicker films or even in bulk-like BaO systems? To answer this question, we will here explore the effects of in-plane compressive strain in BaO bulk systems. Compressive in-plane strain, along both x and y directions, will be varied in 0–10% range as well as the c/a (out-of-plane/in-plane) lattice ratio for values of 1.0–1.3. The observed structural phases and associated physical properties will be presented and discussed.

Structural stability, tunable electronic and optical properties of two-dimensional WS2 and GaN heterostructure: First-principles calculations

Based on first-principles calculations, we explore the structural, electronic and optical properties of WS2/GaN heterostructure and its response to strain. Calculations reveal that in-plane compressive strain of 1% can cause the structural instability of WS2/GaN heterostructure while the tensile strain can retain the stability of structure. Under the tensile strain of 10%, the direct bandgap of the WS2/GaN heterostructure can be tuned significantly and its type-II nature of band alignment is preserved. The band offsets at the WS2-GaN interface coupled with the interlayer polarized field can effectively reduce the recombination of the photoinduced electron-hole pairs. Also, the increasing biaxial tensile strain can rouse the optical absorption for near-infrared light, widening the light harvesting of the WS2/GaN heterostructure. In addition, vertical strain can also engineer the electronic properties and optical absorption of the WS2/GaN heterostructure by changing the interlayer coupling. These results indicate that the WS2/GaN heterostructure has potential applications in optoelectronics.

Strain-Work Function Relationship in Single-Crystal Tetracene.

Understanding the impact of strain on organic semiconductors is important for the development of electronic devices and sensors that are subject to environmental changes and mechanical stimuli; it is also important for understanding the fundamental mechanisms of charge trapping. Following our previous study on the strain effects in rubrene, we present here only the second example of the strain-work function relationship in an organic semiconductor; in this case, the benchmark material tetracene. Thin, platelike single crystals of tetracene with large (001) facets were laminated onto silicon and rubber substrates having significantly different coefficients of thermal expansion; mechanical strain in tetracene was subsequently induced by varying the temperature of the assembly. Tensile and compressive strains parallel to the (001) major facet were measured by grazing incidence X-ray diffraction, and the corresponding shifts in the electronic work functions were recorded via scanning Kelvin probe microscopy (SKPM). The work function of the tetracene (001) crystal surface directly correlated with the net mechanical strain and increased by ∼100 meV for in-plane tensile strains of 0.1% and decreased by approximately the same amount for in-plane compressive strains of -0.1%. This work provides evidence of the general and important impact of strain on the electrical properties of van der Waals bonded crystalline organic semiconductors and thereby supports the hypothesis that heterogeneous strains, for example in thin films, can be a major source of static electronic disorder.

Effects of Strain and Film Thickness on the Stability of the Rhombohedral Phase of HfO2

The discovery of ferroelectric polarization in HfO2-based ultrathin films has spawned a lot of interest due to their potential applications in data storage. Recently, a new R3m rhombohedral phase was proposed to be responsible for the emergence of ferroelectricity in the [111]-oriented Hf0.5Zr0.5O2 thin films, but the fundamental mechanism of ferroelectric polarization in such films remains poorly understood. In this paper, we employ density-functional-theory calculations to investigate structural and polarization properties of the R3m HfO2 phase. We find that the film thickness and in-plane compressive strain effects play a key role in stabilizing the R3m phase leading to robust ferroelectricity of [111]-oriented R3m HfO2.

Anomalous lattice shrinking of LaTiO3 thin film on SrTiO3 (111) induced by three-dimensional electron transfer

High-quality LaTiO3 films in the pseudocubic phase have been grown on SrTiO3 (111) using oxide molecular beam epitaxy. The in situ electron diffractions and ex situ scanning transmission electron microscopy reveal that the films with the thickness up to 60 monolayers bear the in-plane compressive strain, while the ex situ x-ray diffractions indicate the contracted out-of-plane interspacing. Such an anomalous lattice shrinking may be related to the formation of oxidized LaTiO3 + δ and, more importantly, the three-dimensional electron transfer from the film to the substrate. This is supported by spatial electron energy loss spectroscopy that shows the homogeneous distribution of Ti with increased valence in the film.

Deposition-Temperature-Mediated Selective Phase Transition Mechanism of VO2 Films

A clear experimental explanation of the contribution of Mott and Peierls transitions to the insulator–metal transition (IMT) characteristics in vanadium dioxide (VO2) is still lacking. Examining the crystal and electronic structures of epitaxial VO2 films grown at various deposition temperatures, a Mott or a Peierls transition was observed. The VO2 film deposited at 500 °C showed suppressed Peierls transition characteristics because of the large in-plane compressive strain in the insulating phase. The VO2 films deposited at 600 and 650 °C had a higher IMT temperature because of the relaxation of both the in-plane and out-of-plane strain, and there were abundant V4+ states. Therefore, it was related to a collaborative Mott–Peierls transition. Finally, the VO2 film deposited at 720 °C showed a suppressed Mott transition because of the abundance of V3+ states in the insulating phase. Furthermore, an analysis of the electronic structure of the insulating and metallic phases using in situ X-ray photoelectron spectroscopy and X-ray absorption spectroscopy provide a complete band diagram to support the above explanation of the deposition-temperature-dependent IMT characteristics.

Strain effect on magnetic-exchange-induced phonon splitting in NiO films

NiO thin films with various strains were grown on SrTiO3 (STO) and MgO substrates using a pulsed laser deposition technique. The films were characterized using an x-ray diffraction, atomic force microscopy, and infrared reflectance spectroscopy. The films grown on STO (001) substrate show a compressive in-plane strain which increases as the film thickness is reduced resulting in an increase of the NiO phonon frequency. On the other hand, a tensile strain was detected in the NiO film grown on MgO (001) substrate which induces a softening of the phonon frequency. Overall, the variation of in-plane strain from −0.36% (compressive) to 0.48% (tensile) yields the decrease of the phonon frequency from 409.6 cm−1 to 377.5 cm−1 which occurs due to the ∼1% change of interatomic distances. The magnetic exchange-driven phonon splitting Δω in three different samples, with relaxed (i.e. zero) strain, 0.36% compressive strain and 0.48% tensile strain, was measured as a function of temperature. The Δω increases on cooling in NiO relaxed film as in the previously published work on a bulk crystal. The splitting increases on cooling also in 0.48% tensile strained film, but Δω is systematically 3–4 cm−1 smaller than in relaxed film. Since the phonon splitting is proportional to the non-dominant magnetic exchange interaction J1, the reduction of phonon splitting in tensile-strained film was explained by a diminishing of J1 with lattice expansion. Increase of Δω on cooling can be also explained by rising of J1 with reduced temperature.

Enhancing the critical temperature of strained Niobium films

The study of the high critical temperature (Tc) of hydrogen compounds under high pressure has resulted in a considerable focus on Bardeen–Cooper–Schrieffer superconductors. Nb has the highest Tc among the elemental metals at ambient pressure, so reviewing Nb films again is worthwhile. In this study, we investigated the factors that determine the Tc of Nb films by strain introduction and carrier doping. We deposited Nb films of various thicknesses onto Si substrates and evaluated the Tc variation with thickness. In-plane compressive strain in the (110) plane due to residual stress reduced the Tc. First-principles calculations showed that adjusting the density of states at the Fermi level is key for both strain-induced suppression and doping-induced enhancement of the Nb Tc. The application of hydrostatic pressure compensated for the intrinsic strain of the film and increased its Tc, which could also be enhanced by increasing the hole concentration with an electric double-layer transistor. A liquid electrolyte should be used as a pressure medium for applying hydrostatic pressure to increase the Tc of correlated materials, where this increase results from changes in material structure and carrier concentration.

Epitaxial strain and thickness dependent structural, electrical and magnetic properties of La0.67Sr0.33MnO3 films

The crystal structural quality and the strain induced by the substrate strictly impose the magnetic and transport properties of La0.67Sr0.33MnO3 (LSMO) films. In particular, the magnetic anisotropy (MA) of epitaxial LSMO can be finely tuned by varying its thickness and by choosing single crystal substrates with a suitable lattice mismatch with the film. Here, we have deposited LSMO films with thicknesses in the 12–50 nm range by pulsed laser deposition on different single crystal substrates inducing either compressive or tensile in-plane strain on the manganites. The epitaxial quality of films was quantified by ω-scans around a (002) peak with full-width half-maximum values as low as 0.08° for films on the nearly matched NGO (110) substrate to 1.4° films on the high mismatched MgO (001) substrate. As the epitaxial strain in a thin-film increases, a significant reduction in the metal-insulation transition temperature (Tp) was observed. The magnetic properties of the films probed by Kerr magnetometry show that the symmetry of the room temperature MA varies significantly as a function of both strain and thickness. Specifically, we observed pure uniaxial MA on NGO (110) and pure biaxial MA on STO buffered MgO (001), whereas a spin reorientation from uniaxial in-plane to out-of-plane on LSAT (001) and uniaxial to nearly isotropic in-plane on STO (001) substrate as the film thickness is increased. We provide an efficient tool to tune the MA according to the specific spintronic application targeted.

Use of a Be-dome holder for texture and strain characterization of Li metal thin films via sin2(ψ) methodology

Residual strain in electrodeposited Li films may affect safety and performance in Li metal battery anodes, so it is important to understand how to detect residual strain in electrodeposited Li and the conditions under which it arises. To explore this Li films, electrodeposited onto Cu metal substrates, were prepared under an applied pressure of either 10 or 1000 kPa and subsequently tested for the presence or absence of residual strain via sin2(ψ) analysis. X-ray diffraction (XRD) analysis of Li films required preparation and examination within an inert environment; hence, a Be-dome sample holder was employed during XRD characterization. Results show that the Li film grown under 1000 kPa displayed a detectable presence of in-plane compressive strain (−0.066%), whereas the Li film grown under 10 kPa displayed no detectable in-plane strain. The underlying Cu substrate revealed an in-plane residual strain near zero. Texture analysis via pole figure determination was also performed for both Li and Cu and revealed a mild fiber texture for Li metal and a strong bi-axial texture of the Cu substrate. Experimental details concerning sample preparation, alignment, and analysis of the particularly air-sensitive Li films have also been detailed. This work shows that Li metal exhibits residual strain when electrodeposited under compressive stress and that XRD can be used to quantify that strain.

Quantum spin Hall effect, thermoelectric performance, and optical properties of XBi (X = Sc, Y) monolayers

Abstract The investigations of quantum spin Hall effect and the edge states manipulation of two dimensional topological insulators are very salient for the practical applications and fundamental sciences. The high thermoelectric efficiency of these materials has also been confirmed, recently. So, in this study the first-principles calculations are implemented based on density functional theory in the presence of spin orbit interaction, to evaluate the topological phase transition and thermoelectric performance of XBi (X = Sc, Y) monolayers under in-plane strains. It is found that the compressive in-plane strains and spin orbit interaction, which host considerable effects on the electronic structure, can cause the topologically nontrivial phase and high thermoelectric performance. The topological state of XBi monolayers is confirmed with the Z2 topological invariant calculation. These results provide these monolayers for the novel thermoelectric and nanoelectronics quantum devices and topological phenomena. Also some optical properties of XBi monolayers are calculated and investigated under in-plane strains. The results show the considerable transparency and reflectivity of these monolayers near zero energy.

Radius-dependent homogeneous strain in uncoalesced GaN nanowires

We investigate the strain state of ensembles of thin and nearly coalescence-free self-assembled GaN nanowires prepared by plasma-assisted molecular beam epitaxy on Ti/Al2O3(0001) substrates. The shifts of Bragg peaks in high-resolution X-ray diffraction profiles reveal the presence of a homogeneous tensile strain in the out-of-plane direction. This strain is inversely proportional to the average nanowire radius and results from the surface stress acting on the nanowire sidewalls. The superposition of strain from nanowires with different radii in the same ensemble results in a broadening of the Bragg peaks that mimics an inhomogeneous strain on a macroscopic scale. The nanowire ensembles show a small blueshift of the bound-exciton transitions in photoluminescence spectra, reflecting the existence of a compensating in-plane compressive strain, as further supported by grazing incidence X-ray diffraction measurements carried out at a synchrotron. By combining X-ray diffraction and photoluminescence spectroscopy, the surface stress components fx and fz of the air-exposed GaN{11¯00} planes that constitute the nanowire sidewalls are determined experimentally to be 2.25 and −0.7 N/m, respectively.

Deposition-environment-dependent structural and magnetic property modification of [111]-oriented epitaxial CoFe2O4 films

We examined the magnetic, structural, and chemical properties of CoFe2O4 films deposited on Al2O3 substrates using RF-magnetron sputtering under Ar and Ar + O2 environments. Perpendicular magnetic anisotropy with large magnetization was observed in the case of the film deposited under the Ar + O2 environment. However, no significant anisotropy appeared in the film deposited under the Ar environment. Interestingly, depth-dependent X-ray photoelectron spectra showed nearly identical stoichiometric ratios of Co/Fe and oxygen vacancies regardless of the deposition environment. Structurally, high-quality crystallinity with in-plane compressive strain, and lower surface and interfacial roughness were induced in the film deposited under the Ar + O2 environment. Cross-sectional transmission electron microscopy and X-ray reflectivity measurements showed that oxygen not only suppressed the deposition rate but also increased the electron density of the film, resulting in better crystallinity. Hence, the presence of oxygen during deposition should be considered as one of the essential parameters for improving the structural and magnetic properties of ferrite films.

Lattice Vibrational Modes in Epitaxial Metastable Germanium-Tin Semiconductors

Group IV semiconductors, such as GeSn and SiGeSn, are of growing interest for silicon-compatible electronic and photonic applications, notably due to the possibility of tuning the bandgap directness and energy to cover a broad range from the short-wave infrared (SWIR) to the mid-infrared (MIR) [1]. While germanium is an indirect-bandgap semiconductor, an indirect-to-direct transition in GeSn alloys is expected for Sn incorporation higher than 9%, but the residual compressive strain typical to epitaxial layers increases the concentration of Sn to obtain a direct-bandgap [2]. Therefore, an understanding of the interplay of strain and composition on the band structure of GeSn semiconductors is of compelling importance in order to precisely control their physical properties. This calls for accurate analysis of the as-grown material properties and its correlation with the measured optical and electrical performance. Raman spectroscopy is a non-destructive method commonly used to assess the role of the effects of composition and strain in shaping the lattice vibrations. The use of a 633 nm excitation enables a clear detection of all Raman modes in GeSn layers independently of their composition, comparatively to the more restricted results associated with the use of 488 nm[3] or 532 nm [4] excitation. This is plausibly attributed to the fact that this wavelength might be close to resonance with the alloy’s E1 gap [5]. There are very few reports on the identification of the Ge-Sn mode [6] and the disorder-activated mode coinciding with the maximum in the one-phonon density of states in Ge [7], but quantitative analyses of the effects of composition and strain on these modes remain conspicuously missing in literature.With this perspective, this work presents a detailed study of Raman vibrational modes in Sn-rich GeSn semiconductors and how their behavior is affected by temperature. The use of samples of various degrees of Sn composition and in-plane compressive strain allows the decoupling of these effects on the Ge-Ge, DA and Ge-Sn vibrational modes, thus providing an effective platform to test the current predictive models. Samples were grown in a chemical vapor deposition (CVD) reactor using monogermane (GeH4) and tin-tetrachloride (SnCl4) precursors [8]. The GeSn multi-layer heterostuctures with a Sn content in the 4-16 at.% range were grown on a Ge virtual substrate (VS) on a Si wafer. Using X-ray diffraction (XRD), reciprocal space mappings (RSM) were performed on all samples to retrieve the Sn composition and the strain. Temperature-dependent Raman measurements were performed from room temperature (300K) and down to 77 K.In GeSn, the asymmetrical profile of the observed Raman modes is due to alloying as substitutional Sn atoms break the translational symmetry and lead to a relaxation of the momentum selection rule [9]. Therefore, the use of exponentially modified gaussian functions (EMG) as a fitting curve can reflect the asymmetrical profile of the different analyzed Raman modes with much better accuracy [10]. The measurement temperature has a clear effect on the fitting parameters, such as the integrated area of the peak , the full width at half maximum (FWHM), the asymmetry, and its position. For example, the Ge-Sn and Ge-Ge peaks increase in wavenumber with decreasing temperature, while the DA peak shows the opposite trend. Ongoing work focuses on establishing the behavior of the characteristics of each mode. Based on these detailed Raman studies, an exhaustive discussion of the influence of temperature, lattice strain and Sn content on GeSn vibrational modes will be presented. Acknowledgements The authors thank J. Bouchard for the technical support, and NSERC Canada (Discovery, SPG, and CRD Grants), Canada Research Chairs, Canada Foundation for Innovation, Mitacs. Reference [1] S. Gupta, et al., J. Appl. Phys. 113, 7 (2013).[2] A. Attiaoui and O. Moutanabbir, J. Appl. Phys., 116, 6 (2014).[3] R. R. Lieten et al., ECS J. Solid State Sci. Technol., 3, 12, pp. P403–P408 (2014).[4] A. Gassenq et al., Appl. Phys. Lett., 110, 11, p. 112101 (2017).[5] R. Chen et al., ECS J. Solid State Sci. Technol., 2, 4, pp. P138–P145 (2013).[6] D. Zhang et al., J. Alloys Compd., 684, pp. 643–648 (2016).[7] V. D’Costa et al., Solid State Communications, 144, 5-6, pp. 240-244 (2007).[8] S. Assali, J. Nicolas and O. Moutanabbir, Journal of Applied Physics, 125, 2, p. 025304 (2019).[9] P. Parayanthal and F. H. Pollak, Phys. Rev. Lett., 52, 20, pp. 1822–1825 (1984).[10] É. Bouthillier et al., “Decoupling the effects of composition and strain on the vibrational modes of GeSn”, https://arxiv.org/abs/1901.00436, 2019.

Evaluation of in-plane compressive densification strain of honeycomb paperboard

The compressive densification strain of honeycomb paperboard is one of the important parameters which affect the energy absorption property of honeycomb paperboard. The in-plane bearing mechanism of honeycomb paperboard in plastic zone was analyzed based on compression tests of the single row and multi-rows of honeycomb paperboard. The result indicates that the core layer plays a supporting role in resisting the buckling of the face layer. The double inner folds are formed in machine direction and the symmetrical inner folds are formed in cross direction in a honeycomb core. The core single wall and the face layer play a critical role in the load bearing in machine direction, and the core double walls, core single wall, and face layer all play critical roles in the load bearing in cross direction. On this basis, the evaluation equation of the compressive densification strain was obtained based on the energy absorption efficiency method and geometric scale effect, which are verified so that the experiment and test results are in good agreement.

Direct observation of large strain through van der Waals gaps on epitaxial Bi2Te3 /graphite: Pseudomorphic relaxation and the role of Bi2 layers on the BixTey topological insulator series

Layered materials can usually grow without strain on top of distinct substrates if the only interaction between them is due to van der Waals forces. In such a scenario it would be expected that the heterointerface made up of weak bounds would not affect the overlayed material significantly for several large lattice-mismatched systems. Here we have studied the first stages of the heteroepitaxial growth of layered bismuth telluride topological insulator on top of highly oriented pyrolitic graphite (HOPG) by molecular beam epitaxy. Samples were investigated by atomic force microscopy (AFM), synchrotron x-ray diffraction (XRD), and micro-Raman spectroscopy. AFM images show hexagonal/triangular flat islands with exposed HOPG areas for the low coverage regime, and the lattice parameter of these $\mathrm{B}{\mathrm{i}}_{2}\mathrm{T}{\mathrm{e}}_{3}$ structures were measured by XRD. The existence of pseudomorphic strain at the initial $\mathrm{B}{\mathrm{i}}_{2}\mathrm{T}{\mathrm{e}}_{3}$ layers was retrieved by both XRD and Raman spectroscopy. We have found evidence that $\mathrm{B}{\mathrm{i}}_{2}\mathrm{T}{\mathrm{e}}_{3}$ layers near the interface are subject to an in-plane compressive strain, leading to a pseudomorphic out-of-plane lattice expansion. Furthermore, the presence of $\mathrm{B}{\mathrm{i}}_{2}\mathrm{T}{\mathrm{e}}_{3}$ islands locally distorts the topmost layer of HOPG, resulting in tensile strain which was measured by Raman spectroscopy. The observed relaxation of 0.1--0.2% for each van der Waals gap is used to calculate elastic constants of $\mathrm{B}{\mathrm{i}}_{2}$ bilayers, which are crucial building blocks for the formation of other $\mathrm{B}{\mathrm{i}}_{x}\mathrm{T}{\mathrm{e}}_{y}$ topological insulator compounds. Finally, the impact of such a strain in $\mathrm{B}{\mathrm{i}}_{2}\mathrm{T}{\mathrm{e}}_{3}$ electronic structure was investigated by density functional theory calculations. The results show that the band structure of this strained material remains unchanged at the center of the Brillouin zone, confirming the robustness of surface states, but it is consistently affected at the M and K zone edges.

Electronic structure of bulk and multilayer black phosphorus under strain: a minimal model study

The tight-binding method is an important theoretical tool to investigate the physics of black phosphorus. Recently, a fifteen-parameter tight-binding model was proposed to precisely describe the band features. The large number of parameters is an obstacle for understanding the physics, and a minimal model for black phosphorus that grasps the main physics with satisfied precision is quite valuable. We use a tight banding model with only three parameters, that can reproduce the correct energy gap for monolayer, few-layer and bulk black phosphorus, to study the band structure black phosphorus under strain. The energy gap can be decreased by tensile strain normal to the phosphorus layers or in-plane compressive strain. At a certain critical strain, the gap of the phosphorus (either bulk, multilayer, or monolayer) is closed, and the dispersion is linear in the armchair direction and is parabolic in the other two principle directions. The simplicity of our model allows us to analytically investigate the energy gap and the critical strain as functions of the layer number of multilayer black phosphorus. When the strain exceeds the critical value, the valence and conduction bands evolve into two tents with their ridges touching each other.