Electronic and mechanical responses of two-dimensional HfS₂, HfSe₂, ZrS₂, and ZrSe₂ from first-principles

Abstract

During the last decade, numerous high-quality two-dimensional (2D) materials with semiconducting electronic character have been synthesized. Recent experimental study (Sci. Adv. 2017;3: e1700481) nevertheless confirmed that 2D ZrSe₂ and HfSe₂ are among the best candidates to replace the silicon in nanoelectronics owing to their moderate band-gap. We accordingly conducted first-principles calculations to explore the mechanical and electronic responses of not only ZrSe₂ and HfSe₂, but also ZrS₂ and HfS₂ in their single-layer and free-standing form. We particularly studied the possibility of engineering of the electronic properties of these attractive 2D materials using the biaxial or uniaxial tensile loadings. The comprehensive insight provided concerning the intrinsic properties of HfS₂, HfSe₂, ZrS₂, and ZrSe₂ can be useful for their future applications in nanodevices.

Keywords

2D materials / mechanical / electronic / DFT

Introduction

The great success of graphene [¹,²], emerged the two-dimensional (2D) materials as a new class of materials suitable for numerous and diverse applications ranging from nanoelectronics to aerospace structures. For the applications in nanoelectronics, presenting semiconducting electronic character with moderate and tuneable band-gap is highly desirable. Nevertheless, graphene in its natural form presents zero-band-gap semiconducting property and such that opening a band-gap in graphene requires complex physical or chemical modifications such as chemical doping or defect engineering [³–⁶]. As an alternative, synthesize of other 2D material with inherent semiconducting character such as transition metal dichalcogenides [⁷–⁹] has been considered as a more reliable approach to exploit in post-silicon electronics.

The success of silicon in electronics is not only due to its moderate band-gap of 1.1 eV, but also because of the existence of SiO₂ as a high-quality “native” insulator. In this case, other competitors of silicon lack stable oxides and must rely on deposited insulators [¹⁰], which imposes compatibility challenges. Recent experimental study confirmed that 2D ZrSe₂ and HfSe₂ semiconductors with band-gaps ranging from 0.9 to 1.1 eV can serve as very promising candidates to replace the silicon in electronic devices. These 2D materials are highly technologically desirable because of the existence of “high-κ” native dielectrics of HfO₂ and ZrO₂ [¹⁰].

For the engineering design of novel nanodevices using the 2D materials, comprehensive understanding of electronic, mechanical and thermal properties of 2D components play crucial roles. It should be noted that for the materials at nanoscale such as the 2D materials, experimental techniques for the evaluation of properties are complicated, time consuming and expensive as well. In these cases classical and first-principles theoretical approaches can be considered as fast viable alternatives to explore various material properties such as thermal conductivity, mechanical response, electrochemical performance and electronic properties in low cost and trustable level of precision [¹¹–³⁰]. The material properties predicted by the first-principles simulations can be later employed in the hierarchical, semi-concurrent and concurrent multiscale approaches [³¹–³⁵] in order to design real nanodevices. The developement of advances multiscale techniques can play critical roles in the improvement of the engineering deisgn and performance of the future nanodevices.

Motivated by the high technological prospects for the 2D HfS₂, HfSe₂, ZrS₂, and ZrSe₂, in the present investigation we accordingly intend to investigate their properties in single-layer and free-standing form by conducting extensive first-principles density functional theory (DFT) simulations. In this case, we analyzed the structural and electronic properties of these 2D materials at their minimum energy conditions. We applied uniaxial tensile loading conditions to elaborately study the mechanical properties and deformation process as well. Because of the highly attractive electronic properties of these 2D materials, we particularly explored the possibility of the tuning of the band-gap using the uniaxial or biaxial loading conditions. This study addresses some critical properties of 2D HfS₂, HfSe₂, ZrS₂, and ZrSe₂ and therefore may act as a useful guide for their practical applications in nanodevices.

Methods

The DFT calculations in this study were performed using the Vienna ab-initio simulation package (VASP) [³⁶–³⁸]. The plane wave basis set with an energy cut-off of 500 eV and the gradient approximation exchange-correlation functional proposed by Perdew et al. [³⁹] were employed. VMD [⁴⁰] and VESTA [⁴¹] packages were also used for the visualization of atomic structures. Figure 1 illustrates the hexagonal lattice of HfS₂, HfSe₂, ZrS₂, and ZrSe₂ atomic structure which shows ABA atomic stacking sequence.

In this work we analyzed the anisotropy in the mechanical response by uniaxially stretching the structures along the armchair and zigzag directions. We applied periodic boundary conditions along all three Cartesian directions and such that the obtained results represent the properties of large-area single-layer films and not the nanoribbons. Since the dynamical effects such as the temperature are not taken into consideration and periodic boundary conditions were also applied along the planar directions, only a unit-cell modelling is accurate enough for the evaluation of mechanical properties, and such that we only used a unit-cell consisting of 3 atoms. We considered a vacuum layer of 20 Å to avoid image-image interactions along the sheets normal direction. After obtaining the minimized structure, we applied loading conditions to evaluate the mechanical properties. For this purpose, we increased the periodic simulation box size along the loading direction in a multistep procedure, every step with a small engineering strain of 0.001. For the uniaxial loading conditions, upon the stretching along the loading direction the stress along the transverse direction should be negligible. To satisfy this condition, after applying the loading strain, the simulation box size along the transverse direction of the loading was changed accordingly in a way that the transverse stress remained negligible in comparison with the stress along the loading direction. For the biaxial loading condition, the equal loading strain was applied simultaneously along the both planar directions. After applying the changes in the simulation box size, the atomic positions were rescaled to avoid any sudden void formation or bond stretching as well. We then used the conjugate gradient method for the geometry optimizations, with strict termination criteria of 10⁻⁵ eV and 0.005 eV/Å for the energy and the forces, respectively, using a 19×19×1 Monkhorst-Pack [⁴²] k-point mesh size. The final stress values after the termination of energy minimization process were calculated to obtain the stress-strain curves. The ground state electronic properties were first calculated using the PBE functional. Due to underestimation of experimental band-gap values using the PBE functional, we also used the screened hybrid functional HSE06 [⁴³] and quasi-particle many-body perturbation theory (MBPT) via G₀W₀ approximation [⁴⁴,⁴⁵] to evaluate the electronic properties of these materials. A 14 × 14× 1 G centered Monkhorst-Pack k-point mesh is used for PBE and HSE06 calculations.

Results and discussion

We first study the atomic structure of HfS₂, HfSe₂, ZrS₂, and ZrSe₂, which can be well defined by the hexagonal lattice constant (a) and transition metal-chalcogen atom bond length (bl). In Table 1 the lattice constants and the bond lengths for the considered 2D structures at their minimum energy condition are mentioned. We next study the mechanical responses of these 2D structures by conducting the uniaxial tensile simulations. In Fig. 2, the DFT predictions for the uniaxial stress-strain responses of HfS₂, HfSe₂, ZrS₂, and ZrSe₂, elongated along the armchair and zigzag directions are plotted. In all cases, the stress-strain responses present an initial linear relation which is followed by a nonlinear trend up to the ultimate tensile strength point, a point at which the material illustrates its maximum load bearing. The slope of the first initial linear section of the stress-strain response is equal to the elastic modulus. In this work we therefore fitted a line to the stress-strain values for the strain levels below 0.02 to report the elastic modulus. For these initial strain levels within the elastic limit, the strain along the traverse direction (s_t) with respect to the loading strain (s_l) is acceptably constant and can be used to obtain the Poisson’s ratio using the −s_t /s_l. Our results shown in Fig. 2 reveal that the initial linear response of the considered 2D structures match closely which means that their elastic properties are close. In another side, the non-linear part of the stress-strain curves are different depending on the atomic structure and loading direction as well. Our DFT results reveal that the tensile response is not isotropic and along the armchair direction the single-layer HfS₂, HfSe₂, ZrS₂, and ZrSe₂, are considerably stronger as compared with the zigzag. Such an observation is in agreement with earlier studies for the mechanical properties of transition metal dichalcogenides [⁴⁶] and group IV 2D materials [⁴⁷]. The obtained mechanical properties of considered 2D structures are summarized in Table 1. In general, the elastic modulus is around 6% higher when the structure is elongated along the armchair in comparison with zigzag. For the both tensile strength and elastic modulus we found the anisotropy in the mechanical response is almost inversely correlated with the stiffness: such that the lower the elastic modulus or tensile strength, the higher is the anisotropy in the mechanical response. However such a correlation for HfSe₂ and ZrS₂ is not consistent since they show very close mechanical properties and their anisotropy in mechanical properties are also very close. Interestingly, the calculated Poisson’s ratios were found to be almost independent of the structure and slightly higher Poisson’s ratios were acquired along the armchair. For the at ultimate tensile strength point we predict that along the zigzag direction the studied 2D films yield slightly higher strain values.

We next conduct the electronic structure analysis to investigate the origin of differences in the mechanical properties of studied structures. To this aim we first calculated the electron localization function (ELF) [⁴⁸]. In Fig. 3(a) a sample of obtained ELF for HfSe is illustrated which reveals that the electron localization is more concentrated around the Se atoms. This result suggests the charge transfer from the transition metal to the chalcogen atoms. To characterize this charge transfer from transition metal to every chalcogen atom, we conducted the Bader charge analysis [⁴⁹] and the obtained results are included in Table 1. As a general trend, by increasing the charge transfer both the elastic modulus and tensile strength values increase, which is in agreement with the results for transition metal dichalcogenides [⁴⁶]. Nevertheless, an exception exist and for the HfSe₂ and ZrS₂ the values of charge transfer are very close and in this case the lighter structure with slightly lower charge transfer presents negligibly higher elastic modulus and tensile strength.

To better understand the underlying mechanism that results in anisotropic tensile response of HfS₂, HfSe₂, ZrS₂, and ZrSe₂, we analyzed the deformation process. Due to the similarities in the deformation behaviour of the studied 2D structures, in this case we only examine the ZrSe₂ structure. For the deformation during the uniaxial loading, two different bonds can be distinguished, the bond along the armchair (bl_arm) and the bond oriented along the zigzag (bl_zig) direction, which are depicted in Fig. 1. Figure 3, compares the change in the bond lengths as a function of strain for the uniaxial loading along the armchair and zigzag directions. As a general observation, during the uniaxial tensile loading, the bond oriented along the loading direction elongates and the other bond oriented along the transverse direction of loading contracts. Based on our results for the both loading directions, during the stretching the structures contract slightly along the sheet thickness, however the intensity of this contraction remains almost below 3%. For the stretching along the armchair direction, one bond is exactly along the loading direction and directly involves in the load bearing and such that by increasing the strain level this bond increases substantially. In this case, the bond oriented along the transverse direction remains almost unchanged and only after the strain level of ~0.15 starts to contract slightly. On the other side, for the uniaxial stretching along the zigzag, one bond is almost oriented along the loading direction and the other bond is exactly along the transverse direction. In this case, based on our results shown in Fig. 3(b), since the bond involving in the load transfer is not exactly inline of the loading by increasing the strain level the bond stretching is moderate. In this case the contraction of the bond along the transverse direction is more considerable, which helps the material to flow easier along the loading direction. As it is clear, the distinctly higher tensile strength as well as the slightly higher elastic modulus along the armchair direction can be attributed to the fact that during the stretching along the armchair half of the bonds are exactly aligned to the loading direction and such that the deformation is achieved mainly by the bond elongation.

Next, we shift our attention to explore the evolution of band-gap of aforementioned 2D materials under different loading conditions. In Fig. 4, the band structure and total DOS of HfS₂, HfSe₂, ZrS₂, and ZrSe₂ monolayers for different biaxial or uniaxial tensile strains by the PBE are compared. The band structure and DOS for unstrained systems and maximum magnitudes of strain are reported. It is well visible that the free strained nanostructures present indirect band-gaps in which the valence band maximum (VBM) coincides along the K-G direction and the conduction band minimum (CBM) locates at M-point which is in a good agreement with that reported previously for HfSe₂ monolayer [¹⁰]. The acquired results show that the band-gaps of the HfS₂, HfSe₂, ZrS₂, and ZrSe₂ sheets, using PBE, are 1.15, 0.95, 1.0, and 0.85 eV, respectively, which are slightly larger than those found for transition-metal dichalcogenides monolayers in the T phase [⁵⁰]. For all monolayers, their band-gap decreases when the biaxial or uniaxial tensile loading is applied. Note that the reduced band-gap for biaxial strains is lower than uniaxial ones. In order to obtain accurate results of electronic band-gap, the screened hybrid functional HSE06 and quasi-particle G₀W₀ approaches have also been used. Figures 5 and 6 illustrate total DOS of 2D HfS₂, HfSe₂, ZrS₂, and ZrSe₂ monolayers using HSE06 and G₀W₀ approaches, respectively. The HSE06 functional and GWA approaches gave remarkably larger band-gaps than the PBE functional for the all studied structures. The corresponding values within HSE06 for 2D HfS₂, HfSe₂, ZrS₂, and ZrSe₂ monolayers are 1.72, 1.50, 1.45, and 1.22 eV. The band-gap value for HfSe₂ monolayer is in excellent agreement with previous calculation [¹⁰]. Taking the electron-electron interactions into account in G₀W₀ reduces screening, resulting in an increase over the PBE band-gap. The calculated indirect band-gap from G₀W₀ is 1.90, 1.70, 1.65, and 1.30 eV for the unstrained HfS₂, HfSe₂, ZrS₂, and ZrSe₂ sheets, respectively. In Table 2 the band-gap of the strained and unstrained systems obtained by different methods are summarized.

Conclusions

We conducted extensive first-principles DFT calculations to explore the mechanical and electronic responses of pristine single-layer HfS₂, HfSe₂, ZrS₂, and ZrSe₂ monolayers. We first investigated the mechanical responses of these 2D structures by conducting the uniaxial tensile simulations. It was found that the charge transfer from the transition metal to chalcogen atoms correlates directly to elastic modulus and tensile strength. Our DFT results reveal that the mechanical responses of these 2D systems are not isotropic and along the armchair direction the single-layer HfS₂, HfSe₂, ZrS₂, and ZrSe₂, are considerably stronger as compared with the zigzag. It was found that during the stretching along the armchair the deformation evolves more by the bond elongation which explains the higher stress values at any certain strain value as compared with the loading along the zigzag. According to the G₀W₀ method estimations, indirect band-gaps of 1.90, 1.70, 1.65, and 1.30 eV were predicted for unstrained single-layer HfS₂, HfSe₂, ZrS₂, and ZrSe₂, respectively. Based on the PBE, HSE06, and G₀W₀ methods predictions, it was found that the band-gap decreases through applying the biaxial or uniaxial tensile loading which notably confirms the tunability of electronic properties of these 2D structures. It was found that biaxial strains can be employed as a more effective approach for tuning the electronic response of HfS₂, HfSe₂, ZrS₂, and ZrSe₂.

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