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Study of mechanical and electronic properties of single-layer FeB2
Accepted Manuscript Study of mechanical and electronic properties of single-layer FeB2 Homayoun Jafari, Maryam Masoudi, Narges Taghizade, Aidin Ahmadi, Mahdi Faghihnasiri PII:
S1386-9477(18)31258-X
DOI:
https://doi.org/10.1016/j.physe.2019.03.011
Reference:
PHYSE 13497
To appear in:
Physica E: Low-dimensional Systems and Nanostructures
Received Date: 19 August 2018 Revised Date:
1 February 2019
Accepted Date: 15 March 2019
Please cite this article as: H. Jafari, M. Masoudi, N. Taghizade, A. Ahmadi, M. Faghihnasiri, Study of mechanical and electronic properties of single-layer FeB2, Physica E: Low-dimensional Systems and Nanostructures (2019), doi: https://doi.org/10.1016/j.physe.2019.03.011. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Department of Physics, Iran University of Science and Technology, Tehran, Iran Department of Physics, Payame Noor University, PO BOX 19395-3697, Tehran, Iran c Department of Electronic and Electrical Engineering, Faculty of Engineering, University of Birjand, Birjand 97174-34765, Iran d Computational Materials Science Laboratory, Nano Research and Training Center, NRTC, Iran
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* Corresponding Author Email: [email protected]
Abstract: Recent experimental advances for the fabrication of various two dimensional (2D) materials introduced new structures with a wide prospect of applications. In this work, under in-plane uniaxial and biaxial strain, we employed first-principles density functional theory calculations to investigate the mechanical and electronic properties of Hexagonal Iron dibromide (FeB2) sheet. In mechanical section, we investigate deeply the mechanical analyzing of 2D structures to study elastic constants. Based on the results of our modelling, single layer FeB2 depending on the strain direction can yield remarkable elastic modulus in the range of 141.18 N/m for Young’s modulus and also hold on ultimate stress up to 7.08×10-4 N/m, 7.96×10-4 N/m and 8.82×10-4 N/m at the corresponding strains 26.5%, 34% and 19.5% for uniaxial strain along x, y and biaxial strain, respectively. In this paper, second, third and fourth orders of elastic constants beside Young, Bulk and Shear’s moduli and Poisson ratio of single layer of FeB2 have been determined, for the first time. By analyzing electronic band structure, we did not observe any band gaps in terms of strains. However, applying directional strain based on deformation matrices, bands get apart from Fermi level so that Dirac point has been vanished.
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Homayoun Jafari a, Maryam Masoudi b, Narges Taghizade a, Aidin Ahmadi c, Mahdi Faghihnasiri d,*
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Study of Mechanical and Electronic Properties of single-layer FeB2
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Keywords: Density Functional Theory; High order elastic constants; FeB2
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INTRODUCTION Nowadays, studying two dimensional materials are considered among the most attractive and interesting research fields owing to their vast properties and applications. Interests to this category of materials were created through synthesis and production of graphene1 possessing surface structure with honeycomb arrangement of carbon atoms. Graphene is a zero-gap semiconductor that reveals unique electronic2, 3, mechanical4 and thermal5 properties. After the successful synthesis of graphene, other 2D materials including hexagonal structure of boron-nitride6, 7, graphite like structure carbon-nitride8, titaniumaluminum-nitride9, and also lower dimensional materials like silicone nanoribbon10, aluminum-nitride nanotube11 and quantum dots12 were synthesized. 2D materials can exhibit exotic properties which are different from their bulk counterparts due to the quantum confinement effect, making them more attractive for designing hard protector films13-15, novel electronic and optoelectronic devices. In this regard, interests to predicate and investigate the properties of this type of materials were never stopped. Through the continuous developments in synthesis of newer 2D materials, motivated by not only technological applications but also the fundamental research in the current scientific community, they have triggered an enormous interest for their tunable mechanical, optoelectronic, and magnetic properties, significantly enriching the family of 2D materials. Graphene like structure Borophene16, 17, replaced carbon with boron, has been attracted considerable attention due to its unique thermal18, 19, chemical20, 21 and mechanical22-24 features, recently. Studies the properties of different phases of Borophene continued till 2016, when Zhang and coworkers25 designed a new Dirac included material, called FeB2 monolayer which contains embedded planar hexacoordinate Fe atoms in the 2D honeycomb lattice of boron. For bulk configuration of Fe-B, there have been reported ultra-stable phase under pressure and also investigated its special magnetic, mechanical and superconductivity properties27-29. Current challenge in the development of two dimensional materials which researchers facing, is their low stability and so many efforts have been made to overcome this issue by using appropriate substrates. Substrate and the main layer lattice parameter mismatch could cause external strain to the layer and consequently could apply corruptive influence to the material properties30. Therefore, having a comprehensive knowledge about the elastic properties of these materials and their response to the external stress should be a great deal in the fabrication and application of mono-layers31. In this study we investigate the mechanical parameters such as young’s, bulk and shear modulus using second order linear elastic constants32, 33 which can be achieved by fitting the second order polynomial functions with the Stress-Energy curve. Furthermore, we evaluated high orders nonlinear elastic constants such as third and fourth order elastic constants by merging density functional theory and continuum approach34. These parameters have an important role in the phase transition which can be described by Landau’s phase transition theory35. These elastic constants not only being used in mechanical properties investigation, but also they are important in studying and analyzing structural phenomenon such as Grüneisen parameters36, piezoelectric properties37, and trapped-energy resonators38. In this work, we study the accurate elastic constants of FeB2 single layer, understand its behavior under external load and know which variables would affect their elastic response and how these variables would affect those constants. The total energy of the
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systems after structure relaxation was calculated with good accuracy using the density functional theory (DFT) at any applied strain. Finally, we provide a systematic study of the electronic properties of FeB2 under external physical stimuli.
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COMPUTATIONAL DETAILS First-principle study by DFT calculations were carried out with the Quantum ESPRESSO package39. The exchange-correlation term is described by adopting generalized gradient approximation using Perdew– Burke– Ernzerhof of revised for solids (PBEsol) functional40 along with the projector-augmented wave (PAW) potentials for the self-consistent total energy calculations and geometry optimization. The kinetic energy cutoff for the plane wave basis set was chosen to be 700 eV. The Brillion zone was sampled with 14×14×1 k-points using the standard Monkhorst–Pack special grids41. Atomic positions were relaxed until the energy differences were converged within 10-6 eV and the maximum Hellmann– Feynman force on any atom was below 10-5 eV/Å. Periodic boundary conditions for two directions (x and y) were invoked in the simulations and a vacuum region of 15 Å along the z direction was included to safely avoid interaction between adjacent layers.
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RESULTS AND DISCUSSION Atomic Structure. Figure 1 presents the top and side views of atomic structure for FeB2 2D system, containing two B atoms and one Fe atom in its hexagonal unit-cell, in which B atoms are arranged in a honeycomb lattice and Fe atoms are 0.6 Å above the center of six-membered boron rings. This monolayer sheet can be considered as a one-layer Fe−B sheet exfoliated from bulk FeB2. In this quasi-planar FeB2 sheet, each Fe atom coordinates with six boron atoms around it, thus forming planar hexacoordination. Following the geometry optimization of the unit-cell using PBEsol functional, the lattice constants along a and b directions are 3.15 Å. The B−B bond length (1.82 Å) is about the same as that in the bulk FeB2 (1.76 Å)42 but slightly larger than that in the B monolayers (1.67 Å ∼ 1.71 Å)43-46. The length of Fe−B bond in FeB2 monolayer (1.93 Å) is noticeably smaller than that in bulk FeB2 crystal (2.32 Å), suggesting a much stronger interaction between the Fe atoms and boron layer. All of our geometric optimized parameters like B-B and Fe-B bond lengths of 1.82 and 1.93 Å are in good agreement with recent reported results25 of 1.83 and 1.94 Å, respectively.
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Fe B Figure 1. Top and side view of FeB2 monolayer.
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1 Thermal stability of FeB2 using ab-initio molecular dynamics analysis has been investigated. Calculations has been performed with NVT ensemble at room temperature up to 1 ps with 1 fs step. Conducted calculations at room temperature reveals that no deformation occurs and the structure maintain its configuration (Figure 2-a)
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Furthermore, the stability of FeB2 has been investigated at its equilibrium state and can be seen in phonon dispersion graphs shown in Figure 2-b. Since, any negative modes have not been observed in this graph, they elucidate the dynamically stability properties of our structure.
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Figure 2. Total energy fluctuation during AIMD simulation of FeB2 sheet at 300 K (a) and Phonon dispersion (b).
Mechanical properties of the FeB2 monolayer. In order to obtain energy-strain curves, the total energy of the system was calculated under various deformation tensors. The structure is exposed to three types of deformations, namely uniaxial strain along x direction ( or D1), uniaxial strain along y direction ( or D2) and biaxial strain ( or D3). In this study, to investigate the effect of deformation on the structure, we have changed the lattice parameters by 1% of stable unit-cell along compressive and tensile directions until we reached -10% and 30%, respectively. FeB2 is highly similar to borophene, structurally which has received great attention in recent years, except the position of atom boron which is inside hexagonally shaped iron atoms. Borophene, in addition to its big Young’s module (389 N/m), has high flexibility so that enters elastic phase at 16% strain which is comparable to graphene23. Flexibility of two dimensional materials is an interesting property which should be examined under large tensile and compressive strains. That’s why mechanical properties of FeB2 has been investigated
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under -10% compressive up to 30% tensile strains. According to Leonard-Jones curve, as atoms get close to each other, inter-atom repulsion increases significantly. Hence, compressive strains more than -10% is not applicable, structurally. Tensile strains have been applied on the structure till it enters irreversibly from elastic to plastic regions.
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In each deformation tensor, by applying different amounts of mechanical loadings, a potential well is realized. The minimum of this well is associated with the strain-free structure. We define the energy of strained structure as: /. Here, is the total energy of the deformed unitcell, is the total energy of undeformed unitcell and n is the number of atoms in the unitcell. Figure 3 depicts energy-strain (Es-) curves for FeB2 monolayer.
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Figure 3. Energy-strain curves of FeB2 monolayer (a) under uniaxial strain along x direction (D1), y direction (D2), biaxial (D3) and within harmonic region of each (b) D1, (c) D2 and (d) D3 strains.
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Energy appears to be anisotropic for tensile and compressive strains for our considered structure. In the case of uniaxial strain along x direction, energy-strain curve is approximately quadratic function of strain in the range from -5% to 5%. This harmonic region is within the ranges of -4% 4% for uniaxial strain along y direction and biaxial strain. Once an external strain is applied, the total energy of the system increases from its minimum value for the strain-free state. 5
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In general, a sudden change of energy in the energy-strain curve is an indicative of structural changes. Here in FeB2 monolayer, any structural phase transition has not been observed.
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Strain-Stress Relationship. For monolayer FeB2, stress must be expressed as 2D forces per length with units of N/m. Therefore, the stresses obtained from DFT predictions are multiplied by the unit cell thickness (15 Å). Furthermore, the second Piola-Kirchhoff (P-K) stress Σ is defined as47: Σ = JF −1σ ( F −1 )T , (1) For each deformation, F is the corresponding deformation gradient tensor47. Also, ?? denotes the determinant of F and ?? is the true stress with units of N/m. In continuum elastic theory and finite element method, the second P-K stress is employed to explore the impact of large deformations on the material behavior48. Here, our aim is to investigate the non-linear behavior of this structure at large strains and therefore P-K strain-stress curves are plotted and accordingly, the second-, third-, and fourth-order elastic constants are calculated by polynomial fitting of the resultant second P-K strain-stress curves (DFT results) to the continuum elastic theory. Figure 4 illustrates the stress-strain relations obtained from the DFT calculations, as well as the diagrams reproduced by fitting of the DFT results to the equations of continuum elastic theory. These diagrams are drawn for three types of strain: , and .
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Figure 4. (Color online) Stress-strain relationships for three types of strain, namely (a) , (b) , (c)
fitted with Σ1 and (d) fitted with Σ2. Σ1 and Σ2 denote the x and y components of the stress, respectively, while "Cont" corresponds to fitting the DFT results to the continuum elastic theory.
Table 1. Ultimate strains ( ) and ultimate stresses (Σm) for three types of strains: , and .
(N/m) (%)
Uniaxial (x) m=1 m=2 -4 7.08×10 3.61×10-4 26.5 20
Uniaxial (y) m=1 m=2 -4 7.96×10 3.86×10-4 34 12.5
Biaxial m=1 8.82×10-4 19.5
m=2 8.63×10-4 18.5
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Beyond the ultimate strain, the system is in a metastable state. Systems in the metastable state can be easily destroyed by vacancy defects or high temperature effects49. The DFT results for strains below the ultimate strain are used to determine the higher order elastic constants. Within the harmonic regions, specified in the last section for the three types of strains, the stress increases linearly with the strain. Under larger strains, the system goes outside the harmonic region in which higher order terms must be considered as well, for the prediction of stress-strain curves. When exposed to even higher strains, the system enters the plastic region. The structure undergoes irreversible changes when it is in the plastic region. Consequently, if the strain is removed, the system will not go back to its initial state. As aforementioned, in order to determine the elastic constants, the strainstress curves from the DFT calculations are fitted to the equations from the continuum elastic theory. The second-order elastic constants model the linear elastic response of the structure and the higher order ones are essential to the study of non-linear elasticity, as Wei50 and Peng47 have described it completely. In order to determine the elastic constants, suitable deformations should be selected to facilitate the calculation of these constants directly from the stress-strain curves. To that end, different modes of deformation are considered. Hence, three types of strain tensors ( , and ) previously defined by Wei50 and Peng47 have been used in this work.
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In these diagrams, the maxima of the curves reveal the stress in which a material withstands under the respective strain (the ultimate stress Σm) and the corresponding strain is the ultimate strain . The values obtained for the ultimate stress and ultimate strain are given in Table 1.
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Ultimately, the nonzero second-, third- and fourth- order elastic constants (SOEC, TOEC and FOEC, respectively) for single layer FeB2 are presented in Table 2. Table 2. Second-, third- and fourth order elastic constants (in N/m) of FeB2 monolayer.
TOEC -1484.70 -439.62 -2133.60 97.35 -2017.23 -1447.49
C1111 C1112 C1122 C2222 C1222 C6666 C1166 C1266 C2266
FOEC 6937.83 1488.23 768.67 12202.28 -2869.79 17518.84 3894.72 1046.13 2713.73
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C111 C112 C222 C122 C166 C266
As we can see from Table 2, the elastic constants here for FeB2 monolayer are large values; however, since this study is for the first time for our considered FeB2 monolayer there is not reported data to compare our results with. Yet, for overall compare of the magnitude for our obtained elastic constants (in addition to elastic properties such as Young’ modulus, bulk modulus, shear modulus and Poisson ratio), other mechanical investigations of 2D TiN 51, Stanene 52 and Silicene53 have been considered, as can be observed in Table 3.
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SOEC C11 C12 C22 C66
AlN51 Stanene5
C11 182.8 4 31.6
C12 88.2 9 15.8
C22 173.8 5 ---
C66 94.3 5 ---
K 139.8 3 ---
G 94.3 5 ---
Yx 137.6 4 23.5
Yy 131.1 3 25.2
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34.1
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---
---
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νx 0.5 0 0.4 2 0.3 3
νy 0.4 8 0.3 6 0.2 9
2D Young's moduli (in-plane stiffness) along x and y directions (Yx and Yy), Poisson's ratio along x and y directions (νx and νy), shear modulus (G) and the bulk modulus (K) are listed in Table 4. These values can be calculated using the following formulas54, 55:
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Table 3. Second order elastic constants Cij, bulk modulus K, shear modulus G, Young’s modulus Y (in N/m) and Poisson’s ratio ν for 2D AlN, Stanene and Silicene obtained in other literatures.
Yx =
c11c22 − c122 c c c2 and Yy = 11 22− 12 , (7) c22 c11 c c ν x = 12 and ν y = 12 , (8) c22 c11
G = c66 , (9) K =
Y , (10) 2(1 −ν ) 8
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1 Table 4. 2D Young's moduli, Poisson's ratio, 2D shear modulus, and 2D bulk modulus of 2D FeB2. /
/
136.049
141.183
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/ 69.1358
/ 129.1326
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2 In FeB2 nanosheet, each iron atom has located at center of hexagonally shaped 6 bor atoms. Bond length of B-B is 1.82 angstrom which is significantly close to that of bulk structure (1.76 angstrom 56). While bond length of Fe-B in single layer is 1.92 angstrom and for bulk is 2.32 angstrom. This shows that Fe-B bond length is 0.4 bigger for single layer FeB2 in comparison to bulk one. This factor leads to the stronger interaction and bonding between iron and bor which causes increase in mechanical strength and elasticity. Additionally, previous reports of binding energies for this structure shows 4.87 eV 25. This huge binding energy points this fact that FeB2 has great bonds and all of the mentioned factors show its outstanding elasticity and mechanical properties.
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Electronic Properties. Figure 4 (a) displays of partial density of states (PDOS) for B and Fe orbitals arranged beside band structure of FeB2 monolayer in equilibrium condition within -15 to 15 eV interval. Symmetry points where electrons move through this path is Γ-M-K-Γ within Brillouin zone of hexagonal FeB2 monolayer. Graphene like Dirac point in symmetry point K of Brillouin zone can be vividly observed. Hence, FeB2 monolayer is a zero gap semiconductor the same as graphene. Also, orbital contribution of s, p and d of atom Fe and s, p for atom B in occupying total density of states have been drawn in Figure 5. Comparison of band structure and partial density of states indicates that below the Fermi level orbital d of atom Fe has the most share in electron occupation of valence band beside tiny contribution of orbital p of atom B. On the other hand, above the Fermi level although orbital p of atom B has undeniably contribution, orbital d of atom Fe plays the main role. As a glance to the Figure 5, the main responsible of Dirac phenomena is on orbital 5d of atom Fe.
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Figure 5. Band structure of FeB2 monolayer at equilibrium state beside PDOS of atoms Fe and B.
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As the strain effects on structural and mechanical properties of FeB2 monolayer have been determined in this study, we also investigated the electronic features as the strain changes in uniaxial deformations along x, y and biaxial. To have better understanding of the electrons trajectories, we selected R-Γ-K-R-S-Γ-M k-path within deformed hexagonal Brillouin zone of FeB2 monolayer, as illustrated in Figure 6 (a). As can be seen in Figure 6 (b) we have brought band structures of FeB2 monolayer under uniaxial strain along x direction within compressive (-10%) to tensile (30%) strain. Likewise, Figure 6 (c) illustrates the same investigation along y direction. Moreover, comparison between equilibrium band structure of FeB2 monolayer with that of under biaxial compressive and tensile strain inside the same k-path Γ-M-K-Γ of Brillouin zone has been conducted in Figure 6 (d).
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Figure 6. (a) Strain dependent symmetrical points in reciprocal space of hexagonal lattice, (b) band structure of FeB2 monolayer under uniaxial strain along x direction (D1), (c) under uniaxial strain along y direction (D2), (d) under biaxial strain (D3). All energies have been shifted to Fermi level.
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According to Figure 6 (b-d), we have not observed any generated band gaps at all, despite different directionally high amount of strains applied on FeB2 monolayer. Only the Dirac point has been vanished through these strains and also the band structure is degenerated at point K of Brillouin zone. Hence, after applying strain on FeB2 monolayer, the obtained configurations illustrate metallic behavior which is due to the distance between Fe and B atoms. Our electronic investigation results are in good agreement with recent study which has been done by Zhang and coworkers25.
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CONCLUSION. In summary, using first-principles calculations combined with DFT we study mechanical and electronic properties of stable hexagonal FeB2 monolayer under different types of strains. We investigated 6 responses for stress-strain relationships of FeB2 monolayer which are categorized in two different types of axial and shear deformations. Three deformation modes from three axial strains (along x, y and biaxial in compressive and tensile) have been applied on the structure. Energy-strain curves have been studied primarily in three axial types indicating that energy curve against strain percentage is a function of second order polynomial within -5% to 5% for uniaxial strain along x 11
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direction. This harmonic region for uniaxial strain along y direction is in the range of 4%ɛy4% which is also for biaxial strain. Analyzing stress-strain relationships for FeB2 monolayer under distinguished strain based disfigurements, ultimate strain and ultimate stress in addition to second order elastic constants have been calculated. Ultimate stress for FeB2 monolayer under uniaxial strain along x, y and biaxial strains have been determined 3.61E-4, 7.96E-4 and 8.82E-4 N/m, respectively. Also, obtained elastic constants are large values such as 176.05 N/m for C11 and 1169.65 N/m for C22. Moreover, mechanical properties like shear modulus, bulk modulus, Young’s modulus and Poisson ratio have been determined for FeB2 monolayer which are all a bit bigger in y direction in comparison with x direction. Finally, we studied the band structure and PDOS of Iron diboride monolayer in the absence of any types of strain. This study, clearly resulted in existence of a graphene like Dirac point at K symmetry point in Brillouin zone. Comparing band structure and PDOS in equilibrium state illustrate the main responsible of Dirac point phenomena which is on orbital d of atom Iron at most and also a little on orbital p of atom Boron. In fact, interactions between orbitals 5d of atom Fe cause the Dirac point. Band structures of FeB2 monolayer have been computed under different axial strains leading to metallic configurations, still degeneracy at point K, and no band gaps have been generated.
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ACKNOWLEDGMENT The authors gratefully acknowledge the support of the Nano Research and Training Center of Iran (www.NRTC.ir) under Grant No.12.1397.05.15. Computational resources were provided by the Davami Computing Cluster belong to Tafresh University which is supported by the Nano Research and Training Center of Iran (www.NRTC.ir).
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Highlights
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• High order elastic constants of single layer of FeB2. • Young, Bulk and Shear’s moduli and Poisson ratio of single layer of FeB2 have been determined. • Dirac point in band structure of FeB2 vanished under strain.
S1386-9477(18)31258-X
DOI:
https://doi.org/10.1016/j.physe.2019.03.011
Reference:
PHYSE 13497
To appear in:
Physica E: Low-dimensional Systems and Nanostructures
Received Date: 19 August 2018 Revised Date:
1 February 2019
Accepted Date: 15 March 2019
Please cite this article as: H. Jafari, M. Masoudi, N. Taghizade, A. Ahmadi, M. Faghihnasiri, Study of mechanical and electronic properties of single-layer FeB2, Physica E: Low-dimensional Systems and Nanostructures (2019), doi: https://doi.org/10.1016/j.physe.2019.03.011. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Department of Physics, Iran University of Science and Technology, Tehran, Iran Department of Physics, Payame Noor University, PO BOX 19395-3697, Tehran, Iran c Department of Electronic and Electrical Engineering, Faculty of Engineering, University of Birjand, Birjand 97174-34765, Iran d Computational Materials Science Laboratory, Nano Research and Training Center, NRTC, Iran
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* Corresponding Author Email: [email protected]
Abstract: Recent experimental advances for the fabrication of various two dimensional (2D) materials introduced new structures with a wide prospect of applications. In this work, under in-plane uniaxial and biaxial strain, we employed first-principles density functional theory calculations to investigate the mechanical and electronic properties of Hexagonal Iron dibromide (FeB2) sheet. In mechanical section, we investigate deeply the mechanical analyzing of 2D structures to study elastic constants. Based on the results of our modelling, single layer FeB2 depending on the strain direction can yield remarkable elastic modulus in the range of 141.18 N/m for Young’s modulus and also hold on ultimate stress up to 7.08×10-4 N/m, 7.96×10-4 N/m and 8.82×10-4 N/m at the corresponding strains 26.5%, 34% and 19.5% for uniaxial strain along x, y and biaxial strain, respectively. In this paper, second, third and fourth orders of elastic constants beside Young, Bulk and Shear’s moduli and Poisson ratio of single layer of FeB2 have been determined, for the first time. By analyzing electronic band structure, we did not observe any band gaps in terms of strains. However, applying directional strain based on deformation matrices, bands get apart from Fermi level so that Dirac point has been vanished.
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Homayoun Jafari a, Maryam Masoudi b, Narges Taghizade a, Aidin Ahmadi c, Mahdi Faghihnasiri d,*
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Study of Mechanical and Electronic Properties of single-layer FeB2
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Keywords: Density Functional Theory; High order elastic constants; FeB2
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INTRODUCTION Nowadays, studying two dimensional materials are considered among the most attractive and interesting research fields owing to their vast properties and applications. Interests to this category of materials were created through synthesis and production of graphene1 possessing surface structure with honeycomb arrangement of carbon atoms. Graphene is a zero-gap semiconductor that reveals unique electronic2, 3, mechanical4 and thermal5 properties. After the successful synthesis of graphene, other 2D materials including hexagonal structure of boron-nitride6, 7, graphite like structure carbon-nitride8, titaniumaluminum-nitride9, and also lower dimensional materials like silicone nanoribbon10, aluminum-nitride nanotube11 and quantum dots12 were synthesized. 2D materials can exhibit exotic properties which are different from their bulk counterparts due to the quantum confinement effect, making them more attractive for designing hard protector films13-15, novel electronic and optoelectronic devices. In this regard, interests to predicate and investigate the properties of this type of materials were never stopped. Through the continuous developments in synthesis of newer 2D materials, motivated by not only technological applications but also the fundamental research in the current scientific community, they have triggered an enormous interest for their tunable mechanical, optoelectronic, and magnetic properties, significantly enriching the family of 2D materials. Graphene like structure Borophene16, 17, replaced carbon with boron, has been attracted considerable attention due to its unique thermal18, 19, chemical20, 21 and mechanical22-24 features, recently. Studies the properties of different phases of Borophene continued till 2016, when Zhang and coworkers25 designed a new Dirac included material, called FeB2 monolayer which contains embedded planar hexacoordinate Fe atoms in the 2D honeycomb lattice of boron. For bulk configuration of Fe-B, there have been reported ultra-stable phase under pressure and also investigated its special magnetic, mechanical and superconductivity properties27-29. Current challenge in the development of two dimensional materials which researchers facing, is their low stability and so many efforts have been made to overcome this issue by using appropriate substrates. Substrate and the main layer lattice parameter mismatch could cause external strain to the layer and consequently could apply corruptive influence to the material properties30. Therefore, having a comprehensive knowledge about the elastic properties of these materials and their response to the external stress should be a great deal in the fabrication and application of mono-layers31. In this study we investigate the mechanical parameters such as young’s, bulk and shear modulus using second order linear elastic constants32, 33 which can be achieved by fitting the second order polynomial functions with the Stress-Energy curve. Furthermore, we evaluated high orders nonlinear elastic constants such as third and fourth order elastic constants by merging density functional theory and continuum approach34. These parameters have an important role in the phase transition which can be described by Landau’s phase transition theory35. These elastic constants not only being used in mechanical properties investigation, but also they are important in studying and analyzing structural phenomenon such as Grüneisen parameters36, piezoelectric properties37, and trapped-energy resonators38. In this work, we study the accurate elastic constants of FeB2 single layer, understand its behavior under external load and know which variables would affect their elastic response and how these variables would affect those constants. The total energy of the
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systems after structure relaxation was calculated with good accuracy using the density functional theory (DFT) at any applied strain. Finally, we provide a systematic study of the electronic properties of FeB2 under external physical stimuli.
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COMPUTATIONAL DETAILS First-principle study by DFT calculations were carried out with the Quantum ESPRESSO package39. The exchange-correlation term is described by adopting generalized gradient approximation using Perdew– Burke– Ernzerhof of revised for solids (PBEsol) functional40 along with the projector-augmented wave (PAW) potentials for the self-consistent total energy calculations and geometry optimization. The kinetic energy cutoff for the plane wave basis set was chosen to be 700 eV. The Brillion zone was sampled with 14×14×1 k-points using the standard Monkhorst–Pack special grids41. Atomic positions were relaxed until the energy differences were converged within 10-6 eV and the maximum Hellmann– Feynman force on any atom was below 10-5 eV/Å. Periodic boundary conditions for two directions (x and y) were invoked in the simulations and a vacuum region of 15 Å along the z direction was included to safely avoid interaction between adjacent layers.
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RESULTS AND DISCUSSION Atomic Structure. Figure 1 presents the top and side views of atomic structure for FeB2 2D system, containing two B atoms and one Fe atom in its hexagonal unit-cell, in which B atoms are arranged in a honeycomb lattice and Fe atoms are 0.6 Å above the center of six-membered boron rings. This monolayer sheet can be considered as a one-layer Fe−B sheet exfoliated from bulk FeB2. In this quasi-planar FeB2 sheet, each Fe atom coordinates with six boron atoms around it, thus forming planar hexacoordination. Following the geometry optimization of the unit-cell using PBEsol functional, the lattice constants along a and b directions are 3.15 Å. The B−B bond length (1.82 Å) is about the same as that in the bulk FeB2 (1.76 Å)42 but slightly larger than that in the B monolayers (1.67 Å ∼ 1.71 Å)43-46. The length of Fe−B bond in FeB2 monolayer (1.93 Å) is noticeably smaller than that in bulk FeB2 crystal (2.32 Å), suggesting a much stronger interaction between the Fe atoms and boron layer. All of our geometric optimized parameters like B-B and Fe-B bond lengths of 1.82 and 1.93 Å are in good agreement with recent reported results25 of 1.83 and 1.94 Å, respectively.
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Fe B Figure 1. Top and side view of FeB2 monolayer.
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1 Thermal stability of FeB2 using ab-initio molecular dynamics analysis has been investigated. Calculations has been performed with NVT ensemble at room temperature up to 1 ps with 1 fs step. Conducted calculations at room temperature reveals that no deformation occurs and the structure maintain its configuration (Figure 2-a)
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Furthermore, the stability of FeB2 has been investigated at its equilibrium state and can be seen in phonon dispersion graphs shown in Figure 2-b. Since, any negative modes have not been observed in this graph, they elucidate the dynamically stability properties of our structure.
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Figure 2. Total energy fluctuation during AIMD simulation of FeB2 sheet at 300 K (a) and Phonon dispersion (b).
Mechanical properties of the FeB2 monolayer. In order to obtain energy-strain curves, the total energy of the system was calculated under various deformation tensors. The structure is exposed to three types of deformations, namely uniaxial strain along x direction ( or D1), uniaxial strain along y direction ( or D2) and biaxial strain ( or D3). In this study, to investigate the effect of deformation on the structure, we have changed the lattice parameters by 1% of stable unit-cell along compressive and tensile directions until we reached -10% and 30%, respectively. FeB2 is highly similar to borophene, structurally which has received great attention in recent years, except the position of atom boron which is inside hexagonally shaped iron atoms. Borophene, in addition to its big Young’s module (389 N/m), has high flexibility so that enters elastic phase at 16% strain which is comparable to graphene23. Flexibility of two dimensional materials is an interesting property which should be examined under large tensile and compressive strains. That’s why mechanical properties of FeB2 has been investigated
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under -10% compressive up to 30% tensile strains. According to Leonard-Jones curve, as atoms get close to each other, inter-atom repulsion increases significantly. Hence, compressive strains more than -10% is not applicable, structurally. Tensile strains have been applied on the structure till it enters irreversibly from elastic to plastic regions.
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In each deformation tensor, by applying different amounts of mechanical loadings, a potential well is realized. The minimum of this well is associated with the strain-free structure. We define the energy of strained structure as: /. Here, is the total energy of the deformed unitcell, is the total energy of undeformed unitcell and n is the number of atoms in the unitcell. Figure 3 depicts energy-strain (Es-) curves for FeB2 monolayer.
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Figure 3. Energy-strain curves of FeB2 monolayer (a) under uniaxial strain along x direction (D1), y direction (D2), biaxial (D3) and within harmonic region of each (b) D1, (c) D2 and (d) D3 strains.
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Energy appears to be anisotropic for tensile and compressive strains for our considered structure. In the case of uniaxial strain along x direction, energy-strain curve is approximately quadratic function of strain in the range from -5% to 5%. This harmonic region is within the ranges of -4% 4% for uniaxial strain along y direction and biaxial strain. Once an external strain is applied, the total energy of the system increases from its minimum value for the strain-free state. 5
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In general, a sudden change of energy in the energy-strain curve is an indicative of structural changes. Here in FeB2 monolayer, any structural phase transition has not been observed.
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Strain-Stress Relationship. For monolayer FeB2, stress must be expressed as 2D forces per length with units of N/m. Therefore, the stresses obtained from DFT predictions are multiplied by the unit cell thickness (15 Å). Furthermore, the second Piola-Kirchhoff (P-K) stress Σ is defined as47: Σ = JF −1σ ( F −1 )T , (1) For each deformation, F is the corresponding deformation gradient tensor47. Also, ?? denotes the determinant of F and ?? is the true stress with units of N/m. In continuum elastic theory and finite element method, the second P-K stress is employed to explore the impact of large deformations on the material behavior48. Here, our aim is to investigate the non-linear behavior of this structure at large strains and therefore P-K strain-stress curves are plotted and accordingly, the second-, third-, and fourth-order elastic constants are calculated by polynomial fitting of the resultant second P-K strain-stress curves (DFT results) to the continuum elastic theory. Figure 4 illustrates the stress-strain relations obtained from the DFT calculations, as well as the diagrams reproduced by fitting of the DFT results to the equations of continuum elastic theory. These diagrams are drawn for three types of strain: , and .
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Figure 4. (Color online) Stress-strain relationships for three types of strain, namely (a) , (b) , (c)
fitted with Σ1 and (d) fitted with Σ2. Σ1 and Σ2 denote the x and y components of the stress, respectively, while "Cont" corresponds to fitting the DFT results to the continuum elastic theory.
Table 1. Ultimate strains ( ) and ultimate stresses (Σm) for three types of strains: , and .
(N/m) (%)
Uniaxial (x) m=1 m=2 -4 7.08×10 3.61×10-4 26.5 20
Uniaxial (y) m=1 m=2 -4 7.96×10 3.86×10-4 34 12.5
Biaxial m=1 8.82×10-4 19.5
m=2 8.63×10-4 18.5
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Beyond the ultimate strain, the system is in a metastable state. Systems in the metastable state can be easily destroyed by vacancy defects or high temperature effects49. The DFT results for strains below the ultimate strain are used to determine the higher order elastic constants. Within the harmonic regions, specified in the last section for the three types of strains, the stress increases linearly with the strain. Under larger strains, the system goes outside the harmonic region in which higher order terms must be considered as well, for the prediction of stress-strain curves. When exposed to even higher strains, the system enters the plastic region. The structure undergoes irreversible changes when it is in the plastic region. Consequently, if the strain is removed, the system will not go back to its initial state. As aforementioned, in order to determine the elastic constants, the strainstress curves from the DFT calculations are fitted to the equations from the continuum elastic theory. The second-order elastic constants model the linear elastic response of the structure and the higher order ones are essential to the study of non-linear elasticity, as Wei50 and Peng47 have described it completely. In order to determine the elastic constants, suitable deformations should be selected to facilitate the calculation of these constants directly from the stress-strain curves. To that end, different modes of deformation are considered. Hence, three types of strain tensors ( , and ) previously defined by Wei50 and Peng47 have been used in this work.
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In these diagrams, the maxima of the curves reveal the stress in which a material withstands under the respective strain (the ultimate stress Σm) and the corresponding strain is the ultimate strain . The values obtained for the ultimate stress and ultimate strain are given in Table 1.
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Ultimately, the nonzero second-, third- and fourth- order elastic constants (SOEC, TOEC and FOEC, respectively) for single layer FeB2 are presented in Table 2. Table 2. Second-, third- and fourth order elastic constants (in N/m) of FeB2 monolayer.
TOEC -1484.70 -439.62 -2133.60 97.35 -2017.23 -1447.49
C1111 C1112 C1122 C2222 C1222 C6666 C1166 C1266 C2266
FOEC 6937.83 1488.23 768.67 12202.28 -2869.79 17518.84 3894.72 1046.13 2713.73
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C111 C112 C222 C122 C166 C266
As we can see from Table 2, the elastic constants here for FeB2 monolayer are large values; however, since this study is for the first time for our considered FeB2 monolayer there is not reported data to compare our results with. Yet, for overall compare of the magnitude for our obtained elastic constants (in addition to elastic properties such as Young’ modulus, bulk modulus, shear modulus and Poisson ratio), other mechanical investigations of 2D TiN 51, Stanene 52 and Silicene53 have been considered, as can be observed in Table 3.
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SOEC C11 C12 C22 C66
AlN51 Stanene5
C11 182.8 4 31.6
C12 88.2 9 15.8
C22 173.8 5 ---
C66 94.3 5 ---
K 139.8 3 ---
G 94.3 5 ---
Yx 137.6 4 23.5
Yy 131.1 3 25.2
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---
---
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νx 0.5 0 0.4 2 0.3 3
νy 0.4 8 0.3 6 0.2 9
2D Young's moduli (in-plane stiffness) along x and y directions (Yx and Yy), Poisson's ratio along x and y directions (νx and νy), shear modulus (G) and the bulk modulus (K) are listed in Table 4. These values can be calculated using the following formulas54, 55:
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Table 3. Second order elastic constants Cij, bulk modulus K, shear modulus G, Young’s modulus Y (in N/m) and Poisson’s ratio ν for 2D AlN, Stanene and Silicene obtained in other literatures.
Yx =
c11c22 − c122 c c c2 and Yy = 11 22− 12 , (7) c22 c11 c c ν x = 12 and ν y = 12 , (8) c22 c11
G = c66 , (9) K =
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1 Table 4. 2D Young's moduli, Poisson's ratio, 2D shear modulus, and 2D bulk modulus of 2D FeB2. /
/
136.049
141.183
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/ 129.1326
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2 In FeB2 nanosheet, each iron atom has located at center of hexagonally shaped 6 bor atoms. Bond length of B-B is 1.82 angstrom which is significantly close to that of bulk structure (1.76 angstrom 56). While bond length of Fe-B in single layer is 1.92 angstrom and for bulk is 2.32 angstrom. This shows that Fe-B bond length is 0.4 bigger for single layer FeB2 in comparison to bulk one. This factor leads to the stronger interaction and bonding between iron and bor which causes increase in mechanical strength and elasticity. Additionally, previous reports of binding energies for this structure shows 4.87 eV 25. This huge binding energy points this fact that FeB2 has great bonds and all of the mentioned factors show its outstanding elasticity and mechanical properties.
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Electronic Properties. Figure 4 (a) displays of partial density of states (PDOS) for B and Fe orbitals arranged beside band structure of FeB2 monolayer in equilibrium condition within -15 to 15 eV interval. Symmetry points where electrons move through this path is Γ-M-K-Γ within Brillouin zone of hexagonal FeB2 monolayer. Graphene like Dirac point in symmetry point K of Brillouin zone can be vividly observed. Hence, FeB2 monolayer is a zero gap semiconductor the same as graphene. Also, orbital contribution of s, p and d of atom Fe and s, p for atom B in occupying total density of states have been drawn in Figure 5. Comparison of band structure and partial density of states indicates that below the Fermi level orbital d of atom Fe has the most share in electron occupation of valence band beside tiny contribution of orbital p of atom B. On the other hand, above the Fermi level although orbital p of atom B has undeniably contribution, orbital d of atom Fe plays the main role. As a glance to the Figure 5, the main responsible of Dirac phenomena is on orbital 5d of atom Fe.
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Figure 5. Band structure of FeB2 monolayer at equilibrium state beside PDOS of atoms Fe and B.
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As the strain effects on structural and mechanical properties of FeB2 monolayer have been determined in this study, we also investigated the electronic features as the strain changes in uniaxial deformations along x, y and biaxial. To have better understanding of the electrons trajectories, we selected R-Γ-K-R-S-Γ-M k-path within deformed hexagonal Brillouin zone of FeB2 monolayer, as illustrated in Figure 6 (a). As can be seen in Figure 6 (b) we have brought band structures of FeB2 monolayer under uniaxial strain along x direction within compressive (-10%) to tensile (30%) strain. Likewise, Figure 6 (c) illustrates the same investigation along y direction. Moreover, comparison between equilibrium band structure of FeB2 monolayer with that of under biaxial compressive and tensile strain inside the same k-path Γ-M-K-Γ of Brillouin zone has been conducted in Figure 6 (d).
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Figure 6. (a) Strain dependent symmetrical points in reciprocal space of hexagonal lattice, (b) band structure of FeB2 monolayer under uniaxial strain along x direction (D1), (c) under uniaxial strain along y direction (D2), (d) under biaxial strain (D3). All energies have been shifted to Fermi level.
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According to Figure 6 (b-d), we have not observed any generated band gaps at all, despite different directionally high amount of strains applied on FeB2 monolayer. Only the Dirac point has been vanished through these strains and also the band structure is degenerated at point K of Brillouin zone. Hence, after applying strain on FeB2 monolayer, the obtained configurations illustrate metallic behavior which is due to the distance between Fe and B atoms. Our electronic investigation results are in good agreement with recent study which has been done by Zhang and coworkers25.
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CONCLUSION. In summary, using first-principles calculations combined with DFT we study mechanical and electronic properties of stable hexagonal FeB2 monolayer under different types of strains. We investigated 6 responses for stress-strain relationships of FeB2 monolayer which are categorized in two different types of axial and shear deformations. Three deformation modes from three axial strains (along x, y and biaxial in compressive and tensile) have been applied on the structure. Energy-strain curves have been studied primarily in three axial types indicating that energy curve against strain percentage is a function of second order polynomial within -5% to 5% for uniaxial strain along x 11
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direction. This harmonic region for uniaxial strain along y direction is in the range of 4%ɛy4% which is also for biaxial strain. Analyzing stress-strain relationships for FeB2 monolayer under distinguished strain based disfigurements, ultimate strain and ultimate stress in addition to second order elastic constants have been calculated. Ultimate stress for FeB2 monolayer under uniaxial strain along x, y and biaxial strains have been determined 3.61E-4, 7.96E-4 and 8.82E-4 N/m, respectively. Also, obtained elastic constants are large values such as 176.05 N/m for C11 and 1169.65 N/m for C22. Moreover, mechanical properties like shear modulus, bulk modulus, Young’s modulus and Poisson ratio have been determined for FeB2 monolayer which are all a bit bigger in y direction in comparison with x direction. Finally, we studied the band structure and PDOS of Iron diboride monolayer in the absence of any types of strain. This study, clearly resulted in existence of a graphene like Dirac point at K symmetry point in Brillouin zone. Comparing band structure and PDOS in equilibrium state illustrate the main responsible of Dirac point phenomena which is on orbital d of atom Iron at most and also a little on orbital p of atom Boron. In fact, interactions between orbitals 5d of atom Fe cause the Dirac point. Band structures of FeB2 monolayer have been computed under different axial strains leading to metallic configurations, still degeneracy at point K, and no band gaps have been generated.
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ACKNOWLEDGMENT The authors gratefully acknowledge the support of the Nano Research and Training Center of Iran (www.NRTC.ir) under Grant No.12.1397.05.15. Computational resources were provided by the Davami Computing Cluster belong to Tafresh University which is supported by the Nano Research and Training Center of Iran (www.NRTC.ir).
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Highlights
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• High order elastic constants of single layer of FeB2. • Young, Bulk and Shear’s moduli and Poisson ratio of single layer of FeB2 have been determined. • Dirac point in band structure of FeB2 vanished under strain.