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Shear-induced Structural Transformation for Tetragonal BC

Baobing Zheng, Meiguang Zhang, Shaomei Chang, Yaru Zhao, and Hong-Gang Luo J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.5b10863 • Publication Date (Web): 09 Dec 2015 Downloaded from http://pubs.acs.org on December 20, 2015

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The Journal of Physical Chemistry

Shear-induced Structural Transformation for Tetragonal BC4 Baobing Zheng1,*, Meiguang Zhang1, Shaomei Chang1, Yaru Zhao1, Hong-Gang Luo2,3,* 1

College of Physics and optoelectronics Technology, Nonlinear Research Institute, Baoji University of Arts and

Sciences, Baoji 721016, China 2

Center of Interdisciplinary Studies and Key Laboratory for Magnetism and Magnetic Materials of the Ministry

of Education, Lanzhou University, Lanzhou 730000, China 3

Beijing Computational Science Research Center, Beijing 100084, China

ABSTRACT

The hardness and ideal strength of the recent predicted tetragonal t-BC4 were fundamentally investigated by using first-principles calculations within the framework of density-functional theory. The obtained results show that the t-BC4 exhibits a substantially low ideal shear strength of 5 GPa along (001)[100] directions, suggesting that the t-BC4 does not belong to the class of superhard materials. Interestingly, the t-BC4 undergoes a structural transformation upon shear deformation with the remarkable changes of the Fermi surfaces, and changes into the orthogonal o-BC4 phase with a lower total energy than the t-BC4 phase indicative of its more preferable stability. It is found that the structural transformation stems from the bonding of the C5 and C7 atoms in the t-BC4 phase. Keywords: BC4, Hardness, Ideal shear strength, Structural transformation

*

Author to whom correspondence should be addressed. Electronic mail: [email protected] and [email protected] 1

ACS Paragon Plus Environment


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I. INTRODUCTION Upon shear load, materials generally tend to undergo a sliding failure along a plane that is parallel to the direction of the load. The crystal is distorted due to stretching or compressing of the atomic bonds from their equilibrium state under elastic shear loading before slip occurs and then returns to its undistorted shape and no permanent deformation exists when the load is removed, whereas plastic deformation that is mediated by dislocation slip and deformation twinning occurs once beyond the yield strength1,2. Consequently, shear leads to the identification of novel behavior of solids and the exploration of potential technological materials, such as the transformation from hcp phase to bcc phase in pure zirconium3, the formation of amorphous band in boron carbide2,4, the conductor-insulator transition in melt-mixed polypropylene-carbon nanotube dispersions5, gap opening in graphene6, all of which reflect the mechanical response of the materials. Due to excellent resistance to oxidation and interesting electrical properties7-11, the boron-carbon (B-C) compounds, possessing extreme hardness, high mechanical and thermal stability, and high electron density, are always regarded as good candidates of alternate hard materials such as diamond. Thus, great efforts have been devoted to synthesizing and designing various compounds constituted by boron and carbon elements. Typically, boron carbide (B4C) is suitable to used as abrasive or shielding material due to its exceptional hardness (Vickers Hardness of 41–42 GPa12), outstanding elastic properties (bulk modulus of 240 GPa13), and the advantage of being easily synthesized. Experimental attempts to synthesize B-C materials have achieved great success, such as the syntheses of cubic BC5 with extreme Vickers hardness (71 GPa)14 and cubic BC3 phase with Vickers hardness 62 GPa10,15. To complement experiments, the first-principle calculations combined with state-of-the-art structure prediction are developed and are deemed as the most powerful approach to design the novel functional materials16,17. The efficiency of these unbiased approaches have been verified in a number of fields, such as the predictions of crystal structure18,19, surface reconstruction20,21, two-dimensional compounds22, 23, molecular crystal24, 25, and the discovery of structure possessing optimal properties26, 27. Recently, using ab initio particle warm optimization (PSO) algorithm16, Wang et al.28 explored a new P42/mmc structure for BC4 that is mechanically and dynamically stable. The predicted tetragonal BC4 phase (t-BC4) has two adjacent boron-carbon cages along the c direction linked by only one C-C bond, which induce a weak ability to against shear elastic deformation and thus small shear modulus (155 GPa) and low hardness (13 GPa). 2


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However, the essential reason of such a low hardness for t-BC4 is incomprehensible compared with BC3 and BC5. In order to further clarify the microscopic mechanism of low hardness of t-BC4, the origin of the lattice instability t-BC4 cell under large shear strain that occurs at the atomic level during plastic deformation should be fully investigated. In the present work, we applied different empirical hardness models and the stress-strain method combined with first-principles calculations to extensively explore the hardness and ideal strengths of t-BC4, respectively. The obtained results demonstrated that the relatively low hardness of t-BC4 originates from the occurrence of shear-induced structural transformation under the increasing shear loadings. Interestingly, the new phase after structural transformation is the most energetically stable BC4 phase at ambient pressure. We have thus performed first-principles calculations to characterize the structural, mechanical, thermodynamic, and electronic properties of this novel BC4 phase. II. COMPUTATIONAL METHODS The self-consistent calculations were performed using density functional theory within the Perdew-Burke-Ernzerhof (PBE) exchange-correlation and employing the generalized gradient approximation (GGA), as implemented in the Vienna ab initio simulation package (VASP)29, 30. The frozen-core all-electron projector augmented wave (PAW) method31, where the 2s22p1 and 2s22p2 were treated as valence electrons for B and C, was adopted to describe the interactions between electron and core. The cutoff energy of the plane-wave expansion is 520 eV for each structure, and the Monkhorst-Pack k-point meshes in the Brillouin zone32 are 12×12×3 and 2×12×12 for the tetragonal and orthorhombic BC4, respectively, which are enough to ensure a good precision in calculation and analysis of structures. The elastic constants were determined in term of the strain-stress method, and an array of +0.1 and −0.1 precent of distortion are employed to apply the strain for the optimized structure, the elastic constants are deduced from the stress of the strained structure after fully optimizing the atomic position and keeping the cell shape. The bulk modulus, shear modulus, and Young’s modulus were estimated using the Voigt–Reuss–Hill approximation33. The t-BC4 cell was deformed to obtain the stress-strain relationships and ideal strengths according to increasing the displacement in the direction of the applied strain34, 35. The phonon-dispersion curve of the orthorhombic BC4 was calculated using direct supercell approach with the finite 3



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displacement method using the PHONOPY code36. Supercells containing 2×2×2 unit cells and 1×5×5 k-point meshes are used. The full potential linearized augmented plane wave (LAPW) method37 through the WIEN2K code38 were performed to calculate the Fermi surfaces. In addition, the 3D electron localization function and Fermi-surface are generated by VESTA39 and XcrySDen40 softwares, respectively. III. RESULTS AND DISCUSSION The calculated elastic constants Cij, bulk modulus B, shear modulus G, and Young’s modulus E of t-BC4 are listed in Table 1. Although the elastic constants show slight differences with the theoretical values of Ref. 28, the excellent agreement of bulk modulus, shear modulus, and Young’s modulus between present work and Ref. 28 confirm the accuracy and reliability of the present calculations. The hardness of t-BC4 are estimated in term of three different models which proposed by Šimůnek41, Lyakhov and Oganov42, and Chen43, the corresponding hardness are denoted as HŠimůnek, HLyakhov-Oganov, and HChen, respectively. It can be clearly seen that the values of HŠimůnek and HLyakhov-Oganov show a considerable difference with that of HChen, which demonstrates that diverse models of hardness lead to contradictory results for assessing whether the t-BC4 is superhard. Note that the hardness of t-BC4 calculated by Chen’s model is only 13 GPa, which is much less than 40 GPa. Thus, the evaluation criterion of hardness is invalid for the t-BC4. Previous studies have revealed that the ultimate hardness of a material may be assessed from its ideal shear strength and bonding nature, which also appears to correlate with the onset of dislocation formation in an ideal, defect-free crystal. Physically, the plastic deformation of materials occurs far from equilibrium at finite strains in practical measurements of hardness. Therefore, the ideal tensile and shear strengths calculations according to strain-stress relationship are more appropriate and stringent for measuring the hardness of given materials than the calculated value employing various theoretical models of hardness. The calculated stress-strain relationships upon tension of t-BC4 along various directions are plotted in Fig. 1(a). The ideal tensile strengths along [100], [111], [001], and [110] directions are 73 GPa, 78 GPa, 116 GPa, and 138 GPa, respectively. The minimum tensile strength is found in the direction of [100], larger than 40 GPa. The stress-strain relationships and the dependence of total energy on the shear strain under the weakest shear deformation direction of (001)[100] are

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illustrated in Fig. 1(b) for t-BC4. Obviously, the ideal shear strength along (001)[100] direction has a significantly low value of 5 GPa, which indicates that the shear mode under this direction dominates the plastic deformation of t-BC4. The ideal shear strength much less than 40 GPa shows that the t-BC4 is not a superhard material essentially. It should be noted that the total energy of t-BC4 becomes larger gradually as the increase of strain. However, once the strain is beyond the critical value (i.e. 0.08), the total energy of t-BC4 decreases abruptly and reduces to a value lower than the total energy of the unstrained t-BC4. Generally, the total energy of equilibrium structure for given material under the strain would continuously increase up to the critical strain, and then follow by a decrease indicating a lattice instability44. The collapse of the structure accompanies by the breaking of the weaker chemical bonds, but the total energy of unstable lattice is usually larger than that of equilibrium structure because of the effect of the strain. However, the strained t-BC4 does not present any breaking bond and possesses the lower total energy than the unstrained t-BC4. Simultaneously, the structure of BC4 beyond the critical shear strain is identified in the low symmetric orthogonal structure (denoted as o-BC4, space group Cmmm, 24 atoms/cell). In order to further confirm the o-BC4, we re-optimize the o-BC4 phase by full relaxations of both lattice constants and internal atomic coordinations. The optimized o-BC4 still holds identical space group of Cmmm and Wyckoff positions, which further verify the o-BC4 phase. The structures of o-BC4 phase and t-BC4 phase are shown in Fig. 2. Note that the o-BC4 phase is remarkably similar with the t-BC4 phase, and the structure of o-BC4 phase can be viewed as the 1×1×2 supercell of the t-BC4 where some of carbon atoms forms the planar C6 ring. The only difference is that part of the carbon atoms in the adjacent C6 rings become bonding in the o-BC4 phase (marked by red circle), which can be attributed to the stretching of the atomic bonds between carbon atoms under shear stress. Combined with Fig. 3 [the electron localization function (ELF) distributions of t-BC4 under two selected shear strains], we can easily find that the bonding between carbon atoms stem from the C5 and C7 atoms in the adjacent C6 rings (denoted as red arrow). That is to say, the t-BC4 phase undergoes a structural transformation under shear load and change into the orthogonal o-BC4 phase with Cmmm space group. To gain the deeper insight on the structural transformation upon shear load, the energy bands for t-BC4 and o-BC4 are shown in Fig. 4(a) and (b) to illustrate the variation of electronic properties accompanying with structural transformation. Three bands across the Fermi level and strong band 5



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overlap implies that the t-BC4 and o-BC4 phase both hold the metallicity. 3D Fermi-surface plots are present in Fig. 4(c) and (d). Clearly, the Fermi surfaces near the boundary of the first Brillouin zone for the two phase, which can be identified as the contribution of two bands, are very similar. The remarkable changes of the Fermi surfaces between t-BC4 and o-BC4 phase, resulting from another band, are centered at the zone center. Note that the Fermi surfaces are stretched along Γ-Z and Y-T directions of o-BC4 phase, the corresponding stretched directions in t-BC4 phase are Γ-X and Z-R directions, respectively. Generally, this physical phenomenon is reasonably related with the band structure description, particularly the band near the Fermi level. It is found that one of the bands in t-BC4 phase along Γ-X and Z-R directions cross the Fermi level (indicated by red arrows), but this band shifts above Fermi level in o-BC4 phase along the corresponding Γ-Z and Y-T directions (also indicated by red arrows), which makes prefect sense for the changes of the Fermi surfaces between t-BC4 and o-BC4 phase. In order to verify the mechanical and dynamical stability, the elastic constants of the o-BC4 phase are calculated in term of stress-strain relations and the phonon spectra is determined by using direct supercell approach with the finite displacement method. Physically, the o-BC4 phase with orthogonal structure has eight independent elastic constants, as listed in Table 2. According to Born stability criteria, the elastic constants for a stable orthogonal structure are supposed to satisfy the generalized elastic stability criteria as follows45: C11 > 0 , C44 > 0 , C55 > 0 , C66 > 0 , C11C22 > C122 ,

C11C22C33 + 2C12C13C23 − C11C232 − C22C132 − C33C122 > 0 .Obviously, the elastic constants of the o-BC4 phase satisfy all the mechanical stability criteria, thus suggesting the o-BC4 phase is mechanically stable. The phonon-dispersion curve of o-BC4 phase is shown in Fig. 5(a). Clearly, the absence of imaginary phonon frequency in the Brillouin zone demonstrates that the o-BC4 phase is dynamically stable. The calculated bulk modulus (B) and shear modulus (G) of o-BC4 phase are 343 GPa and 244 GPa, respectively, 4.3% and 57.4% larger than that of the estimated results of t-BC4 phase. It should be noted that the significant enlargement of G appears in the o-BC4 phase, which can be ascribed to the formation of new C-C bonds in the o-BC4 phase and thus enhance the ability of resisting shear deformation. The density of states (DOS) including total and partial DOS are shown in Fig. 5(b). The finite value at the Fermi level indicates that the o-BC4 phase should have a metallicity as 6


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expected, which may be resulted from the fact that the electron deficiency of boron atoms would lead to the formation of some empty orbits above the Fermi level when the carbon and boron atoms generate sp3-hybridized orbits. The main contribution of DOS near the Fermi level for o-BC4 phase is derived from the B-p states and the C-p states, suggesting that the boron and carbon atoms in o-BC4 collectively contribute to the formation of the empty orbits IV. CONCLUSIONS In summary, the hardness and the ideal strength for the t-BC4 compound have been fully investigated by using ab initio method. The lowest ideal strength of t-BC4 is found to be 5 GPa along (001)[100] directions, which indicates the t-BC4 can not be considered as a novel intrinsic superhard material eventually despite the high estimated hardness of HŠimůnek and HLyakhov-Oganov. Futhermore, under shear deformation, the t-BC4 phase undergoes a structural transformation and change into a more stable o-BC4 phase with higher shear modulus than the t-BC4 phase. We hope that the present calculations can provide the fundamental information for better understanding of structural transformation for this interesting material, and broaden our insight of searching the novel superhard material.

ACKNOWLEDGMENTS The work was supported partly by NSFC (Grant Nos. 11174115, 11325417, 11204007 and 11404008), PCSIRT (Grant No. IRT1251), national program for basic research of China, Scientific Research Plan Projects of Education Department of Shaanxi province of China (Grant No. 14JK1044) and the Baoji University of Arts and Sciences Key Research (Grant No. ZK0915). REFERENCES (1) Boresi, A. P.; Schmidt, R. J.; Sidebottom, O. M. Advanced Mechanics of Materials. 6th ed.; Wiley: New York, 2003. (2) An, Q.; Goddard, W. A.; Cheng, T. Atomistic Explanation of Shear-Induced Amorphous Band Formation in Boron Carbide. Phys. Rev. Lett. 2014, 113, 095501. (3) Pérez-Prado, M. T.; Zhilyaev, A. P. First Experimental Observation of Shear Induced hcp to bcc Transformation in Pure Zr. Phys. Rev. Lett. 2009, 102, 175504. (4) Fanchini, G.; McCauley, J. W.; Chhowalla, M. Behavior of Disordered Boron Carbide under Stress. Phys. Rev. Lett. 2006, 97, 035502. (5) Obrzut, J.; Douglas, J. F.; Kharchenko, S. B.; Migler, K. B. Shear-induced Conductor-insulator Transition in

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Table 1. Calculated Elastic Constants Cij, Bulk Modulus B, Shear Modulus G, Young’s Modulus E, and hardness H of t-BC4 (in units of GPa). Compound

Source

C11

C12

C13

C33

C44

C66

B

G

E

HŠimůnek

HLyakhov-Oganov

HChen

t-BC4

Present

759

22

137

877

66

146

329

156

403

58

61

13

730

40

147

858

62

140

329

155

402

[28]

Theory

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Table 2. Calculated Elastic Constants Cij of o-BC4. (in units of GPa). Compound

C11

C12

C13

C22

C23

C33

C44

C55

C66

o-BC4

831

87

198

663

122

812

96

424

224

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FIGURE CAPTIONS Fig. 1 (Color online) Calculated stress-strain relations for t-BC4 in various tensile directions (a), the dependence of the stress and total energy on the shear strain for t-BC4 along (001)[100] slip system (b). Fig. 2 (Color online) Crystal structures of o-BC4 (a) and t-BC4 (b), the blue and black spheres represent B and C atoms, respectively, the red ellipses indicate the structural changes of the BC4 following with structural transformation. Fig. 3 (Color online) Developments of ELF for t-BC4 during shear in the (001)[100] slip before (a) and after (b) structural transformation, the bonding of the C5 and C7 atoms are marked by the red arrows. Fig. 4 (Color online) Electronic band structures of t-BC4 (a) and o-BC4 (b), the dashed lines represent Fermi level. 3D Fermi-surface plots for t-BC4 (c) and o-BC4 (d). Fig. 5 (Color online) Calculated phonon dispersion curves of o-BC4 (a), DOS of o-BC4, the dashed lines represent Fermi level (b).

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Fig. 1

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Fig. 2

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Fig. 3

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Fig. 4

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Fig. 5

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Table of Contents (TOC) Image

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(Color online) Calculated stress-strain relations for t-BC4 in various tensile directions (a), the dependence of the stress and total energy on the shear strain for t-BC4 along (001)[100] slip system (b). 23x9mm (600 x 600 DPI)



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(Color online) Crystal structures of o-BC4 (a) and t-BC4 (b), the blue and black spheres represent B and C atoms, respectively. 87x62mm (600 x 600 DPI)


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(Color online) Developments of ELF for t-BC4 during shear in the (001)[100] slip at different strains. 33x27mm (600 x 600 DPI)



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(Color online) Electronic band structures of t-BC4 (a) and o-BC4 (b), the dashed lines represent Fermi level. 3D Fermi-surface plots for t-BC4 (c) and o-BC4 (d). 25x20mm (600 x 600 DPI)


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(Color online) Calculated phonon dispersion curves of o-BC4 (a), DOS of o-BC4, the dashed lines represent Fermi level (b). 20x7mm (600 x 600 DPI)



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Table of Contents (TOC) Image 41x28mm (600 x 600 DPI)


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Shear-Induced Structural Transformation for Tetragonal BC4 - The

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