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In this article, we investigate the predictions of the first principles on structural stability, electronic and mechanical properties of 2D nanostructures: graphene, silicene, germanene and stenane. The electronic band structure and density of states in all these 2D materials are found to be generic in nature. A small band gap is generated in all the reported materials other than graphene. The linearity at the Dirac cone changes to quadratic, from graphene to stenane and a perfect semimetalicity is exhibited only by graphene. All other 2D structures tend to become semiconductors with an infinitesimal band gap. Bonding characteristics are revealed by density of states histogram, charge density contour, and Mulliken population analysis. Among all 2D materials graphene exhibits exotic mechanical properties. Analysis by born stability criteria and the calculation of formation enthalpies envisages the structural stability of all the structures in the 2D form. The calculated second order elastic stiffness tensor is used to determine the moduli of elasticity in turn to explore the mechanical properties of all these structures for the prolific use in engineering science. Graphene is found to be the strongest material but brittle in nature. Germanene and stenane exhibit ductile nature and hence could be easily incorporated with the existing technology in the semiconductor industry on substrates.

Hard materials are always of scientific interest due to their significance in fundamental science and industrial applications. It is also a well known fact that the covalently bonded light elements like carbon, boron, silicon etc. are very hard due to its nature of bonding. The number of publications exposes the enormous amount of interest shown by the research community on the noteworthy properties of 2D structures. The need for miniaturization of electronic devices also calls for extensive research among the 2D nanostructures that has obtained maximum hold in materials science and condensed matter physics.

Graphene has been documented as one of the thinnest, strongest, and stiffest materials ever discovered with a wide range of applications. The advances in the study have served as the motivating factor towards the extension of study in other 2D materials like silicene, germanane and stenane. Among these 2D nanostructures, graphene is the perfect semi metal which exhibits zero band gap due to the presence of massless Dirac Fermions. The orbital overlap of p_{z} orbitals is maximum in graphene and so the hopping strength.

The other 2D materials are found to be stable only with buckling; silicene is next to graphene among the group IV elements and has an additional advantage of integrating it into Si based electronics and hence in nanoelectronics [

The 2D structures of single layer graphene, silicene, germane and stenane as shown in ^{−7} eV and the forces per atom were reduced to 5 × 10^{−5} eV. The calculations were carried out in the irreducible Brillouin zone with 44 × 44 × 1 k point mesh Monkhorst-pack scheme [

The pristine form of the 2D structures under study other than graphene is not experimentally manufactured successfully. It was predicted that the monolayer hexagonal structure of tin cannot be stable in freestanding form, whereas its binary compounds SnC, SnSi, and SnGe are stable even in free-standing form. There are a few reports on the synthesis of single crystals of silicon nanocrystals by thermal crystallization [

The unit cell of these 2D structures consists of two atoms. The interlayer distance is made reasonably large to ensure that there is no interaction between the layers. The schematic view of all 2D materials is shown in

to the buckling distortions in the individual six membered rings. Hence they can be easily functionalized through chemical reactions [^{2} hybridization between one s orbital and two p orbitals leads to the formation of σ bond between carbon atoms that are separated by 1.42 Å. The planar σ bond is responsible for the robustness of the lattice structure. The unhibridized p_{z} orbital which is perpendicular to the planar structure, can bind covalently with neighbouring carbon atoms, leading to the formation of a π bond. The π bond is not involved in covalent planar σ bond in the case of graphene [

The short range puckering seems to have pronounced effect in electronic properties. We find that it produces enormous effect in mechanical properties as well. The calculated total energies _{6h} and changes to C_{3V} in all other 2D materials. This is because the six-fold rotational and two mirror plane symmetries are missing in all materials other than graphene [

Bond order gives the stability of a bond and graphene is reported to have the highest bond order and hence proves its stability in the 2D form [

Mulliken population analysis is performed to analyse the bonding behaviour. The overlap population may be used to assess the covalent or ionic nature of a bond [^{3} hybridization. The bond angle in stenane is 110˚ which is very much closer to the bond angle of sp^{3} hybridized orbitals, 109.5˚. This results in the weakening of π bond due to the 5 p_{z} orbital. This leads to the localization of electrons and hence will lose its Dirac like massless nature. This is reflected in energy dispersion curve, density of states histogram, and charge density contours.

The formation enthalpy refers to the thermodynamic stability. Negative formation enthalpy means an exothermic process and lower formation energy indicates the stability with respect to the decomposition to elemental constituents [

The band structure is calculated along the high symmetry directions Г, M, K, Г in the first Brillouin zone. The electronic band structure of all the four is generic in nature and agrees reasonably well with that reported in the literature. The sp^{2} hybridization and the presence of p_{z} orbitals perpendicular to the plane in graphene give rise to the Dirac cone at the Fermi energy. The electrons in p_{z} orbitals behave like massless fermions. Whereas in the

Lattice Parameter | Total Energy (eV) | Bond Angle (˚) | Buckling Distance (Å) | Bond Length (Å) | Bond Order | |||||
---|---|---|---|---|---|---|---|---|---|---|

2D | Pr. | Ref. | Pr. | Ref. | Pr. | Ref. | Pr. | Ref. | ||

G | 2.46 | 2.46 [ | −310.28844 | 120 | 120 [ | 1.42 | 1.42 | 3.05 | ||

Si | 3.88 | 3.87 [ | −213.55118 | 116.90 | 115.4 [ | 0.41 | 0.44 [ | 2.27 | 2.30 [ | 2.75 |

Ge | 4.03 | 4.03 [ | −213.79328 | 112.56 | 0.68 | 0.64 [ | 2.42 | 2.44 [^{ } 2.48 [ | 2.03 | |

Sn | 4.66 | 4.62 [ | −190.06948 | 110.39 | 110 [ | 0.9 | 0.92 [ | 2.83 | 2.82 [ | 1.99 |

Material | Atomic Populations (Mulliken) | Bond Population | Formation Enthalpy (eV) | Electronegativity | |
---|---|---|---|---|---|

S | P | ||||

Graphene | 1.05 | 2.95 | 3.05 | −3.10E+02 | 2.55 |

Silicene | 1.4 | 2.6 | 2.75 | −2.14E+02 | 1.9 |

Germanene | 1.57 | 2.43 | 2.03 | −2.14E+02 | 2.01 |

Stenane | 1.65 | 2.35 | 1.99 | −1.90E+02 | 1.96 |

case of other 2D materials like silicene, germanene, and stenane the hybridization tend to change from sp^{2} to sp^{3} and results in the formation of sp^{2}/sp^{3} mixed orbital due to buckling. Buckling reduces the overlap area of p_{z} orbitals in turn increases the overlapping of p_{z} orbitals with the planar σ orbitals. Eventually the free movement of electrons as in the case of graphene is reduced. The linearity observed in graphene slowly changes to quadratic with an infinitesimal gap along the K direction at the Fermi level in the electronic dispersion curve. Dirac-like electronic structures are also seen in silicene, germanene and stenane that has been predicted already [_{x}, and 2 p_{y} orbitals in valence bands are highly localized and do not contribute towards the electronic properties. The conduction is mainly due to the electrons in p_{z} orbitals. Dirac point in the energy dispersion curve endorses the behaviour of electrons as massless Fermions and it explicitly shows zero band gap in the case of Graphene. In other 2D materials the mobility of the electrons is reduced and hence effective mass is increased. Due to which a small bang gap is generated in turn the linear behaviour of energy dispersion at the K point changes to quadratic.

The energy of the hybridized s orbital along Г direction is very low in the case of graphene and is around 20 eV. The energy is increased to −11 eV in silicene and germanene and to −9 eV in stenane. The energy of the p_{z} orbital in graphene is around −8 eV whereas in silicene and germanene it lies between −3 eV and −4 eV, which is close to zero in stenane. The energy gap shows a decreasing trend along the Γ direction: 7 eV in graphene, 3 eV in silicene, 1 eV in germanene and 0.6 eV in stenane. This proves that the energy of the orbitals decreases as the ionic radius increases. These values are in perfect agreement with the reported results.

The density of states histogram (_{z} orbital which leads to the zero gap in energy dispersion curves, whereas the hybridization between s and p at different degrees is found to occur in all other 2D materials near the Fermi level. The σ orbitals of buckled atom (Sublattice 2) overlap with the p_{z} orbitals of planar atoms (sublattice 1). Hence s states are found near the Fermi level. This reduces the free nature of electrons in the p_{z} orbital as in the case of graphene and hence linearity at the Dirac point becomes quadratic with a small band gap. The peaks due to the hybridized states in the case of silicene, germanene and stanene are found to be stronger than the peaks due to unhybridized states of p_{z} orbitals in graphene near the Fermi level. Hence we can infer that the electrons in silicene, germanene and stenane will be more localized than the electrons in graphene near the Fermi level. Therefore the strength of C-C bond of graphene is lesser than the Si-Si, Ge-Ge, and Sn-Sn bonds. The DOS curves of all four are generic in nature except stenane. It is noted that the Fermi level has p states in stenane. This may be because of the influence of electrons in d orbitals which is beyond the scope of this work as this is done using pseudo code where only valence electrons are considered for calculation. The absence of band gap in the neighbourhood of the Fermi Level indicates that all the four have semi metallic character. The valence band starting from zero up to ~4 eV is mainly due to p_{z} electrons. The main contribution from s electrons to the valence band is only from above ~4 eV. The states at the Fermi level are from p_{z} electrons in all four materials. Hence we can conclude that the metallic behaviour of the materials is due to the p_{z} electrons. The _{ }states dominate the bottom of the conduction bands ranging from zero to 10 eV in graphene, 6 eV in silicene, 4 eV in germanene and 4 eV in stenane. Besides the distribution from ^{*} states. The π^{*} and σ^{*} bands are split into several peaks: 8 π^{*} peaks and 6 σ^{*} peaks in graphene, 10 π^{*} peaks and 9 σ^{*} peaks in silicene, 12 π^{*} peaks and 10 σ^{*} peaks in germanene, 4 π^{*} peaks and σ^{*} peaks in silicene. In principle the energy difference between the peaks

corresponds to the separation between the peaks in the K edges [_{z} orbitals at Fermi level in stanene shows reduced stability.

_{z} orbital which are responsible for conduction in other 2D materials is low due to the overlap of sp^{2} hybridized orbital with the p_{z} orbital. The puckering effect is also seen clearly in the electron density contour of silicene where the atoms in the raised sublattice are seen prominently than the atoms in the other sublattice. It is more visible in stenane that the puckered atoms are not seen in the charge density contour.

The elastic constants indicate the response of materials to an externally applied stress and offer important information regarding the bonding characteristics, anisotropy and hardness etc. [

are calculated from the applied strain and the computed stress. The calculation involves second order derivatives of the total energy with respect to lattice distortion. Young’s modulus, Poisson’s ratio, Bulk modulus and Shear modulus are derived from the elastic stiffness tensor and is mentioned in the following sections. The Young’s modulus, bulk modulus and shear modulus of graphene are very high compared to silicene, Germanene and stenane.

Elastic constants are critical parameters for analysing the mechanical properties of any material. The second order elastic constants exhibit the linear elastic response. The higher (>2) order elastic constants are important to characterize the nonlinear elastic response. The unit cell construction is done with 3D model by keeping the interlayer distance sufficiently large along the z-direction to minimize the van der Waals interaction.

The five independent elastic constants for a hexagonal structure C_{ij} are C_{11}, C_{12}, C_{13}, C_{33}, C_{44} and C_{66}. These are based on symmetry. C_{11}, C_{22}, & C_{33} are uniaxial elastic constants. C_{44}, C_{55}, and C_{66} are shear elastic constants. The born stability criteria [_{33} along the z axis is found to be very small which shows that the interactions along the z axis is minimized and it behaves as a perfect 2D structure.

Hence for 2D materials, the stress-strain relation can be expressed in matrix form [

Besides the formation enthalpies, the elastic constants also endorse that they are mechanically stable.

The relationship between 3D elastic constants and 2D tension coefficients [

between two layers. The in-plane Young’s modulus Y_{s} and Poison’s ratio [

relationships.

The calculation of elastic constants shows that the Young’s modulus of graphene is 337 N/m along x-direction and y direction and agrees well with the already reported value of 340 N/m [

Bulk modulus and shear modulus can also be obtained from the calculated elastic constants. Bulk modulus describes the resistance of a material to volume change and Shear modulus determines the resistance due to shape change. Three approximations are mainly used to obtain these values: Voigt, Reuss, and Hill. Voigt’s approach gives the upper bound of elastic properties in terms of uniform strain, the Reuss’s approach gives the lower bound in terms of uniform stress and Hill’s approach [

Voigt Approximation [

Young’s Modulus (N/m) | Bulk Modulus | Shear Modulus | ||||
---|---|---|---|---|---|---|

Present | Reported | Present | Reported | Present | Reported | |

Graphene | 337.1418 | 340 [ | 91.8603 | 23.58246 | ||

Silicene | 61.33517 | 63 [ | 20.16984 | 13.5 [ | 4.1069 | |

Germanene | 42.05487 | 13.36239 | 2.55942 | |||

Stenane | 24.46117 | 8.99832 | 1.62622 |

where

The in plane stiffness of all the four is reported to decrease from graphene to stenane. The in-plane stiffness of Si, Ge, and Sn is 20%, 14% and 9% of that of graphene and can be explained with the help of bonding characteristics. In the periodic table Si, Ge, and Sn are found below C in the IV group and the bond length of all increases correspondingly from carbon. The atomic radius increases as we move down from carbon to tin and hence the bond length increases. Bond length increases from 1.42 (graphene) to 2.83 Å (stenane) in the present calculations and agree well with the reported values. This in turn reduces the bond strength. Bond length is inversely proportional to bond strength and the bond dissociation energy; a stronger bond will be shorter. Also all these structures form puckers in order to stabilize the honeycomb structure. This tends to the formation of sp^{3} hybridization results in a mixture of sp^{2}/sp^{3} hybrid orbitals and hence weakens the π bond as against graphene. Thus the in-plane stiffness is lesser in all the structures other than graphene. Furthermore the electronegativity of carbon atoms is higher than all the other providing high mechanical strength to Graphene.

The ratio of shear to bulk modulus (

Poisson’s ratio υ is associated with the volume change during the uniaxial deformation, which usually ranges from −1 to 0.5. The bigger the value of Poisson’s ratio, the better is the plasticity of the material. Typically, the value of υ is small for pure covalent materials (υ = 0.1), which is the case for graphene. The values are shown in

The hardness of materials is calculated using the expression

The elastic anisotropy of crystals is an important parameter for engineering science since it correlates the possibility of microcracks appearance in materials [

G/B | Poisson Ratio | Hardness in GPa | Hardness in N/m | ||
---|---|---|---|---|---|

Present | Reported | ||||

Graphene | 0.770164 | 0.18361 | 0.186 [ | 6.796 | 20.388 |

Silicene | 0.610848 | 0.318878 | 0.3 [ | 0.3872 | 1.1616 |

Germanene | 0.574617 | 0.332063 | 0.1657 | 0.4971 | |

Stenane | 0.542175 | 0.390127 | 0.02124 | 0.06372 |

A1 = C_{44}/(C_{11} + C_{33} − 2C_{13}) | A2 = 4C_{55}/(C_{22} + C_{33} − 2C_{23}) | A3 = 4C_{66}/(C_{11} + C_{22} − 2C_{12}) | |
---|---|---|---|

Graphene | −0.00087 | −0.00087 | 1 |

Silicene | 0.0024 | 0.002398 | 0.999999 |

Germanene | −0.1669 | −0.1669 | 1.000001 |

Stenane | −0.01893 | −0.01893 | 1.000002 |

isotropic crystal, while any value smaller or greater than one provides information on the degree of shear anisotropy possessed by the crystal [

Finally the strain energy per atom in-plane and out of plane is calculated considering compressions and tensions within the range −0.1 to +0.1. The strain energy per atom is calculated using the formula,

In summary, we have studied the structural stability and the electronic and mechanical properties of 2D nanostructures using DFT based First Principles Calculations. The structural analysis reveals that the sp^{2} hybridization occurs only in graphene. In all other materials hybridization tends to change from sp^{2} to sp^{3} due to buckling. The bond angle in stenane shows that the hybridization has almost become sp^{3}. This buckling effect results in the weakening of π bond leading to a greater overlap between σ and π orbitals which in turn increases the bond energy. The buckling in the structure provides stability to form the hexagonal honeycomb structure in silicene, germanene and stenane. The negative values of formation enthalpy endorse the stable structural formation of all 2D structures under ambient conditions. However, the fabrication of free standing form of stenane still remains as a challenge to experimentalist. The electronic band structure calculations exhibits semimetallic behaviour of having zero band gap only for graphene. The other 2D structures under study posses an infinitesimal band gap at the K point due to the buckling in the structure.

The mechanical properties of all four 2D structures have been studied extensively. The strain energy per atom is isotropic along strain direction (along x). It is also symmetrical during tensile and compressive strains. This is not the same when strain is applied along other directions. This is due to the non-uniform stretching of bonds between the planar and raised sublattices which confirms the puckering in the structures. Graphene exhibits

extreme mechanical properties compared to other 2D structures due to the pure covalent bonding between atoms. The in-plane stiffness is found to be very high for graphene whereas silicene, germanene and stenane have 20%, 14% and 9% graphene. Graphene and silicene are found to be brittle in nature, while germanene and stenane are ductile. Hence the latter two can be used extensively in semiconductor industry where substrates can add on to enhance the conduction properties. Crystal isotropy is observed in the basal plane of all four structures. This work concludes that not only graphene, other 2D structures: silicene, germanene, and stenane can also be used extensively in semiconductor and optical industries due to their similar structure and appreciable electronic and mechanical properties.

The computational facility offered by UGC-DAE Consortium of Indhira Gandhi Centre for Atomic Research is deeply acknowledged.

Rita John,Benita Merlin, (2016) Theoretical Investigation of Structural, Electronic, and Mechanical Properties of Two Dimensional C, Si, Ge, Sn. Crystal Structure Theory and Applications,05,43-55. doi: 10.4236/csta.2016.5304