Article pubs.acs.org/JPCC
Strain-Mediated Modification of Phagraphene Dirac Cones Alejandro Lopez-Bezanilla* Argonne National Laboratory, 9700 South Cass Avenue, Lemont, Illinois 60439, United States ABSTRACT: A first-principles study on the electronic and dynamical properties of phagraphene [Nano Lett. 2015, 15 (9), p 6182] is presented. This carbon allotrope exhibits a square unit cell, Dirac cones, and robustness against uniaxial deformation. By analyzing the contribution of each carbon atom orbital in the formation of the electronic states, we conclude that only the pz orbitals of 8 out of the 20 atoms in the square unit cell are responsible of the formation of the nanostructure Dirac cones. Spatial symmetry breaking of the underlying honeycomb-like network upon shear stress application leads to a band gap opening. The analysis of the phonon spectra demonstrates that the dynamical stability of phagraphene is guaranteed for small distortion angles. Phagraphene is identified here as the first all-C graphitic monolayer with Dirac cones modifiable by a small and realistic physical deformation. The analysis and conclusions of this study can be applied to other monolayered materials exhibiting Dirac cones in square lattices.
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INTRODUCTION
An interesting approach toward the alteration of the electronic properties of π-conjugated two-dimensional materials is to modify the overlap between neighboring orbitals via tensile or compressive strain. Monolayered materials can resist large strain values and support elastic deformations before breakdown, allowing strain engineering to tune their physical properties.3 In a two-dimensional material exhibiting a puckered structure such as phosphorene,4 compressive strain in the transverse direction leads to band gap reduction and band crossing. Uniaxial stress was observed to split the 2D mode of graphene and induce Dirac cones shifts,5 and only by applying stress with triangular symmetry a uniform quantizing pseudomagnetic field could be generated, which would yield small gaps observable at room temperature.6 In ref 2, tensile uniaxial strain of up to 5% applied normal to phagraphene edges was shown to drift the Dirac points and induce small changes in the slope of the bands forming the Dirac cones. This result is expected since, similarly to graphene, modifications in the bond lengths only leads to different hopping amplitudes between neighboring orbitals, which in turn modify the dispersion of the electronic bands. In the absence of spin−orbit coupling or a magnetic field to remove time-reversal symmetry, to lift the degeneracy of the Dirac points, and open a band gap spatial symmetry breaking is required. Expressed differently, while the topological structure remains unchanged the electronic band gap would remain closed. The flat geometry of the nanostructure prevents homogeneous normal stress from being introduced into phagraphene the required elements that could break the unit cell symmetry. Shearing stress is proposed here as a suitable method to break the internal symmetry of phagraphene and to
Graphene summarizes the outstanding mechanical and electronic properties that tomorrow’s electronic devices need for surpassing silicon’s performance while reducing energy consumption. However, one physical property of graphene represents a fundamental impediment for its massive integration as a building block in an electronic device, namely, the lack of an electronic band gap, which prevents the creation of on/off states. The conduction and the valence bands of graphene touch each other in six points of the Brillouin zone (BZ) forming the so-called Dirac cones. In the vicinity of the Dirac points, the energy is directly proportional to the electron momentum, and the electrons behave as relativistic charge carriers with a negligible mass. Dirac cones are typically associated with hexagonal lattices or structures exhibiting an equivalent topology. The opposite is rather peculiar and deserves further analysis.1 In a recent paper by Wang and co-workers,2 a one-atomthick nanostructure named phagraphene and composed of pentagons, hexagons, and heptagons was proposed as a carbon allotrope exhibiting Dirac cones in a rectangular lattice. The zero band gap of the nanostructure is described as robust against external stress. Indeed, uniaxial strain was observed to shift the Dirac point in the BZ without affecting the metallicity of the material and modify the effective mass of the charge carriers while preserving the Dirac points. In this paper we provide theoretical evidence that the Dirac cones of phagraphene are actually modifiable under external force application. Shearing stress is demonstrated to break the internal lattice symmetry, modify the strength of the chemical C−C bonds, and develop an electronic band gap. Moreover, an underlying honeycomb-like network of π-bonds yielding the Dirac cones is unveiled, demonstrating that they are actually created from a hexagonal distribution of C atoms pz-orbitals. © XXXX American Chemical Society
Received: June 3, 2016 Revised: July 6, 2016
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DOI: 10.1021/acs.jpcc.6b05593 J. Phys. Chem. C XXXX, XXX, XXX−XXX
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BZ, as shown in Figure 2. It is worth noting that there are actually two Dirac cones in the BZ, related by symmetry operations and of which only one is shown, being the second one in the Γ → −Y line. The fragility of the Dirac cones is manifested by a band gap opening at small shearing tension values in both x- and y-directions. For a fixed cell vector in the x-direction, ax, the x-component of the other in-plane cell vector, ay, is modified gradually to mimic shear stress, yielding a relative angle θy (see Figure 1b). A small reduction of θy from 90° to 89.9° opens a meV-large band gap, as shown in Figure 2. Further application of shearing stress reducing θy in ∼1° vanishes the Dirac cones and the nanostructure become semiconducting. Similarly, continuous modification of the ycomponent of the ax vector, while keeping ay fixed, induces a distortion of the C-network that is able to remove the Dirac cones from the band structure, as shown in the two leftmost panels of Figure 2a. The same results are obtained if the shear angle is reversed. The size of the band gap increases as the shear angle is modified well beyond the limit of the nanostructure dynamical stability. Figure 2b shows the phonon spectra of phagraphene ground state, and the distorted structures whose bands are displayed in the panels of Figure 2a. 60 modes are extended over a frequency range of ∼56 THz. The in-plane acoustic branches are characterized by linear dispersions at low q near the center of the BZ. The out-of-plane phonon branches exhibit nonlinear energy dispersions at both the zone edge and zone center. Note that whereas most acoustic modes exhibit large dispersive branches, optical modes in the upper half of the spectrum are rather flat, in a similar fashion to the lowest acoustic branch. In comparison to uniaxial strain,2 shearing deformation is more restrictive in the amount of stress that can be applied before the nanostructure becomes metastable and long wave perturbations destroy the structure. The dynamical stability of the slightly deformed structures is guaranteed by the absence of
exploit the relation between its mechanical and electronic properties, which lead to a band gap opening.
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RESULTS AND DISCUSSION Shear Stress. Whereas normal stress is the result of a force being applied perpendicularly to the structure edges, shear stress consists on a force applied tangentially or parallel to the structure edges. The former changes the volume of the material, while the latter results in a shape change. Figure 1a shows a
Figure 1. (a) Square unit cell of phagraphene defined by 20 atoms forming pentagons, hexagons, and heptagons. The six inequivalent C atoms are labeled from A to F. In (b) two types of shear angles, θx and θy, are distinguished according to the modification of the x or y component of the y- or x-unit cell vector, respectively.
phagraphene rectangular unit cell. The in-plane cell vectors forming an angle θ = 90° in the ground state can be progressively deformed to mimic an external force acting parallel to the x- or y-axis (Figure 1b shows the deformation angles). Such a deformation is able to break the internal symmetry of the unit cell and induce various types of electronic changes, ranging from the distortion of the Dirac cones to the removal of the degeneracy of bands at different points in the
Figure 2. (a) Electronic band diagrams of various sheared phagraphene structures. Depending on the cell vector that is modified (see Figure 1b), different shearing stress may be applied before the nanostructure becomes dynamically unstable, as the phonon spectra shown in (b) demonstrate with the absence of negative frequencies. Beyond these deformation limits, some instability may appear as pointed by the red circle. Electronic bands and phonon modes are plotted versus its propagation direction along the high-symmetry lines Γ → X → S → Γ → Y → X of the Brillouin zone, as depicted in the diagram. B
DOI: 10.1021/acs.jpcc.6b05593 J. Phys. Chem. C XXXX, XXX, XXX−XXX
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Origin of the Dirac Cones. To unveil the origin of the Dirac cone and the band gap opening, an analysis of the contribution of each atomic orbital to the formation of the πbands is performed. Color-weighted bands of the pz orbitals of the inequivalent C atoms (see Figure 1) are plotted in Figure 4. Solely these orbitals of the CA- and CD-type carbon atoms have a significant contribution to the formation of the electronic bands in the vicinity of the Fermi level that touch to each other forming the Dirac points. The rest of the atomic orbitals hybridize in sp2 orbitals with energies well below the Fermi level. Therefore, phagraphene is preeminently a π-conjugated material in which only 8 out of the 20 pz orbitals in the unit cell have a relevant contribution in the formation of the Dirac cones. The origin of the Dirac cones can thus be explained by means of a simplified model based on a lattice that mimics a distorted hexagonal network of C atoms similar to that of graphene. Figure 5 shows the effective network of CA and CD atoms yielding the honeycomb-like lattice superimposed to the actual lattice of phagraphene. Contrary to what could be expected a priori from the rectangular unit cell and the low space-inversion symmetry of the atom network, this lattice is topologically equivalent to the graphene honeycomb lattice, and as such, it yields Dirac cones in the BZ. Unlike the gapless Dirac spectrum in graphene, which is robust and topologically protected against small perturbations, this distorted honeycomb-like lattice is sensitive to deformations such as shear stress and allows for strainmediated band gap control. To additionally confirm the origin of the Dirac cones, we resort to maximally localized Wannier functions (MLWFs),8 obtained using the Wannier90 code9 from the first-principles ground state to derive an equivalent description of the system but with only eight orbitals in the unit cell. In particular, the four valence and four conduction bands in the vicinity of the Fermi level are reproduced by projecting the Bloch states onto atomic-like pz orbitals of the CA- and CD-type atoms. Minimizing the MLWF spread, the band structure obtained using the Wannier90 interpolation method compared to the DFT calculation is in excellent agreement. Figure 5 shows the Wannier interpolated band diagram. This band diagram corresponds to a simplified tight-binding model where the 8 × 8 Hamiltonian matrix is constructed by considering only the pz orbitals of the eight contributing C atoms within the unit cell, demonstrating that the honeycomblike lattice is able to reproduce the Dirac cones.
negative branches in the phonon spectrum. It is worth noting though that at the deformation value of θy = 88.3° some dynamical instability may arise, as demonstrated by a nonnegligeble negative component of the out-of-plane flexural phonon (pointed by a circle in Figure 2b). Indeed, in the region X →S the 2-fold degenerate branch splits for any applied deformation, and one of the branches acquire lower frequencies until it becomes negative. For larger deformations this branch becomes softer and destabilizes the nanostructure, since the associate force cannot reestablish the atoms to the original position. Note that this response of phagraphene structure to strain introduces a notable difference with respect to graphene, in which the Dirac cones of the honeycomb lattice are protected by symmetry and from any perturbation that does not violate parity and time reversal. Only under extreme tensile strain the Dirac point has been predicted to vanish in graphene,6,7 whereas a small distortion of the shearing angle is enough to open a band gap in phagraphene. Uniaxial Stress. Two-dimensional materials can withstand large uniaxial deformation stress values which, in the case of phagraphene, is of up to 4.5% in the ax vector direction.2 To analyze the difference of uniaxial strain with respect to shearing stress an evolution of the phonon spectrum of phagraphene when tensile strain is applied in such a direction is on display in Figure 3. For the sake of clarity, only a frequency range
Figure 3. Evolution of the phonon spectrum of phagraphene for uniaxial tensile strain in the ax vector. Soft modes representing dynamical instability appear at a tensile strain of 5% of the ax vector length.
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CONCLUSIONS Supported by density functional theory based calculations, theoretical evidence of the possibility to open an electronic band gap in a graphitic layer has been provided. The Dirac cones of phagraphene can be mechanically altered by the action of shearing stress applied along both unit cell vector directions. A phonon spectrum analysis allowed to determine the degree of deformation that can be imposed to the nanostructure before it becomes dynamically unstable. An examination of the individual contribution of each C atom orbital enabled us to unveil the origin of the Dirac cones. This distinctive formation typical of hexagonal lattices and exhibited by the phagraphene square lattice is explained by an underlying honeycomb-like network created by some particular orbitals of the C network. This analysis and conclusions can be easily translatable to other monolayered structures exhibiting Dirac cones in rectangular
between 0 and 10 THz is plotted. First remarkable consequence of the 2% strained structure is the lift of the degeneracy for phonon branches in the X → S region of the BZ. One branch is practically unaltered, while the other shifts to higher energies and meets another branch at ∼4 THz at the Γ point, both becoming linear in the vicinity of q = 0. Several gaps are opened throughout the spectrum although the largely dispersive branches remain linear up to the tensile value of 4.5%. A parabolic branch with a minimum at ∼6 THz in the Y → X BZ region softens as tensile strain is increased, loosing the quadratic shape at the strain value of 4.5%. This represents the onset of the instability, as demonstrated by the soft modes arising at a strain of 5% at the X point. C
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Figure 4. Color-weighted band diagrams showing the independent contribution of the pz orbitals of the inequivalent atoms of phagraphene, as labeled in Figure 1a. Only CA- and CD-type carbon atoms have a significant contribution in the formation of the Dirac cones.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Phone: +1 630 252 0748. Notes
The author declares no competing financial interest.
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ACKNOWLEDGMENTS I am thankful to Ivar Martin for fruitful discussions. I acknowledge DOE BES Glue funding through Grant No. FWP#70081 and the computing resources provided on Blues high-performance computing cluster operated by the Laboratory Computing Resource Center at Argonne National Laboratory. Work at Argonne is supported by DOE-BES under Contract No. DE-AC02-06CH11357.
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Figure 5. On the left, effective honeycomb-like lattice formed by the CA- and CD-type atoms (red) yielding the Dirac cones of phagraphene (gray). On the right, Wannier interpolated electronic band diagram constructed by projecting onto the pz-orbitals of the CA- and CD-type atoms the Bloch states of the DFT calculation.
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unit cells.1 The lack of mirror symmetry upon modification of the angle formed by the unit cell vectors is responsible of distortion of the Dirac cones. Due to the tunability of its band gap, this graphitic structure and other such as graphynes may suppose an alternative to graphene as an all-C building block of electronic devices.
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COMPUTATIONAL METHODOLOGY First-principles calculations were performed within the generalized gradient approximation to the density functional theory (DFT) and the projector-augmented-wave method as implemented in VASP.10−12 The electronic wave functions were computed with plane waves up to a kinetic-energy cutoff of 500 eV. The integration in the k-space was performed using a 20 × 20 × 1 Monkhorst−Pack k-point mesh centered at Γpoint. Lattice constants and atomic coordinates were fully relaxed until the residual forces were smaller than 10−3 eV/Å. The force-constant method was used in the phonon calculations, and the dynamical matrices were computed using the finite differences method in large supercells. The phonon spectra was obtained from the PHONOPY package.13 D
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