Directional Control of the Electronic and Transport Properties of

Article pubs.acs.org/JPCC

Directional Control of the Electronic and Transport Properties of Graphynes J. E. Padilha,*,† A. Fazzio,*,† and Antônio J. R. da Silva*,†,‡ †

Instituto de Física, Universidade de São Paulo, CP 66318, São Paulo, SP 05315-970, Brazil Laboratório Nacional de Luz Síncrotron, CP 6192, Campinas, SP 13083-970, Brazil

‡

S Supporting Information *

ABSTRACT: The Dirac energy dispersion, presented by graphene and other 2D materials, provides exquisite properties that are very useful for the design of new devices. A new class of 2D materials consisting only of carbon atoms and that presents Dirac cones in its electronic structure are the graphynes. Through electronic transport results based on first-principles calculation combined with nonequilibrium Green’s functions, we show that the conductivity of the α, β, and 6,6,12-graphyne is higher than that for graphene. For 6,6,12-graphyne we predict that via an anisotropic strain both the electronic energy gap and the current can be modulated, presenting a strong directional dependence of its properties.

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INTRODUCTION

istic might lead to a directional dependent electronic property, which could potentially be used for nanoelectronic devices. In this work we show for the 6,6,12-graphyne that (i) it presents a strong directional dependence of the current; (ii) both the gap and the current can be controlled by strain; and (iii) it has a conductivity higher than graphene. Moreover, we also show that the conductivity of the α and β graphyne is always higher than graphene. These results open up new routes to construct devices, where not only their intrinsic properties present a directional dependence but also the transport can be tuned by an external directional strain.

A topic that for more than six decades has raised a great interest in the condensed matter community is the physics of linear dispersion bands around the Fermi level or, as they are called today, Dirac cones. Even though they were first calculated in a tight binding model by Wallace1 in 1947 when studying the properties of graphite and used by McClure2 to investigate its diamagnetic properties in 1956, the notion of massless Dirac fermions was first proposed independently by Semenoff3 and DiVincenzo and Mele4 in 1984, which became an experimental reality only in 2004, when Geim and Novoselov5−7 isolated a graphene layer. After this experimental realization, a frantic search for other 2D materials intensified.8−10 Besides the fact that these 2D systems present new physical phenomena, they exhibit electronic properties with great potential for applications in electronic devices,11 which are closely related to the presence of Dirac cones in their electronic structure.12−22 Thus, finding ways to manipulate this Dirac cone is a window to manipulate the properties of these materials. Many new materials have been proposed seeking similar properties of graphene. In particular, a class of materials consisting only of carbon atoms and not restricted to a hexagonal lattice are the graphynes and graphdyines, which are made of a mixing of sp and sp2 hybridization.12−22 A great effort has been devoted to synthesize these structures, and polymeric building blocks have already been obtained,23−28 suggesting that soon it will be possible to obtain layers of these 2D materials. The presence of Dirac cones in these materials was clearly pointed out by Malko et al.21 when studying the α, β, and 6,6,12-graphyne. Due to the pmm symmetry of the 6,6,12graphyne, the structure of this material has a rectangular symmetry and presents two different Dirac cones in the Brillouin zone at two nonequivalent positions. This character© XXXX American Chemical Society

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COMPUTATIONAL DETAILS Geometry Optimizations and Electronic Structure Calculations. Geometry optimizations and electronic structure calculations were obtained through ab initio total energy calculations based on density functional theory as implemented in the SIESTA code.29−31 For the exchange correlation functional we consider the GGA-PBE approximation.32 Norm-conserved Troullier−Martins pseudopotentials33 were used to describe the interaction between the valence and core electrons. We used a double-ζ basis plus a polarization function (DZP) to describe the Kohn−Sham orbitals and 300 Ry of mesh cutoff to describe the density on the grid. The structures were considered relaxed when the residual forces on the atoms were smaller than 0.01 eV/Å. The STM images were generated using the Tersoff−Hamann procedure34 with a voltage between the tip and the sample of 0.5 V for the unoccupied levels, represented by the shaded regions in the band structures (middle panel) in Figure 1. Received: June 24, 2014 Revised: July 23, 2014

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dx.doi.org/10.1021/jp5062804 | J. Phys. Chem. C XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry C

Article

Figure 1. Schematic representation (left panel), band structure, and Brillouin zone (mid panel) and STM images (right panel) for (a) 6,6,12graphyne, (b) α-graphyne, and (c) β-graphyne. In each structure we show the unit cell defined by the lattice vectors a1 and a2 and the scale in the STM images is given in Å.

where f(E − μL[R]) are the Fermi−Dirac distributions for the left (L)[right (R)] leads. To take into account the 2D structure of the graphyne layer we consider a set of 1000 k-points transversal to the transport direction, using the Monkhorst−Pack scheme.39 All transport calculations were made self-consistently.

Electronic Transport Calculations. The electronic transport calculations were based on the Landauer−Bütiker formalism,35 where the Hamiltonian of the system is obtained via the nonequilibrium Green’s function method combined with DFT (DFT-NEGF), as implemented on the TRANSAMPA code.36−38 As we are dealing with 2D materials, the transport properties have to be calculated with the inclusion of k-points transversal to the transport direction (k⊥). For each k⊥ in reciprocal space we obtain a k⊥-dependent density matrix, Dμν(k⊥), by the integration of the lesser Green’s function,37 so that the total density is given by the integration of Dμν(k⊥) in all reciprocal space Dμν =

1 (2π )2

∫ dk⊥D(k⊥)μν

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RESULTS AND DISCUSSION The 2D form of carbon, named graphynes, can be viewed as derivatives of graphene, where the hexagons of the graphene lattice are connected through acetylenic linkages.12,18 Inside the enormous family of materials that can be formed with those links, we choose for our study the 6,6,12-, the α-, and the βgraphyne. They present exquisite properties that are very similar, or even superior, to graphene.22,40,41 In Figure 1(a), (b), and (c) we show the atomic structure (left panel), the energy dispersion with its respective Brillouin zone (mid panel), and the STM images of the 6,6,12-, α-, and β-graphyne. Through the STM images we clearly see that all carbon atoms remain on the plane, a property presented by almost all of the sp−sp2 carbon structures. The 6,6,12-graphyne presents a pmm symmetry with a rectangular lattice defined by the unit vectors a1 and a2, as shown in Figure 1(a) (left panel). Due to this reduced symmetry it features also a rectangular Brillouin zone. This material presents two Dirac cones located at different points in the Brillouin zone, named I and II. The orbitals that come from the acetylene group located on the direction a2 do not contribute to the Dirac cone I (Figure 1(c)), located between Γ and X′ on the Brillouin zone. However, the orbitals that contribute to the Dirac cone II, located between M and X of the

(1)

Once convergence is achieved, we can obtain a k⊥ transmission function, T(E,k⊥), which is integrated over the whole Brillouin zone to obtain the total transmission function, T(E) T (E , k⊥) = Tr[Γ kL⊥(E)Ga(E , k⊥)Γ kR⊥(E)Gr(E , k⊥)] T (E ) =

1 (2π )2

∫ dk⊥2T(E , k⊥)

(2)

where Ga[r](E,k⊥) is the advanced [retarded] Green’s function, and Γk⊥ L[R](E) are the coupling matrices between the left [right] leads with the scattering region. With the total transmission function, T(E), one can obtain the current I, by I=

2e h

∞

∫−∞ T(E)[f (E − μL ) − f (E − μR )]dE

(3) B



Article

Figure 2. Schematic representation (left panel) of the unit cell, defined by the vectors X1 and X2, used on the transport calculations and transmission function for (a) 6,6,12-graphyne, (b) α-graphyne, and (c) β-graphyne. In (d) we show the Ids × Vds for the α, β, and 6,6,12-graphynes. For the α and β we show only the results along the X1 direction, because in X2 the results are the same. We also show the transport properties for a graphene layer for the sake of comparison. All calculations were made in a self-consistent fashion.

Figure 3. Band structure for a stretch (top panel) and compress (bottom panel) along the (a) X1 and (b) X2 direction. (c) Evolution of the Fermi velocity for the Dirac cones I and II as a function of strain. (d) Evolution of the band gap of the second Dirac cone and distance d in the inset as a function of strain along the X2 direction.

Brillouin zone, are delocalized in the whole system21(see Figure S1, Supporting Information). This fact implies that 6,6,12graphyne should present a different transport behavior, depending on the direction of the applied bias voltage. Consequently, the electronic transport could be tuned by a directional-dependent perturbation, like strain. The α-graphyne presents an electronic structure very similar to graphene, showing a Dirac cone at the K and K′ points of the Brillouin zone, as shown in Figure 1(b). The β-graphyne also presents a Dirac cone in its electronic structure, but the Dirac point is located in a line between Γ and M.21

To assess the functionality and properties of a material, it is very useful to understand its intrinsic electronic transport properties. This information can also be helpful in the design of new devices. To this end we present the results for the electronic transport properties of the graphynes considered in this work in Figure 2. The transport setup is constructed using the unit cell defined by the dotted box in each figure, and as we are dealing with a 2D material, this system could present two distinct transport directions. In our simulations the transport direction considered is X1(X2) with periodicity in X2(X1). The transport calculations were obtained through a combination of C



Article

Figure 4. Current voltage characteristic for: (a) transport on the X1 direction with strain on X1; (b) transport on the X1 direction with strain on X2; (c) transport on the X2 direction with strain on X1; (d) transport on the X2 direction with strain on X2. (e) Schematic representation of the device used, showing the X1 and X2 directions. The strength of the strain in all results shown are 1%.

always increases independently of the sign of the applied strain along the X1 direction. Stretching along the X2 direction, there is an opening of a bang gap around the X point of the Brillouin zone (Dirac cone II) (Figure 3(b) (top panel)) that can be continuously tuned (Figure 3(d) (circle blue curve)). This gap opening is a result of a decoupling of the acetylenic group evidenced by the linear increase of the distance d between the acetylenic group and the hexagon (shown in Figure 3(d) (square green curve)). This is consistent with the effect that when we eliminate the acetylene group and saturate with hydrogen atoms the Dirac cone II disappears with an opening of a band gap (see Figure S3, Supporting Information). 19 In addition, in this strain configuration, the Dirac cone I remains unaltered and the Fermi velocity almost constant, around 5 × 105 m/s. If we compress the system along X2, as shown in Figure 3(b) (bottom panel), there is no band gap opening as expected since there is no decoupling of the acetylene group. The linear dispersion relation of the Dirac cone II increases, and its Fermi velocity goes from 1 × 105 to 4 × 105 m/s. We carried out calculations for a biaxial and uniform strain, and the results are equivalent to the uniaxial stress (see Figure S4, Supporting Information). For the α and β structures the electronic band structures remain unaltered for strains up to 6% (see Figure S5, Supporting Information). We now investigate how these changes in the electronic structure under different strain conditions will manifest themselves in the transport properties of the 6,6,12-graphyne. We perform electronic transport calculations for strains going from −6% to 6% in both directions X1 and X2 and calculate the current also along X1 and X2. Depicted in Figure 4(a) is the current along the X1 direction applying the stress on the same direction. We see that the current is almost unaltered when the system is stretched, whereas the current increases by approximately 37% when the system is compressed. With the current still along X1 but stressing along X2, a different behavior is observed, as shown in Figure 4(b). The current decreases when the system is stretched due to opening of the band gap on the Dirac cone II and consequent decrease in the number of

density functional theory with the nonequilibrium Green’s function method (NEGF-DFT). For the 6,6,12-graphyne, the X1 and X2 directions are not equivalent, so the transport properties of such a system now depend on the chosen direction, as we can see on the transmission function in Figure 2(a) (right panel) (blue solid line for X1 and orange dashed line for X2). Looking to the current for the 6,6,12 system in Figure 2(d), we see that for the X1(X2) direction the current is almost 3(4) times bigger, when compared to the other structures, α, β, and graphene. For the α and β structures only the transport along the X1 direction with periodicity on X2 is presented because both X1 and X2 give the same result. All of the graphyne structures present electronic transmission functions around the Fermi level characteristic of band structure with Dirac cones, such as graphene. However, the transmittance is always bigger than graphene (black dashed lines). Consequently, the electronic current, Ids, flowing on the systems will always be higher than graphene, as we can see in Figure 2(d). All of these results are consistent with the conclusions obtained by Chen et al.,41 where the authors predict that the carrier mobility of graphyne will be larger than graphene. Since the 6,6,12-graphyne presents this directional feature, we could use a directional external perturbation to tune its band structure. To this end, we apply an anisotropic strain in the 6,6,12 system. All strain configurations were made in the elastic regime (see Figure S2, Supporting Information). Also it is very important to note that we used strain intensities that do not compromise the stability of the structures, as pointed out by Zhang et al.42 In Figure 3(a) and (b) we show the band structure for a 6,6,12-graphyne stretched (top panel) and compressed (bottom) along the X1 and X2 direction, respectively. Around the Dirac point I, the dispersion relation decreases when we stretch along the X1 direction (top panel in Figure 3(a)). The opposite behavior is observed when we compress along the X1 direction (bottom panel in Figure 3(a)). At the Dirac cone I, the Fermi velocity goes from ≈5.2 × 105 to ≈2.3 × 105 m/s for a strain going from −6% to 6%, as shown in Figure 3(c). For the second Dirac cone the Fermi velocity D



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quantum channels in this direction. However, this decrease is not so expressive due to the main contribution coming from the Dirac cone I that remains unaltered. Compressing the system, there is an increase of the density of state close to the Fermi level, resulting in a slight increase of the current. We now analyze the current along the X2 direction, as presented in Figure 4(c). The current increases (decreases) when the system is stretched (compressed) along the X1 direction due to an increase (decrease) in the density of states close to the Fermi level. This increase (decrease) is a result of changes in the Dirac cone I (II). When compressing along the X2, there is again a small increase in the current as shown in Figure 4(c), due to an increase in the density of states. Finally, there is a significant decrease in the current when the system is stretched along X2. Since when the current is along X2 the main contribution comes from the Dirac cone II, and the stretch causes a band gap opening at this point, the current drop is now ∼45% for a strain of 1%. This current−strain behavior could be very useful in the design of electronic devices that could be controlled by strain.

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CONCLUSION In conclusion, we have shown that α, β, and 6,6,12-graphyne might have superior transport properties to graphene. In particular, the presence of two Dirac cones in the 6,6,12 structure leads to a strong directional transport property, which can be manipulated by the application of an external strain. It is important to note that we can modulate the conductivity preserving their high mobility, a very important feature for electronic devices.

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ASSOCIATED CONTENT

S Supporting Information *

(i) Band structure and squared wave function for the 6,6,12graphyne, (ii) mechanical properties of the 6,6,12-graphyne under strain conditions, (iii) hydrogenated 6,6,12-graphyne, (iv) electronic properties of 6,6,12-graphyne under biaxial strain conditions, and (v) electronic properties of strained α and β graphynes. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] *E-mail: [email protected] *E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We would like to thank the funding agencies of Brazil, CNPq/ INCT, CAPES, and FAPESP. REFERENCES

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Directional Control of the Electronic and Transport Properties of

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