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First principle study of inducing superconductivity in α-graphyne by hole-doping and biaxial tensile strain
Computational Materials Science 124 (2016) 183–189
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First principle study of inducing superconductivity in a-graphyne by hole-doping and biaxial tensile strain Toktam Morshedloo a, Mahmood Rezaee Roknabadi a,⇑, Mohammad Behdani a, Mohsen Modarresi a, Ali Kazempour b a b
Department of Physics, Ferdowsi University of Mashhad, Mashhad, Iran Department of Physics, Payame Noor University, PO Box 119395-3697, Tehran, Iran
a r t i c l e
i n f o
Article history: Received 29 February 2016 Received in revised form 5 June 2016 Accepted 4 July 2016
a b s t r a c t First-principle calculations within Eliashberg theory framework have been utilized to investigate theoretically the effect of hole-doping along with applying strain on superconductivity in a-graphyne and graphene as a benchmark. We show how the electronic properties, electron-phonon interaction strength and superconducting critical temperature T C of a-graphyne and graphene are affected by hole-doping and biaxial tensile strain. Although, intrinsic a-graphyne is semi-metal but it is found that increasing holedoped concentration of 12.5–25% changes its electronic properties from metal to semiconductor with a band gap energy of 0.46 eV. Furthermore, we demonstrate that it is not possible to induce superconductivity in a-graphyne by the conspiracy of 12.5% hole-doping and critical strain (8%) contrary to 12.5% hole-doped graphene under critical strain (12%) which becomes superconductor with T C 12 K. It seems that the presence of C„C bond suppresses the effect of strain significantly and hole-doping slightly on improving electron-phonon coupling constant and T C in contrast to graphene. Ó 2016 Elsevier B.V. All rights reserved.
1. Introduction Two dimensional (2D) materials have been attracting enormous interest for experimentalists and theoreticians [1,2]. Among these materials, graphene as a 2D single layer of sp2-bonded carbon atoms arranged in a honeycomb crystal lattice has been the center of research attention and efforts recently due to its superior properties [3–5]. Robustness, stability, flexibility, zero-gap electronic structure, massless Dirac fermion behavior and half integer quantum Hall effect make graphene a promising material for the next generation of electronic devices with no silicon based electronic limitations [2,5,6,8]. Different techniques have been used for controlling the properties of graphene, such as intentionally introducing different defects, strains, dopants and employing different methods of chemical functionalization [3,5,9]. These research efforts have also generated an increasing interest in exploring new graphene-like carbon allotropes which could possibly share the Dirac cones of electronic band structure and chiral nature of electronic carriers in graphene, along with extraordinary properties [10,11]. An intriguing candidate is graphyne that is formed by inserting AC„CA triple bonds into the honeycomb lattice of graphene and contains sp2 and sp hybridized carbon atoms ⇑ Corresponding author. E-mail address: [email protected] (M.R. Roknabadi). http://dx.doi.org/10.1016/j.commatsci.2016.07.002 0927-0256/Ó 2016 Elsevier B.V. All rights reserved.
[4,11,12]. Due to the different distributions of the sp and sp2 hybridized carbon atoms, five typical structures of graphyne have been proposed as a-, b-, c-, d-, (6, 6, 12)- graphyne [13]. In addition, recently a new family of graphynes were also predicted as square graphynes (S-graphynes) that exhibit highly anisotropic Dirac fermions [14]. Graphyne was suggested by Bauphman et al. in 1987 and has been attracted attention after discovery of fullerenes [4,12]. The presence of acetylenic linkage (AC„CA) in these structures could lead to a rich variety of electronic properties. Therefore, graphyne can have a great potential application as the host material [15]. Although, only building blocks and cutouts of finite-size graphyne have been synthesized so far, but experimentalists are taking the very first steps toward the fabrication of crystalline forms of them [1,5,10,12]. On the other side, in-depth theoretical investigations and numerical characterizations have been performed for determining lattice stability of different graphyne structures and understanding their optical, mechanical and thermoelectric properties [9,10]. The promising theoretical results might encourage experimentalists to try more for making them. Among different possible graphyne lattices, a-graphyne is the most symmetric modification of graphene where the carbon triple bond (AC„CA) is inserted into every carbon bond in graphene [10,16]. It has electronic band structure with Dirac cone and point located at high symmetry K-point in the irreducible Brillouin zone, similar to that of graphene [12]. Therefore, it is expected to have physical
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properties comparable to those of graphene and superior chemical properties more than it as a result of having intrinsic components suitable for chemical substitution reactions [17]. Despite the remarkable properties of graphene, electron-phonon coupling strength is very weak in it and phonon-mediated superconductivity does not occur. It might be arisen from the disappearance of the density of states at the Fermi level (EF ) [1,18,19]. a-graphyne is predicted not to exhibit a superconducting state as the same reason. If a way were found to induce superconducting in them, it could promote their diverse applications in nano-scale superconducting quantum interference devices, single-electron superconductor-quantum dot devices, nanometer-scale superconducting transistors, cryogenic solidstate cooler [18], superconducting hot electron bolometers, superconducting single-photon detectors and Josephson junctions [1]. Several ideas have been proposed to induce superconductivity in graphene within the phononmediated mechanism as a Bardeen-Cooper-Schrieffer (BCS) superconductor [1,2,6,18–21]. For instance, adsorption of alkali metal atoms on graphene has been found to enhance density of state at Fermi level as well as increase the electron-phonon coupling potential. In this approach, the highest superconducting critical temperature T C which could be observed in Lithium-covered graphene LiC6, is around 8 K [1,18,21]. Another approach shows that graphene could be a BCS superconductor by combining the effect of charge doping and biaxial strain. Doping increases density of states at Fermi level and strain promotes the electro-phonon coupling strength. The value of T C is determined around 30 K at the experimental accessible hole-doping ( 4 1014 cm2 ) and strain (16%) levels [1,6,7]. Furthermore, hole-doped fully hydrogenated graphene (graphane) is predicted to be a superconductor with T C above 80 K. In this system the electron-phonon coupling strength is increased considerably due to a strong sp3 bonding enhanced by a giant Kohn anomaly in the optical phonon dispersion [19]. Motivated by these recent results about graphene, we assess the effect of the hole-doping and biaxial tensile strain on electronphonon coupling strength and T C of a-graphyne. The calculated lower mobility and relaxation time of a-graphyne in compared with graphene which are due to the stronger longitude acoustic (LA) phonon scatterings of graphyne [11,12], inspire more us for studying the possibility of inducing phonon-mediated superconductivity in a-graphyne by chemical and mechanical modifications. Our calculations have been performed for a-graphyne modified simultaneously by hole-doping and biaxial tensile strain. For comparison, electron-phonon coupling strength and T C of graphene in the same conditions were calculated as a benchmark. Since the experimental accessible boron doping (hole-doping) concentration for graphene is around 4 1014 cm2 (12.5%) [6,22], two hole-concentrations, 2:4 1014 cm2 (12.5%) and 4:7 1014 cm2 (25%) are considered for a-graphyne in this study.
by a boron (B) atom generates a system with 12.5% hole-doping (4:5 1014 cm2 ) as shown in Fig. 1(a). Since a-graphyne unit cell area is almost twice 2 2 super-cell area of graphene, 12.5% holedoping produces 2:4 1014 cm2 hole-doping concentration (see Fig. 1(b)). Moreover, 4:7 1014 cm2 hole-doping concentration (25%) is investigated for a-graphyne which is obtained by substituting two C atoms with two B atoms in unit cell (in Fig. 1(b), C atom at position 3 should be replaced by B atom, too). The position of B atom in the graphene super-cell and a-graphyne unit cell are chosen based on the results reported by Servati et al. [22] and Dai et al. [23], respectively where the configuration has the lowest energy. The electronic band structures are calculated by using a 15 15 1 Monkhorst Pack k-point grid for sampling the first Brillouin zone (BZ) and a cold smearing of 0.02 Ry for the selfconsistent cycles. A plane-wave cutoff energy of 60 and 65 Ry are adopted for graphene and a-graphyne, respectively. All atomic positions and lattice constants are optimized by Broyden-FletcherGoldfarb-Shanno (BFGS) method and geometry relaxation are performed until the total force on each atom becomes smaller than 106 Ry/Bohr. After cell and atomic relaxation, all B-doped structures remain hexagonal which a and b lattice constants are equal to each other. The convergence of total energy is considered to be lower than 1012 Ry. The dynamical matrix, phonon dispersion spectra and electron-phonon coupling parameters k are calculated based on density functional perturbation theory (DFPT) within the linear response on a 25 25 1 uniform k-point grid and 5 5 1 q-point mesh. For investigating the effect of strain, we apply a series of biaxial tensile strains on hole-doped graphene and a-graphyne and simultaneously relax every strained structure to optimize the positions of atoms by minimizing the total force and energy. The biaxial tensile strain is simulated by increasing the length of two lattice basis vectors without changing the angle between them. The strain value is enhanced until the phonon frequency becomes negative at a specific q wave vector which indicates instability of system and is known as ‘‘phonon instability” [24,25]. The largest value of strain which has not lead to negative phonon frequency, is determines as critical strain. The Eliashberg function a2 FðxÞ is calculated as following to evaluated the strength of electron-phonon coupling,
a2 FðxÞ ¼
X 1 jg v j2 dðenk Þdðemkþq Þdðx xvq Þ; Nð0ÞNk Nq nk;mq;v nk;mkþq ð1Þ
where N(0) is the total density of states per spin and N k and Nq the total numbers of k and q points, respectively. The electron eigenvalues are labeled with the band index (n and m) and the wave vector (k and k + q), while phonon frequencies with the mode number ðmÞ and the wave vector (q). g vnk;mkþqðnk;mkþqÞ is the electron-phonon matrix element. The electron-phonon coupling kðxÞ is defined as,
2. Computational methods First-principles calculations are performed within density functional theory (DFT) framework to obtain the electronic properties of intrinsic and hole-doped a-graphyne and graphene under different biaxial tensile strains. Quantum-espresso code is used with ultra-soft pseudo-potential and the generalized gradient approximation (GGA) for exchange-correlation functional in the Perdew BurkeErnzerhof (PBE) scheme. We ignored using hybrid functionals because they overestimate phonon related properties as mentioned in [28,29]. The van der Waals interaction is included using Grimme’s DFT-D2 approach to calculate the optimized lattice constants. a-graphyne and graphene layers are separated by a large vacuum region of 15 Å to eliminate the interlayer interactions. A 2 2 supercell of graphene with one carbon (C) atom replaced
Fig. 1. Crystal structure of 12.5% hole-doping graphene (a) and 12.5% hole-doping a-graphyne (b). Yellow (purple) spheres represent carbon (boron) atoms. The 2 2 super-cell of graphene and a-graphyne unit cell are indicated by dashed red lines. The different sites of carbon atoms are referred to as C1, C2 and C3. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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Z kðxÞ ¼ 2
0
x
dx0
a2 Fðx0 Þ ; x0
ð2Þ
185
(a)
and total electron-phonon coupling constant is kðx ! 1Þ. Finally, we use Allen-Dynes formula to estimate the superconducting critical temperature with l ¼ 0:1 [18]. 3. Results and discussion 3.1. Electronic properties After fully relaxation, the lattice parameter of graphene supercell is obtained 4.92 Å with a CAC bond length of 1.42 Å, in good agreement with experimental results and other theoretical studies [22,26] as shown in Table 1. For a-graphyne, the optimized lattice constant is 6.94 Å with single and triple bond length of 1.39 and 1.22 Å, respectively which are nearly the same as previous reported values [9,15] (see Table 1). In the equilibrium geometry of 12.5% hole-doped or B-doped graphene referred as GB, 12.5% (a-GB) and 25% (a-GBB) hole-doped or B-doped a-graphyne, the lattice constant and the distance between atoms are calculated and presented in Table 1. The presence of B atom increases lattice constant as expected. It enhances the bond length between C2 and C3 atoms, but has no effect on C1AC2 bond length with sp2 hybridization in GB and sp hybridization in a-GB and a-GBB. Fig. 2(a) illustrates the electronic band structure of graphene (G) compared with the electronic band structures of 12.5% B-doped graphene (GB) and GB under maximum possible biaxial tensile strain of 12% (GBS12%). It reveals that hole-doping displaces the Fermi level downward with respect to the top of the valence band whereas p electronic band crosses EF . Therefore, graphene as a semi-metal is converted to metal. The p electronic band approaches EF around C symmetry point and the area of Fermi surface increases as a result of this hole-doping concentration (4:5 1014 cm2 ). It can also be seen that the band edges at the M and K points are influenced significantly by hole-doping which leads to symmetry breaking and lifts band degeneracies. The critical strain of GB is determined 12% and it is referred as GBS12% in Fig. 2(a). Applying biaxial strain smoothens the electronic bands and further enlarges the Fermi surface around C point. Decreasing the slope of tangent line to the band dispersion at EF (@Ek =@k) due to smoothing, indicates that the velocity or mobility of electrones (or holes) decreases with the increase of the cell volume and therefore, the possibility of electron scattering by lattice vibrations (or phonons) increases. The electronic band structures of intrinsic a-graphyne (a-G) and a-G with two different hole-doping concentrations and under maximum possible strain are displayed in Fig. 2(b) and (c). In 12.5% B-doped a-graphyne (a-GB), Fermi surface shifts downward and a-G is changed from semi-metal to metal same as graphene. In both cases of GB and a-GB, the doping has the effect of lowering EF below the top of the valence band and does not change the slope of p electronic band close to EF compared with intrinsic G and a-G, Table 1 The optimized lattice constant and distance between atoms (Å) for G 2 2 super-cell, GB,a-G, a-GB and a-GBB. Values in parentheses referred to previous studies.
G GB
a-G a-GB a-GBB
Lattice constant(a)
dC1AC2
dC2AC3
dC1AB
4.92 (4.92 [22]) 5.03 6.94 (6.97 [9]) 7.15 7.38
1.42 (1.42 [22,26]) 1.42 1.22 (1.23 [9]) 1.23 1.23
1.47 1.39 (1.40 [9]) 1.39 –
1.50
1.51 1.52
(b)
(c)
Fig. 2. The electronic band structures of (a) intrinsic graphene (G) and 12.5% holedoped graphene (GB) and GB under critical biaxial strain of 12%, (b) intrinsic (a-G) and 12.5% hole-doped a-graphyne (a-GB) and a-GB under critical biaxial strain of 8% and (c) intrinsic a-graphyne (a-G) and 25% hole-doped a-graphyne (a-GBB) and a-GBB under critical biaxial strain of 12%.
respectively. For a-GB, the critical strain is obtained 8%. Its electronic band structure is referred as a-GBS8% in Fig. 2(b) and it can be seen that applying critical strain smoothens electronic bands same as GBS12%. The displacement of Fermi surface increases with enhancement of hole concentration as EF is located under p band in 25% hole-doped a-graphyne (a-GBB) and a band gap energy of 0.46 eV is appeared at C point. Therefore, electronic characteristic of a-GB is changed from metal to semiconductor in
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a-GBB. Applying strain smoothens electronic bands of a-GBB, too and leads an increase in band gap energy. a-GBB under critical biaxial strain of 14% (a-GBBS14%) has a band gap of 1.27 eV at C point. It is important to note that increasing hole concentration from 12.5% to 25% enhances critical strain from 8% to 14% which means that it becomes substantially more robust by increasing hole doping. It is almost the same result what was predicted for graphene by hole or electron doping [25]. However, according to Bardeen-Cooper-Schrieffer (BCS) theory, since there are not any electronic states at Fermi level in a-GBB and a-GBBS14%, superconductivity could not occur in these two structures. Therefore, we do not study them for inducing superconductivity, in the following. Fig. 3(a) and (b) shows the evolution of total electronic density of states (TEDOS) as a function of strain for GB and a-GB, respectively, compared with G and a-G. Hole-doping increases significantly TEDOS at Fermi level in both structures, but this enhancement is smaller for a-GB than GB. Applying strain increases TEDOS gradually at Fermi surface until the strain value reaches to critical strain where system becomes unstable. Calculating partial density of states (PDOS) determines that pz orbitals of B and C have the most contribution in TEDOS at EF . The effect of biaxial strain on this contribution is shown in Fig. 4(a) and (b) for GB and a-GB, respectively. It can be seen that the influence of stain on pz orbital of C3 atom is the greatest among other atoms, in both structures. Strain does not affect noticeably the pz orbitals of other atoms. As it is clear, the existence of C„C bond with sp hybridization suppresses the effect of strain on increasing contribution of pz orbitals in TEDOS at EF . The above results indicate that hole-doping
and strain increase TEDOS at EF and based on BCS theory, help to induce superconductivity in G and a-G. It is in good agreement with other reports for graphene [6]. 3.2. Lattice dynamics and phonon softening The calculated phonon spectrum confirms the dynamical stability of GB and a-GB under different biaxial strains and determines their critical strains as mentioned in previous. Fig. 5 shows phonon dispersions of intrinsic and hole-doped G and a-G as well as phonon dispersions of GB and a-GB under medium strain (4%) and critical strain of 12% and 8%, respectively. The lattice vibrations of G with 8 atoms in its supercell and a-G with 8 atoms in unit cell are characterized by 24 phonon branches. Since G and a-G possess the D6h point group symmetry, the phonon modes at C point are decomposed according to CG ¼ A1g þ A2g þ 3B2g þ E1g þ 2E2g þ 2A2u þ B1u þ B2u þ 3E1u þ E2u for G and CaG ¼ A1g þ A2g þ 2B1g þ E1g þ 3E2g þ 2A2u þ B1u þ B2u þ 3E1u þ E2u for a-G. While TA and LA acoustic modes have linear dispersions, the out-of-plane ZA mode shows a quadratic dispersion in the vicinity of q ¼ 0 in G and a-G (insets of Fig. 5(a) and (e)), which is a characteristic feature of the phonon dispersions in layered crystals as observed experimentally for graphene [8]. The highest optical frequency ( 2253 cm1 ) of a-G is noticeably higher in energy than highest optical frequency of G ( 1558 cm1 ) at C point as seen in Fig. 5(a) and (e). The calculated highest frequency of G is negligibly smaller than the experimental result of 1587 cm1 obtained by inelastic X-ray
(a) (a)
(b)
Fig. 3. The evolution of total electronic density of states (TEDOS) as a function of strain for (a) GB and (b) a-GB in compared with G and a-G.
(b)
Fig. 4. The effect of biaxial tensile strain on the density of states arising from pz orbitals of C and B atoms at Fermi level For GB and a-GB.
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(a)
(b)
(c)
(d)
(e)
(f)
(g)
187
(h)
Fig. 5. Phonon dispersion of intrinsic and hole-doped G and a-G under different strains. The red arrows indicate shifting of optical branches. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
scattering measurements [8]. Three highest optical branches are separated from other branches in a-G due to existence C„C bond. Fig. 5(b) and (f) shows the effect of hole-doping on phonon dispersion. After doping, as a result of breaking symmetry, where D6h point group symmetry turns to D3h , the composition of the vibration representation at the C point changes to C ¼ 2A01 þ 2A02 þ
6E0 þ 4A002 þ 2E00 for GB and a-GB. Hole doping shifts optical branches downward and softens them, especially around K symmetry point in both structures. Moreover, the degeneracies of optical modes disappear at the most points of Brillouin zone. The effect of strain on phonon spectrum is more than hole-doping. As a general trend for both hole-doped structures,
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increasing strain leads to enhancement of softening in all branches, even acoustic branches. Under critical strain, all rather optical branches become completely flat. It is observed that hole-doping and applying critical strain simultaneously shift optical branches downward around 600 and 440 cm1 for G and a-G, respectively which are indicated in figures by red arrows. For investigating and comparing accurately the effect of doping and then applying strain on optical modes, the nonzero phonon frequencies of G, GB, GBS12%, a-G, a-GB and a-GBS8% at C point are summarized in Table 2. Insets of Fig. 5 display the effect of hole doping on the acoustic modes and as well as the evaluation of these modes as a function of applied strain near to the C point toward the M symmetry point for G and a-G. It can be seen in insets of Fig. 5(b)– (d) that doping softens ZT in GB but applying strain hardens it in GBS12%. However, LA has not been affected by doping, but considerably softened under applied critical strain. Doping slightly and strain substantially softens TA mode around C point. Based on insets of Fig. 5(f)–(h) doping softens TA mode and hardens ZA mode of a-G slightly and strain hardens both of TA and ZA modes. LA mode is not affected noticeably by doping and softens slightly by strain in a-G. Interestingly, it is also seen that the low-energy flexural phonon (ZA) of GB and a-GB turn almost in a linear dependence of wave vector under 12% and 8%, respectively. Near the C point, the ratio of phonon group velocity (@ x=@k) in GB and GBS12% to G are found to be v GB =v G ¼ 0:91; v GBS12% =v G ¼ 0:45 for LA mode and v GB =v G ¼ 0:83 and v GBS12% =v G ¼ 0:39 for TA mode, respectively. The same calculation has been done for a-G and obtained v aGB =v aG ¼ 0:93, v aGBS8% =v aG ¼ 0:70 for LA and v aGB =v aG ¼ 0:96 and v aGBS8% =v aG ¼ 1:43 for TA, respectively. 3.3. Electron-phonon coupling and superconductive properties
The behavior of a2 FðxÞ turns out that hole-doping increases electron-phonon interaction due to enhancement of TEDOS at Fermi level. However, electron-phonon coupling kðxÞ is increased after applying strain and reaching to critical strain in both structures due to softening optical modes. Softening and shifting optical modes to lower frequencies give rise to a lower logarithmic frequency average (xlog ) and hence increasing kðxÞ. xlog determines frequency which dominates the electron-phonon coupling. xlog is obtained 813, 1361 and 279 cm1 for G, GB and GBS12% and 110, 1389 and 1002 cm1 for a-G, a-GB and a-GBS8%, respectively. It is observed that hole-doping and applying strain simultaneously increase electron-phonon coupling. For a-GBS8%, it is expected that weaker electron-phonon interaction and higher xlog prevent inducing superconductivity in it. The superconducting critical temperature T C is estimated by using Allen-Dynes formula with l ¼ 0:1. Since total electron-phonon coupling constant k indicates T C , we illustrate these two characteristics as a function of different strains for GB and a-GB in Fig. 7. It is found that GB and a-GB are not superconductor in unstrained state which confirms that hole-doping is a necessary but insufficient condition to induce superconductivity in these two structures. GB under critical strain experience a dramatic enhancement in T C due to increasing k. The general trend of k and T C with respect to biaxial tensile strain for B-doped graphene in 4:5 1014 cm2 concentration are in relatively good agreement with the results of Si et al. We reproduce k 0:75 and T C 12 K under 12% biaxial tensile strain whereas Si et al. reported similar results under 14% strain. This difference may be due to using different exchange-correlation functional in calculations. T C remains zero as strain is increasing and reaching to critical value in a-GB. It can means that C„C bond suppresses
Based on Eq. (2), the displacement of phonon frequencies to low energies due to applying tensile strain can give rise to increasing the electron-phonon coupling. Also, according to other reports, applying biaxial tensile strain on graphene can enhance electronphonon interaction due to softening the in-plane optical modes [6,27]. Therefore, we have investigated the effect of strain on electron-phonon coupling strength in hole-doped G and a-G by calculating the Eliashberg function a2 FðxÞ and electron-phonon coupling kðxÞ for different strain. The calculated results have been shown for GB and a-GB under medium and critical strain compared with unstrained G, GB, a-G and a-GB in Fig. 6, respectively. Table 2 The optical phonon frequencies (in cm1 ) of G, GB, GBS12%, a-G, a-GB and a-GBS8% at C point. G
GB
GBS12%
a-G
a-GB
a-GBS8%
470.87 470.87 470.87 626.35 626.35 626.35 634.95 634.95 634.95 880.84 1329.61 1329.61 1329.61 1342.78 1342.78 1342.78 1392.49 1392.49 1392.49 1557.91 1557.91
381.89 381.89 436.19 554.59 599.79 599.79 612.30 612.84 612.84 870.97 1190.53 1190.53 1210.08 1246.49 1246.49 1270.29 1319.50 1326.39 1326.39 1508.72 1508.72
457.27 457.27 482.34 482.34 505.10 523.9 552.82 552.82 630.08 655.04 655.04 676.49 688.78 755.92 755.92 809.89 916.99 916.99 957.69 1005.83 1005.83
167.73 205.54 213.36 213.36 391.07 441.53 441.53 528.55 528.55 554.31 554.31 607.73 635.33 1043.24 1043.24 1044.00 1481.96 1481.96 2065.07 2065.07 2252.60
151.43 162.17 162.17 192.84 377.98 443.65 443.65 475.53 475.53 535.47 535.47 594.84 669.69 940.87 998.44 998.44 1356.5 1356.5 2098.94 2098.94 2180.64
263.40 270.65 270.65 331.24 448.64 448.64 509.66 525.65 525.65 549.97 549.97 654.73 655.48 704.4 717.33 717.33 940.75 940.75 1733.96 1733.96 1833.11
Fig. 6. The Eliashberg function a2 FðxÞ and electron-phonon coupling kðxÞ for 12.5% hole-doped graphene (GB) and a-G (a-GB) under medium and critical strain in compared with unstrained G, GB, a-G and a-GB.
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graphene as a benchmark in the framework of Eliashberg formalism. 25% hole-doping makes a-graphyne as a semiconductor with a band gap energy of 0.46 eV respect to 12.5% hole-doping which changes it from semi-metal to metal. Our results indicate holedoping increase TEDOS at Fermi level and applying biaxial tensile strain, especially critical strain, shifts optical phonon modes downward to low energy and softens optical branches dramatically and acoustic branches slightly as a general trend for both structures. However, it is observed the effect of hole-doping and biaxial critical strain on k is negligible in a-graphyne against graphene due to existence of C„C bond. T C of hole-doped a-graphyne under critical strain (8%) remains zero while hole-doped graphene under critical strain (12%) becomes superconductor with T C 12 K. It seems hole-doping and applying critical strain could not make a superconducting state in a-graphyne. Acknowledgments We thank Matthias Scheffler, Igor Ying Zhang, S.J. Hashemifar and Alireza Salimi for useful discussions. Calculations were performed at Fritz Haber Institute of the Max Planck Society, Theory department, by using QUANTUM ESPRESSO. References
Fig. 7. Total electron-phonon coupling constant k and T C as a function of different strains for GB and a-GB.
strain effect on electron-phonon coupling constant and does not allow any enhancement in T C . Therefore, combination of holedoping and applying strain is not suitable method for inducing superconductivity in a-G. As reported by Si et al. there are very similar trends in k and T C for the electron and hole doped graphene under tensile strain, expect that the T C in the electron doped graphene is slightly lower than T C in the hole doped graphene at the same doping and strain level. Since it is as a result of the high electron-hole symmetry around the Dirac point in graphene, hence we expect same behavior for a-graphyne due to the same reason which can be investigated. 4. Conclusion We have performed DFPT calculations to study theoretically the possibility of induced superconductivity in a-graphyne and
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Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
First principle study of inducing superconductivity in a-graphyne by hole-doping and biaxial tensile strain Toktam Morshedloo a, Mahmood Rezaee Roknabadi a,⇑, Mohammad Behdani a, Mohsen Modarresi a, Ali Kazempour b a b
Department of Physics, Ferdowsi University of Mashhad, Mashhad, Iran Department of Physics, Payame Noor University, PO Box 119395-3697, Tehran, Iran
a r t i c l e
i n f o
Article history: Received 29 February 2016 Received in revised form 5 June 2016 Accepted 4 July 2016
a b s t r a c t First-principle calculations within Eliashberg theory framework have been utilized to investigate theoretically the effect of hole-doping along with applying strain on superconductivity in a-graphyne and graphene as a benchmark. We show how the electronic properties, electron-phonon interaction strength and superconducting critical temperature T C of a-graphyne and graphene are affected by hole-doping and biaxial tensile strain. Although, intrinsic a-graphyne is semi-metal but it is found that increasing holedoped concentration of 12.5–25% changes its electronic properties from metal to semiconductor with a band gap energy of 0.46 eV. Furthermore, we demonstrate that it is not possible to induce superconductivity in a-graphyne by the conspiracy of 12.5% hole-doping and critical strain (8%) contrary to 12.5% hole-doped graphene under critical strain (12%) which becomes superconductor with T C 12 K. It seems that the presence of C„C bond suppresses the effect of strain significantly and hole-doping slightly on improving electron-phonon coupling constant and T C in contrast to graphene. Ó 2016 Elsevier B.V. All rights reserved.
1. Introduction Two dimensional (2D) materials have been attracting enormous interest for experimentalists and theoreticians [1,2]. Among these materials, graphene as a 2D single layer of sp2-bonded carbon atoms arranged in a honeycomb crystal lattice has been the center of research attention and efforts recently due to its superior properties [3–5]. Robustness, stability, flexibility, zero-gap electronic structure, massless Dirac fermion behavior and half integer quantum Hall effect make graphene a promising material for the next generation of electronic devices with no silicon based electronic limitations [2,5,6,8]. Different techniques have been used for controlling the properties of graphene, such as intentionally introducing different defects, strains, dopants and employing different methods of chemical functionalization [3,5,9]. These research efforts have also generated an increasing interest in exploring new graphene-like carbon allotropes which could possibly share the Dirac cones of electronic band structure and chiral nature of electronic carriers in graphene, along with extraordinary properties [10,11]. An intriguing candidate is graphyne that is formed by inserting AC„CA triple bonds into the honeycomb lattice of graphene and contains sp2 and sp hybridized carbon atoms ⇑ Corresponding author. E-mail address: [email protected] (M.R. Roknabadi). http://dx.doi.org/10.1016/j.commatsci.2016.07.002 0927-0256/Ó 2016 Elsevier B.V. All rights reserved.
[4,11,12]. Due to the different distributions of the sp and sp2 hybridized carbon atoms, five typical structures of graphyne have been proposed as a-, b-, c-, d-, (6, 6, 12)- graphyne [13]. In addition, recently a new family of graphynes were also predicted as square graphynes (S-graphynes) that exhibit highly anisotropic Dirac fermions [14]. Graphyne was suggested by Bauphman et al. in 1987 and has been attracted attention after discovery of fullerenes [4,12]. The presence of acetylenic linkage (AC„CA) in these structures could lead to a rich variety of electronic properties. Therefore, graphyne can have a great potential application as the host material [15]. Although, only building blocks and cutouts of finite-size graphyne have been synthesized so far, but experimentalists are taking the very first steps toward the fabrication of crystalline forms of them [1,5,10,12]. On the other side, in-depth theoretical investigations and numerical characterizations have been performed for determining lattice stability of different graphyne structures and understanding their optical, mechanical and thermoelectric properties [9,10]. The promising theoretical results might encourage experimentalists to try more for making them. Among different possible graphyne lattices, a-graphyne is the most symmetric modification of graphene where the carbon triple bond (AC„CA) is inserted into every carbon bond in graphene [10,16]. It has electronic band structure with Dirac cone and point located at high symmetry K-point in the irreducible Brillouin zone, similar to that of graphene [12]. Therefore, it is expected to have physical
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properties comparable to those of graphene and superior chemical properties more than it as a result of having intrinsic components suitable for chemical substitution reactions [17]. Despite the remarkable properties of graphene, electron-phonon coupling strength is very weak in it and phonon-mediated superconductivity does not occur. It might be arisen from the disappearance of the density of states at the Fermi level (EF ) [1,18,19]. a-graphyne is predicted not to exhibit a superconducting state as the same reason. If a way were found to induce superconducting in them, it could promote their diverse applications in nano-scale superconducting quantum interference devices, single-electron superconductor-quantum dot devices, nanometer-scale superconducting transistors, cryogenic solidstate cooler [18], superconducting hot electron bolometers, superconducting single-photon detectors and Josephson junctions [1]. Several ideas have been proposed to induce superconductivity in graphene within the phononmediated mechanism as a Bardeen-Cooper-Schrieffer (BCS) superconductor [1,2,6,18–21]. For instance, adsorption of alkali metal atoms on graphene has been found to enhance density of state at Fermi level as well as increase the electron-phonon coupling potential. In this approach, the highest superconducting critical temperature T C which could be observed in Lithium-covered graphene LiC6, is around 8 K [1,18,21]. Another approach shows that graphene could be a BCS superconductor by combining the effect of charge doping and biaxial strain. Doping increases density of states at Fermi level and strain promotes the electro-phonon coupling strength. The value of T C is determined around 30 K at the experimental accessible hole-doping ( 4 1014 cm2 ) and strain (16%) levels [1,6,7]. Furthermore, hole-doped fully hydrogenated graphene (graphane) is predicted to be a superconductor with T C above 80 K. In this system the electron-phonon coupling strength is increased considerably due to a strong sp3 bonding enhanced by a giant Kohn anomaly in the optical phonon dispersion [19]. Motivated by these recent results about graphene, we assess the effect of the hole-doping and biaxial tensile strain on electronphonon coupling strength and T C of a-graphyne. The calculated lower mobility and relaxation time of a-graphyne in compared with graphene which are due to the stronger longitude acoustic (LA) phonon scatterings of graphyne [11,12], inspire more us for studying the possibility of inducing phonon-mediated superconductivity in a-graphyne by chemical and mechanical modifications. Our calculations have been performed for a-graphyne modified simultaneously by hole-doping and biaxial tensile strain. For comparison, electron-phonon coupling strength and T C of graphene in the same conditions were calculated as a benchmark. Since the experimental accessible boron doping (hole-doping) concentration for graphene is around 4 1014 cm2 (12.5%) [6,22], two hole-concentrations, 2:4 1014 cm2 (12.5%) and 4:7 1014 cm2 (25%) are considered for a-graphyne in this study.
by a boron (B) atom generates a system with 12.5% hole-doping (4:5 1014 cm2 ) as shown in Fig. 1(a). Since a-graphyne unit cell area is almost twice 2 2 super-cell area of graphene, 12.5% holedoping produces 2:4 1014 cm2 hole-doping concentration (see Fig. 1(b)). Moreover, 4:7 1014 cm2 hole-doping concentration (25%) is investigated for a-graphyne which is obtained by substituting two C atoms with two B atoms in unit cell (in Fig. 1(b), C atom at position 3 should be replaced by B atom, too). The position of B atom in the graphene super-cell and a-graphyne unit cell are chosen based on the results reported by Servati et al. [22] and Dai et al. [23], respectively where the configuration has the lowest energy. The electronic band structures are calculated by using a 15 15 1 Monkhorst Pack k-point grid for sampling the first Brillouin zone (BZ) and a cold smearing of 0.02 Ry for the selfconsistent cycles. A plane-wave cutoff energy of 60 and 65 Ry are adopted for graphene and a-graphyne, respectively. All atomic positions and lattice constants are optimized by Broyden-FletcherGoldfarb-Shanno (BFGS) method and geometry relaxation are performed until the total force on each atom becomes smaller than 106 Ry/Bohr. After cell and atomic relaxation, all B-doped structures remain hexagonal which a and b lattice constants are equal to each other. The convergence of total energy is considered to be lower than 1012 Ry. The dynamical matrix, phonon dispersion spectra and electron-phonon coupling parameters k are calculated based on density functional perturbation theory (DFPT) within the linear response on a 25 25 1 uniform k-point grid and 5 5 1 q-point mesh. For investigating the effect of strain, we apply a series of biaxial tensile strains on hole-doped graphene and a-graphyne and simultaneously relax every strained structure to optimize the positions of atoms by minimizing the total force and energy. The biaxial tensile strain is simulated by increasing the length of two lattice basis vectors without changing the angle between them. The strain value is enhanced until the phonon frequency becomes negative at a specific q wave vector which indicates instability of system and is known as ‘‘phonon instability” [24,25]. The largest value of strain which has not lead to negative phonon frequency, is determines as critical strain. The Eliashberg function a2 FðxÞ is calculated as following to evaluated the strength of electron-phonon coupling,
a2 FðxÞ ¼
X 1 jg v j2 dðenk Þdðemkþq Þdðx xvq Þ; Nð0ÞNk Nq nk;mq;v nk;mkþq ð1Þ
where N(0) is the total density of states per spin and N k and Nq the total numbers of k and q points, respectively. The electron eigenvalues are labeled with the band index (n and m) and the wave vector (k and k + q), while phonon frequencies with the mode number ðmÞ and the wave vector (q). g vnk;mkþqðnk;mkþqÞ is the electron-phonon matrix element. The electron-phonon coupling kðxÞ is defined as,
2. Computational methods First-principles calculations are performed within density functional theory (DFT) framework to obtain the electronic properties of intrinsic and hole-doped a-graphyne and graphene under different biaxial tensile strains. Quantum-espresso code is used with ultra-soft pseudo-potential and the generalized gradient approximation (GGA) for exchange-correlation functional in the Perdew BurkeErnzerhof (PBE) scheme. We ignored using hybrid functionals because they overestimate phonon related properties as mentioned in [28,29]. The van der Waals interaction is included using Grimme’s DFT-D2 approach to calculate the optimized lattice constants. a-graphyne and graphene layers are separated by a large vacuum region of 15 Å to eliminate the interlayer interactions. A 2 2 supercell of graphene with one carbon (C) atom replaced
Fig. 1. Crystal structure of 12.5% hole-doping graphene (a) and 12.5% hole-doping a-graphyne (b). Yellow (purple) spheres represent carbon (boron) atoms. The 2 2 super-cell of graphene and a-graphyne unit cell are indicated by dashed red lines. The different sites of carbon atoms are referred to as C1, C2 and C3. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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Z kðxÞ ¼ 2
0
x
dx0
a2 Fðx0 Þ ; x0
ð2Þ
185
(a)
and total electron-phonon coupling constant is kðx ! 1Þ. Finally, we use Allen-Dynes formula to estimate the superconducting critical temperature with l ¼ 0:1 [18]. 3. Results and discussion 3.1. Electronic properties After fully relaxation, the lattice parameter of graphene supercell is obtained 4.92 Å with a CAC bond length of 1.42 Å, in good agreement with experimental results and other theoretical studies [22,26] as shown in Table 1. For a-graphyne, the optimized lattice constant is 6.94 Å with single and triple bond length of 1.39 and 1.22 Å, respectively which are nearly the same as previous reported values [9,15] (see Table 1). In the equilibrium geometry of 12.5% hole-doped or B-doped graphene referred as GB, 12.5% (a-GB) and 25% (a-GBB) hole-doped or B-doped a-graphyne, the lattice constant and the distance between atoms are calculated and presented in Table 1. The presence of B atom increases lattice constant as expected. It enhances the bond length between C2 and C3 atoms, but has no effect on C1AC2 bond length with sp2 hybridization in GB and sp hybridization in a-GB and a-GBB. Fig. 2(a) illustrates the electronic band structure of graphene (G) compared with the electronic band structures of 12.5% B-doped graphene (GB) and GB under maximum possible biaxial tensile strain of 12% (GBS12%). It reveals that hole-doping displaces the Fermi level downward with respect to the top of the valence band whereas p electronic band crosses EF . Therefore, graphene as a semi-metal is converted to metal. The p electronic band approaches EF around C symmetry point and the area of Fermi surface increases as a result of this hole-doping concentration (4:5 1014 cm2 ). It can also be seen that the band edges at the M and K points are influenced significantly by hole-doping which leads to symmetry breaking and lifts band degeneracies. The critical strain of GB is determined 12% and it is referred as GBS12% in Fig. 2(a). Applying biaxial strain smoothens the electronic bands and further enlarges the Fermi surface around C point. Decreasing the slope of tangent line to the band dispersion at EF (@Ek =@k) due to smoothing, indicates that the velocity or mobility of electrones (or holes) decreases with the increase of the cell volume and therefore, the possibility of electron scattering by lattice vibrations (or phonons) increases. The electronic band structures of intrinsic a-graphyne (a-G) and a-G with two different hole-doping concentrations and under maximum possible strain are displayed in Fig. 2(b) and (c). In 12.5% B-doped a-graphyne (a-GB), Fermi surface shifts downward and a-G is changed from semi-metal to metal same as graphene. In both cases of GB and a-GB, the doping has the effect of lowering EF below the top of the valence band and does not change the slope of p electronic band close to EF compared with intrinsic G and a-G, Table 1 The optimized lattice constant and distance between atoms (Å) for G 2 2 super-cell, GB,a-G, a-GB and a-GBB. Values in parentheses referred to previous studies.
G GB
a-G a-GB a-GBB
Lattice constant(a)
dC1AC2
dC2AC3
dC1AB
4.92 (4.92 [22]) 5.03 6.94 (6.97 [9]) 7.15 7.38
1.42 (1.42 [22,26]) 1.42 1.22 (1.23 [9]) 1.23 1.23
1.47 1.39 (1.40 [9]) 1.39 –
1.50
1.51 1.52
(b)
(c)
Fig. 2. The electronic band structures of (a) intrinsic graphene (G) and 12.5% holedoped graphene (GB) and GB under critical biaxial strain of 12%, (b) intrinsic (a-G) and 12.5% hole-doped a-graphyne (a-GB) and a-GB under critical biaxial strain of 8% and (c) intrinsic a-graphyne (a-G) and 25% hole-doped a-graphyne (a-GBB) and a-GBB under critical biaxial strain of 12%.
respectively. For a-GB, the critical strain is obtained 8%. Its electronic band structure is referred as a-GBS8% in Fig. 2(b) and it can be seen that applying critical strain smoothens electronic bands same as GBS12%. The displacement of Fermi surface increases with enhancement of hole concentration as EF is located under p band in 25% hole-doped a-graphyne (a-GBB) and a band gap energy of 0.46 eV is appeared at C point. Therefore, electronic characteristic of a-GB is changed from metal to semiconductor in
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a-GBB. Applying strain smoothens electronic bands of a-GBB, too and leads an increase in band gap energy. a-GBB under critical biaxial strain of 14% (a-GBBS14%) has a band gap of 1.27 eV at C point. It is important to note that increasing hole concentration from 12.5% to 25% enhances critical strain from 8% to 14% which means that it becomes substantially more robust by increasing hole doping. It is almost the same result what was predicted for graphene by hole or electron doping [25]. However, according to Bardeen-Cooper-Schrieffer (BCS) theory, since there are not any electronic states at Fermi level in a-GBB and a-GBBS14%, superconductivity could not occur in these two structures. Therefore, we do not study them for inducing superconductivity, in the following. Fig. 3(a) and (b) shows the evolution of total electronic density of states (TEDOS) as a function of strain for GB and a-GB, respectively, compared with G and a-G. Hole-doping increases significantly TEDOS at Fermi level in both structures, but this enhancement is smaller for a-GB than GB. Applying strain increases TEDOS gradually at Fermi surface until the strain value reaches to critical strain where system becomes unstable. Calculating partial density of states (PDOS) determines that pz orbitals of B and C have the most contribution in TEDOS at EF . The effect of biaxial strain on this contribution is shown in Fig. 4(a) and (b) for GB and a-GB, respectively. It can be seen that the influence of stain on pz orbital of C3 atom is the greatest among other atoms, in both structures. Strain does not affect noticeably the pz orbitals of other atoms. As it is clear, the existence of C„C bond with sp hybridization suppresses the effect of strain on increasing contribution of pz orbitals in TEDOS at EF . The above results indicate that hole-doping
and strain increase TEDOS at EF and based on BCS theory, help to induce superconductivity in G and a-G. It is in good agreement with other reports for graphene [6]. 3.2. Lattice dynamics and phonon softening The calculated phonon spectrum confirms the dynamical stability of GB and a-GB under different biaxial strains and determines their critical strains as mentioned in previous. Fig. 5 shows phonon dispersions of intrinsic and hole-doped G and a-G as well as phonon dispersions of GB and a-GB under medium strain (4%) and critical strain of 12% and 8%, respectively. The lattice vibrations of G with 8 atoms in its supercell and a-G with 8 atoms in unit cell are characterized by 24 phonon branches. Since G and a-G possess the D6h point group symmetry, the phonon modes at C point are decomposed according to CG ¼ A1g þ A2g þ 3B2g þ E1g þ 2E2g þ 2A2u þ B1u þ B2u þ 3E1u þ E2u for G and CaG ¼ A1g þ A2g þ 2B1g þ E1g þ 3E2g þ 2A2u þ B1u þ B2u þ 3E1u þ E2u for a-G. While TA and LA acoustic modes have linear dispersions, the out-of-plane ZA mode shows a quadratic dispersion in the vicinity of q ¼ 0 in G and a-G (insets of Fig. 5(a) and (e)), which is a characteristic feature of the phonon dispersions in layered crystals as observed experimentally for graphene [8]. The highest optical frequency ( 2253 cm1 ) of a-G is noticeably higher in energy than highest optical frequency of G ( 1558 cm1 ) at C point as seen in Fig. 5(a) and (e). The calculated highest frequency of G is negligibly smaller than the experimental result of 1587 cm1 obtained by inelastic X-ray
(a) (a)
(b)
Fig. 3. The evolution of total electronic density of states (TEDOS) as a function of strain for (a) GB and (b) a-GB in compared with G and a-G.
(b)
Fig. 4. The effect of biaxial tensile strain on the density of states arising from pz orbitals of C and B atoms at Fermi level For GB and a-GB.
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(a)
(b)
(c)
(d)
(e)
(f)
(g)
187
(h)
Fig. 5. Phonon dispersion of intrinsic and hole-doped G and a-G under different strains. The red arrows indicate shifting of optical branches. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
scattering measurements [8]. Three highest optical branches are separated from other branches in a-G due to existence C„C bond. Fig. 5(b) and (f) shows the effect of hole-doping on phonon dispersion. After doping, as a result of breaking symmetry, where D6h point group symmetry turns to D3h , the composition of the vibration representation at the C point changes to C ¼ 2A01 þ 2A02 þ
6E0 þ 4A002 þ 2E00 for GB and a-GB. Hole doping shifts optical branches downward and softens them, especially around K symmetry point in both structures. Moreover, the degeneracies of optical modes disappear at the most points of Brillouin zone. The effect of strain on phonon spectrum is more than hole-doping. As a general trend for both hole-doped structures,
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increasing strain leads to enhancement of softening in all branches, even acoustic branches. Under critical strain, all rather optical branches become completely flat. It is observed that hole-doping and applying critical strain simultaneously shift optical branches downward around 600 and 440 cm1 for G and a-G, respectively which are indicated in figures by red arrows. For investigating and comparing accurately the effect of doping and then applying strain on optical modes, the nonzero phonon frequencies of G, GB, GBS12%, a-G, a-GB and a-GBS8% at C point are summarized in Table 2. Insets of Fig. 5 display the effect of hole doping on the acoustic modes and as well as the evaluation of these modes as a function of applied strain near to the C point toward the M symmetry point for G and a-G. It can be seen in insets of Fig. 5(b)– (d) that doping softens ZT in GB but applying strain hardens it in GBS12%. However, LA has not been affected by doping, but considerably softened under applied critical strain. Doping slightly and strain substantially softens TA mode around C point. Based on insets of Fig. 5(f)–(h) doping softens TA mode and hardens ZA mode of a-G slightly and strain hardens both of TA and ZA modes. LA mode is not affected noticeably by doping and softens slightly by strain in a-G. Interestingly, it is also seen that the low-energy flexural phonon (ZA) of GB and a-GB turn almost in a linear dependence of wave vector under 12% and 8%, respectively. Near the C point, the ratio of phonon group velocity (@ x=@k) in GB and GBS12% to G are found to be v GB =v G ¼ 0:91; v GBS12% =v G ¼ 0:45 for LA mode and v GB =v G ¼ 0:83 and v GBS12% =v G ¼ 0:39 for TA mode, respectively. The same calculation has been done for a-G and obtained v aGB =v aG ¼ 0:93, v aGBS8% =v aG ¼ 0:70 for LA and v aGB =v aG ¼ 0:96 and v aGBS8% =v aG ¼ 1:43 for TA, respectively. 3.3. Electron-phonon coupling and superconductive properties
The behavior of a2 FðxÞ turns out that hole-doping increases electron-phonon interaction due to enhancement of TEDOS at Fermi level. However, electron-phonon coupling kðxÞ is increased after applying strain and reaching to critical strain in both structures due to softening optical modes. Softening and shifting optical modes to lower frequencies give rise to a lower logarithmic frequency average (xlog ) and hence increasing kðxÞ. xlog determines frequency which dominates the electron-phonon coupling. xlog is obtained 813, 1361 and 279 cm1 for G, GB and GBS12% and 110, 1389 and 1002 cm1 for a-G, a-GB and a-GBS8%, respectively. It is observed that hole-doping and applying strain simultaneously increase electron-phonon coupling. For a-GBS8%, it is expected that weaker electron-phonon interaction and higher xlog prevent inducing superconductivity in it. The superconducting critical temperature T C is estimated by using Allen-Dynes formula with l ¼ 0:1. Since total electron-phonon coupling constant k indicates T C , we illustrate these two characteristics as a function of different strains for GB and a-GB in Fig. 7. It is found that GB and a-GB are not superconductor in unstrained state which confirms that hole-doping is a necessary but insufficient condition to induce superconductivity in these two structures. GB under critical strain experience a dramatic enhancement in T C due to increasing k. The general trend of k and T C with respect to biaxial tensile strain for B-doped graphene in 4:5 1014 cm2 concentration are in relatively good agreement with the results of Si et al. We reproduce k 0:75 and T C 12 K under 12% biaxial tensile strain whereas Si et al. reported similar results under 14% strain. This difference may be due to using different exchange-correlation functional in calculations. T C remains zero as strain is increasing and reaching to critical value in a-GB. It can means that C„C bond suppresses
Based on Eq. (2), the displacement of phonon frequencies to low energies due to applying tensile strain can give rise to increasing the electron-phonon coupling. Also, according to other reports, applying biaxial tensile strain on graphene can enhance electronphonon interaction due to softening the in-plane optical modes [6,27]. Therefore, we have investigated the effect of strain on electron-phonon coupling strength in hole-doped G and a-G by calculating the Eliashberg function a2 FðxÞ and electron-phonon coupling kðxÞ for different strain. The calculated results have been shown for GB and a-GB under medium and critical strain compared with unstrained G, GB, a-G and a-GB in Fig. 6, respectively. Table 2 The optical phonon frequencies (in cm1 ) of G, GB, GBS12%, a-G, a-GB and a-GBS8% at C point. G
GB
GBS12%
a-G
a-GB
a-GBS8%
470.87 470.87 470.87 626.35 626.35 626.35 634.95 634.95 634.95 880.84 1329.61 1329.61 1329.61 1342.78 1342.78 1342.78 1392.49 1392.49 1392.49 1557.91 1557.91
381.89 381.89 436.19 554.59 599.79 599.79 612.30 612.84 612.84 870.97 1190.53 1190.53 1210.08 1246.49 1246.49 1270.29 1319.50 1326.39 1326.39 1508.72 1508.72
457.27 457.27 482.34 482.34 505.10 523.9 552.82 552.82 630.08 655.04 655.04 676.49 688.78 755.92 755.92 809.89 916.99 916.99 957.69 1005.83 1005.83
167.73 205.54 213.36 213.36 391.07 441.53 441.53 528.55 528.55 554.31 554.31 607.73 635.33 1043.24 1043.24 1044.00 1481.96 1481.96 2065.07 2065.07 2252.60
151.43 162.17 162.17 192.84 377.98 443.65 443.65 475.53 475.53 535.47 535.47 594.84 669.69 940.87 998.44 998.44 1356.5 1356.5 2098.94 2098.94 2180.64
263.40 270.65 270.65 331.24 448.64 448.64 509.66 525.65 525.65 549.97 549.97 654.73 655.48 704.4 717.33 717.33 940.75 940.75 1733.96 1733.96 1833.11
Fig. 6. The Eliashberg function a2 FðxÞ and electron-phonon coupling kðxÞ for 12.5% hole-doped graphene (GB) and a-G (a-GB) under medium and critical strain in compared with unstrained G, GB, a-G and a-GB.
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graphene as a benchmark in the framework of Eliashberg formalism. 25% hole-doping makes a-graphyne as a semiconductor with a band gap energy of 0.46 eV respect to 12.5% hole-doping which changes it from semi-metal to metal. Our results indicate holedoping increase TEDOS at Fermi level and applying biaxial tensile strain, especially critical strain, shifts optical phonon modes downward to low energy and softens optical branches dramatically and acoustic branches slightly as a general trend for both structures. However, it is observed the effect of hole-doping and biaxial critical strain on k is negligible in a-graphyne against graphene due to existence of C„C bond. T C of hole-doped a-graphyne under critical strain (8%) remains zero while hole-doped graphene under critical strain (12%) becomes superconductor with T C 12 K. It seems hole-doping and applying critical strain could not make a superconducting state in a-graphyne. Acknowledgments We thank Matthias Scheffler, Igor Ying Zhang, S.J. Hashemifar and Alireza Salimi for useful discussions. Calculations were performed at Fritz Haber Institute of the Max Planck Society, Theory department, by using QUANTUM ESPRESSO. References
Fig. 7. Total electron-phonon coupling constant k and T C as a function of different strains for GB and a-GB.
strain effect on electron-phonon coupling constant and does not allow any enhancement in T C . Therefore, combination of holedoping and applying strain is not suitable method for inducing superconductivity in a-G. As reported by Si et al. there are very similar trends in k and T C for the electron and hole doped graphene under tensile strain, expect that the T C in the electron doped graphene is slightly lower than T C in the hole doped graphene at the same doping and strain level. Since it is as a result of the high electron-hole symmetry around the Dirac point in graphene, hence we expect same behavior for a-graphyne due to the same reason which can be investigated. 4. Conclusion We have performed DFPT calculations to study theoretically the possibility of induced superconductivity in a-graphyne and
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