The electronic transport properties of porous zigzag graphene clusters

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The electronic transport properties of porous zigzag graphen clusters Hamidreza Simchi, Mahdi Esmaeilzadeh, Hossein Mazidabadi

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S1386-9477(13)00230-0 http://dx.doi.org/10.1016/j.physe.2013.06.021 PHYSE11304

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Received date: 26 February 2013 Revised date: 9 June 2013 Accepted date: 18 June 2013 Cite this article as: Hamidreza Simchi, Mahdi Esmaeilzadeh, Hossein Mazidabadi, The electronic transport properties of porous zigzag graphen clusters, Physica E, http://dx.doi.org/10.1016/j.physe.2013.06.021 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

The electronic transport properties of porous zigzag graphen clusters  Hamidreza Simchi*,1,2, Mahdi Esmaeilzadeh**,1, and Hossein Mazidabadi1 1

Department of Physics, Iran University of Science and Technology, Narrmak, Tehran 16844, Iran

2

Semiconductor Technology Center, Tehran Iran

Abstract By omitting the some carbon atoms from middle of a zigzag graphene cluster, and Hydrogen termination of sp2 orbital, we make the different porous zigzag graphene clusters, and investigate the electron transport properties of the structures by non-equilibrium Green function method at zero bias regime. It is shown that, the conductance of porous clusters depends on the final symmetry of porous cluster and the local imbalance number (nA-nB), which nA and nB is number of omitted atoms from Asublattice and B-sublattice respectively. Also it is shown that, if three carbon atoms (one type-A and two type-B sites) are omitted the conductance for |E-Ef|≥5 eV are significantly higher than the conductance of original zigzag graphene cluster due to the increment in less affected conducting channels. We show that, spin flipping is happened under Rashba spin orbit interaction at E=Ef , when three atoms are omitted from the original cluster. Therefore the local imbalance number and final symmetry of porous graphene cluster can be used as a rule for designing porous graphene devices and the device can be used in spintronic applications. PACS number(s): 85.65.+h, 85.75.‐d, 73.63.‐b, 61.46.‐w   Keywords:  Molecular  charge  transport,  graphene  leads,  graphene  clusters,  pore,  Rashba  spin‐orbit  interaction    * E‐mail: [email protected], Phone: 0098‐21‐77240477, Fax: 77240497  ** E‐mail: [email protected], Phone: 0098‐21‐77240477, Fax: 77240497 

  

 1  

1. Introduction Band-gap engineering is a powerful technique for the design of new semiconductor materials and devices. A new generation of devices with unique capabilities, ranging from solid-state photomultipliers to resonant tunneling transistors, is emerging from this approach [1]. Band-gap engineering of carbon nanotubes [2] and graphene nanoribbons (GNR) [3] has been studied. The density functional theory studies have predicted that, the energy gap of a GNR scales inversely with the channel width (w) as, Egap=α/w, with a corresponding α value between 0.2-1.5 eV nm [4, 5]. In experimental study α=0.2 eV nm has been found [3]. Zeng et al., have considered three dimensional (one limited dimension including six carbon atoms and two Oxygen (Hydrogen) atoms and two infinite dimension) 8ZGNR-H/ZGNR-O heterostructure and shown that the zero bias conductance of both hydrogen and oxygen terminated zigzag graphene nanoribbon is equal to 3G0 and zero bias conductance of heterostructure is equal to 0.3G0 at Fermi energy [6].The Fermi wavelength of electrons in graphene (or equivalently the carrier de Broglie wavelength), λF, is proportional to (n)-0.5 which, n is the electron density per Cm2 [7] and it has been shown that, it is typically equal to 0.74 nm (i.e., 7.4A0) [8]. From a two dimensional (2D) graphene sheet, a GNR can be constructed with quasi1D structure if the width of GNR be at order of λF. Li et al., have studied the conductance of GNRs with single and weak disorder at their edges by using tight binding method and have shown that, the zigzag GNRs will change from metallic to semiconducting due to the Anderson localization [9]. The width of their zigzag and armchair GNR included four and seven carbon atoms respectively and they used one dimensional (1D) model to explain the conductance [9]. Lin et al., have studied not only experimentally but also theoretically the conductance quantization in GNRs, where 1D transport subband are formed due to the lateral quantum confinement [10]. They have shown that, the conductance curve of the GNR is in "V" shape that reflects symmetric hole and electron transport at negative and positive gate voltage respectively [10]. Using four-orbital-per-atom tight binding method and Green function approach, Gorjizadeh et al., have studied the effects of vacancy and impurity-vacancy defects on the quantum conductance of armchair and zigzag ribbons [11].They have shown that, the effect of the vacancy on the π bands, which are responsible for conduction near Fermi energy, is

 2  

significant [11]. Also they have shown that near the Fermi energy, the conductance behavior clearly depends on the position of the vacancy, and the number of conducting channels decrease when there is a vacancy in the ribbon [11]. Ahmadi et al., have introduced an analytical model of GNR conductance when its width is very smaller than λF and its length is very greater than λF [12]. Sahin et al., have shown that, a spin polarized current can be produced by pure hydrogenated rectangular graphene clusters [13]. Also they have shown that, by adding Vanadium atom to the structure a sharp discontinuity in the transmission spectra is seen which arise from Fano resonances [13]. The electronic structure of porous nanocarbons and graphene has been studied [14, 15]. It has been shown that, by positioning standard pores (i.e., honeycomb-shaped pore) at the center of AGNR, a band gap opens in its band structure, while ZGNR remains metallic [15]. Also Russo et al., have demonstrated a scalable method for creating extremely small structures in graphene with atomic precision and created graphene nonporous with radii as small as 3 A0 [16]. The band gap engineering of GNR [3] which is made by patterning technique, scalable method for creating extremely small graphene nonporous [16], and 1D quantum conductance of GNRs [10-12] motivated us to study the conductance of porous zigzag graphene nano-cluster (ZGNC) connected to zigzag graphene leads, when not only its width but also its length is at order of λF.           In the article, we study the electron transport properties of ZGNC including different number of pores. Geometry optimization and non-equilibrium Green function (i.e., DFT-NEGF) calculation are based on our previous published article [17]. We show that, the conductance of porous cluster depends on the final symmetry of porous cluster and the local imbalance number (nA-nB), which nA and nB is number of omitted atoms from A-sublattice and B-sublattice respectively. Also it is shown that, if three carbon atoms (one type-A and two type-B sites) are omitted the conductance for |E-Ef|≥5 eV are significantly higher than the conductance of original zigzag graphene cluster. We show that, spin flipping is happened under Rashba spin orbit interaction at E=EF , when three atoms are omitted from the original cluster. The paper is organized as follows: In Sec. 2, the calculation method and in Sec. 3, the results and discussion are presented. The summary of our work is given in Sec. 4.    3  

  2. Calculation method Geometry optimization and zero bias non-equilibrium Green function (i.e., DFT-NEGF) calculation are done based on our previous published article [17]. The semi-infinite graphene ribbonZGNC- semi-infinite graphene ribbon molecule (called extended molecule) is divided into three parts: the left electrode, the extended channel and the right electrode [see Fig.1 (a)]. As Fig.1 (a) shows main scattering region includes four super cells and five extra carbon atoms while each lead includes two super cells. Before doing the DFT and transport calculations, it is necessary we specify the basis sets and functional. In our calculations we use 3-21G Split Valence Basis and B3LYP as functional [18, 19]. Before doing the DFT calculation, we find the complete relaxed structure using B3LYP 321G Opt test command of Gaussian Code [17, 18]. Using non-standard commands of Gaussian code i.e., iop(5/33=3) and iop(3/33=1), one can find the Fock and Overlap matrices, respectively [17,18]. Since Gaussian basis functions are not orthogonal we define an effective Hamiltonian, H=Fock + E*(I – Overlap), where E is the energy of electrons and I is unit matrix [17]. For calculating the electronic Density of States (DOS) and Transmission Function (TF) it is necessary we calculate the surface green function of left and right leads. The surface green function is calculated using the standard procedure [17, 20-22]. The Green function is defined as G= ((E+η) * I – H – ΣL - ΣR)-1, where η is an infinitesimal imaginary number and ΣL,R is the self energy of left (L) and right (R) leads [17]. The transmission is found using T (E) = real (trace (ΓL * G * ΓR * G')) where ΓL, R is coupling matrix between left and right leads [17]. The conductance is found using Ğ=Ğ0 * T (E) where Ğ0 is quanta of conductance (e2/h) [17].The electronic DOS can be found using DOS=Imaginary (G)/π. When Rashba spin-orbit interaction is considered, the total Hamiltonian is given by [17, 23]

+↑ i

H = (C C

+↓ i

⎛H )⎜ 0 ⎝ 0

↑ 0 ⎞ ⎛C j ⎞ 0 +↑ +↓ ⎛ ⎟ ⎜⎜ ↓ ⎟⎟ + (C i C i ) ⎜ H0 ⎠ C j ⎝ −2i λRashba ⎝ ⎠

2i λRashba ⎞ ⎛C i↑+1 ⎞ ⎟ ⎜⎜ ↓ ⎟⎟ (1) 0 ⎠ ⎝C i +1 ⎠

 4  

where H0 is Hamiltonian without Rashba spin–orbit interaction, C i+↑( ↓ ) and C i↑( ↓ ) are creation and annihilation of spin up, ↑ , (spin down, ↓ ) operators, respectively. λRashba is the Rashba spin–orbit strength. Min et al., have used second-order perturbation theory and derived explicit expressions for the intrinsic and Rashba spin-orbit interaction induced gap in energy band structure of an isolated graphene sheet [24]. Kochan et al., have introduced a realistic multiband tight binding model to explain the effect of d-orbital in the spin orbit coupling at K-point [25]. Ding et al., have studied spin-polarized electron transport through ZGNR [26] and Ahamdi et al., have shown that, when the strength of Rashba spin orbit interaction, λR, is equal to 0.493 and incident angle, θ, is at range -450 to +450, spin flipping is happened in graphene sheet while if λR=0.8 the spin flipping is happened only at θ=480[27]. We assume that λR is equal to 0.8 eV [27]. Using Eq. (1), we can calculate the Green function of extended channel in presence of Rashba spin–orbit interaction. Finally Green function matrix is decomposed into four parts as

⎛G ↑↑ G ↑↓ ⎞ G =⎜ ⎟ ⎜G ↓↑ G ↓↓ ⎟ ⎝ ⎠

(2)

which is spin-dependent and ↑ means spin up and ↓ means spin down. The total spin up (down) conductance is equal to Ğ↑(↓) (E) = Ğ0( real (trace (ΓL * G↑↑(↓↓) * ΓR * G↑↑(↓↓'))+ real (trace (ΓL * G↑↓(↓↑) * ΓR * G↑↓(↓↑)')))

(3)

3. Results and discussion Fig.1 (a) shows the optimized schematic configuration of a hydrogen terminated zigzag graphene cluster connected to zigzag graphene leads. Its width and length is equal to 8.71A0 and 19.62 A0 respectively. The region of leads is specified with non- hydrogen terminated edge carbon atoms schematically. It is noted that the region of left and right leads includes two super cells of ZGNR but the main scattering region includes four super cells of ZGNR and extra five carbon atoms. The atoms in blue color will be omitted for creating pores in graphen cluster. As Fig. 1(b) shows the zero bias conductance of graphene cluster without pore, is in "V" shape but there is an asymmetry between E>EF (electrons) and E
 5  

agreement with the result of Ref. [10] but asymmetry is caused by second size quantization, i.e., decreasing the length of ribbon which is in order of λF now. It has been shown that all ZGNR have a typical peak in their electronic DOS slightly below the Fermi energy which is due to the edge states [15, 16, 28, 29]. Therefore as Fig.1 (b) shows near the EF, the peak in DOS can be attributed to the edge states of two semi-infinite ZGNRs which are connected to ZGNC from left and right. For calculating the transmission and DOS of GNRs, an extended scattering channel including a unit cell (or super cell) and a part of its left and right neighborhood unit cells (or super cells) is considered [9, 30-32]. Therefore it is expected that the DOS be symmetric respect to EF approximately [9, 30-32]. But as mentioned above, the main scattering region of ZGNC includes four super cells of ZGNR and five extra carbon atoms and therefore the main scattering region has an asymmetric structure. Then it is expected that DOS be asymmetric respect to EF. Also it is noted that for EEF the DOS is attributed to the unoccupied molecular orbitals (UMO). The difference between the highest OMO (called HOMO) and the lowest UMO (called LUMO) is band gap energy. Therefore our results show that the bad gap energy is zero due to the zigzag structure of ZGNC, and there are differences between OMO and UMO of ZGNC connected to ZGNRs. Now we start to omit the carbon atoms which are placed at the middle of the ZGNC and shown in blue color in Fig.1 (a) one by one, and calculate not only the conductance but also the electronic density of states (DOS) of the ZGNC. The numbers on the blue carbon atoms are attributed to the number of atom in ZGNC including leads. It is noted that, after omitting the carbon atoms, a Hydrogen atom is attached to their neighboring carbon atoms and therefore the broken sp2 bonds, which is produced by removing one carbon atom, do not create three dangling bonds at the neighboring atoms i.e., each of the neighboring carbon atom is passivated by one Hydrogen atom. Therefore the status is not similar to the considered status by Ref. [11]. The fourth and two last columns of table one shows the geometry of omitted carbon atoms global lattice imbalance and the local lattice imbalance (nA-nB) respectively. If nA and nB are the number of atoms of sublattices A and B in cluster respectively, (nA-nB) is called global lattice imbalance and if nA and nB are number of

 6  

omitted atoms from site A and B respectively, (nA-nB) is called local lattice imbalance.. The first omitted carbon atom which is placed at A-type site is 5th atom from the left (right) side of the carbon atom strip. It is noted that, some of them belongs to A-type sites and some of them belongs to B-type sites. Table 1 shows the zero bias conductance of different structures at Fermi energy, EF. It has been shown that, the HOMO and LUMO bands in ZGNRs are localized at the ribbon edges. Consequently when the pores are positioned at the center of the ribbons its band-structure remains metallic and no significant change is observed when the pore is closer to the edge of ZGNRs [15]. As the table 1 shows, when the local imbalance is not equal to zero (except for cluster including a pore) the conductance of device at Ef decreases and be equal to zero for some cases. It has been shown that, graphene-based systems have a bi-partite lattice, with nA and nB sites per unit cell in the A and B sublattice, leading to nA-nB flat bands at Fermi level [33]. Therefore it can be concluded that, local imbalance in the porous graphene cluster perturbs edge-localized states and consequently the conductance at Ef decreases similar to the other graphene-based systems [34]. In addition, by deleting the odd number of carbon atoms, the symmetry of structure is sustained, while by omitting the even number of carbon atoms, the symmetry is broken. Therefore the difference in the value of conductance at Fermi energy not only can be attributed to the local imbalance, but also, it may be attributed to the final symmetry of the structure. Fig.2 (a) to (c) shows the zero bias conductance (in blue color) and DOS (in green color and arbitrary unit) of ZGNC including 1, 2, 4 vacancies. It is note that, all zigzag ribbons have a typical peak in their DOS slightly below the Fermi energy which is due to the edge states [15, 16, 28, 29]. The peak is seen in Fig.2 (a) to (c) also. As they show generally, by increasing (decreasing) the DOS the conductance increase (decreases) but the value of the conductance is near to the conductance of ZGNC without pore approximately at other energies. Of course the conductance of structure including a pore is smaller than ZGNC without pore. Therefore it can be concluded that, since the local imbalance is equal to zero, the conductance at other energies does not change approximately. Fig.3 (a) and (b) show the zero bias conductance (in blue color) and DOS (in green color and arbitrary unit) of ZGNC including 3, and 5 vacancies. As they show the conductance of structure including three vacancies is significantly higher than the conductance of ZGNC without pore for |E-Ef|≥5 eV approximately. It is clear that by increasing the odd number of  7  

vacancies, the original structure is transferred to two paths structures gradually and consequently, the carriers have little electronic transport channel for moving from left lead to right lead and vice versa. Therefore it is expected that, the conductance decreases. Then the conductance decrement can be attributed not only to the local imbalance but also to the damage of electronic transport channel by the presence of pores. But the structure including three vacancies beaks the rule and the conductance increases significantly for |E-Ef|≥5 eV approximately. By attention to the above description, it can be concluded that, when the number of the vacancies is equal to three, the number of less affected conducting channels increases and in consequence the conductance increases. As a summary, Fig.4 shows the normalized zero bias conductance of the ZGNC when (E-Ef) is equal to -10 eV, -5 eV, 0 eV, 5 eV and 10 eV for different number of vacancies. The normalized conductance is defined as: conductance of ZGNC including vacancy/ZGNC without vacancy for each (E-Ef). As the figure shows, when ZGNC includes three vacancies the normalized conductance is maximum, and at Fermi energy, it increases for even number of vacancies, and decreases for odd number of vacancies (except one vacancy). Of course one can consider other configurations and studied their conductance. But since our main aim is to studying the conductance of zigzag graphene cluster when its width and length is at order of λF, we consider only the above specific configurations. We use first nearest-neighbor tight binding method [25, 27, and 34], and the introduced effective spin-orbit matrix by Ref. [24] to study the Rashba spin-orbit interaction effects on porous ZGNC including three vacancies. As Fig.5 shows, when λR is equal to 0.8 eV [27], spin up electrons flips to spin down electrons i.e., spin flipping is happened. Therefore the device can be used in spintronic applications. 4. Summary In summary, our first-principle studies on the transport properties of zigzag graphene nano cluster (ZGNC) including different type of pores showed that, by decreasing the length of ZGNC without vacancy when its width is at order of the Fermi wavelength of electron in graphene, an asymmetry is appeared in "V" shape curve of zero bias conductance. It has been shown that, the zero

 8  

bias conductance at Fermi energy of structure including 2, and 4-vacacies increases because, not only the local imbalance is zero but also the symmetry of structure is broken. Also we showed the zero bias conductance decreases when 3, and 5-vacancies are included in the structure because, not only the local imbalance is not zero but also the symmetry of structure is not broken. For other energies it has been shown that, the conductance of structure including 2, and 4-vacacies is near to the conductance of original structure while, the conductance of structure including 5-vacancies decreases because, the local imbalance is zero for former case and is not zero for later case. It has been shown that the conductance of structure including 3-vacancies is significantly higher than the conductance of the original structure for |E-Ef|≥5 eV which can be attributed to the number of less affected conducting channels compared to the other number of vacancies. We showed that, Rashba spin-orbit interaction flips spin up electrons to spin down electrons. Therefore the local imbalance number and final symmetry of porous graphene cluster can be used as a rule for designing porous graphene devices and the device can be used in spintronic applications. References [1] F. Capasso, " Band-Gap Engineering: From Physics and Materials to New Semiconductor Devices",Science 235, 4785, (1987), pp. 172-176. [2] V. H. Crespi, M. L. Cohen, and Angel Rubio," In Situ Band Gap Engineering of Carbon Nanotubes", Phys. Rev. Lett. 79, (1997), pp. 2093-2096. [3] M. Y. Han, B. Ozyilmaz, Y.Zhang, and P. Kim, "Energy Band-Gap Engineering of Graphene Nanoribbons", Phys. Rev. Lett. 98, (2007), pp. 206805 (1-4). [4] Y. W. Son, M. L. Cohen, and S. G. Louie," Energy Gaps in Graphene Nanoribbons" Phys. Rev. Lett. 97, (2006), 216803(1-4). [5] V. Barone, O. Hod, and G. E. Scuseria, " Electronic Structure and Stability of Semiconducting Graphene Nanoribbons", Nano Lett. 6, (2006), pp. 2748–2754,. [6] M. Zeng, L. Shen, M. Yang, C. Zhang, and Y. Feng, "Charge and spin transport in graphene-based heterostructure", App. Phys. Lett. 98, (2011), pp. 053101(1-3).

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[7] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov and A. K. Geim, "The electronic properties of graphene", Rev. Mod. Phys. 81, (2009), pp. 109–162. [8] L. Tapaszto, G. Dobrik, P. Nemes-Incze, G. Vertesy, Ph. Lambin, and L. P. Biro, "Tuning the electronic structure of graphene by ion irradiation", Phys. Rev. B 78, (2008), pp. 233407(1-4). [9] T. C. Li, and Shao-Ping Lu, "Quantum conductance of graphene nanoribbons with edge defects" arXiv:cond-mat/0609009v1 (2006), pp. 1-8. [10] Y. M. Lin, V. Perebeinos, Z. Chen and P. Avouis, "

Conductance Quantization in Graphene

Nanoribbons", arXiv:cond-mat/0805.0035v2 (2008), pp. 1-5. [11] N. Gorjizadeh, A. A. Farajian and Y. Kawazoe, "The effects of defects on the conductance of graphene nanoribbons", Nanotech. 20, (2009), pp. 015201. [12] M. T. Ahmadi, Z. Johari, N. Aziziah Amin, A. H. Fallahpour, and R. Ismail," Graphene nanoribbon conductance model in parabolic band structure", Hindawi Publ. Corp. J. Nanomaterial, (2010), pp. 753738 (1-4). [13] H. Sahin and R. T. Senger, "First-principles calculations of spin-dependent conductance of graphene flakes", Phys. Rev. B 78, (2008), pp. 205423 (1-8). [14] M. Hatanaka, " Spin preference of small metal clusters", Chem. Phys. Lett., 492, (2010), pp. 109114. [15]

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Nanocarbons",

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[19] J.S. Binkley, J.A. Pople, W.J. Hehre, Self-consistent molecular orbital methods. 21. Small splitvalence basis sets for first-row elements, J. Am. Chem. Soc., 102, (1980)  pp. 939-947.  [20] M.P. Lopez Sancho, J.M. Lopez Sancho and J. Rubio, J. Phys. Quick iterative scheme for the calculation of transfer matrices: application to Mo(100), F: Met. Phys. 14, (1984) pp. 1205-1215. [21] P.S. Krstic, D.J. Dean, X. G. Zhang, D.Keffer, Y.S. Leng, P.T. Cummings, and J.C. Wells, Computational chemistry for molecular electronics, Computational Material Science 28, (2003) pp. 321-341. [22] Y. Wu, P.A. Childs, Conductance of Graphene Nanoribbon Junction and the Tight Binding Model, Nanoscale Research Letter, 6, (2011) pp. 62. [23] S. Konschuh, M. Gmitra, and J. Fabin, Tight-binding theory of the spin-orbut coupling in graphene, Phys. Rev. B 82, (2010) 245412. [24] H. Min, J. E. Hill, N. A. Sinitsyn, B. R. Sahu, Leonard Kleinman, and A. H. MacDonald, "Intrinsic and Rashba spin-orbit interactions in graphene sheets", Phys. Rev. B 74, (2006) 165310. [25] D. Kochan, M. Gmitra, and J. Fabian, "Theory of the ac Spin-Valve Effect" Phys. Rev. Lett. 107, (2011) 176604. [26] G. H. Ding and C. T. Chan, J. Phys. Cond. Matt. 23, (2011), pp. 205304. [27] S. Ahmadi, M. Esmaeilzadeh, E. Namvar, and G. Pan," Spin-inversion in nanoscale graphene sheets with a Rashba spin-orbit barrier", AIP Advances 2, (2012), pp. 012130 (1-9). [28] D. J. Klein Chem. Phys. Lett. Graphitic polymer strips with edge states, 217, (1994), 261. [29] M. Fujita, K. Wakabayashi, K. Nakada, and K. Kusakabe, Peculiar Localized State at Zigzag Graphite Edge, J. Phys. Soc. Japan 65, (1996), 1920. [30] F. Cervantes-Sodi, G. Csanyi, S. Piscanec, and A. C. Ferrari, Edge-functionalized and substitutionally doped graphene nanoribbons: Electronic and spin properties, Phy. Rev. B 77, (2008), 165427. [31] M. Ezawa, Peculiar width dependence of the electronic properties of carbon nanoribbons, Phy. Rev. B 73, (2006) 045432. [32]A. Liping and L. Nianhua, First-principle study on transport properties of zigzag graphene nanoribbon with different spin-configurations, J. Semicon. 32, (2011), 052001-6. [33] M. Vanevic, M. S. Stojanovic and, M. Kindermann, "Character of electronic states in graphene antidot lattices: Flat bands and spatial localization", Phys. Rev. B 80, (2009), pp. 045410(1-8). [34] Y. Wu and P. A. Childs, " Conductance of Graphene Nanoribbon Junctions and the Tight Binding Model", Nanoscale Research letter, 6, 62 (2011), pp 1-5.   11  

Figure caption  Fig.1‐(Color Online) (a) Optimized schematic configuration of a ZGNC-H connected to zigzag graphene leads and (b) its conductance (in blur color) and DOS (in green color and arbitrary unit) . The atoms in blue color will be omitted for making vacancies in the cluster. Fig.2‐(Color Online) Conductance (in blue color) and DOS (in green color and arbitrary unit) of  graphene cluster including (a) 1‐vacancy, (b)2‐vacancies,(c) 4‐vancancies Fig.3‐ (Color Online) Conductance (in blue color) and DOS (in green color and arbitrary unit) of   graphene cluster including (a) 3‐vacancies, (b)5‐vacancies Fig.4‐ (Color Online) Normalized conductance when E‐Ef is equal to ‐10 eV (in red color), ‐5 eV (in    blue color), 0 eV (in green color), 5 eV (in cross‐blue color) and 10 eV (in cross‐red color)   Fig.5 (Color Online) The Conductance of porous ZGNC including three vacancies without (a) and in  presence of Rashba spin‐orbit interaction (b)

  12  

Table 1- Different graphene clusters, their transmission at Fermi energy and geometry of omitted carbon atoms Item

Graphene

T(EF)

Geometry of omitted carbon atoms

nano cluster

Global lattice

Local lattice

imbalance (nA-nB)

imbalance (nA-nB)

1

without

0.27

vacancy

2

including

23-22=1

0-0=0

A

22-22=0

1-0=1

A

22-21=1

1-1=0

22-20=2

1-2=-1

A

21-20=1

2-2=0

A

19-20=-1

3-2=1

0.44

one vacancy

3

including

0.19

B

two vacancies 4

including

0.0

B

B

three A

vacancies 5

including

0.36

B

B

four A

vacancies 6

including

0.06

B

B

five vacancies

A

A

  13  

  (a)

  (b) Fig.1‐(Color Online) (a) Optimized schematic configuration of a ZGNC-H connected to zigzag graphene leads and (b) its conductance (in blur color) and DOS (in green color and arbitrary unit) . The atoms in blue color will be omitted for making vacancies in the cluster.   14  

  (a)

  (b)

(c)

Fig.2‐(Color Online) Conductance (in blue color) and DOS (in green color and arbitrary unit) of graphene  cluster including (a) 1‐vacancy, (b)2‐vacancies,(c) 4‐vancancies

  15  

  (a)

(b)

Fig.3‐ (Color Online) Conductance (in blue color) and DOS (in green color and arbitrary unit) of     graphene cluster including (a) 3‐vacancies, (b)5‐vacancies

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Fig.4‐ (Color Online) Normalized conductance when E‐Ef is equal to ‐10 eV (in red color), ‐5 eV (in    blue color), 0 eV (in green color), 5 eV (in cross‐blue color) and 10 eV (in cross‐red color)  

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  (a)

(b)

Fig.5 (Color Online) The Conductance of porous ZGNC including three vacancies without (a)    and in presence of Rashba spin‐orbit interaction (b)

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Highlights of article's reference number (PHYSE 11304) 1‐The conductance of porous clusters depends on the final symmetry of porous cluster and the local imbalance number. 2 The conductance of porous cluster can be significantly higher than original zigzag graphene cluster. 3- Under Rashba spin orbit interaction, spin flipping is seen in porous cluster.

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